pyerrors.correlators

   1import warnings
   2from itertools import permutations
   3import numpy as np
   4import autograd.numpy as anp
   5import matplotlib.pyplot as plt
   6import scipy.linalg
   7from .obs import Obs, reweight, correlate, CObs
   8from .misc import dump_object, _assert_equal_properties
   9from .fits import least_squares
  10from .roots import find_root
  11
  12
  13class Corr:
  14    """The class for a correlator (time dependent sequence of pe.Obs).
  15
  16    Everything, this class does, can be achieved using lists or arrays of Obs.
  17    But it is simply more convenient to have a dedicated object for correlators.
  18    One often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient
  19    to iterate over all timeslices for every operation. This is especially true, when dealing with matrices.
  20
  21    The correlator can have two types of content: An Obs at every timeslice OR a GEVP
  22    matrix at every timeslice. Other dependency (eg. spatial) are not supported.
  23
  24    """
  25
  26    def __init__(self, data_input, padding=[0, 0], prange=None):
  27        """ Initialize a Corr object.
  28
  29        Parameters
  30        ----------
  31        data_input : list or array
  32            list of Obs or list of arrays of Obs or array of Corrs
  33        padding : list, optional
  34            List with two entries where the first labels the padding
  35            at the front of the correlator and the second the padding
  36            at the back.
  37        prange : list, optional
  38            List containing the first and last timeslice of the plateau
  39            region indentified for this correlator.
  40        """
  41
  42        if isinstance(data_input, np.ndarray):
  43
  44            # This only works, if the array fulfills the conditions below
  45            if not len(data_input.shape) == 2 and data_input.shape[0] == data_input.shape[1]:
  46                raise Exception("Incompatible array shape")
  47            if not all([isinstance(item, Corr) for item in data_input.flatten()]):
  48                raise Exception("If the input is an array, its elements must be of type pe.Corr")
  49            if not all([item.N == 1 for item in data_input.flatten()]):
  50                raise Exception("Can only construct matrix correlator from single valued correlators")
  51            if not len(set([item.T for item in data_input.flatten()])) == 1:
  52                raise Exception("All input Correlators must be defined over the same timeslices.")
  53
  54            T = data_input[0, 0].T
  55            N = data_input.shape[0]
  56            input_as_list = []
  57            for t in range(T):
  58                if any([(item.content[t] is None) for item in data_input.flatten()]):
  59                    if not all([(item.content[t] is None) for item in data_input.flatten()]):
  60                        warnings.warn("Input ill-defined at different timeslices. Conversion leads to data loss!", RuntimeWarning)
  61                    input_as_list.append(None)
  62                else:
  63                    array_at_timeslace = np.empty([N, N], dtype="object")
  64                    for i in range(N):
  65                        for j in range(N):
  66                            array_at_timeslace[i, j] = data_input[i, j][t]
  67                    input_as_list.append(array_at_timeslace)
  68            data_input = input_as_list
  69
  70        if isinstance(data_input, list):
  71
  72            if all([isinstance(item, (Obs, CObs)) for item in data_input]):
  73                _assert_equal_properties(data_input)
  74                self.content = [np.asarray([item]) for item in data_input]
  75            if all([isinstance(item, (Obs, CObs)) or item is None for item in data_input]):
  76                _assert_equal_properties([o for o in data_input if o is not None])
  77                self.content = [np.asarray([item]) if item is not None else None for item in data_input]
  78                self.N = 1
  79
  80            elif all([isinstance(item, np.ndarray) or item is None for item in data_input]) and any([isinstance(item, np.ndarray) for item in data_input]):
  81                self.content = data_input
  82                noNull = [a for a in self.content if not (a is None)]  # To check if the matrices are correct for all undefined elements
  83                self.N = noNull[0].shape[0]
  84                if self.N > 1 and noNull[0].shape[0] != noNull[0].shape[1]:
  85                    raise Exception("Smearing matrices are not NxN")
  86                if (not all([item.shape == noNull[0].shape for item in noNull])):
  87                    raise Exception("Items in data_input are not of identical shape." + str(noNull))
  88            else:
  89                raise Exception("data_input contains item of wrong type")
  90        else:
  91            raise Exception("Data input was not given as list or correct array")
  92
  93        self.tag = None
  94
  95        # An undefined timeslice is represented by the None object
  96        self.content = [None] * padding[0] + self.content + [None] * padding[1]
  97        self.T = len(self.content)
  98        self.prange = prange
  99
 100    def __getitem__(self, idx):
 101        """Return the content of timeslice idx"""
 102        if self.content[idx] is None:
 103            return None
 104        elif len(self.content[idx]) == 1:
 105            return self.content[idx][0]
 106        else:
 107            return self.content[idx]
 108
 109    @property
 110    def reweighted(self):
 111        bool_array = np.array([list(map(lambda x: x.reweighted, o)) for o in [x for x in self.content if x is not None]])
 112        if np.all(bool_array == 1):
 113            return True
 114        elif np.all(bool_array == 0):
 115            return False
 116        else:
 117            raise Exception("Reweighting status of correlator corrupted.")
 118
 119    def gamma_method(self, **kwargs):
 120        """Apply the gamma method to the content of the Corr."""
 121        for item in self.content:
 122            if not (item is None):
 123                if self.N == 1:
 124                    item[0].gamma_method(**kwargs)
 125                else:
 126                    for i in range(self.N):
 127                        for j in range(self.N):
 128                            item[i, j].gamma_method(**kwargs)
 129
 130    def projected(self, vector_l=None, vector_r=None, normalize=False):
 131        """We need to project the Correlator with a Vector to get a single value at each timeslice.
 132
 133        The method can use one or two vectors.
 134        If two are specified it returns v1@G@v2 (the order might be very important.)
 135        By default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to
 136        """
 137        if self.N == 1:
 138            raise Exception("Trying to project a Corr, that already has N=1.")
 139
 140        if vector_l is None:
 141            vector_l, vector_r = np.asarray([1.] + (self.N - 1) * [0.]), np.asarray([1.] + (self.N - 1) * [0.])
 142        elif (vector_r is None):
 143            vector_r = vector_l
 144        if isinstance(vector_l, list) and not isinstance(vector_r, list):
 145            if len(vector_l) != self.T:
 146                raise Exception("Length of vector list must be equal to T")
 147            vector_r = [vector_r] * self.T
 148        if isinstance(vector_r, list) and not isinstance(vector_l, list):
 149            if len(vector_r) != self.T:
 150                raise Exception("Length of vector list must be equal to T")
 151            vector_l = [vector_l] * self.T
 152
 153        if not isinstance(vector_l, list):
 154            if not vector_l.shape == vector_r.shape == (self.N,):
 155                raise Exception("Vectors are of wrong shape!")
 156            if normalize:
 157                vector_l, vector_r = vector_l / np.sqrt((vector_l @ vector_l)), vector_r / np.sqrt(vector_r @ vector_r)
 158            newcontent = [None if _check_for_none(self, item) else np.asarray([vector_l.T @ item @ vector_r]) for item in self.content]
 159
 160        else:
 161            # There are no checks here yet. There are so many possible scenarios, where this can go wrong.
 162            if normalize:
 163                for t in range(self.T):
 164                    vector_l[t], vector_r[t] = vector_l[t] / np.sqrt((vector_l[t] @ vector_l[t])), vector_r[t] / np.sqrt(vector_r[t] @ vector_r[t])
 165
 166            newcontent = [None if (_check_for_none(self, self.content[t]) or vector_l[t] is None or vector_r[t] is None) else np.asarray([vector_l[t].T @ self.content[t] @ vector_r[t]]) for t in range(self.T)]
 167        return Corr(newcontent)
 168
 169    def item(self, i, j):
 170        """Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.
 171
 172        Parameters
 173        ----------
 174        i : int
 175            First index to be picked.
 176        j : int
 177            Second index to be picked.
 178        """
 179        if self.N == 1:
 180            raise Exception("Trying to pick item from projected Corr")
 181        newcontent = [None if (item is None) else item[i, j] for item in self.content]
 182        return Corr(newcontent)
 183
 184    def plottable(self):
 185        """Outputs the correlator in a plotable format.
 186
 187        Outputs three lists containing the timeslice index, the value on each
 188        timeslice and the error on each timeslice.
 189        """
 190        if self.N != 1:
 191            raise Exception("Can only make Corr[N=1] plottable")
 192        x_list = [x for x in range(self.T) if not self.content[x] is None]
 193        y_list = [y[0].value for y in self.content if y is not None]
 194        y_err_list = [y[0].dvalue for y in self.content if y is not None]
 195
 196        return x_list, y_list, y_err_list
 197
 198    def symmetric(self):
 199        """ Symmetrize the correlator around x0=0."""
 200        if self.N != 1:
 201            raise Exception('symmetric cannot be safely applied to multi-dimensional correlators.')
 202        if self.T % 2 != 0:
 203            raise Exception("Can not symmetrize odd T")
 204
 205        if np.argmax(np.abs(self.content)) != 0:
 206            warnings.warn("Correlator does not seem to be symmetric around x0=0.", RuntimeWarning)
 207
 208        newcontent = [self.content[0]]
 209        for t in range(1, self.T):
 210            if (self.content[t] is None) or (self.content[self.T - t] is None):
 211                newcontent.append(None)
 212            else:
 213                newcontent.append(0.5 * (self.content[t] + self.content[self.T - t]))
 214        if (all([x is None for x in newcontent])):
 215            raise Exception("Corr could not be symmetrized: No redundant values")
 216        return Corr(newcontent, prange=self.prange)
 217
 218    def anti_symmetric(self):
 219        """Anti-symmetrize the correlator around x0=0."""
 220        if self.N != 1:
 221            raise Exception('anti_symmetric cannot be safely applied to multi-dimensional correlators.')
 222        if self.T % 2 != 0:
 223            raise Exception("Can not symmetrize odd T")
 224
 225        test = 1 * self
 226        test.gamma_method()
 227        if not all([o.is_zero_within_error(3) for o in test.content[0]]):
 228            warnings.warn("Correlator does not seem to be anti-symmetric around x0=0.", RuntimeWarning)
 229
 230        newcontent = [self.content[0]]
 231        for t in range(1, self.T):
 232            if (self.content[t] is None) or (self.content[self.T - t] is None):
 233                newcontent.append(None)
 234            else:
 235                newcontent.append(0.5 * (self.content[t] - self.content[self.T - t]))
 236        if (all([x is None for x in newcontent])):
 237            raise Exception("Corr could not be symmetrized: No redundant values")
 238        return Corr(newcontent, prange=self.prange)
 239
 240    def is_matrix_symmetric(self):
 241        """Checks whether a correlator matrices is symmetric on every timeslice."""
 242        if self.N == 1:
 243            raise Exception("Only works for correlator matrices.")
 244        for t in range(self.T):
 245            if self[t] is None:
 246                continue
 247            for i in range(self.N):
 248                for j in range(i + 1, self.N):
 249                    if self[t][i, j] is self[t][j, i]:
 250                        continue
 251                    if hash(self[t][i, j]) != hash(self[t][j, i]):
 252                        return False
 253        return True
 254
 255    def matrix_symmetric(self):
 256        """Symmetrizes the correlator matrices on every timeslice."""
 257        if self.N == 1:
 258            raise Exception("Trying to symmetrize a correlator matrix, that already has N=1.")
 259        if self.is_matrix_symmetric():
 260            return 1.0 * self
 261        else:
 262            transposed = [None if _check_for_none(self, G) else G.T for G in self.content]
 263            return 0.5 * (Corr(transposed) + self)
 264
 265    def GEVP(self, t0, ts=None, sort="Eigenvalue", **kwargs):
 266        r'''Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
 267
 268        The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the
 269        largest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing
 270        ```python
 271        C.GEVP(t0=2)[0]  # Ground state vector(s)
 272        C.GEVP(t0=2)[:3]  # Vectors for the lowest three states
 273        ```
 274
 275        Parameters
 276        ----------
 277        t0 : int
 278            The time t0 for the right hand side of the GEVP according to $G(t)v_i=\lambda_i G(t_0)v_i$
 279        ts : int
 280            fixed time $G(t_s)v_i=\lambda_i G(t_0)v_i$ if sort=None.
 281            If sort="Eigenvector" it gives a reference point for the sorting method.
 282        sort : string
 283            If this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.
 284            - "Eigenvalue": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice.
 285            - "Eigenvector": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.
 286              The reference state is identified by its eigenvalue at $t=t_s$.
 287
 288        Other Parameters
 289        ----------------
 290        state : int
 291           Returns only the vector(s) for a specified state. The lowest state is zero.
 292        '''
 293
 294        if self.N == 1:
 295            raise Exception("GEVP methods only works on correlator matrices and not single correlators.")
 296        if ts is not None:
 297            if (ts <= t0):
 298                raise Exception("ts has to be larger than t0.")
 299
 300        if "sorted_list" in kwargs:
 301            warnings.warn("Argument 'sorted_list' is deprecated, use 'sort' instead.", DeprecationWarning)
 302            sort = kwargs.get("sorted_list")
 303
 304        if self.is_matrix_symmetric():
 305            symmetric_corr = self
 306        else:
 307            symmetric_corr = self.matrix_symmetric()
 308
 309        if sort is None:
 310            if (ts is None):
 311                raise Exception("ts is required if sort=None.")
 312            if (self.content[t0] is None) or (self.content[ts] is None):
 313                raise Exception("Corr not defined at t0/ts.")
 314            G0, Gt = np.empty([self.N, self.N], dtype="double"), np.empty([self.N, self.N], dtype="double")
 315            for i in range(self.N):
 316                for j in range(self.N):
 317                    G0[i, j] = symmetric_corr[t0][i, j].value
 318                    Gt[i, j] = symmetric_corr[ts][i, j].value
 319
 320            reordered_vecs = _GEVP_solver(Gt, G0)
 321
 322        elif sort in ["Eigenvalue", "Eigenvector"]:
 323            if sort == "Eigenvalue" and ts is not None:
 324                warnings.warn("ts has no effect when sorting by eigenvalue is chosen.", RuntimeWarning)
 325            all_vecs = [None] * (t0 + 1)
 326            G0 = np.vectorize(lambda x: x.value)(symmetric_corr[t0])
 327            for t in range(t0 + 1, self.T):
 328                try:
 329                    Gt = np.vectorize(lambda x: x.value)(symmetric_corr[t])
 330                    all_vecs.append(_GEVP_solver(Gt, G0))
 331                except Exception:
 332                    all_vecs.append(None)
 333            if sort == "Eigenvector":
 334                if (ts is None):
 335                    raise Exception("ts is required for the Eigenvector sorting method.")
 336                all_vecs = _sort_vectors(all_vecs, ts)
 337
 338            reordered_vecs = [[v[s] if v is not None else None for v in all_vecs] for s in range(self.N)]
 339        else:
 340            raise Exception("Unkown value for 'sort'.")
 341
 342        if "state" in kwargs:
 343            return reordered_vecs[kwargs.get("state")]
 344        else:
 345            return reordered_vecs
 346
 347    def Eigenvalue(self, t0, ts=None, state=0, sort="Eigenvalue"):
 348        """Determines the eigenvalue of the GEVP by solving and projecting the correlator
 349
 350        Parameters
 351        ----------
 352        state : int
 353            The state one is interested in ordered by energy. The lowest state is zero.
 354
 355        All other parameters are identical to the ones of Corr.GEVP.
 356        """
 357        vec = self.GEVP(t0, ts=ts, sort=sort)[state]
 358        return self.projected(vec)
 359
 360    def Hankel(self, N, periodic=False):
 361        """Constructs an NxN Hankel matrix
 362
 363        C(t) c(t+1) ... c(t+n-1)
 364        C(t+1) c(t+2) ... c(t+n)
 365        .................
 366        C(t+(n-1)) c(t+n) ... c(t+2(n-1))
 367
 368        Parameters
 369        ----------
 370        N : int
 371            Dimension of the Hankel matrix
 372        periodic : bool, optional
 373            determines whether the matrix is extended periodically
 374        """
 375
 376        if self.N != 1:
 377            raise Exception("Multi-operator Prony not implemented!")
 378
 379        array = np.empty([N, N], dtype="object")
 380        new_content = []
 381        for t in range(self.T):
 382            new_content.append(array.copy())
 383
 384        def wrap(i):
 385            while i >= self.T:
 386                i -= self.T
 387            return i
 388
 389        for t in range(self.T):
 390            for i in range(N):
 391                for j in range(N):
 392                    if periodic:
 393                        new_content[t][i, j] = self.content[wrap(t + i + j)][0]
 394                    elif (t + i + j) >= self.T:
 395                        new_content[t] = None
 396                    else:
 397                        new_content[t][i, j] = self.content[t + i + j][0]
 398
 399        return Corr(new_content)
 400
 401    def roll(self, dt):
 402        """Periodically shift the correlator by dt timeslices
 403
 404        Parameters
 405        ----------
 406        dt : int
 407            number of timeslices
 408        """
 409        return Corr(list(np.roll(np.array(self.content, dtype=object), dt)))
 410
 411    def reverse(self):
 412        """Reverse the time ordering of the Corr"""
 413        return Corr(self.content[:: -1])
 414
 415    def thin(self, spacing=2, offset=0):
 416        """Thin out a correlator to suppress correlations
 417
 418        Parameters
 419        ----------
 420        spacing : int
 421            Keep only every 'spacing'th entry of the correlator
 422        offset : int
 423            Offset the equal spacing
 424        """
 425        new_content = []
 426        for t in range(self.T):
 427            if (offset + t) % spacing != 0:
 428                new_content.append(None)
 429            else:
 430                new_content.append(self.content[t])
 431        return Corr(new_content)
 432
 433    def correlate(self, partner):
 434        """Correlate the correlator with another correlator or Obs
 435
 436        Parameters
 437        ----------
 438        partner : Obs or Corr
 439            partner to correlate the correlator with.
 440            Can either be an Obs which is correlated with all entries of the
 441            correlator or a Corr of same length.
 442        """
 443        if self.N != 1:
 444            raise Exception("Only one-dimensional correlators can be safely correlated.")
 445        new_content = []
 446        for x0, t_slice in enumerate(self.content):
 447            if _check_for_none(self, t_slice):
 448                new_content.append(None)
 449            else:
 450                if isinstance(partner, Corr):
 451                    if _check_for_none(partner, partner.content[x0]):
 452                        new_content.append(None)
 453                    else:
 454                        new_content.append(np.array([correlate(o, partner.content[x0][0]) for o in t_slice]))
 455                elif isinstance(partner, Obs):  # Should this include CObs?
 456                    new_content.append(np.array([correlate(o, partner) for o in t_slice]))
 457                else:
 458                    raise Exception("Can only correlate with an Obs or a Corr.")
 459
 460        return Corr(new_content)
 461
 462    def reweight(self, weight, **kwargs):
 463        """Reweight the correlator.
 464
 465        Parameters
 466        ----------
 467        weight : Obs
 468            Reweighting factor. An Observable that has to be defined on a superset of the
 469            configurations in obs[i].idl for all i.
 470        all_configs : bool
 471            if True, the reweighted observables are normalized by the average of
 472            the reweighting factor on all configurations in weight.idl and not
 473            on the configurations in obs[i].idl.
 474        """
 475        if self.N != 1:
 476            raise Exception("Reweighting only implemented for one-dimensional correlators.")
 477        new_content = []
 478        for t_slice in self.content:
 479            if _check_for_none(self, t_slice):
 480                new_content.append(None)
 481            else:
 482                new_content.append(np.array(reweight(weight, t_slice, **kwargs)))
 483        return Corr(new_content)
 484
 485    def T_symmetry(self, partner, parity=+1):
 486        """Return the time symmetry average of the correlator and its partner
 487
 488        Parameters
 489        ----------
 490        partner : Corr
 491            Time symmetry partner of the Corr
 492        partity : int
 493            Parity quantum number of the correlator, can be +1 or -1
 494        """
 495        if self.N != 1:
 496            raise Exception("T_symmetry only implemented for one-dimensional correlators.")
 497        if not isinstance(partner, Corr):
 498            raise Exception("T partner has to be a Corr object.")
 499        if parity not in [+1, -1]:
 500            raise Exception("Parity has to be +1 or -1.")
 501        T_partner = parity * partner.reverse()
 502
 503        t_slices = []
 504        test = (self - T_partner)
 505        test.gamma_method()
 506        for x0, t_slice in enumerate(test.content):
 507            if t_slice is not None:
 508                if not t_slice[0].is_zero_within_error(5):
 509                    t_slices.append(x0)
 510        if t_slices:
 511            warnings.warn("T symmetry partners do not agree within 5 sigma on time slices " + str(t_slices) + ".", RuntimeWarning)
 512
 513        return (self + T_partner) / 2
 514
 515    def deriv(self, variant="symmetric"):
 516        """Return the first derivative of the correlator with respect to x0.
 517
 518        Parameters
 519        ----------
 520        variant : str
 521            decides which definition of the finite differences derivative is used.
 522            Available choice: symmetric, forward, backward, improved, log, default: symmetric
