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Feature/corr matrix and inverse cov matrix as input in least squares function for correlated fits (#223)
* feat: corr_matrix kwargs as input for least squares fit * feat/tests: inverse covariance matrix and correlation matrix kwargs as input for least squares function * feat/tests/example: reduced new kwargs to 'inv_chol_cov_matrix' and outsourced the inversion & cholesky decomposition of the covariance matrix (function 'invert_corr_cov_cholesky(corr, covdiag)') * tests: added tests for inv_chol_cov_matrix kwarg for the case of combined fits * fix: renamed covdiag to inverrdiag needed for the cholesky decomposition and corrected its documentation * examples: added an example of a correlated combined fit to the least_squares documentation * feat/tests/fix(of typos): added function 'sort_corr()' (and a test of it) to sort correlation matrix according to a list of alphabetically sorted keys * docs: added more elaborate documentation/example of sort_corr(), fixed typos in documentation of invert_corr_cov_cholesky()
This commit is contained in:
Pia Leonie Jones Petrak
GitHub
parent
3830e3f777
commit
1d6f7f65c0
5 changed files with 596 additions and 46 deletions
File diff suppressed because one or more lines are too long
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@ -14,7 +14,7 @@ from autograd import hessian as auto_hessian
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from autograd import elementwise_grad as egrad
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from numdifftools import Jacobian as num_jacobian
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from numdifftools import Hessian as num_hessian
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from .obs import Obs, derived_observable, covariance, cov_Obs
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from .obs import Obs, derived_observable, covariance, cov_Obs, invert_corr_cov_cholesky
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class Fit_result(Sequence):
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@ -151,6 +151,14 @@ def least_squares(x, y, func, priors=None, silent=False, **kwargs):
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For details about how the covariance matrix is estimated see `pyerrors.obs.covariance`.
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In practice the correlation matrix is Cholesky decomposed and inverted (instead of the covariance matrix).
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This procedure should be numerically more stable as the correlation matrix is typically better conditioned (Jacobi preconditioning).
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inv_chol_cov_matrix [array,list], optional
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array: shape = (no of y values) X (no of y values)
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list: for an uncombined fit: [""]
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for a combined fit: list of keys belonging to the corr_matrix saved in the array, must be the same as the keys of the y dict in alphabetical order
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If correlated_fit=True is set as well, can provide an inverse covariance matrix (y errors, dy_f included!) of your own choosing for a correlated fit.
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The matrix must be a lower triangular matrix constructed from a Cholesky decomposition: The function invert_corr_cov_cholesky(corr, inverrdiag) can be
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used to construct it from a correlation matrix (corr) and the errors dy_f of the data points (inverrdiag = np.diag(1 / np.asarray(dy_f))). For the correct
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ordering the correlation matrix (corr) can be sorted via the function sort_corr(corr, kl, yd) where kl is the list of keys and yd the y dict.
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expected_chisquare : bool
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If True estimates the expected chisquare which is
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corrected by effects caused by correlated input data (default False).
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@ -165,6 +173,57 @@ def least_squares(x, y, func, priors=None, silent=False, **kwargs):
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-------
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output : Fit_result
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Parameters and information on the fitted result.
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Examples
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------
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>>> # Example of a correlated (correlated_fit = True, inv_chol_cov_matrix handed over) combined fit, based on a randomly generated data set
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>>> import numpy as np
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>>> from scipy.stats import norm
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>>> from scipy.linalg import cholesky
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>>> import pyerrors as pe
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>>> # generating the random data set
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>>> num_samples = 400
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>>> N = 3
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>>> x = np.arange(N)
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>>> x1 = norm.rvs(size=(N, num_samples)) # generate random numbers
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>>> x2 = norm.rvs(size=(N, num_samples)) # generate random numbers
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>>> r = r1 = r2 = np.zeros((N, N))
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>>> y = {}
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>>> for i in range(N):
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>>> for j in range(N):
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>>> r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix
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>>> errl = np.sqrt([3.4, 2.5, 3.6]) # set y errors
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>>> for i in range(N):
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>>> for j in range(N):
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>>> r[i, j] *= errl[i] * errl[j] # element in covariance matrix
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>>> c = cholesky(r, lower=True)
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>>> y = {'a': np.dot(c, x1), 'b': np.dot(c, x2)} # generate y data with the covariance matrix defined
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>>> # random data set has been generated, now the dictionaries and the inverse covariance matrix to be handed over are built
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>>> x_dict = {}
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>>> y_dict = {}
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>>> chol_inv_dict = {}
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>>> data = []
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>>> for key in y.keys():
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>>> x_dict[key] = x
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>>> for i in range(N):
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>>> data.append(pe.Obs([[i + 1 + o for o in y[key][i]]], ['ens'])) # generate y Obs from the y data
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>>> [o.gamma_method() for o in data]
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>>> corr = pe.covariance(data, correlation=True)
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>>> inverrdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))
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>>> chol_inv = pe.obs.invert_corr_cov_cholesky(corr, inverrdiag) # gives form of the inverse covariance matrix needed for the combined correlated fit below
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>>> y_dict = {'a': data[:3], 'b': data[3:]}
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>>> # common fit parameter p[0] in combined fit
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>>> def fit1(p, x):
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>>> return p[0] + p[1] * x
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>>> def fit2(p, x):
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>>> return p[0] + p[2] * x
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>>> fitf_dict = {'a': fit1, 'b':fit2}
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>>> fitp_inv_cov_combined_fit = pe.least_squares(x_dict,y_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,['a','b']])
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Fit with 3 parameters
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Method: Levenberg-Marquardt
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`ftol` termination condition is satisfied.
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chisquare/d.o.f.: 0.5388013574561786 # random
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fit parameters [1.11897846 0.96361162 0.92325319] # random
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'''
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output = Fit_result()
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@ -297,15 +356,19 @@ def least_squares(x, y, func, priors=None, silent=False, **kwargs):
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return anp.sum(general_chisqfunc_uncorr(p, y_f, p_f) ** 2)
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if kwargs.get('correlated_fit') is True:
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if 'inv_chol_cov_matrix' in kwargs:
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chol_inv = kwargs.get('inv_chol_cov_matrix')
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if (chol_inv[0].shape[0] != len(dy_f)):
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raise TypeError('The number of columns of the inverse covariance matrix handed over needs to be equal to the number of y errors.')
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if (chol_inv[0].shape[0] != chol_inv[0].shape[1]):
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raise TypeError('The inverse covariance matrix handed over needs to have the same number of rows as columns.')
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if (chol_inv[1] != key_ls):