 523        """
 524        if self.N != 1:
 525            raise Exception("deriv only implemented for one-dimensional correlators.")
 526        if variant == "symmetric":
 527            newcontent = []
 528            for t in range(1, self.T - 1):
 529                if (self.content[t - 1] is None) or (self.content[t + 1] is None):
 530                    newcontent.append(None)
 531                else:
 532                    newcontent.append(0.5 * (self.content[t + 1] - self.content[t - 1]))
 533            if (all([x is None for x in newcontent])):
 534                raise Exception('Derivative is undefined at all timeslices')
 535            return Corr(newcontent, padding=[1, 1])
 536        elif variant == "forward":
 537            newcontent = []
 538            for t in range(self.T - 1):
 539                if (self.content[t] is None) or (self.content[t + 1] is None):
 540                    newcontent.append(None)
 541                else:
 542                    newcontent.append(self.content[t + 1] - self.content[t])
 543            if (all([x is None for x in newcontent])):
 544                raise Exception("Derivative is undefined at all timeslices")
 545            return Corr(newcontent, padding=[0, 1])
 546        elif variant == "backward":
 547            newcontent = []
 548            for t in range(1, self.T):
 549                if (self.content[t - 1] is None) or (self.content[t] is None):
 550                    newcontent.append(None)
 551                else:
 552                    newcontent.append(self.content[t] - self.content[t - 1])
 553            if (all([x is None for x in newcontent])):
 554                raise Exception("Derivative is undefined at all timeslices")
 555            return Corr(newcontent, padding=[1, 0])
 556        elif variant == "improved":
 557            newcontent = []
 558            for t in range(2, self.T - 2):
 559                if (self.content[t - 2] is None) or (self.content[t - 1] is None) or (self.content[t + 1] is None) or (self.content[t + 2] is None):
 560                    newcontent.append(None)
 561                else:
 562                    newcontent.append((1 / 12) * (self.content[t - 2] - 8 * self.content[t - 1] + 8 * self.content[t + 1] - self.content[t + 2]))
 563            if (all([x is None for x in newcontent])):
 564                raise Exception('Derivative is undefined at all timeslices')
 565            return Corr(newcontent, padding=[2, 2])
 566        elif variant == 'log':
 567            newcontent = []
 568            for t in range(self.T):
 569                if (self.content[t] is None) or (self.content[t] <= 0):
 570                    newcontent.append(None)
 571                else:
 572                    newcontent.append(np.log(self.content[t]))
 573            if (all([x is None for x in newcontent])):
 574                raise Exception("Log is undefined at all timeslices")
 575            logcorr = Corr(newcontent)
 576            return self * logcorr.deriv('symmetric')
 577        else:
 578            raise Exception("Unknown variant.")
 579
 580    def second_deriv(self, variant="symmetric"):
 581        """Return the second derivative of the correlator with respect to x0.
 582
 583        Parameters
 584        ----------
 585        variant : str
 586            decides which definition of the finite differences derivative is used.
 587            Available choice: symmetric, improved, log, default: symmetric
 588        """
 589        if self.N != 1:
 590            raise Exception("second_deriv only implemented for one-dimensional correlators.")
 591        if variant == "symmetric":
 592            newcontent = []
 593            for t in range(1, self.T - 1):
 594                if (self.content[t - 1] is None) or (self.content[t + 1] is None):
 595                    newcontent.append(None)
 596                else:
 597                    newcontent.append((self.content[t + 1] - 2 * self.content[t] + self.content[t - 1]))
 598            if (all([x is None for x in newcontent])):
 599                raise Exception("Derivative is undefined at all timeslices")
 600            return Corr(newcontent, padding=[1, 1])
 601        elif variant == "improved":
 602            newcontent = []
 603            for t in range(2, self.T - 2):
 604                if (self.content[t - 2] is None) or (self.content[t - 1] is None) or (self.content[t] is None) or (self.content[t + 1] is None) or (self.content[t + 2] is None):
 605                    newcontent.append(None)
 606                else:
 607                    newcontent.append((1 / 12) * (-self.content[t + 2] + 16 * self.content[t + 1] - 30 * self.content[t] + 16 * self.content[t - 1] - self.content[t - 2]))
 608            if (all([x is None for x in newcontent])):
 609                raise Exception("Derivative is undefined at all timeslices")
 610            return Corr(newcontent, padding=[2, 2])
 611        elif variant == 'log':
 612            newcontent = []
 613            for t in range(self.T):
 614                if (self.content[t] is None) or (self.content[t] <= 0):
 615                    newcontent.append(None)
 616                else:
 617                    newcontent.append(np.log(self.content[t]))
 618            if (all([x is None for x in newcontent])):
 619                raise Exception("Log is undefined at all timeslices")
 620            logcorr = Corr(newcontent)
 621            return self * (logcorr.second_deriv('symmetric') + (logcorr.deriv('symmetric'))**2)
 622        else:
 623            raise Exception("Unknown variant.")
 624
 625    def m_eff(self, variant='log', guess=1.0):
 626        """Returns the effective mass of the correlator as correlator object
 627
 628        Parameters
 629        ----------
 630        variant : str
 631            log : uses the standard effective mass log(C(t) / C(t+1))
 632            cosh, periodic : Use periodicitiy of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.
 633            sinh : Use anti-periodicitiy of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.
 634            See, e.g., arXiv:1205.5380
 635            arccosh : Uses the explicit form of the symmetrized correlator (not recommended)
 636            logsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
 637        guess : float
 638            guess for the root finder, only relevant for the root variant
 639        """
 640        if self.N != 1:
 641            raise Exception('Correlator must be projected before getting m_eff')
 642        if variant == 'log':
 643            newcontent = []
 644            for t in range(self.T - 1):
 645                if ((self.content[t] is None) or (self.content[t + 1] is None)) or (self.content[t + 1][0].value == 0):
 646                    newcontent.append(None)
 647                elif self.content[t][0].value / self.content[t + 1][0].value < 0:
 648                    newcontent.append(None)
 649                else:
 650                    newcontent.append(self.content[t] / self.content[t + 1])
 651            if (all([x is None for x in newcontent])):
 652                raise Exception('m_eff is undefined at all timeslices')
 653
 654            return np.log(Corr(newcontent, padding=[0, 1]))
 655
 656        elif variant == 'logsym':
 657            newcontent = []
 658            for t in range(1, self.T - 1):
 659                if ((self.content[t - 1] is None) or (self.content[t + 1] is None)) or (self.content[t + 1][0].value == 0):
 660                    newcontent.append(None)
 661                elif self.content[t - 1][0].value / self.content[t + 1][0].value < 0:
 662                    newcontent.append(None)
 663                else:
 664                    newcontent.append(self.content[t - 1] / self.content[t + 1])
 665            if (all([x is None for x in newcontent])):
 666                raise Exception('m_eff is undefined at all timeslices')
 667
 668            return np.log(Corr(newcontent, padding=[1, 1])) / 2
 669
 670        elif variant in ['periodic', 'cosh', 'sinh']:
 671            if variant in ['periodic', 'cosh']:
 672                func = anp.cosh
 673            else:
 674                func = anp.sinh
 675
 676            def root_function(x, d):
 677                return func(x * (t - self.T / 2)) / func(x * (t + 1 - self.T / 2)) - d
 678
 679            newcontent = []
 680            for t in range(self.T - 1):
 681                if (self.content[t] is None) or (self.content[t + 1] is None) or (self.content[t + 1][0].value == 0):
 682                    newcontent.append(None)
 683                # Fill the two timeslices in the middle of the lattice with their predecessors
 684                elif variant == 'sinh' and t in [self.T / 2, self.T / 2 - 1]:
 685                    newcontent.append(newcontent[-1])
 686                elif self.content[t][0].value / self.content[t + 1][0].value < 0:
 687                    newcontent.append(None)
 688                else:
 689                    newcontent.append(np.abs(find_root(self.content[t][0] / self.content[t + 1][0], root_function, guess=guess)))
 690            if (all([x is None for x in newcontent])):
 691                raise Exception('m_eff is undefined at all timeslices')
 692
 693            return Corr(newcontent, padding=[0, 1])
 694
 695        elif variant == 'arccosh':
 696            newcontent = []
 697            for t in range(1, self.T - 1):
 698                if (self.content[t] is None) or (self.content[t + 1] is None) or (self.content[t - 1] is None) or (self.content[t][0].value == 0):
 699                    newcontent.append(None)
 700                else:
 701                    newcontent.append((self.content[t + 1] + self.content[t - 1]) / (2 * self.content[t]))
 702            if (all([x is None for x in newcontent])):
 703                raise Exception("m_eff is undefined at all timeslices")
 704            return np.arccosh(Corr(newcontent, padding=[1, 1]))
 705
 706        else:
 707            raise Exception('Unknown variant.')
 708
 709    def fit(self, function, fitrange=None, silent=False, **kwargs):
 710        r'''Fits function to the data
 711
 712        Parameters
 713        ----------
 714        function : obj
 715            function to fit to the data. See fits.least_squares for details.
 716        fitrange : list
 717            Two element list containing the timeslices on which the fit is supposed to start and stop.
 718            Caution: This range is inclusive as opposed to standard python indexing.
 719            `fitrange=[4, 6]` corresponds to the three entries 4, 5 and 6.
 720            If not specified, self.prange or all timeslices are used.
 721        silent : bool
 722            Decides whether output is printed to the standard output.
 723        '''
 724        if self.N != 1:
 725            raise Exception("Correlator must be projected before fitting")
 726
 727        if fitrange is None:
 728            if self.prange:
 729                fitrange = self.prange
 730            else:
 731                fitrange = [0, self.T - 1]
 732        else:
 733            if not isinstance(fitrange, list):
 734                raise Exception("fitrange has to be a list with two elements")
 735            if len(fitrange) != 2:
 736                raise Exception("fitrange has to have exactly two elements [fit_start, fit_stop]")
 737
 738        xs = [x for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]
 739        ys = [self.content[x][0] for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]
 740        result = least_squares(xs, ys, function, silent=silent, **kwargs)
 741        return result
 742
 743    def plateau(self, plateau_range=None, method="fit", auto_gamma=False):
 744        """ Extract a plateau value from a Corr object
 745
 746        Parameters
 747        ----------
 748        plateau_range : list
 749            list with two entries, indicating the first and the last timeslice
 750            of the plateau region.
 751        method : str
 752            method to extract the plateau.
 753                'fit' fits a constant to the plateau region
 754                'avg', 'average' or 'mean' just average over the given timeslices.
 755        auto_gamma : bool
 756            apply gamma_method with default parameters to the Corr. Defaults to None
 757        """
 758        if not plateau_range:
 759            if self.prange:
 760                plateau_range = self.prange
 761            else:
 762                raise Exception("no plateau range provided")
 763        if self.N != 1:
 764            raise Exception("Correlator must be projected before getting a plateau.")
 765        if (all([self.content[t] is None for t in range(plateau_range[0], plateau_range[1] + 1)])):
 766            raise Exception("plateau is undefined at all timeslices in plateaurange.")
 767        if auto_gamma:
 768            self.gamma_method()
 769        if method == "fit":
 770            def const_func(a, t):
 771                return a[0]
 772            return self.fit(const_func, plateau_range)[0]
 773        elif method in ["avg", "average", "mean"]:
 774            returnvalue = np.mean([item[0] for item in self.content[plateau_range[0]:plateau_range[1] + 1] if item is not None])
 775            return returnvalue
 776
 777        else:
 778            raise Exception("Unsupported plateau method: " + method)
 779
 780    def set_prange(self, prange):
 781        """Sets the attribute prange of the Corr object."""
 782        if not len(prange) == 2:
 783            raise Exception("prange must be a list or array with two values")
 784        if not ((isinstance(prange[0], int)) and (isinstance(prange[1], int))):
 785            raise Exception("Start and end point must be integers")
 786        if not (0 <= prange[0] <= self.T and 0 <= prange[1] <= self.T and prange[0] < prange[1]):
 787            raise Exception("Start and end point must define a range in the interval 0,T")
 788
 789        self.prange = prange
 790        return
 791
 792    def show(self, x_range=None, comp=None, y_range=None, logscale=False, plateau=None, fit_res=None, ylabel=None, save=None, auto_gamma=False, hide_sigma=None, references=None, title=None):
 793        """Plots the correlator using the tag of the correlator as label if available.
 794
 795        Parameters
 796        ----------
 797        x_range : list
 798            list of two values, determining the range of the x-axis e.g. [4, 8].
 799        comp : Corr or list of Corr
 800            Correlator or list of correlators which are plotted for comparison.
 801            The tags of these correlators are used as labels if available.
 802        logscale : bool
 803            Sets y-axis to logscale.
 804        plateau : Obs
 805            Plateau value to be visualized in the figure.
 806        fit_res : Fit_result
 807            Fit_result object to be visualized.
 808        ylabel : str
 809            Label for the y-axis.
 810        save : str
 811            path to file in which the figure should be saved.
 812        auto_gamma : bool
 813            Apply the gamma method with standard parameters to all correlators and plateau values before plotting.
 814        hide_sigma : float
 815            Hides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
 816        references : list
 817            List of floating point values that are displayed as horizontal lines for reference.
 818        title : string
 819            Optional title of the figure.
 820        """
 821        if self.N != 1:
 822            raise Exception("Correlator must be projected before plotting")
 823
 824        if auto_gamma:
 825            self.gamma_method()
 826
 827        if x_range is None:
 828            x_range = [0, self.T - 1]
 829
 830        fig = plt.figure()
 831        ax1 = fig.add_subplot(111)
 832
 833        x, y, y_err = self.plottable()
 834        if hide_sigma:
 835            hide_from = np.argmax((hide_sigma * np.array(y_err[1:])) > np.abs(y[1:])) - 1
 836        else:
 837            hide_from = None
 838        ax1.errorbar(x[:hide_from], y[:hide_from], y_err[:hide_from], label=self.tag)
 839        if logscale:
 840            ax1.set_yscale('log')
 841        else:
 842            if y_range is None:
 843                try:
 844                    y_min = min([(x[0].value - x[0].dvalue) for x in self.content[x_range[0]: x_range[1] + 1] if (x is not None) and x[0].dvalue < 2 * np.abs(x[0].value)])
 845                    y_max = max([(x[0].value + x[0].dvalue) for x in self.content[x_range[0]: x_range[1] + 1] if (x is not None) and x[0].dvalue < 2 * np.abs(x[0].value)])
 846                    ax1.set_ylim([y_min - 0.1 * (y_max - y_min), y_max + 0.1 * (y_max - y_min)])
 847                except Exception:
 848                    pass
 849            else:
 850                ax1.set_ylim(y_range)
 851        if comp:
 852            if isinstance(comp, (Corr, list)):
 853                for corr in comp if isinstance(comp, list) else [comp]:
 854                    if auto_gamma:
 855                        corr.gamma_method()
 856                    x, y, y_err = corr.plottable()
 857                    if hide_sigma:
 858                        hide_from = np.argmax((hide_sigma * np.array(y_err[1:])) > np.abs(y[1:])) - 1
 859                    else:
 860                        hide_from = None
 861                    ax1.errorbar(x[:hide_from], y[:hide_from], y_err[:hide_from], label=corr.tag, mfc=plt.rcParams['axes.facecolor'])
 862            else:
 863                raise Exception("'comp' must be a correlator or a list of correlators.")
 864
 865        if plateau:
 866            if isinstance(plateau, Obs):
 867                if auto_gamma:
 868                    plateau.gamma_method()
 869                ax1.axhline(y=plateau.value, linewidth=2, color=plt.rcParams['text.color'], alpha=0.6, marker=',', ls='--', label=str(plateau))
 870                ax1.axhspan(plateau.value - plateau.dvalue, plateau.value + plateau.dvalue, alpha=0.25, color=plt.rcParams['text.color'], ls='-')
 871            else:
 872                raise Exception("'plateau' must be an Obs")
 873
 874        if references:
 875            if isinstance(references, list):
 876                for ref in references:
 877                    ax1.axhline(y=ref, linewidth=1, color=plt.rcParams['text.color'], alpha=0.6, marker=',', ls='--')
 878            else:
 879                raise Exception("'references' must be a list of floating pint values.")
 880
 881        if self.prange:
 882            ax1.axvline(self.prange[0], 0, 1, ls='-', marker=',')
 883            ax1.axvline(self.prange[1], 0, 1, ls='-', marker=',')
 884
 885        if fit_res:
 886            x_samples = np.arange(x_range[0], x_range[1] + 1, 0.05)
 887            ax1.plot(x_samples,
 888                     fit_res.fit_function([o.value for o in fit_res.fit_parameters], x_samples),
 889                     ls='-', marker=',', lw=2)
 890
 891        ax1.set_xlabel(r'$x_0 / a$')
 892        if ylabel:
 893            ax1.set_ylabel(ylabel)
 894        ax1.set_xlim([x_range[0] - 0.5, x_range[1] + 0.5])
 895
 896        handles, labels = ax1.get_legend_handles_labels()
 897        if labels:
 898            ax1.legend()
 899
 900        if title:
 901            plt.title(title)
 902
 903        plt.draw()
 904
 905        if save:
 906            if isinstance(save, str):
 907                fig.savefig(save, bbox_inches='tight')
 908            else:
 909                raise Exception("'save' has to be a string.")
 910
 911    def spaghetti_plot(self, logscale=True):
 912        """Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.
 913
 914        Parameters
 915        ----------
 916        logscale : bool
 917            Determines whether the scale of the y-axis is logarithmic or standard.
 918        """
 919        if self.N != 1:
 920            raise Exception("Correlator needs to be projected first.")
 921
 922        mc_names = list(set([item for sublist in [o[0].mc_names for o in self.content if o is not None] for item in sublist]))
 923        x0_vals = [n for (n, o) in zip(np.arange(self.T), self.content) if o is not None]
 924
 925        for name in mc_names:
 926            data = np.array([o[0].deltas[name] + o[0].r_values[name] for o in self.content if o is not None]).T
 927
 928            fig = plt.figure()
 929            ax = fig.add_subplot(111)
 930            for dat in data:
 931                ax.plot(x0_vals, dat, ls='-', marker='')
 932
 933            if logscale is True:
 934                ax.set_yscale('log')
 935
 936            ax.set_xlabel(r'$x_0 / a$')
 937            plt.title(name)
 938            plt.draw()
 939
 940    def dump(self, filename, datatype="json.gz", **kwargs):
 941        """Dumps the Corr into a file of chosen type
 942        Parameters
 943        ----------
 944        filename : str
 945            Name of the file to be saved.
 946        datatype : str
 947            Format of the exported file. Supported formats include
 948            "json.gz" and "pickle"
 949        path : str
 950            specifies a custom path for the file (default '.')
 951        """
 952        if datatype == "json.gz":
 953            from .input.json import dump_to_json
 954            if 'path' in kwargs:
 955                file_name = kwargs.get('path') + '/' + filename
 956            else:
 957                file_name = filename
 958            dump_to_json(self, file_name)
 959        elif datatype == "pickle":
 960            dump_object(self, filename, **kwargs)
 961        else:
 962            raise Exception("Unknown datatype " + str(datatype))
 963
 964    def print(self, print_range=None):
 965        print(self.__repr__(print_range))
 966
 967    def __repr__(self, print_range=None):
 968        if print_range is None:
 969            print_range = [0, None]
 970
 971        content_string = ""
 972        content_string += "Corr T=" + str(self.T) + " N=" + str(self.N) + "\n"  # +" filled with"+ str(type(self.content[0][0])) there should be a good solution here
 973
 974        if self.tag is not None:
 975            content_string += "Description: " + self.tag + "\n"
 976        if self.N != 1:
 977            return content_string
 978
 979        if print_range[1]:
 980            print_range[1] += 1
 981        content_string += 'x0/a\tCorr(x0/a)\n------------------\n'
 982        for i, sub_corr in enumerate(self.content[print_range[0]:print_range[1]]):
 983            if sub_corr is None:
 984                content_string += str(i + print_range[0]) + '\n'
 985            else:
 986                content_string += str(i + print_range[0])
 987                for element in sub_corr:
 988                    content_string += '\t' + ' ' * int(element >= 0) + str(element)
 989                content_string += '\n'
 990        return content_string
 991
 992    def __str__(self):
 993        return self.__repr__()
 994
 995    # We define the basic operations, that can be performed with correlators.
 996    # While */+- get defined here, they only work for Corr*Obs and not Obs*Corr.
 997    # This is because Obs*Corr checks Obs.__mul__ first and does not catch an exception.
 998    # One could try and tell Obs to check if the y in __mul__ is a Corr and
 999
1000    def __add__(self, y):
1001        if isinstance(y, Corr):
1002            if ((self.N != y.N) or (self.T != y.T)):
1003                raise Exception("Addition of Corrs with different shape")
1004            newcontent = []
1005            for t in range(self.T):
1006                if _check_for_none(self, self.content[t]) or _check_for_none(y, y.content[t]):
1007                    newcontent.append(None)
1008                else:
1009                    newcontent.append(self.content[t] + y.content[t])
1010            return Corr(newcontent)
1011
1012        elif isinstance(y, (Obs, int, float, CObs)):
1013            newcontent = []
1014            for t in range(self.T):
1015                if _check_for_none(self, self.content[t]):
1016                    newcontent.append(None)
1017                else:
1018                    newcontent.append(self.content[t] + y)
1019            return Corr(newcontent, prange=self.prange)
1020        elif isinstance(y, np.ndarray):
1021            if y.shape == (self.T,):
1022                return Corr(list((np.array(self.content).T + y).T))
1023            else:
1024                raise ValueError("operands could not be broadcast together")
1025        else:
1026            raise TypeError("Corr + wrong type")
1027
1028    def __mul__(self, y):
1029        if isinstance(y, Corr):
1030            if not ((self.N == 1 or y.N == 1 or self.N == y.N) and self.T == y.T):
1031                raise Exception("Multiplication of Corr object requires N=N or N=1 and T=T")
1032            newcontent = []
1033            for t in range(self.T):
1034                if _check_for_none(self, self.content[t]) or _check_for_none(y, y.content[t]):
1035                    newcontent.append(None)
1036                else:
1037                    newcontent.append(self.content[t] * y.content[t])
1038            return Corr(newcontent)
1039
1040        elif isinstance(y, (Obs, int, float, CObs)):
1041            newcontent = []
1042            for t in range(self.T):
1043                if _check_for_none(self, self.content[t]):
1044                    newcontent.append(None)
1045                else:
1046                    newcontent.append(self.content[t] * y)
1047            return Corr(newcontent, prange=self.prange)
1048        elif isinstance(y, np.ndarray):
1049            if y.shape == (self.T,):
1050                return Corr(list((np.array(self.content).T * y).T))
1051            else:
1052                raise ValueError("operands could not be broadcast together")
1053        else:
1054            raise TypeError("Corr * wrong type")
1055
1056    def __truediv__(self, y):
1057        if isinstance(y, Corr):
1058            if not ((self.N == 1 or y.N == 1 or self.N == y.N) and self.T == y.T):
1059                raise Exception("Multiplication of Corr object requires N=N or N=1 and T=T")
1060            newcontent = []
1061            for t in range(self.T):
1062                if _check_for_none(self, self.content[t]) or _check_for_none(y, y.content[t]):
1063                    newcontent.append(None)
1064                else:
1065                    newcontent.append(self.content[t] / y.content[t])
1066            for t in range(self.T):
1067                if _check_for_none(self, newcontent[t]):
1068                    continue
1069                if np.isnan(np.sum(newcontent[t]).value):
1070                    newcontent[t] = None
1071
1072            if all([item is None for item in newcontent]):
1073                raise Exception("Division returns completely undefined correlator")
1074            return Corr(newcontent)
1075
1076        elif isinstance(y, (Obs, CObs)):
1077            if isinstance(y, Obs):
1078                if y.value == 0:
1079                    raise Exception('Division by zero will return undefined correlator')
1080            if isinstance(y, CObs):
1081                if y.is_zero():
1082                    raise Exception('Division by zero will return undefined correlator')
1083
1084            newcontent = []
1085            for t in range(self.T):
1086                if _check_for_none(self, self.content[t]):
1087                    newcontent.append(None)
1088                else:
1089                    newcontent.append(self.content[t] / y)
1090            return Corr(newcontent, prange=self.prange)
1091
1092        elif isinstance(y, (int, float)):
1093            if y == 0:
1094                raise Exception('Division by zero will return undefined correlator')
1095            newcontent = []
1096            for t in range(self.T):
1097                if _check_for_none(self, self.content[t]):
1098                    newcontent.append(None)
1099                else:
1100                    newcontent.append(self.content[t] / y)
1101            return Corr(newcontent, prange=self.prange)
1102        elif isinstance(y, np.ndarray):
1103            if y.shape == (self.T,):
1104                return Corr(list((np.array(self.content).T / y).T))
1105            else:
1106                raise ValueError("operands could not be broadcast together")
1107        else:
1108            raise TypeError('Corr / wrong type')
1109
1110    def __neg__(self):
1111        newcontent = [None if _check_for_none(self, item) else -1. * item for item in self.content]
1112        return Corr(newcontent, prange=self.prange)
1113
1114    def __sub__(self, y):
1115        return self + (-y)
1116
1117    def __pow__(self, y):
1118        if isinstance(y, (Obs, int, float, CObs)):
1119            newcontent = [None if _check_for_none(self, item) else item**y for item in self.content]
1120            return Corr(newcontent, prange=self.prange)
1121        else:
1122            raise TypeError('Type of exponent not supported')
1123
1124    def __abs__(self):
1125        newcontent = [None if _check_for_none(self, item) else np.abs(item) for item in self.content]
1126        return Corr(newcontent, prange=self.prange)
1127
1128    # The numpy functions:
1129    def sqrt(self):
1130        return self ** 0.5
1131
1132    def log(self):
1133        newcontent = [None if _check_for_none(self, item) else np.log(item) for item in self.content]
1134        return Corr(newcontent, prange=self.prange)
1135
1136    def exp(self):
1137        newcontent = [None if _check_for_none(self, item) else np.exp(item) for item in self.content]
1138        return Corr(newcontent, prange=self.prange)
1139
1140    def _apply_func_to_corr(self, func):
1141        newcontent = [None if _check_for_none(self, item) else func(item) for item in self.content]
1142        for t in range(self.T):
1143            if _check_for_none(self, newcontent[t]):
1144                continue
1145            if np.isnan(np.sum(newcontent[t]).value):
1146                newcontent[t] = None
1147        if all([item is None for item in newcontent]):
1148            raise Exception('Operation returns undefined correlator')
1149        return Corr(newcontent)
1150
1151    def sin(self):
1152        return self._apply_func_to_corr(np.sin)
1153
1154    def cos(self):
1155        return self._apply_func_to_corr(np.cos)
1156
1157    def tan(self):
1158        return self._apply_func_to_corr(np.tan)
1159
1160    def sinh(self):
1161        return self._apply_func_to_corr(np.sinh)
1162
1163    def cosh(self):
1164        return self._apply_func_to_corr(np.cosh)
1165
1166    def tanh(self):
1167        return self._apply_func_to_corr(np.tanh)
1168
1169    def arcsin(self):
1170        return self._apply_func_to_corr(np.arcsin)
1171
1172    def arccos(self):
1173        return self._apply_func_to_corr(np.arccos)
1174
1175    def arctan(self):
1176        return self._apply_func_to_corr(np.arctan)
1177
1178    def arcsinh(self):
1179        return self._apply_func_to_corr(np.arcsinh)
1180
1181    def arccosh(self):
1182        return self._apply_func_to_corr(np.arccosh)
1183
1184    def arctanh(self):
1185        return self._apply_func_to_corr(np.arctanh)
1186
1187    # Right hand side operations (require tweak in main module to work)
1188    def __radd__(self, y):
1189        return self + y
1190
1191    def __rsub__(self, y):
1192        return -self + y
1193
1194    def __rmul__(self, y):
1195        return self * y
1196
1197    def __rtruediv__(self, y):
1198        return (self / y) ** (-1)
1199
1200    @property
1201    def real(self):
1202        def return_real(obs_OR_cobs):
1203            if isinstance(obs_OR_cobs, CObs):
1204                return obs_OR_cobs.real
1205            else:
1206                return obs_OR_cobs
1207
1208        return self._apply_func_to_corr(return_real)
1209
1210    @property
1211    def imag(self):
1212        def return_imag(obs_OR_cobs):
1213            if isinstance(obs_OR_cobs, CObs):
1214                return obs_OR_cobs.imag
1215            else:
1216                return obs_OR_cobs * 0  # So it stays the right type
1217
1218        return self._apply_func_to_corr(return_imag)
1219
1220    def prune(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):
1221        r''' Project large correlation matrix to lowest states
1222
1223        This method can be used to reduce the size of an (N x N) correlation matrix
1224        to (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise
1225        is still small.
1226
1227        Parameters
1228        ----------
1229        Ntrunc: int
1230            Rank of the target matrix.
1231        tproj: int
1232            Time where the eigenvectors are evaluated, corresponds to ts in the GEVP method.
1233            The default value is 3.
1234        t0proj: int
1235            Time where the correlation matrix is inverted. Choosing t0proj=1 is strongly
1236            discouraged for O(a) improved theories, since the correctness of the procedure
1237            cannot be granted in this case. The default value is 2.
1238        basematrix : Corr
1239            Correlation matrix that is used to determine the eigenvectors of the
1240            lowest states based on a GEVP. basematrix is taken to be the Corr itself if
1241            is is not specified.
1242
1243        Notes
1244        -----
1245        We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving
1246        the GEVP $$C(t) v_n(t, t_0) = \lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \equiv t_\mathrm{proj}$
1247        and $t_0 \equiv t_{0, \mathrm{proj}}$. The target matrix is projected onto the subspace of the
1248        resulting eigenvectors $v_n, n=1,\dots,N_\mathrm{trunc}$ via
1249        $$G^\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large
1250        correlation matrix and to remove some noise that is added by irrelevant operators.
1251        This may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated
1252        bound $t_0 \leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.
1253        '''
1254
1255        if self.N == 1:
1256            raise Exception('Method cannot be applied to one-dimensional correlators.')
1257        if basematrix is None:
1258            basematrix = self
1259        if Ntrunc >= basematrix.N:
1260            raise Exception('Cannot truncate using Ntrunc <= %d' % (basematrix.N))
1261        if basematrix.N != self.N:
1262            raise Exception('basematrix and targetmatrix have to be of the same size.')
1263
1264        evecs = basematrix.GEVP(t0proj, tproj, sort=None)[:Ntrunc]
1265
1266        tmpmat = np.empty((Ntrunc, Ntrunc), dtype=object)
1267        rmat = []
1268        for t in range(basematrix.T):
1269            for i in range(Ntrunc):
1270                for j in range(Ntrunc):
1271                    tmpmat[i][j] = evecs[i].T @ self[t] @ evecs[j]
1272            rmat.append(np.copy(tmpmat))
1273
1274        newcontent = [None if (self.content[t] is None) else rmat[t] for t in range(self.T)]
1275        return Corr(newcontent)
1276
1277
1278def _sort_vectors(vec_set, ts):
1279    """Helper function used to find a set of Eigenvectors consistent over all timeslices"""
1280    reference_sorting = np.array(vec_set[ts])
1281    N = reference_sorting.shape[0]
1282    sorted_vec_set = []
1283    for t in range(len(vec_set)):
1284        if vec_set[t] is None:
1285            sorted_vec_set.append(None)
1286        elif not t == ts:
1287            perms = [list(o) for o in permutations([i for i in range(N)], N)]
1288            best_score = 0
1289            for perm in perms:
1290                current_score = 1
1291                for k in range(N):
1292                    new_sorting = reference_sorting.copy()
1293                    new_sorting[perm[k], :] = vec_set[t][k]
1294                    current_score *= abs(np.linalg.det(new_sorting))
1295                if current_score > best_score:
1296                    best_score = current_score
1297                    best_perm = perm
1298            sorted_vec_set.append([vec_set[t][k] for k in best_perm])
1299        else:
1300            sorted_vec_set.append(vec_set[t])
1301
1302    return sorted_vec_set
1303
1304
1305def _check_for_none(corr, entry):
1306    """Checks if entry for correlator corr is None"""
1307    return len(list(filter(None, np.asarray(entry).flatten()))) < corr.N ** 2
1308
1309
1310def _GEVP_solver(Gt, G0):
1311    """Helper function for solving the GEVP and sorting the eigenvectors.
1312
1313    The helper function assumes that both provided matrices are symmetric and
1314    only processes the lower triangular part of both matrices. In case the matrices
1315    are not symmetric the upper triangular parts are effectively discarded."""
1316    return scipy.linalg.eigh(Gt, G0, lower=True)[1].T[::-1]
class Corr:
  14class Corr:
  15    """The class for a correlator (time dependent sequence of pe.Obs).
  16
  17    Everything, this class does, can be achieved using lists or arrays of Obs.
  18    But it is simply more convenient to have a dedicated object for correlators.
  19    One often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient
  20    to iterate over all timeslices for every operation. This is especially true, when dealing with matrices.
  21
  22    The correlator can have two types of content: An Obs at every timeslice OR a GEVP
  23    matrix at every timeslice. Other dependency (eg. spatial) are not supported.
  24
  25    """
  26
  27    def __init__(self, data_input, padding=[0, 0], prange=None):
  28        """ Initialize a Corr object.
  29
  30        Parameters
  31        ----------
  32        data_input : list or array
  33            list of Obs or list of arrays of Obs or array of Corrs
  34        padding : list, optional
  35            List with two entries where the first labels the padding
  36            at the front of the correlator and the second the padding
  37            at the back.
  38        prange : list, optional
  39            List containing the first and last timeslice of the plateau
  40            region indentified for this correlator.
  41        """
  42
  43        if isinstance(data_input, np.ndarray):
  44
  45            # This only works, if the array fulfills the conditions below
  46            if not len(data_input.shape) == 2 and data_input.shape[0] == data_input.shape[1]:
  47                raise Exception("Incompatible array shape")
  48            if not all([isinstance(item, Corr) for item in data_input.flatten()]):
  49                raise Exception("If the input is an array, its elements must be of type pe.Corr")
  50            if not all([item.N == 1 for item in data_input.flatten()]):
  51                raise Exception("Can only construct matrix correlator from single valued correlators")
  52            if not len(set([item.T for item in data_input.flatten()])) == 1:
  53                raise Exception("All input Correlators must be defined over the same timeslices.")
  54
  55            T = data_input[0, 0].T
  56            N = data_input.shape[0]
  57            input_as_list = []
  58            for t in range(T):
  59                if any([(item.content[t] is None) for item in data_input.flatten()]):
  60                    if not all([(item.content[t] is None) for item in data_input.flatten()]):
  61                        warnings.warn("Input ill-defined at different timeslices. Conversion leads to data loss!", RuntimeWarning)
  62                    input_as_list.append(None)
  63                else:
  64                    array_at_timeslace = np.empty([N, N], dtype="object")
  65                    for i in range(N):
  66                        for j in range(N):
  67                            array_at_timeslace[i, j] = data_input[i, j][t]
  68                    input_as_list.append(array_at_timeslace)
  69            data_input = input_as_list
  70
  71        if isinstance(data_input, list):
  72
  73            if all([isinstance(item, (Obs, CObs)) for item in data_input]):
  74                _assert_equal_properties(data_input)
  75                self.content = [np.asarray([item]) for item in data_input]
  76            if all([isinstance(item, (Obs, CObs)) or item is None for item in data_input]):
  77                _assert_equal_properties([o for o in data_input if o is not None])
  78                self.content = [np.asarray([item]) if item is not None else None for item in data_input]
  79                self.N = 1
  80
  81            elif all([isinstance(item, np.ndarray) or item is None for item in data_input]) and any([isinstance(item, np.ndarray) for item in data_input]):
  82                self.content = data_input
  83                noNull = [a for a in self.content if not (a is None)]  # To check if the matrices are correct for all undefined elements
  84                self.N = noNull[0].shape[0]
  85                if self.N > 1 and noNull[0].shape[0] != noNull[0].shape[1]:
  86                    raise Exception("Smearing matrices are not NxN")
  87                if (not all([item.shape == noNull[0].shape for item in noNull])):
  88                    raise Exception("Items in data_input are not of identical shape." + str(noNull))
  89            else:
  90                raise Exception("data_input contains item of wrong type")
  91        else:
  92            raise Exception("Data input was not given as list or correct array")
  93
  94        self.tag = None
  95
  96        # An undefined timeslice is represented by the None object
  97        self.content = [None] * padding[0] + self.content + [None] * padding[1]
  98        self.T = len(self.content)
  99        self.prange = prange
 100
 101    def __getitem__(self, idx):
 102        """Return the content of timeslice idx"""
 103        if self.content[idx] is None:
 104            return None
 105        elif len(self.content[idx]) == 1:
 106            return self.content[idx][0]
 107        else:
 108            return self.content[idx]
 109
 110    @property
 111    def reweighted(self):
 112        bool_array = np.array([list(map(lambda x: x.reweighted, o)) for o in [x for x in self.content if x is not None]])
 113        if np.all(bool_array == 1):
 114            return True
 115        elif np.all(bool_array == 0):
 116            return False
 117        else:
 118            raise Exception("Reweighting status of correlator corrupted.")
 119
 120    def gamma_method(self, **kwargs):
 121        """Apply the gamma method to the content of the Corr."""
 122        for item in self.content:
 123            if not (item is None):
 124                if self.N == 1:
 125                    item[0].gamma_method(**kwargs)
 126                else:
 127                    for i in range(self.N):
 128                        for j in range(self.N):
 129                            item[i, j].gamma_method(**kwargs)
 130
 131    def projected(self, vector_l=None, vector_r=None, normalize=False):
 132        """We need to project the Correlator with a Vector to get a single value at each timeslice.
 133
 134        The method can use one or two vectors.
 135        If two are specified it returns v1@G@v2 (the order might be very important.)
 136        By default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to
 137        """
 138        if self.N == 1:
 139            raise Exception("Trying to project a Corr, that already has N=1.")
 140
 141        if vector_l is None:
 142            vector_l, vector_r = np.asarray([1.] + (self.N - 1) * [0.]), np.asarray([1.] + (self.N - 1) * [0.])
 143        elif (vector_r is None):
 144            vector_r = vector_l
 145        if isinstance(vector_l, list) and not isinstance(vector_r, list):
 146            if len(vector_l) != self.T:
 147                raise Exception("Length of vector list must be equal to T")
 148            vector_r = [vector_r] * self.T
 149        if isinstance(vector_r, list) and not isinstance(vector_l, list):
 150            if len(vector_r) != self.T:
 151                raise Exception("Length of vector list must be equal to T")
 152            vector_l = [vector_l] * self.T
 153
 154        if not isinstance(vector_l, list):
 155            if not vector_l.shape == vector_r.shape == (self.N,):
 156                raise Exception("Vectors are of wrong shape!")
 157            if normalize:
 158                vector_l, vector_r = vector_l / np.sqrt((vector_l @ vector_l)), vector_r / np.sqrt(vector_r @ vector_r)
 159            newcontent = [None if _check_for_none(self, item) else np.asarray([vector_l.T @ item @ vector_r]) for item in self.content]
 160
 161        else:
 162            # There are no checks here yet. There are so many possible scenarios, where this can go wrong.
 163            if normalize:
 164                for t in range(self.T):
 165                    vector_l[t], vector_r[t] = vector_l[t] / np.sqrt((vector_l[t] @ vector_l[t])), vector_r[t] / np.sqrt(vector_r[t] @ vector_r[t])
 166
 167            newcontent = [None if (_check_for_none(self, self.content[t]) or vector_l[t] is None or vector_r[t] is None) else np.asarray([vector_l[t].T @ self.content[t] @ vector_r[t]]) for t in range(self.T)]
 168        return Corr(newcontent)
 169
 170    def item(self, i, j):
 171        """Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.
 172
 173        Parameters
 174        ----------
 175        i : int
 176            First index to be picked.
 177        j : int
 178            Second index to be picked.
 179        """
 180        if self.N == 1:
 181            raise Exception("Trying to pick item from projected Corr")
 182        newcontent = [None if (item is None) else item[i, j] for item in self.content]
 183        return Corr(newcontent)
 184
 185    def plottable(self):
 186        """Outputs the correlator in a plotable format.
 187
 188        Outputs three lists containing the timeslice index, the value on each
 189        timeslice and the error on each timeslice.
 190        """
 191        if self.N != 1:
 192            raise Exception("Can only make Corr[N=1] plottable")
 193        x_list = [x for x in range(self.T) if not self.content[x] is None]
 194        y_list = [y[0].value for y in self.content if y is not None]
 195        y_err_list = [y[0].dvalue for y in self.content if y is not None]
 196
 197        return x_list, y_list, y_err_list
 198
 199    def symmetric(self):
 200        """ Symmetrize the correlator around x0=0."""
 201        if self.N != 1:
 202            raise Exception('symmetric cannot be safely applied to multi-dimensional correlators.')
 203        if self.T % 2 != 0:
 204            raise Exception("Can not symmetrize odd T")
 205
 206        if np.argmax(np.abs(self.content)) != 0:
 207            warnings.warn("Correlator does not seem to be symmetric around x0=0.", RuntimeWarning)
 208
 209        newcontent = [self.content[0]]
 210        for t in range(1, self.T):
 211            if (self.content[t] is None) or (self.content[self.T - t] is None):
 212                newcontent.append(None)
 213            else:
 214                newcontent.append(0.5 * (self.content[t] + self.content[self.T - t]))
 215        if (all([x is None for x in newcontent])):
 216            raise Exception("Corr could not be symmetrized: No redundant values")
 217        return Corr(newcontent, prange=self.prange)
 218
 219    def anti_symmetric(self):
 220        """Anti-symmetrize the correlator around x0=0."""
 221        if self.N != 1:
 222            raise Exception('anti_symmetric cannot be safely applied to multi-dimensional correlators.')
 223        if self.T % 2 != 0:
 224            raise Exception("Can not symmetrize odd T")
 225
 226        test = 1 * self
 227        test.gamma_method()
 228        if not all([o.is_zero_within_error(3) for o in test.content[0]]):
 229            warnings.warn("Correlator does not seem to be anti-symmetric around x0=0.", RuntimeWarning)
 230
 231        newcontent = [self.content[0]]
 232        for t in range(1, self.T):
 233            if (self.content[t] is None) or (self.content[self.T - t] is None):
 234                newcontent.append(None)
 235            else:
 236                newcontent.append(0.5 * (self.content[t] - self.content[self.T - t]))
 237        if (all([x is None for x in newcontent])):
 238            raise Exception("Corr could not be symmetrized: No redundant values")
 239        return Corr(newcontent, prange=self.prange)
 240
 241    def is_matrix_symmetric(self):
 242        """Checks whether a correlator matrices is symmetric on every timeslice."""
 243        if self.N == 1:
 244            raise Exception("Only works for correlator matrices.")
 245        for t in range(self.T):
 246            if self[t] is None:
 247                continue
 248            for i in range(self.N):
 249                for j in range(i + 1, self.N):
 250                    if self[t][i, j] is self[t][j, i]:
 251                        continue
 252                    if hash(self[t][i, j]) != hash(self[t][j, i]):
 253                        return False
 254        return True
 255
 256    def matrix_symmetric(self):
 257        """Symmetrizes the correlator matrices on every timeslice."""
 258        if self.N == 1:
 259            raise Exception("Trying to symmetrize a correlator matrix, that already has N=1.")
 260        if self.is_matrix_symmetric():
 261            return 1.0 * self
 262        else:
 263            transposed = [None if _check_for_none(self, G) else G.T for G in self.content]
 264            return 0.5 * (Corr(transposed) + self)
 265
 266    def GEVP(self, t0, ts=None, sort="Eigenvalue", **kwargs):
 267        r'''Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
 268
 269        The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the
 270        largest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing
 271        ```python
 272        C.GEVP(t0=2)[0]  # Ground state vector(s)
 273        C.GEVP(t0=2)[:3]  # Vectors for the lowest three states
 274        ```
 275
 276        Parameters
 277        ----------
 278        t0 : int
 279            The time t0 for the right hand side of the GEVP according to $G(t)v_i=\lambda_i G(t_0)v_i$
 280        ts : int
 281            fixed time $G(t_s)v_i=\lambda_i G(t_0)v_i$ if sort=None.
 282            If sort="Eigenvector" it gives a reference point for the sorting method.
 283        sort : string
 284            If this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.
 285            - "Eigenvalue": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice.
 286            - "Eigenvector": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.
 287              The reference state is identified by its eigenvalue at $t=t_s$.
 288
 289        Other Parameters
 290        ----------------
 291        state : int
 292           Returns only the vector(s) for a specified state. The lowest state is zero.
 293        '''
 294
 295        if self.N == 1:
 296            raise Exception("GEVP methods only works on correlator matrices and not single correlators.")
 297        if ts is not None:
 298            if (ts <= t0):
 299                raise Exception("ts has to be larger than t0.")
 300
 301        if "sorted_list" in kwargs:
 302            warnings.warn("Argument 'sorted_list' is deprecated, use 'sort' instead.", DeprecationWarning)
 303            sort = kwargs.get("sorted_list")
 304
 305        if self.is_matrix_symmetric():
 306            symmetric_corr = self
 307        else:
 308            symmetric_corr = self.matrix_symmetric()
 309
 310        if sort is None:
 311            if (ts is None):
 312                raise Exception("ts is required if sort=None.")
 313            if (self.content[t0] is None) or (self.content[ts] is None):
 314                raise Exception("Corr not defined at t0/ts.")
 315            G0, Gt = np.empty([self.N, self.N], dtype="double"), np.empty([self.N, self.N], dtype="double")
 316            for i in range(self.N):
 317                for j in range(self.N):
 318                    G0[i, j] = symmetric_corr[t0][i, j].value
 319                    Gt[i, j] = symmetric_corr[ts][i, j].value
 320
 321            reordered_vecs = _GEVP_solver(Gt, G0)
 322
 323        elif sort in ["Eigenvalue", "Eigenvector"]:
 324            if sort == "Eigenvalue" and ts is not None:
 325                warnings.warn("ts has no effect when sorting by eigenvalue is chosen.", RuntimeWarning)
 326            all_vecs = [None] * (t0 + 1)
 327            G0 = np.vectorize(lambda x: x.value)(symmetric_corr[t0])
 328            for t in range(t0 + 1, self.T):
 329                try:
 330                    Gt = np.vectorize(lambda x: x.value)(symmetric_corr[t])
 331                    all_vecs.append(_GEVP_solver(Gt, G0))
 332                except Exception:
 333                    all_vecs.append(None)
 334            if sort == "Eigenvector":
 335                if (ts is None):
 336                    raise Exception("ts is required for the Eigenvector sorting method.")
 337                all_vecs = _sort_vectors(all_vecs, ts)
 338
 339            reordered_vecs = [[v[s] if v is not None else None for v in all_vecs] for s in range(self.N)]
 340        else:
 341            raise Exception("Unkown value for 'sort'.")
 342
 343        if "state" in kwargs:
 344            return reordered_vecs[kwargs.get("state")]
 345        else:
 346            return reordered_vecs
 347
 348    def Eigenvalue(self, t0, ts=None, state=0, sort="Eigenvalue"):
 349        """Determines the eigenvalue of the GEVP by solving and projecting the correlator
 350
 351        Parameters
 352        ----------
 353        state : int
 354            The state one is interested in ordered by energy. The lowest state is zero.
 355
 356        All other parameters are identical to the ones of Corr.GEVP.
 357        """
 358        vec = self.GEVP(t0, ts=ts, sort=sort)[state]
 359        return self.projected(vec)
 360
 361    def Hankel(self, N, periodic=False):
 362        """Constructs an NxN Hankel matrix
 363
 364        C(t) c(t+1) ... c(t+n-1)
 365        C(t+1) c(t+2) ... c(t+n)
 366        .................
 367        C(t+(n-1)) c(t+n) ... c(t+2(n-1))
 368
 369        Parameters
 370        ----------
 371        N : int
 372            Dimension of the Hankel matrix
 373        periodic : bool, optional
 374            determines whether the matrix is extended periodically
 375        """
 376
 377        if self.N != 1:
 378            raise Exception("Multi-operator Prony not implemented!")
 379
 380        array = np.empty([N, N], dtype="object")
 381        new_content = []
 382        for t in range(self.T):
 383            new_content.append(array.copy())
 384
 385        def wrap(i):
 386            while i >= self.T:
 387                i -= self.T
 388            return i
 389
 390        for t in range(self.T):
 391            for i in range(N):
 392                for j in range(N):
 393                    if periodic:
 394                        new_content[t][i, j] = self.content[wrap(t + i + j)][0]
 395                    elif (t + i + j) >= self.T:
 396                        new_content[t] = None
 397                    else:
 398                        new_content[t][i, j] = self.content[t + i + j][0]
 399
 400        return Corr(new_content)
 401
 402    def roll(self, dt):
 403        """Periodically shift the correlator by dt timeslices
 404
 405        Parameters
 406        ----------
 407        dt : int
 408            number of timeslices
 409        """
 410        return Corr(list(np.roll(np.array(self.content, dtype=object), dt)))
 411
 412    def reverse(self):
 413        """Reverse the time ordering of the Corr"""
 414        return Corr(self.content[:: -1])
 415
 416    def thin(self, spacing=2, offset=0):
 417        """Thin out a correlator to suppress correlations
 418
 419        Parameters
 420        ----------
 421        spacing : int
 422            Keep only every 'spacing'th entry of the correlator
 423        offset : int
 424            Offset the equal spacing
 425        """
 426        new_content = []
 427        for t in range(self.T):
 428            if (offset + t) % spacing != 0:
 429                new_content.append(None)
 430            else:
 431                new_content.append(self.content[t])
 432        return Corr(new_content)
 433
 434    def correlate(self, partner):
 435        """Correlate the correlator with another correlator or Obs
 436
 437        Parameters
 438        ----------
 439        partner : Obs or Corr
 440            partner to correlate the correlator with.
 441            Can either be an Obs which is correlated with all entries of the
 442            correlator or a Corr of same length.
 443        """
 444        if self.N != 1:
 445            raise Exception("Only one-dimensional correlators can be safely correlated.")
 446        new_content = []
 447        for x0, t_slice in enumerate(self.content):
 448            if _check_for_none(self, t_slice):
 449                new_content.append(None)
 450            else:
 451                if isinstance(partner, Corr):
 452                    if _check_for_none(partner, partner.content[x0]):
 453                        new_content.append(None)
 454                    else:
 455                        new_content.append(np.array([correlate(o, partner.content[x0][0]) for o in t_slice]))
 456                elif isinstance(partner, Obs):  # Should this include CObs?
 457                    new_content.append(np.array([correlate(o, partner) for o in t_slice]))
 458                else:
 459                    raise Exception("Can only correlate with an Obs or a Corr.")
 460
 461        return Corr(new_content)
 462
 463    def reweight(self, weight, **kwargs):
 464        """Reweight the correlator.
 465
 466        Parameters
 467        ----------
 468        weight : Obs
 469            Reweighting factor. An Observable that has to be defined on a superset of the
 470            configurations in obs[i].idl for all i.
 471        all_configs : bool
 472            if True, the reweighted observables are normalized by the average of
 473            the reweighting factor on all configurations in weight.idl and not
 474            on the configurations in obs[i].idl.
 475        """
 476        if self.N != 1:
 477            raise Exception("Reweighting only implemented for one-dimensional correlators.")
 478        new_content = []
 479        for t_slice in self.content:
 480            if _check_for_none(self, t_slice):
 481                new_content.append(None)
 482            else:
 483                new_content.append(np.array(reweight(weight, t_slice, **kwargs)))
 484        return Corr(new_content)
 485
 486    def T_symmetry(self, partner, parity=+1):
 487        """Return the time symmetry average of the correlator and its partner
 488
 489        Parameters
 490        ----------
 491        partner : Corr
 492            Time symmetry partner of the Corr
 493        partity : int
 494            Parity quantum number of the correlator, can be +1 or -1
 495        """
 496        if self.N != 1:
 497            raise Exception("T_symmetry only implemented for one-dimensional correlators.")
 498        if not isinstance(partner, Corr):
 499            raise Exception("T partner has to be a Corr object.")
 500        if parity not in [+1, -1]:
 501            raise Exception("Parity has to be +1 or -1.")
 502        T_partner = parity * partner.reverse()
 503
 504        t_slices = []
 505        test = (self - T_partner)
 506        test.gamma_method()
 507        for x0, t_slice in enumerate(test.content):
 508            if t_slice is not None:
 509                if not t_slice[0].is_zero_within_error(5):
 510                    t_slices.append(x0)
 511        if t_slices:
 512            warnings.warn("T symmetry partners do not agree within 5 sigma on time slices " + str(t_slices) + ".", RuntimeWarning)
 513
 514        return (self + T_partner) / 2
 515
 516    def deriv(self, variant="symmetric"):
 517        """Return the first derivative of the correlator with respect to x0.
 518
 519        Parameters
 520        ----------
 521        variant : str
 522            decides which definition of the finite differences derivative is used.
 523            Available choice: symmetric, forward, backward, improved, log, default: symmetric
 524        """
 525        if self.N != 1:
 526            raise Exception("deriv only implemented for one-dimensional correlators.")
 527        if variant == "symmetric":
 528            newcontent = []
 529            for t in range(1, self.T - 1):
 530                if (self.content[t - 1] is None) or (self.content[t + 1] is None):
 531                    newcontent.append(None)
 532                else:
 533                    newcontent.append(0.5 * (self.content[t + 1] - self.content[t - 1]))
 534            if (all([x is None for x in newcontent])):
 535                raise Exception('Derivative is undefined at all timeslices')
 536            return Corr(newcontent, padding=[1, 1])
 537        elif variant == "forward":
 538            newcontent = []
 539            for t in range(self.T - 1):
 540                if (self.content[t] is None) or (self.content[t + 1] is None):
 541                    newcontent.append(None)
 542                else:
 543                    newcontent.append(self.content[t + 1] - self.content[t])
 544            if (all([x is None for x in newcontent])):
 545                raise Exception("Derivative is undefined at all timeslices")
 546            return Corr(newcontent, padding=[0, 1])
 547        elif variant == "backward":
 548            newcontent = []
 549            for t in range(1, self.T):
 550                if (self.content[t - 1] is None) or (self.content[t] is None):
 551                    newcontent.append(None)
 552                else:
 553                    newcontent.append(self.content[t] - self.content[t - 1])
 554            if (all([x is None for x in newcontent])):
 555                raise Exception("Derivative is undefined at all timeslices")
 556            return Corr(newcontent, padding=[1, 0])
 557        elif variant == "improved":
 558            newcontent = []
 559            for t in range(2, self.T - 2):
 560                if (self.content[t - 2] is None) or (self.content[t - 1] is None) or (self.content[t + 1] is None) or (self.content[t + 2] is None):
 561                    newcontent.append(None)
 562                else:
 563                    newcontent.append((1 / 12) * (self.content[t - 2] - 8 * self.content[t - 1] + 8 * self.content[t + 1] - self.content[t + 2]))
 564            if (all([x is None for x in newcontent])):
 565                raise Exception('Derivative is undefined at all timeslices')
 566            return Corr(newcontent, padding=[2, 2])
 567        elif variant == 'log':
 568            newcontent = []
 569            for t in range(self.T):
 570                if (self.content[t] is None) or (self.content[t] <= 0):
 571                    newcontent.append(None)
 572                else:
 573                    newcontent.append(np.log(self.content[t]))
 574            if (all([x is None for x in newcontent])):
 575                raise Exception("Log is undefined at all timeslices")
 576            logcorr = Corr(newcontent)
 577            return self * logcorr.deriv('symmetric')
 578        else:
 579            raise Exception("Unknown variant.")
 580
 581    def second_deriv(self, variant="symmetric"):
 582        """Return the second derivative of the correlator with respect to x0.
 583
 584        Parameters
 585        ----------
 586        variant : str
 587            decides which definition of the finite differences derivative is used.
 588            Available choice: symmetric, improved, log, default: symmetric
 589        """
 590        if self.N != 1:
 591            raise Exception("second_deriv only implemented for one-dimensional correlators.")
 592        if variant == "symmetric":
 593            newcontent = []
 594            for t in range(1, self.T - 1):
 595                if (self.content[t - 1] is None) or (self.content[t + 1] is None):
 596                    newcontent.append(None)
 597                else:
 598                    newcontent.append((self.content[t + 1] - 2 * self.content[t] + self.content[t - 1]))
 599            if (all([x is None for x in newcontent])):
 600                raise Exception("Derivative is undefined at all timeslices")
 601            return Corr(newcontent, padding=[1, 1])
 602        elif variant == "improved":
 603            newcontent = []
 604            for t in range(2, self.T - 2):
 605                if (self.content[t - 2] is None) or (self.content[t - 1] is None) or (self.content[t] is None) or (self.content[t + 1] is None) or (self.content[t + 2] is None):
 606                    newcontent.append(None)
 607                else:
 608                    newcontent.append((1 / 12) * (-self.content[t + 2] + 16 * self.content[t + 1] - 30 * self.content[t] + 16 * self.content[t - 1] - self.content[t - 2]))
 609            if (all([x is None for x in newcontent])):
 610                raise Exception("Derivative is undefined at all timeslices")
 611            return Corr(newcontent, padding=[2, 2])
 612        elif variant == 'log':
 613            newcontent = []
 614            for t in range(self.T):
 615                if (self.content[t] is None) or (self.content[t] <= 0):
 616                    newcontent.append(None)
 617                else:
 618                    newcontent.append(np.log(self.content[t]))
 619            if (all([x is None for x in newcontent])):
 620                raise Exception("Log is undefined at all timeslices")
 621            logcorr = Corr(newcontent)
 622            return self * (logcorr.second_deriv('symmetric') + (logcorr.deriv('symmetric'))**2)
 623        else:
 624            raise Exception("Unknown variant.")
 625
 626    def m_eff(self, variant='log', guess=1.0):
 627        """Returns the effective mass of the correlator as correlator object
 628
 629        Parameters
 630        ----------
 631        variant : str
 632            log : uses the standard effective mass log(C(t) / C(t+1))
 633            cosh, periodic : Use periodicitiy of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.
 634            sinh : Use anti-periodicitiy of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.
 635            See, e.g., arXiv:1205.5380
 636            arccosh : Uses the explicit form of the symmetrized correlator (not recommended)
 637            logsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
 638        guess : float
 639            guess for the root finder, only relevant for the root variant
 640        """
 641        if self.N != 1:
 642            raise Exception('Correlator must be projected before getting m_eff')
 643        if variant == 'log':
 644            newcontent = []
 645            for t in range(self.T - 1):
 646                if ((self.content[t] is None) or (self.content[t + 1] is None)) or (self.content[t + 1][0].value == 0):
 647                    newcontent.append(None)
 648                elif self.content[t][0].value / self.content[t + 1][0].value < 0:
 649                    newcontent.append(None)
 650                else:
 651                    newcontent.append(self.content[t] / self.content[t + 1])
 652            if (all([x is None for x in newcontent])):
 653                raise Exception('m_eff is undefined at all timeslices')
 654
 655            return np.log(Corr(newcontent, padding=[0, 1]))
 656
 657        elif variant == 'logsym':
 658            newcontent = []
 659            for t in range(1, self.T - 1):
 660                if ((self.content[t - 1] is None) or (self.content[t + 1] is None)) or (self.content[t + 1][0].value == 0):
 661                    newcontent.append(None)
 662                elif self.content[t - 1][0].value / self.content[t + 1][0].value < 0:
 663                    newcontent.append(None)
 664                else:
 665                    newcontent.append(self.content[t - 1] / self.content[t + 1])
 666            if (all([x is None for x in newcontent])):
 667                raise Exception('m_eff is undefined at all timeslices')
 668
 669            return np.log(Corr(newcontent, padding=[1, 1])) / 2
 670
 671        elif variant in ['periodic', 'cosh', 'sinh']:
 672            if variant in ['periodic', 'cosh']:
 673                func = anp.cosh
 674            else:
 675                func = anp.sinh
 676
 677            def root_function(x, d):
 678                return func(x * (t - self.T / 2)) / func(x * (t + 1 - self.T / 2)) - d
 679
 680            newcontent = []
 681            for t in range(self.T - 1):
 682                if (self.content[t] is None) or (self.content[t + 1] is None) or (self.content[t + 1][0].value == 0):
 683                    newcontent.append(None)
 684                # Fill the two timeslices in the middle of the lattice with their predecessors
 685                elif variant == 'sinh' and t in [self.T / 2, self.T / 2 - 1]:
 686                    newcontent.append(newcontent[-1])
 687                elif self.content[t][0].value / self.content[t + 1][0].value < 0:
 688                    newcontent.append(None)
 689                else:
 690                    newcontent.append(np.abs(find_root(self.content[t][0] / self.content[t + 1][0], root_function, guess=guess)))
 691            if (all([x is None for x in newcontent])):
 692                raise Exception('m_eff is undefined at all timeslices')
 693
 694            return Corr(newcontent, padding=[0, 1])
 695
 696        elif variant == 'arccosh':
 697            newcontent = []
 698            for t in range(1, self.T - 1):
 699                if (self.content[t] is None) or (self.content[t + 1] is None) or (self.content[t - 1] is None) or (self.content[t][0].value == 0):
 700                    newcontent.append(None)
 701                else:
 702                    newcontent.append((self.content[t + 1] + self.content[t - 1]) / (2 * self.content[t]))
 703            if (all([x is None for x in newcontent])):
 704                raise Exception("m_eff is undefined at all timeslices")
 705            return np.arccosh(Corr(newcontent, padding=[1, 1]))
 706
 707        else:
 708            raise Exception('Unknown variant.')
 709
 710    def fit(self, function, fitrange=None, silent=False, **kwargs):
 711        r'''Fits function to the data
 712
 713        Parameters
 714        ----------
 715        function : obj
 716            function to fit to the data. See fits.least_squares for details.
 717        fitrange : list
 718            Two element list containing the timeslices on which the fit is supposed to start and stop.
 719            Caution: This range is inclusive as opposed to standard python indexing.
 720            `fitrange=[4, 6]` corresponds to the three entries 4, 5 and 6.
 721            If not specified, self.prange or all timeslices are used.
 722        silent : bool
 723            Decides whether output is printed to the standard output.
 724        '''
 725        if self.N != 1:
 726            raise Exception("Correlator must be projected before fitting")
 727
 728        if fitrange is None:
 729            if self.prange:
 730                fitrange = self.prange
 731            else:
 732                fitrange = [0, self.T - 1]
 733        else:
 734            if not isinstance(fitrange, list):
 735                raise Exception("fitrange has to be a list with two elements")
 736            if len(fitrange) != 2:
 737                raise Exception("fitrange has to have exactly two elements [fit_start, fit_stop]")
 738
 739        xs = [x for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]
 740        ys = [self.content[x][0] for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]
 741        result = least_squares(xs, ys, function, silent=silent, **kwargs)
 742        return result
 743
 744    def plateau(self, plateau_range=None, method="fit", auto_gamma=False):
 745        """ Extract a plateau value from a Corr object
 746
 747        Parameters
 748        ----------
 749        plateau_range : list
 750            list with two entries, indicating the first and the last timeslice
 751            of the plateau region.
 752        method : str
 753            method to extract the plateau.
 754                'fit' fits a constant to the plateau region
 755                'avg', 'average' or 'mean' just average over the given timeslices.
 756        auto_gamma : bool
 757            apply gamma_method with default parameters to the Corr. Defaults to None
 758        """
 759        if not plateau_range:
 760            if self.prange:
 761                plateau_range = self.prange
 762            else:
 763                raise Exception("no plateau range provided")
 764        if self.N != 1:
 765            raise Exception("Correlator must be projected before getting a plateau.")
 766        if (all([self.content[t] is None for t in range(plateau_range[0], plateau_range[1] + 1)])):
 767            raise Exception("plateau is undefined at all timeslices in plateaurange.")
 768        if auto_gamma:
 769            self.gamma_method()
 770        if method == "fit":
 771            def const_func(a, t):
 772                return a[0]
 773            return self.fit(const_func, plateau_range)[0]
 774        elif method in ["avg", "average", "mean"]:
 775            returnvalue = np.mean([item[0] for item in self.content[plateau_range[0]:plateau_range[1] + 1] if item is not None])
 776            return returnvalue
 777
 778        else:
 779            raise Exception("Unsupported plateau method: " + method)
 780
 781    def set_prange(self, prange):
 782        """Sets the attribute prange of the Corr object."""
 783        if not len(prange) == 2:
 784            raise Exception("prange must be a list or array with two values")
 785        if not ((isinstance(prange[0], int)) and (isinstance(prange[1], int))):
 786            raise Exception("Start and end point must be integers")
 787        if not (0 <= prange[0] <= self.T and 0 <= prange[1] <= self.T and prange[0] < prange[1]):
 788            raise Exception("Start and end point must define a range in the interval 0,T")
 789
 790        self.prange = prange
 791        return
 792
 793    def show(self, x_range=None, comp=None, y_range=None, logscale=False, plateau=None, fit_res=None, ylabel=None, save=None, auto_gamma=False, hide_sigma=None, references=None, title=None):
 794        """Plots the correlator using the tag of the correlator as label if available.
 795
 796        Parameters
 797        ----------
 798        x_range : list
 799            list of two values, determining the range of the x-axis e.g. [4, 8].
 800        comp : Corr or list of Corr
 801            Correlator or list of correlators which are plotted for comparison.
 802            The tags of these correlators are used as labels if available.
 803        logscale : bool
 804            Sets y-axis to logscale.
 805        plateau : Obs
 806            Plateau value to be visualized in the figure.
 807        fit_res : Fit_result
 808            Fit_result object to be visualized.
 809        ylabel : str
 810            Label for the y-axis.
 811        save : str
 812            path to file in which the figure should be saved.
 813        auto_gamma : bool
 814            Apply the gamma method with standard parameters to all correlators and plateau values before plotting.
 815        hide_sigma : float
 816            Hides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
 817        references : list
 818            List of floating point values that are displayed as horizontal lines for reference.
 819        title : string
 820            Optional title of the figure.
 821        """
 822        if self.N != 1:
 823            raise Exception("Correlator must be projected before plotting")
 824
 825        if auto_gamma:
 826            self.gamma_method()
 827
 828        if x_range is None:
 829            x_range = [0, self.T - 1]
 830
 831        fig = plt.figure()
 832        ax1 = fig.add_subplot(111)
 833
 834        x, y, y_err = self.plottable()
 835        if hide_sigma:
 836            hide_from = np.argmax((hide_sigma * np.array(y_err[1:])) > np.abs(y[1:])) - 1
 837        else:
 838            hide_from = None
 839        ax1.errorbar(x[:hide_from], y[:hide_from], y_err[:hide_from], label=self.tag)
 840        if logscale:
 841            ax1.set_yscale('log')
 842        else:
 843            if y_range is None:
 844                try:
 845                    y_min = min([(x[0].value - x[0].dvalue) for x in self.content[x_range[0]: x_range[1] + 1] if (x is not None) and x[0].dvalue < 2 * np.abs(x[0].value)])
 846                    y_max = max([(x[0].value + x[0].dvalue) for x in self.content[x_range[0]: x_range[1] + 1] if (x is not None) and x[0].dvalue < 2 * np.abs(x[0].value)])
 847                    ax1.set_ylim([y_min - 0.1 * (y_max - y_min), y_max + 0.1 * (y_max - y_min)])
 848                except Exception:
 849                    pass
 850            else:
 851                ax1.set_ylim(y_range)
 852        if comp:
 853            if isinstance(comp, (Corr, list)):
 854                for corr in comp if isinstance(comp, list) else [comp]:
 855                    if auto_gamma:
 856                        corr.gamma_method()
 857                    x, y, y_err = corr.plottable()
 858                    if hide_sigma:
 859                        hide_from = np.argmax((hide_sigma * np.array(y_err[1:])) > np.abs(y[1:])) - 1
 860                    else:
 861                        hide_from = None
 862                    ax1.errorbar(x[:hide_from], y[:hide_from], y_err[:hide_from], label=corr.tag, mfc=plt.rcParams['axes.facecolor'])
 863            else:
 864                raise Exception("'comp' must be a correlator or a list of correlators.")
 865
 866        if plateau:
 867            if isinstance(plateau, Obs):
 868                if auto_gamma:
 869                    plateau.gamma_method()
 870                ax1.axhline(y=plateau.value, linewidth=2, color=plt.rcParams['text.color'], alpha=0.6, marker=',', ls='--', label=str(plateau))
 871                ax1.axhspan(plateau.value - plateau.dvalue, plateau.value + plateau.dvalue, alpha=0.25, color=plt.rcParams['text.color'], ls='-')
 872            else:
 873                raise Exception("'plateau' must be an Obs")
 874
 875        if references:
 876            if isinstance(references, list):
 877                for ref in references:
 878                    ax1.axhline(y=ref, linewidth=1, color=plt.rcParams['text.color'], alpha=0.6, marker=',', ls='--')
 879            else:
 880                raise Exception("'references' must be a list of floating pint values.")
 881
 882        if self.prange:
 883            ax1.axvline(self.prange[0], 0, 1, ls='-', marker=',')
 884            ax1.axvline(self.prange[1], 0, 1, ls='-', marker=',')
 885
 886        if fit_res:
 887            x_samples = np.arange(x_range[0], x_range[1] + 1, 0.05)
 888            ax1.plot(x_samples,
 889                     fit_res.fit_function([o.value for o in fit_res.fit_parameters], x_samples),
 890                     ls='-', marker=',', lw=2)
 891
 892        ax1.set_xlabel(r'$x_0 / a$')
 893        if ylabel:
 894            ax1.set_ylabel(ylabel)
 895        ax1.set_xlim([x_range[0] - 0.5, x_range[1] + 0.5])
 896
 897        handles, labels = ax1.get_legend_handles_labels()
 898        if labels:
 899            ax1.legend()
 900
 901        if title:
 902            plt.title(title)
 903
 904        plt.draw()
 905
 906        if save:
 907            if isinstance(save, str):
 908                fig.savefig(save, bbox_inches='tight')
 909            else:
 910                raise Exception("'save' has to be a string.")
 911
 912    def spaghetti_plot(self, logscale=True):
 913        """Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.
 914
 915        Parameters
 916        ----------
 917        logscale : bool
 918            Determines whether the scale of the y-axis is logarithmic or standard.
 919        """
 920        if self.N != 1:
 921            raise Exception("Correlator needs to be projected first.")
 922
 923        mc_names = list(set([item for sublist in [o[0].mc_names for o in self.content if o is not None] for item in sublist]))
 924        x0_vals = [n for (n, o) in zip(np.arange(self.T), self.content) if o is not None]
 925
 926        for name in mc_names:
 927            data = np.array([o[0].deltas[name] + o[0].r_values[name] for o in self.content if o is not None]).T
 928
 929            fig = plt.figure()
 930            ax = fig.add_subplot(111)
 931            for dat in data:
 932                ax.plot(x0_vals, dat, ls='-', marker='')
 933
 934            if logscale is True:
 935                ax.set_yscale('log')
 936
 937            ax.set_xlabel(r'$x_0 / a$')
 938            plt.title(name)
 939            plt.draw()
 940
 941    def dump(self, filename, datatype="json.gz", **kwargs):
 942        """Dumps the Corr into a file of chosen type
 943        Parameters
 944        ----------
 945        filename : str
 946            Name of the file to be saved.
 947        datatype : str
 948            Format of the exported file. Supported formats include
 949            "json.gz" and "pickle"
 950        path : str
 951            specifies a custom path for the file (default '.')
 952        """
 953        if datatype == "json.gz":
 954            from .input.json import dump_to_json
 955            if 'path' in kwargs:
 956                file_name = kwargs.get('path') + '/' + filename
 957            else:
 958                file_name = filename
 959            dump_to_json(self, file_name)
 960        elif datatype == "pickle":
 961            dump_object(self, filename, **kwargs)
 962        else:
 963            raise Exception("Unknown datatype " + str(datatype))
 964
 965    def print(self, print_range=None):
 966        print(self.__repr__(print_range))
 967
 968    def __repr__(self, print_range=None):
 969        if print_range is None:
 970            print_range = [0, None]
 971
 972        content_string = ""
 973        content_string += "Corr T=" + str(self.T) + " N=" + str(self.N) + "\n"  # +" filled with"+ str(type(self.content[0][0])) there should be a good solution here
 974
 975        if self.tag is not None:
 976            content_string += "Description: " + self.tag + "\n"
 977        if self.N != 1:
 978            return content_string
 979
 980        if print_range[1]:
 981            print_range[1] += 1
 982        content_string += 'x0/a\tCorr(x0/a)\n------------------\n'
 983        for i, sub_corr in enumerate(self.content[print_range[0]:print_range[1]]):
 984            if sub_corr is None:
 985                content_string += str(i + print_range[0]) + '\n'
 986            else:
 987                content_string += str(i + print_range[0])
 988                for element in sub_corr:
 989                    content_string += '\t' + ' ' * int(element >= 0) + str(element)
 990                content_string += '\n'
 991        return content_string
 992
 993    def __str__(self):
 994        return self.__repr__()
 995
 996    # We define the basic operations, that can be performed with correlators.
 997    # While */+- get defined here, they only work for Corr*Obs and not Obs*Corr.
 998    # This is because Obs*Corr checks Obs.__mul__ first and does not catch an exception.
 999    # One could try and tell Obs to check if the y in __mul__ is a Corr and
1000
1001    def __add__(self, y):
1002        if isinstance(y, Corr):
1003            if ((self.N != y.N) or (self.T != y.T)):
1004                raise Exception("Addition of Corrs with different shape")
1005            newcontent = []
1006            for t in range(self.T):
1007                if _check_for_none(self, self.content[t]) or _check_for_none(y, y.content[t]):
1008                    newcontent.append(None)
1009                else:
1010                    newcontent.append(self.content[t] + y.content[t])
1011            return Corr(newcontent)
1012
1013        elif isinstance(y, (Obs, int, float, CObs)):
1014            newcontent = []
1015            for t in range(self.T):
1016                if _check_for_none(self, self.content[t]):
1017                    newcontent.append(None)
1018                else:
1019                    newcontent.append(self.content[t] + y)
1020            return Corr(newcontent, prange=self.prange)
1021        elif isinstance(y, np.ndarray):
1022            if y.shape == (self.T,):
1023                return Corr(list((np.array(self.content).T + y).T))
1024            else:
1025                raise ValueError("operands could not be broadcast together")
1026        else:
1027            raise TypeError("Corr + wrong type")
1028
1029    def __mul__(self, y):
1030        if isinstance(y, Corr):
1031            if not ((self.N == 1 or y.N == 1 or self.N == y.N) and self.T == y.T):
1032                raise Exception("Multiplication of Corr object requires N=N or N=1 and T=T")
1033            newcontent = []
1034            for t in range(self.T):
1035                if _check_for_none(self, self.content[t]) or _check_for_none(y, y.content[t]):
1036                    newcontent.append(None)
1037                else:
1038                    newcontent.append(self.content[t] * y.content[t])
1039            return Corr(newcontent)
1040
1041        elif isinstance(y, (Obs, int, float, CObs)):
1042            newcontent = []
1043            for t in range(self.T):
1044                if _check_for_none(self, self.content[t]):
1045                    newcontent.append(None)
1046                else:
1047                    newcontent.append(self.content[t] * y)
1048            return Corr(newcontent, prange=self.prange)
1049        elif isinstance(y, np.ndarray):
1050            if y.shape == (self.T,):
1051                return Corr(list((np.array(self.content).T * y).T))
1052            else:
1053                raise ValueError("operands could not be broadcast together")
1054        else:
1055            raise TypeError("Corr * wrong type")
1056
1057    def __truediv__(self, y):
1058        if isinstance(y, Corr):
1059            if not ((self.N == 1 or y.N == 1 or self.N == y.N) and self.T == y.T):
1060                raise Exception("Multiplication of Corr object requires N=N or N=1 and T=T")
1061            newcontent = []
1062            for t in range(self.T):
1063                if _check_for_none(self, self.content[t]) or _check_for_none(y, y.content[t]):
1064                    newcontent.append(None)
1065                else:
1066                    newcontent.append(self.content[t] / y.content[t])
1067            for t in range(self.T):
1068                if _check_for_none(self, newcontent[t]):
1069                    continue
1070                if np.isnan(np.sum(newcontent[t]).value):
1071                    newcontent[t] = None
1072
1073            if all([item is None for item in newcontent]):
1074                raise Exception("Division returns completely undefined correlator")
1075            return Corr(newcontent)
1076
1077        elif isinstance(y, (Obs, CObs)):
1078            if isinstance(y, Obs):
1079                if y.value == 0:
1080                    raise Exception('Division by zero will return undefined correlator')
1081            if isinstance(y, CObs):
1082                if y.is_zero():
1083                    raise Exception('Division by zero will return undefined correlator')
1084
1085            newcontent = []
1086            for t in range(self.T):
1087                if _check_for_none(self, self.content[t]):
1088                    newcontent.append(None)
1089                else:
1090                    newcontent.append(self.content[t] / y)
1091            return Corr(newcontent, prange=self.prange)
1092
1093        elif isinstance(y, (int, float)):
1094            if y == 0:
1095                raise Exception('Division by zero will return undefined correlator')
1096            newcontent = []
1097            for t in range(self.T):
1098                if _check_for_none(self, self.content[t]):
1099                    newcontent.append(None)
1100                else:
1101                    newcontent.append(self.content[t] / y)
1102            return Corr(newcontent, prange=self.prange)
1103        elif isinstance(y, np.ndarray):
1104            if y.shape == (self.T,):
1105                return Corr(list((np.array(self.content).T / y).T))
1106            else:
1107                raise ValueError("operands could not be broadcast together")
1108        else:
1109            raise TypeError('Corr / wrong type')
1110
1111    def __neg__(self):
1112        newcontent = [None if _check_for_none(self, item) else -1. * item for item in self.content]
1113        return Corr(newcontent, prange=self.prange)
1114
1115    def __sub__(self, y):
1116        return self + (-y)
1117
1118    def __pow__(self, y):
1119        if isinstance(y, (Obs, int, float, CObs)):
1120            newcontent = [None if _check_for_none(self, item) else item**y for item in self.content]
1121            return Corr(newcontent, prange=self.prange)
1122        else:
1123            raise TypeError('Type of exponent not supported')
1124
1125    def __abs__(self):
1126        newcontent = [None if _check_for_none(self, item) else np.abs(item) for item in self.content]
1127        return Corr(newcontent, prange=self.prange)
1128
1129    # The numpy functions:
1130    def sqrt(self):
1131        return self ** 0.5
1132
1133    def log(self):
1134        newcontent = [None if _check_for_none(self, item) else np.log(item) for item in self.content]
1135        return Corr(newcontent, prange=self.prange)
1136
1137    def exp(self):
1138        newcontent = [None if _check_for_none(self, item) else np.exp(item) for item in self.content]
1139        return Corr(newcontent, prange=self.prange)
1140
1141    def _apply_func_to_corr(self, func):
1142        newcontent = [None if _check_for_none(self, item) else func(item) for item in self.content]
1143        for t in range(self.T):
1144            if _check_for_none(self, newcontent[t]):
1145                continue
1146            if np.isnan(np.sum(newcontent[t]).value):
1147                newcontent[t] = None
1148        if all([item is None for item in newcontent]):
1149            raise Exception('Operation returns undefined correlator')
1150        return Corr(newcontent)
1151
1152    def sin(self):
1153        return self._apply_func_to_corr(np.sin)
1154
1155    def cos(self):
1156        return self._apply_func_to_corr(np.cos)
1157
1158    def tan(self):
1159        return self._apply_func_to_corr(np.tan)
1160
1161    def sinh(self):
1162        return self._apply_func_to_corr(np.sinh)
1163
1164    def cosh(self):
1165        return self._apply_func_to_corr(np.cosh)
1166
1167    def tanh(self):
1168        return self._apply_func_to_corr(np.tanh)
1169
1170    def arcsin(self):
1171        return self._apply_func_to_corr(np.arcsin)
1172
1173    def arccos(self):
1174        return self._apply_func_to_corr(np.arccos)
1175
1176    def arctan(self):
1177        return self._apply_func_to_corr(np.arctan)
1178
1179    def arcsinh(self):
1180        return self._apply_func_to_corr(np.arcsinh)
1181
1182    def arccosh(self):
1183        return self._apply_func_to_corr(np.arccosh)
1184
1185    def arctanh(self):
1186        return self._apply_func_to_corr(np.arctanh)
1187
1188    # Right hand side operations (require tweak in main module to work)
1189    def __radd__(self, y):
1190        return self + y
1191
1192    def __rsub__(self, y):
1193        return -self + y
1194
1195    def __rmul__(self, y):
1196        return self * y
1197
1198    def __rtruediv__(self, y):
1199        return (self / y) ** (-1)
1200
1201    @property
1202    def real(self):
1203        def return_real(obs_OR_cobs):
1204            if isinstance(obs_OR_cobs, CObs):
1205                return obs_OR_cobs.real
1206            else:
1207                return obs_OR_cobs
1208
1209        return self._apply_func_to_corr(return_real)
1210
1211    @property
1212    def imag(self):
1213        def return_imag(obs_OR_cobs):
1214            if isinstance(obs_OR_cobs, CObs):
1215                return obs_OR_cobs.imag
1216            else:
1217                return obs_OR_cobs * 0  # So it stays the right type
1218
1219        return self._apply_func_to_corr(return_imag)
1220
1221    def prune(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):
1222        r''' Project large correlation matrix to lowest states
1223
1224        This method can be used to reduce the size of an (N x N) correlation matrix
1225        to (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise
1226        is still small.
1227
1228        Parameters
1229        ----------
1230        Ntrunc: int
1231            Rank of the target matrix.
1232        tproj: int
1233            Time where the eigenvectors are evaluated, corresponds to ts in the GEVP method.
1234            The default value is 3.
1235        t0proj: int
1236            Time where the correlation matrix is inverted. Choosing t0proj=1 is strongly
1237            discouraged for O(a) improved theories, since the correctness of the procedure
1238            cannot be granted in this case. The default value is 2.
1239        basematrix : Corr
1240            Correlation matrix that is used to determine the eigenvectors of the
1241            lowest states based on a GEVP. basematrix is taken to be the Corr itself if
1242            is is not specified.
1243
1244        Notes
1245        -----
1246        We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving
1247        the GEVP $$C(t) v_n(t, t_0) = \lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \equiv t_\mathrm{proj}$
1248        and $t_0 \equiv t_{0, \mathrm{proj}}$. The target matrix is projected onto the subspace of the
1249        resulting eigenvectors $v_n, n=1,\dots,N_\mathrm{trunc}$ via
1250        $$G^\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large
1251        correlation matrix and to remove some noise that is added by irrelevant operators.
1252        This may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated
1253        bound $t_0 \leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.
1254        '''
1255
1256        if self.N == 1:
1257            raise Exception('Method cannot be applied to one-dimensional correlators.')
1258        if basematrix is None:
1259            basematrix = self
1260        if Ntrunc >= basematrix.N:
1261            raise Exception('Cannot truncate using Ntrunc <= %d' % (basematrix.N))
1262        if basematrix.N != self.N:
1263            raise Exception('basematrix and targetmatrix have to be of the same size.')
1264
1265        evecs = basematrix.GEVP(t0proj, tproj, sort=None)[:Ntrunc]
1266
1267        tmpmat = np.empty((Ntrunc, Ntrunc), dtype=object)
1268        rmat = []
1269        for t in range(basematrix.T):
1270            for i in range(Ntrunc):
1271                for j in range(Ntrunc):
1272                    tmpmat[i][j] = evecs[i].T @ self[t] @ evecs[j]
1273            rmat.append(np.copy(tmpmat))
1274
1275        newcontent = [None if (self.content[t] is None) else rmat[t] for t in range(self.T)]
1276        return Corr(newcontent)

The class for a correlator (time dependent sequence of pe.Obs).

Everything, this class does, can be achieved using lists or arrays of Obs. But it is simply more convenient to have a dedicated object for correlators. One often wants to add or multiply correlators of the same length at every timeslice and it is inconvenient to iterate over all timeslices for every operation. This is especially true, when dealing with matrices.

The correlator can have two types of content: An Obs at every timeslice OR a GEVP matrix at every timeslice. Other dependency (eg. spatial) are not supported.

Corr(data_input, padding=[0, 0], prange=None)
27    def __init__(self, data_input, padding=[0, 0], prange=None):
28        """ Initialize a Corr object.
29
30        Parameters
31        ----------
32        data_input : list or array
33            list of Obs or list of arrays of Obs or array of Corrs
34        padding : list, optional
35            List with two entries where the first labels the padding
36            at the front of the correlator and the second the padding
37            at the back.
38        prange : list, optional
39            List containing the first and last timeslice of the plateau
40            region indentified for this correlator.
41        """
42
43        if isinstance(data_input, np.ndarray):
44
45            # This only works, if the array fulfills the conditions below
46            if not len(data_input.shape) == 2 and data_input.shape[0] == data_input.shape[1]:
47                raise Exception("Incompatible array shape")
48            if not all([isinstance(item, Corr) for item in data_input.flatten()]):
49                raise Exception("If the input is an array, its elements must be of type pe.Corr")
50            if not all([item.N == 1 for item in data_input.flatten()]):
51                raise Exception("Can only construct matrix correlator from single valued correlators")
52            if not len(set([item.T for item in data_input.flatten()])) == 1:
53                raise Exception("All input Correlators must be defined over the same timeslices.")
54
55            T = data_input[0, 0].T
56            N = data_input.shape[0]
57            input_as_list = []
58            for t in range(T):
59                if any([(item.content[t] is None) for item in data_input.flatten()]):
60                    if not all([(item.content[t] is None) for item in data_input.flatten()]):
61                        warnings.warn("Input ill-defined at different timeslices. Conversion leads to data loss!", RuntimeWarning)
62                    input_as_list.append(None)
63                else:
64                    array_at_timeslace = np.empty([N, N], dtype="object")
65                    for i in range(N):
66                        for j in range(N):
67                            array_at_timeslace[i, j] = data_input[i, j][t]
68                    input_as_list.append(array_at_timeslace)
69            data_input = input_as_list
70
71        if isinstance(data_input, list):
72
73            if all([isinstance(item, (Obs, CObs)) for item in data_input]):
74                _assert_equal_properties(data_input)
75                self.content = [np.asarray([item]) for item in data_input]
76            if all([isinstance(item, (Obs, CObs)) or item is None for item in data_input]):
77                _assert_equal_properties([o for o in data_input if o is not None])
78                self.content = [np.asarray([item]) if item is not None else None for item in data_input]
79                self.N = 1
80
81            elif all([isinstance(item, np.ndarray) or item is None for item in data_input]) and any([isinstance(item, np.ndarray) for item in data_input]):
82                self.content = data_input
83                noNull = [a for a in self.content if not (a is None)]  # To check if the matrices are correct for all undefined elements
84                self.N = noNull[0].shape[0]
85                if self.N > 1 and noNull[0].shape[0] != noNull[0].shape[1]:
86                    raise Exception("Smearing matrices are not NxN")
87                if (not all([item.shape == noNull[0].shape for item in noNull])):
88                    raise Exception("Items in data_input are not of identical shape." + str(noNull))
89            else:
90                raise Exception("data_input contains item of wrong type")
91        else:
92            raise Exception("Data input was not given as list or correct array")
93
94        self.tag = None
95
96        # An undefined timeslice is represented by the None object
97        self.content = [None] * padding[0] + self.content + [None] * padding[1]
98        self.T = len(self.content)
99        self.prange = prange

Initialize a Corr object.

Parameters
  • data_input (list or array): list of Obs or list of arrays of Obs or array of Corrs
  • padding (list, optional): List with two entries where the first labels the padding at the front of the correlator and the second the padding at the back.
  • prange (list, optional): List containing the first and last timeslice of the plateau region indentified for this correlator.
def gamma_method(self, **kwargs):
120    def gamma_method(self, **kwargs):
121        """Apply the gamma method to the content of the Corr."""
122        for item in self.content:
123            if not (item is None):
124                if self.N == 1:
125                    item[0].gamma_method(**kwargs)
126                else:
127                    for i in range(self.N):
128                        for j in range(self.N):
129                            item[i, j].gamma_method(**kwargs)

Apply the gamma method to the content of the Corr.

def projected(self, vector_l=None, vector_r=None, normalize=False):
131    def projected(self, vector_l=None, vector_r=None, normalize=False):
132        """We need to project the Correlator with a Vector to get a single value at each timeslice.
133
134        The method can use one or two vectors.
135        If two are specified it returns v1@G@v2 (the order might be very important.)
136        By default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to
137        """
138        if self.N == 1:
139            raise Exception("Trying to project a Corr, that already has N=1.")
140
141        if vector_l is None:
142            vector_l, vector_r = np.asarray([1.] + (self.N - 1) * [0.]), np.asarray([1.] + (self.N - 1) * [0.])
143        elif (vector_r is None):
144            vector_r = vector_l
145        if isinstance(vector_l, list) and not isinstance(vector_r, list):
146            if len(vector_l) != self.T:
147                raise Exception("Length of vector list must be equal to T")
148            vector_r = [vector_r] * self.T
149        if isinstance(vector_r, list) and not isinstance(vector_l, list):
150            if len(vector_r) != self.T:
151                raise Exception("Length of vector list must be equal to T")
152            vector_l = [vector_l] * self.T
153
154        if not isinstance(vector_l, list):
155            if not vector_l.shape == vector_r.shape == (self.N,):
156                raise Exception("Vectors are of wrong shape!")
157            if normalize:
158                vector_l, vector_r = vector_l / np.sqrt((vector_l @ vector_l)), vector_r / np.sqrt(vector_r @ vector_r)
159            newcontent = [None if _check_for_none(self, item) else np.asarray([vector_l.T @ item @ vector_r]) for item in self.content]
160
161        else:
162            # There are no checks here yet. There are so many possible scenarios, where this can go wrong.
163            if normalize:
164                for t in range(self.T):
165                    vector_l[t], vector_r[t] = vector_l[t] / np.sqrt((vector_l[t] @ vector_l[t])), vector_r[t] / np.sqrt(vector_r[t] @ vector_r[t])
166
167            newcontent = [None if (_check_for_none(self, self.content[t]) or vector_l[t] is None or vector_r[t] is None) else np.asarray([vector_l[t].T @ self.content[t] @ vector_r[t]]) for t in range(self.T)]
168        return Corr(newcontent)

We need to project the Correlator with a Vector to get a single value at each timeslice.

The method can use one or two vectors. If two are specified it returns v1@G@v2 (the order might be very important.) By default it will return the lowest source, which usually means unsmeared-unsmeared (0,0), but it does not have to

def item(self, i, j):
170    def item(self, i, j):
171        """Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.
172
173        Parameters
174        ----------
175        i : int
176            First index to be picked.
177        j : int
178            Second index to be picked.
179        """
180        if self.N == 1:
181            raise Exception("Trying to pick item from projected Corr")
182        newcontent = [None if (item is None) else item[i, j] for item in self.content]
183        return Corr(newcontent)

Picks the element [i,j] from every matrix and returns a correlator containing one Obs per timeslice.

Parameters
  • i (int): First index to be picked.
  • j (int): Second index to be picked.
def plottable(self):
185    def plottable(self):
186        """Outputs the correlator in a plotable format.
187
188        Outputs three lists containing the timeslice index, the value on each
189        timeslice and the error on each timeslice.
190        """
191        if self.N != 1:
192            raise Exception("Can only make Corr[N=1] plottable")
193        x_list = [x for x in range(self.T) if not self.content[x] is None]
194        y_list = [y[0].value for y in self.content if y is not None]
195        y_err_list = [y[0].dvalue for y in self.content if y is not None]
196
197        return x_list, y_list, y_err_list

Outputs the correlator in a plotable format.

Outputs three lists containing the timeslice index, the value on each timeslice and the error on each timeslice.

def symmetric(self):
199    def symmetric(self):
200        """ Symmetrize the correlator around x0=0."""
201        if self.N != 1:
202            raise Exception('symmetric cannot be safely applied to multi-dimensional correlators.')
203        if self.T % 2 != 0:
204            raise Exception("Can not symmetrize odd T")
205
206        if np.argmax(np.abs(self.content)) != 0:
207            warnings.warn("Correlator does not seem to be symmetric around x0=0.", RuntimeWarning)
208
209        newcontent = [self.content[0]]
210        for t in range(1, self.T):
211            if (self.content[t] is None) or (self.content[self.T - t] is None):
212                newcontent.append(None)
213            else:
214                newcontent.append(0.5 * (self.content[t] + self.content[self.T - t]))
215        if (all([x is None for x in newcontent])):
216            raise Exception("Corr could not be symmetrized: No redundant values")
217        return Corr(newcontent, prange=self.prange)

Symmetrize the correlator around x0=0.

def anti_symmetric(self):
219    def anti_symmetric(self):
220        """Anti-symmetrize the correlator around x0=0."""
221        if self.N != 1:
222            raise Exception('anti_symmetric cannot be safely applied to multi-dimensional correlators.')
223        if self.T % 2 != 0:
224            raise Exception("Can not symmetrize odd T")
225
226        test = 1 * self
227        test.gamma_method()
228        if not all([o.is_zero_within_error(3) for o in test.content[0]]):
229            warnings.warn("Correlator does not seem to be anti-symmetric around x0=0.", RuntimeWarning)
230
231        newcontent = [self.content[0]]
232        for t in range(1, self.T):
233            if (self.content[t] is None) or (self.content[self.T - t] is None):
234                newcontent.append(None)
235            else:
236                newcontent.append(0.5 * (self.content[t] - self.content[self.T - t]))
237        if (all([x is None for x in newcontent])):
238            raise Exception("Corr could not be symmetrized: No redundant values")
239        return Corr(newcontent, prange=self.prange)

Anti-symmetrize the correlator around x0=0.

def is_matrix_symmetric(self):
241    def is_matrix_symmetric(self):
242        """Checks whether a correlator matrices is symmetric on every timeslice."""
243        if self.N == 1:
244            raise Exception("Only works for correlator matrices.")
245        for t in range(self.T):
246            if self[t] is None:
247                continue
248            for i in range(self.N):
249                for j in range(i + 1, self.N):
250                    if self[t][i, j] is self[t][j, i]:
251                        continue
252                    if hash(self[t][i, j]) != hash(self[t][j, i]):
253                        return False
254        return True

Checks whether a correlator matrices is symmetric on every timeslice.

def matrix_symmetric(self):
256    def matrix_symmetric(self):
257        """Symmetrizes the correlator matrices on every timeslice."""
258        if self.N == 1:
259            raise Exception("Trying to symmetrize a correlator matrix, that already has N=1.")
260        if self.is_matrix_symmetric():
261            return 1.0 * self
262        else:
263            transposed = [None if _check_for_none(self, G) else G.T for G in self.content]
264            return 0.5 * (Corr(transposed) + self)

Symmetrizes the correlator matrices on every timeslice.

def GEVP(self, t0, ts=None, sort='Eigenvalue', **kwargs):
266    def GEVP(self, t0, ts=None, sort="Eigenvalue", **kwargs):
267        r'''Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.
268
269        The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the
270        largest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing
271        ```python
272        C.GEVP(t0=2)[0]  # Ground state vector(s)
273        C.GEVP(t0=2)[:3]  # Vectors for the lowest three states
274        ```
275
276        Parameters
277        ----------
278        t0 : int
279            The time t0 for the right hand side of the GEVP according to $G(t)v_i=\lambda_i G(t_0)v_i$
280        ts : int
281            fixed time $G(t_s)v_i=\lambda_i G(t_0)v_i$ if sort=None.
282            If sort="Eigenvector" it gives a reference point for the sorting method.
283        sort : string
284            If this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.
285            - "Eigenvalue": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice.
286            - "Eigenvector": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state.
287              The reference state is identified by its eigenvalue at $t=t_s$.
288
289        Other Parameters
290        ----------------
291        state : int
292           Returns only the vector(s) for a specified state. The lowest state is zero.
293        '''
294
295        if self.N == 1:
296            raise Exception("GEVP methods only works on correlator matrices and not single correlators.")
297        if ts is not None:
298            if (ts <= t0):
299                raise Exception("ts has to be larger than t0.")
300
301        if "sorted_list" in kwargs:
302            warnings.warn("Argument 'sorted_list' is deprecated, use 'sort' instead.", DeprecationWarning)
303            sort = kwargs.get("sorted_list")
304
305        if self.is_matrix_symmetric():
306            symmetric_corr = self
307        else:
308            symmetric_corr = self.matrix_symmetric()
309
310        if sort is None:
311            if (ts is None):
312                raise Exception("ts is required if sort=None.")
313            if (self.content[t0] is None) or (self.content[ts] is None):
314                raise Exception("Corr not defined at t0/ts.")
315            G0, Gt = np.empty([self.N, self.N], dtype="double"), np.empty([self.N, self.N], dtype="double")
316            for i in range(self.N):
317                for j in range(self.N):
318                    G0[i, j] = symmetric_corr[t0][i, j].value
319                    Gt[i, j] = symmetric_corr[ts][i, j].value
320
321            reordered_vecs = _GEVP_solver(Gt, G0)
322
323        elif sort in ["Eigenvalue", "Eigenvector"]:
324            if sort == "Eigenvalue" and ts is not None:
325                warnings.warn("ts has no effect when sorting by eigenvalue is chosen.", RuntimeWarning)
326            all_vecs = [None] * (t0 + 1)
327            G0 = np.vectorize(lambda x: x.value)(symmetric_corr[t0])
328            for t in range(t0 + 1, self.T):
329                try:
330                    Gt = np.vectorize(lambda x: x.value)(symmetric_corr[t])
331                    all_vecs.append(_GEVP_solver(Gt, G0))
332                except Exception:
333                    all_vecs.append(None)
334            if sort == "Eigenvector":
335                if (ts is None):
336                    raise Exception("ts is required for the Eigenvector sorting method.")
337                all_vecs = _sort_vectors(all_vecs, ts)
338
339            reordered_vecs = [[v[s] if v is not None else None for v in all_vecs] for s in range(self.N)]
340        else:
341            raise Exception("Unkown value for 'sort'.")
342
343        if "state" in kwargs:
344            return reordered_vecs[kwargs.get("state")]
345        else:
346            return reordered_vecs

Solve the generalized eigenvalue problem on the correlator matrix and returns the corresponding eigenvectors.

The eigenvectors are sorted according to the descending eigenvalues, the zeroth eigenvector(s) correspond to the largest eigenvalue(s). The eigenvector(s) for the individual states can be accessed via slicing

C.GEVP(t0=2)[0]  # Ground state vector(s)
C.GEVP(t0=2)[:3]  # Vectors for the lowest three states
Parameters
  • t0 (int): The time t0 for the right hand side of the GEVP according to $G(t)v_i=\lambda_i G(t_0)v_i$
  • ts (int): fixed time $G(t_s)v_i=\lambda_i G(t_0)v_i$ if sort=None. If sort="Eigenvector" it gives a reference point for the sorting method.
  • sort (string): If this argument is set, a list of self.T vectors per state is returned. If it is set to None, only one vector is returned.
    • "Eigenvalue": The eigenvector is chosen according to which eigenvalue it belongs individually on every timeslice.
    • "Eigenvector": Use the method described in arXiv:2004.10472 to find the set of v(t) belonging to the state. The reference state is identified by its eigenvalue at $t=t_s$.
Other Parameters
  • state (int): Returns only the vector(s) for a specified state. The lowest state is zero.
def Eigenvalue(self, t0, ts=None, state=0, sort='Eigenvalue'):
348    def Eigenvalue(self, t0, ts=None, state=0, sort="Eigenvalue"):
349        """Determines the eigenvalue of the GEVP by solving and projecting the correlator
350
351        Parameters
352        ----------
353        state : int
354            The state one is interested in ordered by energy. The lowest state is zero.
355
356        All other parameters are identical to the ones of Corr.GEVP.
357        """
358        vec = self.GEVP(t0, ts=ts, sort=sort)[state]
359        return self.projected(vec)

Determines the eigenvalue of the GEVP by solving and projecting the correlator

Parameters
  • state (int): The state one is interested in ordered by energy. The lowest state is zero.
  • All other parameters are identical to the ones of Corr.GEVP.
def Hankel(self, N, periodic=False):
361    def Hankel(self, N, periodic=False):
362        """Constructs an NxN Hankel matrix
363
364        C(t) c(t+1) ... c(t+n-1)
365        C(t+1) c(t+2) ... c(t+n)
366        .................
367        C(t+(n-1)) c(t+n) ... c(t+2(n-1))
368
369        Parameters
370        ----------
371        N : int
372            Dimension of the Hankel matrix
373        periodic : bool, optional
374            determines whether the matrix is extended periodically
375        """
376
377        if self.N != 1:
378            raise Exception("Multi-operator Prony not implemented!")
379
380        array = np.empty([N, N], dtype="object")
381        new_content = []
382        for t in range(self.T):
383            new_content.append(array.copy())
384
385        def wrap(i):
386            while i >= self.T:
387                i -= self.T
388            return i
389
390        for t in range(self.T):
391            for i in range(N):
392                for j in range(N):
393                    if periodic:
394                        new_content[t][i, j] = self.content[wrap(t + i + j)][0]
395                    elif (t + i + j) >= self.T:
396                        new_content[t] = None
397                    else:
398                        new_content[t][i, j] = self.content[t + i + j][0]
399
400        return Corr(new_content)

Constructs an NxN Hankel matrix

C(t) c(t+1) ... c(t+n-1) C(t+1) c(t+2) ... c(t+n) ................. C(t+(n-1)) c(t+n) ... c(t+2(n-1))

Parameters
  • N (int): Dimension of the Hankel matrix
  • periodic (bool, optional): determines whether the matrix is extended periodically
def roll(self, dt):
402    def roll(self, dt):
403        """Periodically shift the correlator by dt timeslices
404
405        Parameters
406        ----------
407        dt : int
408            number of timeslices
409        """
410        return Corr(list(np.roll(np.array(self.content, dtype=object), dt)))

Periodically shift the correlator by dt timeslices

Parameters
  • dt (int): number of timeslices
def reverse(self):
412    def reverse(self):
413        """Reverse the time ordering of the Corr"""
414        return Corr(self.content[:: -1])

Reverse the time ordering of the Corr

def thin(self, spacing=2, offset=0):
416    def thin(self, spacing=2, offset=0):
417        """Thin out a correlator to suppress correlations
418
419        Parameters
420        ----------
421        spacing : int
422            Keep only every 'spacing'th entry of the correlator
423        offset : int
424            Offset the equal spacing
425        """
426        new_content = []
427        for t in range(self.T):
428            if (offset + t) % spacing != 0:
429                new_content.append(None)
430            else:
431                new_content.append(self.content[t])
432        return Corr(new_content)

Thin out a correlator to suppress correlations

Parameters
  • spacing (int): Keep only every 'spacing'th entry of the correlator
  • offset (int): Offset the equal spacing
def correlate(self, partner):
434    def correlate(self, partner):
435        """Correlate the correlator with another correlator or Obs
436
437        Parameters
438        ----------
439        partner : Obs or Corr
440            partner to correlate the correlator with.
441            Can either be an Obs which is correlated with all entries of the
442            correlator or a Corr of same length.
443        """
444        if self.N != 1:
445            raise Exception("Only one-dimensional correlators can be safely correlated.")
446        new_content = []
447        for x0, t_slice in enumerate(self.content):
448            if _check_for_none(self, t_slice):
449                new_content.append(None)
450            else:
451                if isinstance(partner, Corr):
452                    if _check_for_none(partner, partner.content[x0]):
453                        new_content.append(None)
454                    else:
455                        new_content.append(np.array([correlate(o, partner.content[x0][0]) for o in t_slice]))
456                elif isinstance(partner, Obs):  # Should this include CObs?
457                    new_content.append(np.array([correlate(o, partner) for o in t_slice]))
458                else:
459                    raise Exception("Can only correlate with an Obs or a Corr.")
460
461        return Corr(new_content)

Correlate the correlator with another correlator or Obs

Parameters
  • partner (Obs or Corr): partner to correlate the correlator with. Can either be an Obs which is correlated with all entries of the correlator or a Corr of same length.
def reweight(self, weight, **kwargs):
463    def reweight(self, weight, **kwargs):
464        """Reweight the correlator.
465
466        Parameters
467        ----------
468        weight : Obs
469            Reweighting factor. An Observable that has to be defined on a superset of the
470            configurations in obs[i].idl for all i.
471        all_configs : bool
472            if True, the reweighted observables are normalized by the average of
473            the reweighting factor on all configurations in weight.idl and not
474            on the configurations in obs[i].idl.
475        """
476        if self.N != 1:
477            raise Exception("Reweighting only implemented for one-dimensional correlators.")
478        new_content = []
479        for t_slice in self.content:
480            if _check_for_none(self, t_slice):
481                new_content.append(None)
482            else:
483                new_content.append(np.array(reweight(weight, t_slice, **kwargs)))
484        return Corr(new_content)

Reweight the correlator.

Parameters
  • weight (Obs): Reweighting factor. An Observable that has to be defined on a superset of the configurations in obs[i].idl for all i.
  • all_configs (bool): if True, the reweighted observables are normalized by the average of the reweighting factor on all configurations in weight.idl and not on the configurations in obs[i].idl.
def T_symmetry(self, partner, parity=1):
486    def T_symmetry(self, partner, parity=+1):
487        """Return the time symmetry average of the correlator and its partner
488
489        Parameters
490        ----------
491        partner : Corr
492            Time symmetry partner of the Corr
493        partity : int
494            Parity quantum number of the correlator, can be +1 or -1
495        """
496        if self.N != 1:
497            raise Exception("T_symmetry only implemented for one-dimensional correlators.")
498        if not isinstance(partner, Corr):
499            raise Exception("T partner has to be a Corr object.")
500        if parity not in [+1, -1]:
501            raise Exception("Parity has to be +1 or -1.")
502        T_partner = parity * partner.reverse()
503
504        t_slices = []
505        test = (self - T_partner)
506        test.gamma_method()
507        for x0, t_slice in enumerate(test.content):
508            if t_slice is not None:
509                if not t_slice[0].is_zero_within_error(5):
510                    t_slices.append(x0)
511        if t_slices:
512            warnings.warn("T symmetry partners do not agree within 5 sigma on time slices " + str(t_slices) + ".", RuntimeWarning)
513
514        return (self + T_partner) / 2

Return the time symmetry average of the correlator and its partner

Parameters
  • partner (Corr): Time symmetry partner of the Corr
  • partity (int): Parity quantum number of the correlator, can be +1 or -1
def deriv(self, variant='symmetric'):
516    def deriv(self, variant="symmetric"):
517        """Return the first derivative of the correlator with respect to x0.
518
519        Parameters
520        ----------
521        variant : str
522            decides which definition of the finite differences derivative is used.
523            Available choice: symmetric, forward, backward, improved, log, default: symmetric
524        """
525        if self.N != 1:
526            raise Exception("deriv only implemented for one-dimensional correlators.")
527        if variant == "symmetric":
528            newcontent = []
529            for t in range(1, self.T - 1):
530                if (self.content[t - 1] is None) or (self.content[t + 1] is None):
531                    newcontent.append(None)
532                else:
533                    newcontent.append(0.5 * (self.content[t + 1] - self.content[t - 1]))
534            if (all([x is None for x in newcontent])):
535                raise Exception('Derivative is undefined at all timeslices')
536            return Corr(newcontent, padding=[1, 1])
537        elif variant == "forward":
538            newcontent = []
539            for t in range(self.T - 1):
540                if (self.content[t] is None) or (self.content[t + 1] is None):
541                    newcontent.append(None)
542                else:
543                    newcontent.append(self.content[t + 1] - self.content[t])
544            if (all([x is None for x in newcontent])):
545                raise Exception("Derivative is undefined at all timeslices")
546            return Corr(newcontent, padding=[0, 1])
547        elif variant == "backward":
548            newcontent = []
549            for t in range(1, self.T):
550                if (self.content[t - 1] is None) or (self.content[t] is None):
551                    newcontent.append(None)
552                else:
553                    newcontent.append(self.content[t] - self.content[t - 1])
554            if (all([x is None for x in newcontent])):
555                raise Exception("Derivative is undefined at all timeslices")
556            return Corr(newcontent, padding=[1, 0])
557        elif variant == "improved":
558            newcontent = []
559            for t in range(2, self.T - 2):
560                if (self.content[t - 2] is None) or (self.content[t - 1] is None) or (self.content[t + 1] is None) or (self.content[t + 2] is None):
561                    newcontent.append(None)
562                else:
563                    newcontent.append((1 / 12) * (self.content[t - 2] - 8 * self.content[t - 1] + 8 * self.content[t + 1] - self.content[t + 2]))
564            if (all([x is None for x in newcontent])):
565                raise Exception('Derivative is undefined at all timeslices')
566            return Corr(newcontent, padding=[2, 2])
567        elif variant == 'log':
568            newcontent = []
569            for t in range(self.T):
570                if (self.content[t] is None) or (self.content[t] <= 0):
571                    newcontent.append(None)
572                else:
573                    newcontent.append(np.log(self.content[t]))
574            if (all([x is None for x in newcontent])):
575                raise Exception("Log is undefined at all timeslices")
576            logcorr = Corr(newcontent)
577            return self * logcorr.deriv('symmetric')
578        else:
579            raise Exception("Unknown variant.")

Return the first derivative of the correlator with respect to x0.

Parameters
  • variant (str): decides which definition of the finite differences derivative is used. Available choice: symmetric, forward, backward, improved, log, default: symmetric
def second_deriv(self, variant='symmetric'):
581    def second_deriv(self, variant="symmetric"):
582        """Return the second derivative of the correlator with respect to x0.
583
584        Parameters
585        ----------
586        variant : str
587            decides which definition of the finite differences derivative is used.
588            Available choice: symmetric, improved, log, default: symmetric
589        """
590        if self.N != 1:
591            raise Exception("second_deriv only implemented for one-dimensional correlators.")
592        if variant == "symmetric":
593            newcontent = []
594            for t in range(1, self.T - 1):
595                if (self.content[t - 1] is None) or (self.content[t + 1] is None):
596                    newcontent.append(None)
597                else:
598                    newcontent.append((self.content[t + 1] - 2 * self.content[t] + self.content[t - 1]))
599            if (all([x is None for x in newcontent])):
600                raise Exception("Derivative is undefined at all timeslices")
601            return Corr(newcontent, padding=[1, 1])
602        elif variant == "improved":
603            newcontent = []
604            for t in range(2, self.T - 2):
605                if (self.content[t - 2] is None) or (self.content[t - 1] is None) or (self.content[t] is None) or (self.content[t + 1] is None) or (self.content[t + 2] is None):
606                    newcontent.append(None)
607                else:
608                    newcontent.append((1 / 12) * (-self.content[t + 2] + 16 * self.content[t + 1] - 30 * self.content[t] + 16 * self.content[t - 1] - self.content[t - 2]))
609            if (all([x is None for x in newcontent])):
610                raise Exception("Derivative is undefined at all timeslices")
611            return Corr(newcontent, padding=[2, 2])
612        elif variant == 'log':
613            newcontent = []
614            for t in range(self.T):
615                if (self.content[t] is None) or (self.content[t] <= 0):
616                    newcontent.append(None)
617                else:
618                    newcontent.append(np.log(self.content[t]))
619            if (all([x is None for x in newcontent])):
620                raise Exception("Log is undefined at all timeslices")
621            logcorr = Corr(newcontent)
622            return self * (logcorr.second_deriv('symmetric') + (logcorr.deriv('symmetric'))**2)
623        else:
624            raise Exception("Unknown variant.")

Return the second derivative of the correlator with respect to x0.

Parameters
  • variant (str): decides which definition of the finite differences derivative is used. Available choice: symmetric, improved, log, default: symmetric
def m_eff(self, variant='log', guess=1.0):
626    def m_eff(self, variant='log', guess=1.0):
627        """Returns the effective mass of the correlator as correlator object
628
629        Parameters
630        ----------
631        variant : str
632            log : uses the standard effective mass log(C(t) / C(t+1))
633            cosh, periodic : Use periodicitiy of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m.
634            sinh : Use anti-periodicitiy of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m.
635            See, e.g., arXiv:1205.5380
636            arccosh : Uses the explicit form of the symmetrized correlator (not recommended)
637            logsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
638        guess : float
639            guess for the root finder, only relevant for the root variant
640        """
641        if self.N != 1:
642            raise Exception('Correlator must be projected before getting m_eff')
643        if variant == 'log':
644            newcontent = []
645            for t in range(self.T - 1):
646                if ((self.content[t] is None) or (self.content[t + 1] is None)) or (self.content[t + 1][0].value == 0):
647                    newcontent.append(None)
648                elif self.content[t][0].value / self.content[t + 1][0].value < 0:
649                    newcontent.append(None)
650                else:
651                    newcontent.append(self.content[t] / self.content[t + 1])
652            if (all([x is None for x in newcontent])):
653                raise Exception('m_eff is undefined at all timeslices')
654
655            return np.log(Corr(newcontent, padding=[0, 1]))
656
657        elif variant == 'logsym':
658            newcontent = []
659            for t in range(1, self.T - 1):
660                if ((self.content[t - 1] is None) or (self.content[t + 1] is None)) or (self.content[t + 1][0].value == 0):
661                    newcontent.append(None)
662                elif self.content[t - 1][0].value / self.content[t + 1][0].value < 0:
663                    newcontent.append(None)
664                else:
665                    newcontent.append(self.content[t - 1] / self.content[t + 1])
666            if (all([x is None for x in newcontent])):
667                raise Exception('m_eff is undefined at all timeslices')
668
669            return np.log(Corr(newcontent, padding=[1, 1])) / 2
670
671        elif variant in ['periodic', 'cosh', 'sinh']:
672            if variant in ['periodic', 'cosh']:
673                func = anp.cosh
674            else:
675                func = anp.sinh
676
677            def root_function(x, d):
678                return func(x * (t - self.T / 2)) / func(x * (t + 1 - self.T / 2)) - d
679
680            newcontent = []
681            for t in range(self.T - 1):
682                if (self.content[t] is None) or (self.content[t + 1] is None) or (self.content[t + 1][0].value == 0):
683                    newcontent.append(None)
684                # Fill the two timeslices in the middle of the lattice with their predecessors
685                elif variant == 'sinh' and t in [self.T / 2, self.T / 2 - 1]:
686                    newcontent.append(newcontent[-1])
687                elif self.content[t][0].value / self.content[t + 1][0].value < 0:
688                    newcontent.append(None)
689                else:
690                    newcontent.append(np.abs(find_root(self.content[t][0] / self.content[t + 1][0], root_function, guess=guess)))
691            if (all([x is None for x in newcontent])):
692                raise Exception('m_eff is undefined at all timeslices')
693
694            return Corr(newcontent, padding=[0, 1])
695
696        elif variant == 'arccosh':
697            newcontent = []
698            for t in range(1, self.T - 1):
699                if (self.content[t] is None) or (self.content[t + 1] is None) or (self.content[t - 1] is None) or (self.content[t][0].value == 0):
700                    newcontent.append(None)
701                else:
702                    newcontent.append((self.content[t + 1] + self.content[t - 1]) / (2 * self.content[t]))
703            if (all([x is None for x in newcontent])):
704                raise Exception("m_eff is undefined at all timeslices")
705            return np.arccosh(Corr(newcontent, padding=[1, 1]))
706
707        else:
708            raise Exception('Unknown variant.')

Returns the effective mass of the correlator as correlator object

Parameters
  • variant (str): log : uses the standard effective mass log(C(t) / C(t+1)) cosh, periodic : Use periodicitiy of the correlator by solving C(t) / C(t+1) = cosh(m * (t - T/2)) / cosh(m * (t + 1 - T/2)) for m. sinh : Use anti-periodicitiy of the correlator by solving C(t) / C(t+1) = sinh(m * (t - T/2)) / sinh(m * (t + 1 - T/2)) for m. See, e.g., arXiv:1205.5380 arccosh : Uses the explicit form of the symmetrized correlator (not recommended) logsym: uses the symmetric effective mass log(C(t-1) / C(t+1))/2
  • guess (float): guess for the root finder, only relevant for the root variant
def fit(self, function, fitrange=None, silent=False, **kwargs):
710    def fit(self, function, fitrange=None, silent=False, **kwargs):
711        r'''Fits function to the data
712
713        Parameters
714        ----------
715        function : obj
716            function to fit to the data. See fits.least_squares for details.
717        fitrange : list
718            Two element list containing the timeslices on which the fit is supposed to start and stop.
719            Caution: This range is inclusive as opposed to standard python indexing.
720            `fitrange=[4, 6]` corresponds to the three entries 4, 5 and 6.
721            If not specified, self.prange or all timeslices are used.
722        silent : bool
723            Decides whether output is printed to the standard output.
724        '''
725        if self.N != 1:
726            raise Exception("Correlator must be projected before fitting")
727
728        if fitrange is None:
729            if self.prange:
730                fitrange = self.prange
731            else:
732                fitrange = [0, self.T - 1]
733        else:
734            if not isinstance(fitrange, list):
735                raise Exception("fitrange has to be a list with two elements")
736            if len(fitrange) != 2:
737                raise Exception("fitrange has to have exactly two elements [fit_start, fit_stop]")
738
739        xs = [x for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]
740        ys = [self.content[x][0] for x in range(fitrange[0], fitrange[1] + 1) if not self.content[x] is None]
741        result = least_squares(xs, ys, function, silent=silent, **kwargs)
742        return result

Fits function to the data

Parameters
  • function (obj): function to fit to the data. See fits.least_squares for details.
  • fitrange (list): Two element list containing the timeslices on which the fit is supposed to start and stop. Caution: This range is inclusive as opposed to standard python indexing. fitrange=[4, 6] corresponds to the three entries 4, 5 and 6. If not specified, self.prange or all timeslices are used.
  • silent (bool): Decides whether output is printed to the standard output.
def plateau(self, plateau_range=None, method='fit', auto_gamma=False):
744    def plateau(self, plateau_range=None, method="fit", auto_gamma=False):
745        """ Extract a plateau value from a Corr object
746
747        Parameters
748        ----------
749        plateau_range : list
750            list with two entries, indicating the first and the last timeslice
751            of the plateau region.
752        method : str
753            method to extract the plateau.
754                'fit' fits a constant to the plateau region
755                'avg', 'average' or 'mean' just average over the given timeslices.
756        auto_gamma : bool
757            apply gamma_method with default parameters to the Corr. Defaults to None
758        """
759        if not plateau_range:
760            if self.prange:
761                plateau_range = self.prange
762            else:
763                raise Exception("no plateau range provided")
764        if self.N != 1:
765            raise Exception("Correlator must be projected before getting a plateau.")
766        if (all([self.content[t] is None for t in range(plateau_range[0], plateau_range[1] + 1)])):
767            raise Exception("plateau is undefined at all timeslices in plateaurange.")
768        if auto_gamma:
769            self.gamma_method()
770        if method == "fit":
771            def const_func(a, t):
772                return a[0]
773            return self.fit(const_func, plateau_range)[0]
774        elif method in ["avg", "average", "mean"]:
775            returnvalue = np.mean([item[0] for item in self.content[plateau_range[0]:plateau_range[1] + 1] if item is not None])
776            return returnvalue
777
778        else:
779            raise Exception("Unsupported plateau method: " + method)

Extract a plateau value from a Corr object

Parameters
  • plateau_range (list): list with two entries, indicating the first and the last timeslice of the plateau region.
  • method (str): method to extract the plateau. 'fit' fits a constant to the plateau region 'avg', 'average' or 'mean' just average over the given timeslices.
  • auto_gamma (bool): apply gamma_method with default parameters to the Corr. Defaults to None
def set_prange(self, prange):
781    def set_prange(self, prange):
782        """Sets the attribute prange of the Corr object."""
783        if not len(prange) == 2:
784            raise Exception("prange must be a list or array with two values")
785        if not ((isinstance(prange[0], int)) and (isinstance(prange[1], int))):
786            raise Exception("Start and end point must be integers")
787        if not (0 <= prange[0] <= self.T and 0 <= prange[1] <= self.T and prange[0] < prange[1]):
788            raise Exception("Start and end point must define a range in the interval 0,T")
789
790        self.prange = prange
791        return

Sets the attribute prange of the Corr object.

def show( self, x_range=None, comp=None, y_range=None, logscale=False, plateau=None, fit_res=None, ylabel=None, save=None, auto_gamma=False, hide_sigma=None, references=None, title=None):
793    def show(self, x_range=None, comp=None, y_range=None, logscale=False, plateau=None, fit_res=None, ylabel=None, save=None, auto_gamma=False, hide_sigma=None, references=None, title=None):
794        """Plots the correlator using the tag of the correlator as label if available.
795
796        Parameters
797        ----------
798        x_range : list
799            list of two values, determining the range of the x-axis e.g. [4, 8].
800        comp : Corr or list of Corr
801            Correlator or list of correlators which are plotted for comparison.
802            The tags of these correlators are used as labels if available.
803        logscale : bool
804            Sets y-axis to logscale.
805        plateau : Obs
806            Plateau value to be visualized in the figure.
807        fit_res : Fit_result
808            Fit_result object to be visualized.
809        ylabel : str
810            Label for the y-axis.
811        save : str
812            path to file in which the figure should be saved.
813        auto_gamma : bool
814            Apply the gamma method with standard parameters to all correlators and plateau values before plotting.
815        hide_sigma : float
816            Hides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
817        references : list
818            List of floating point values that are displayed as horizontal lines for reference.
819        title : string
820            Optional title of the figure.
821        """
822        if self.N != 1:
823            raise Exception("Correlator must be projected before plotting")
824
825        if auto_gamma:
826            self.gamma_method()
827
828        if x_range is None:
829            x_range = [0, self.T - 1]
830
831        fig = plt.figure()
832        ax1 = fig.add_subplot(111)
833
834        x, y, y_err = self.plottable()
835        if hide_sigma:
836            hide_from = np.argmax((hide_sigma * np.array(y_err[1:])) > np.abs(y[1:])) - 1
837        else:
838            hide_from = None
839        ax1.errorbar(x[:hide_from], y[:hide_from], y_err[:hide_from], label=self.tag)
840        if logscale:
841            ax1.set_yscale('log')
842        else:
843            if y_range is None:
844                try:
845                    y_min = min([(x[0].value - x[0].dvalue) for x in self.content[x_range[0]: x_range[1] + 1] if (x is not None) and x[0].dvalue < 2 * np.abs(x[0].value)])
846                    y_max = max([(x[0].value + x[0].dvalue) for x in self.content[x_range[0]: x_range[1] + 1] if (x is not None) and x[0].dvalue < 2 * np.abs(x[0].value)])
847                    ax1.set_ylim([y_min - 0.1 * (y_max - y_min), y_max + 0.1 * (y_max - y_min)])
848                except Exception:
849                    pass
850            else:
851                ax1.set_ylim(y_range)
852        if comp:
853            if isinstance(comp, (Corr, list)):
854                for corr in comp if isinstance(comp, list) else [comp]:
855                    if auto_gamma:
856                        corr.gamma_method()
857                    x, y, y_err = corr.plottable()
858                    if hide_sigma:
859                        hide_from = np.argmax((hide_sigma * np.array(y_err[1:])) > np.abs(y[1:])) - 1
860                    else:
861                        hide_from = None
862                    ax1.errorbar(x[:hide_from], y[:hide_from], y_err[:hide_from], label=corr.tag, mfc=plt.rcParams['axes.facecolor'])
863            else:
864                raise Exception("'comp' must be a correlator or a list of correlators.")
865
866        if plateau:
867            if isinstance(plateau, Obs):
868                if auto_gamma:
869                    plateau.gamma_method()
870                ax1.axhline(y=plateau.value, linewidth=2, color=plt.rcParams['text.color'], alpha=0.6, marker=',', ls='--', label=str(plateau))
871                ax1.axhspan(plateau.value - plateau.dvalue, plateau.value + plateau.dvalue, alpha=0.25, color=plt.rcParams['text.color'], ls='-')
872            else:
873                raise Exception("'plateau' must be an Obs")
874
875        if references:
876            if isinstance(references, list):
877                for ref in references:
878                    ax1.axhline(y=ref, linewidth=1, color=plt.rcParams['text.color'], alpha=0.6, marker=',', ls='--')
879            else:
880                raise Exception("'references' must be a list of floating pint values.")
881
882        if self.prange:
883            ax1.axvline(self.prange[0], 0, 1, ls='-', marker=',')
884            ax1.axvline(self.prange[1], 0, 1, ls='-', marker=',')
885
886        if fit_res:
887            x_samples = np.arange(x_range[0], x_range[1] + 1, 0.05)
888            ax1.plot(x_samples,
889                     fit_res.fit_function([o.value for o in fit_res.fit_parameters], x_samples),
890                     ls='-', marker=',', lw=2)
891
892        ax1.set_xlabel(r'$x_0 / a$')
893        if ylabel:
894            ax1.set_ylabel(ylabel)
895        ax1.set_xlim([x_range[0] - 0.5, x_range[1] + 0.5])
896
897        handles, labels = ax1.get_legend_handles_labels()
898        if labels:
899            ax1.legend()
900
901        if title:
902            plt.title(title)
903
904        plt.draw()
905
906        if save:
907            if isinstance(save, str):
908                fig.savefig(save, bbox_inches='tight')
909            else:
910                raise Exception("'save' has to be a string.")

Plots the correlator using the tag of the correlator as label if available.

Parameters
  • x_range (list): list of two values, determining the range of the x-axis e.g. [4, 8].
  • comp (Corr or list of Corr): Correlator or list of correlators which are plotted for comparison. The tags of these correlators are used as labels if available.
  • logscale (bool): Sets y-axis to logscale.
  • plateau (Obs): Plateau value to be visualized in the figure.
  • fit_res (Fit_result): Fit_result object to be visualized.
  • ylabel (str): Label for the y-axis.
  • save (str): path to file in which the figure should be saved.
  • auto_gamma (bool): Apply the gamma method with standard parameters to all correlators and plateau values before plotting.
  • hide_sigma (float): Hides data points from the first value on which is consistent with zero within 'hide_sigma' standard errors.
  • references (list): List of floating point values that are displayed as horizontal lines for reference.
  • title (string): Optional title of the figure.
def spaghetti_plot(self, logscale=True):
912    def spaghetti_plot(self, logscale=True):
913        """Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.
914
915        Parameters
916        ----------
917        logscale : bool
918            Determines whether the scale of the y-axis is logarithmic or standard.
919        """
920        if self.N != 1:
921            raise Exception("Correlator needs to be projected first.")
922
923        mc_names = list(set([item for sublist in [o[0].mc_names for o in self.content if o is not None] for item in sublist]))
924        x0_vals = [n for (n, o) in zip(np.arange(self.T), self.content) if o is not None]
925
926        for name in mc_names:
927            data = np.array([o[0].deltas[name] + o[0].r_values[name] for o in self.content if o is not None]).T
928
929            fig = plt.figure()
930            ax = fig.add_subplot(111)
931            for dat in data:
932                ax.plot(x0_vals, dat, ls='-', marker='')
933
934            if logscale is True:
935                ax.set_yscale('log')
936
937            ax.set_xlabel(r'$x_0 / a$')
938            plt.title(name)
939            plt.draw()

Produces a spaghetti plot of the correlator suited to monitor exceptional configurations.

Parameters
  • logscale (bool): Determines whether the scale of the y-axis is logarithmic or standard.
def dump(self, filename, datatype='json.gz', **kwargs):
941    def dump(self, filename, datatype="json.gz", **kwargs):
942        """Dumps the Corr into a file of chosen type
943        Parameters
944        ----------
945        filename : str
946            Name of the file to be saved.
947        datatype : str
948            Format of the exported file. Supported formats include
949            "json.gz" and "pickle"
950        path : str
951            specifies a custom path for the file (default '.')
952        """
953        if datatype == "json.gz":
954            from .input.json import dump_to_json
955            if 'path' in kwargs:
956                file_name = kwargs.get('path') + '/' + filename
957            else:
958                file_name = filename
959            dump_to_json(self, file_name)
960        elif datatype == "pickle":
961            dump_object(self, filename, **kwargs)
962        else:
963            raise Exception("Unknown datatype " + str(datatype))

Dumps the Corr into a file of chosen type

Parameters
  • filename (str): Name of the file to be saved.
  • datatype (str): Format of the exported file. Supported formats include "json.gz" and "pickle"
  • path (str): specifies a custom path for the file (default '.')
def print(self, print_range=None):
965    def print(self, print_range=None):
966        print(self.__repr__(print_range))
def sqrt(self):
1130    def sqrt(self):
1131        return self ** 0.5
def log(self):
1133    def log(self):
1134        newcontent = [None if _check_for_none(self, item) else np.log(item) for item in self.content]
1135        return Corr(newcontent, prange=self.prange)
def exp(self):
1137    def exp(self):
1138        newcontent = [None if _check_for_none(self, item) else np.exp(item) for item in self.content]
1139        return Corr(newcontent, prange=self.prange)
def sin(self):
1152    def sin(self):
1153        return self._apply_func_to_corr(np.sin)
def cos(self):
1155    def cos(self):
1156        return self._apply_func_to_corr(np.cos)
def tan(self):
1158    def tan(self):
1159        return self._apply_func_to_corr(np.tan)
def sinh(self):
1161    def sinh(self):
1162        return self._apply_func_to_corr(np.sinh)
def cosh(self):
1164    def cosh(self):
1165        return self._apply_func_to_corr(np.cosh)
def tanh(self):
1167    def tanh(self):
1168        return self._apply_func_to_corr(np.tanh)
def arcsin(self):
1170    def arcsin(self):
1171        return self._apply_func_to_corr(np.arcsin)
def arccos(self):
1173    def arccos(self):
1174        return self._apply_func_to_corr(np.arccos)
def arctan(self):
1176    def arctan(self):
1177        return self._apply_func_to_corr(np.arctan)
def arcsinh(self):
1179    def arcsinh(self):
1180        return self._apply_func_to_corr(np.arcsinh)
def arccosh(self):
1182    def arccosh(self):
1183        return self._apply_func_to_corr(np.arccosh)
def arctanh(self):
1185    def arctanh(self):
1186        return self._apply_func_to_corr(np.arctanh)
def prune(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):
1221    def prune(self, Ntrunc, tproj=3, t0proj=2, basematrix=None):
1222        r''' Project large correlation matrix to lowest states
1223
1224        This method can be used to reduce the size of an (N x N) correlation matrix
1225        to (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise
1226        is still small.
1227
1228        Parameters
1229        ----------
1230        Ntrunc: int
1231            Rank of the target matrix.
1232        tproj: int
1233            Time where the eigenvectors are evaluated, corresponds to ts in the GEVP method.
1234            The default value is 3.
1235        t0proj: int
1236            Time where the correlation matrix is inverted. Choosing t0proj=1 is strongly
1237            discouraged for O(a) improved theories, since the correctness of the procedure
1238            cannot be granted in this case. The default value is 2.
1239        basematrix : Corr
1240            Correlation matrix that is used to determine the eigenvectors of the
1241            lowest states based on a GEVP. basematrix is taken to be the Corr itself if
1242            is is not specified.
1243
1244        Notes
1245        -----
1246        We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving
1247        the GEVP $$C(t) v_n(t, t_0) = \lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \equiv t_\mathrm{proj}$
1248        and $t_0 \equiv t_{0, \mathrm{proj}}$. The target matrix is projected onto the subspace of the
1249        resulting eigenvectors $v_n, n=1,\dots,N_\mathrm{trunc}$ via
1250        $$G^\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large
1251        correlation matrix and to remove some noise that is added by irrelevant operators.
1252        This may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated
1253        bound $t_0 \leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.
1254        '''
1255
1256        if self.N == 1:
1257            raise Exception('Method cannot be applied to one-dimensional correlators.')
1258        if basematrix is None:
1259            basematrix = self
1260        if Ntrunc >= basematrix.N:
1261            raise Exception('Cannot truncate using Ntrunc <= %d' % (basematrix.N))
1262        if basematrix.N != self.N:
1263            raise Exception('basematrix and targetmatrix have to be of the same size.')
1264
1265        evecs = basematrix.GEVP(t0proj, tproj, sort=None)[:Ntrunc]
1266
1267        tmpmat = np.empty((Ntrunc, Ntrunc), dtype=object)
1268        rmat = []
1269        for t in range(basematrix.T):
1270            for i in range(Ntrunc):
1271                for j in range(Ntrunc):
1272                    tmpmat[i][j] = evecs[i].T @ self[t] @ evecs[j]
1273            rmat.append(np.copy(tmpmat))
1274
1275        newcontent = [None if (self.content[t] is None) else rmat[t] for t in range(self.T)]
1276        return Corr(newcontent)

Project large correlation matrix to lowest states

This method can be used to reduce the size of an (N x N) correlation matrix to (Ntrunc x Ntrunc) by solving a GEVP at very early times where the noise is still small.

Parameters
  • Ntrunc (int): Rank of the target matrix.
  • tproj (int): Time where the eigenvectors are evaluated, corresponds to ts in the GEVP method. The default value is 3.
  • t0proj (int): Time where the correlation matrix is inverted. Choosing t0proj=1 is strongly discouraged for O(a) improved theories, since the correctness of the procedure cannot be granted in this case. The default value is 2.
  • basematrix (Corr): Correlation matrix that is used to determine the eigenvectors of the lowest states based on a GEVP. basematrix is taken to be the Corr itself if is is not specified.
Notes

We have the basematrix $C(t)$ and the target matrix $G(t)$. We start by solving the GEVP $$C(t) v_n(t, t_0) = \lambda_n(t, t_0) C(t_0) v_n(t, t_0)$$ where $t \equiv t_\mathrm{proj}$ and $t_0 \equiv t_{0, \mathrm{proj}}$. The target matrix is projected onto the subspace of the resulting eigenvectors $v_n, n=1,\dots,N_\mathrm{trunc}$ via $$G^\prime_{i, j}(t) = (v_i, G(t) v_j)$$. This allows to reduce the size of a large correlation matrix and to remove some noise that is added by irrelevant operators. This may allow to use the GEVP on $G(t)$ at late times such that the theoretically motivated bound $t_0 \leq t/2$ holds, since the condition number of $G(t)$ is decreased, compared to $C(t)$.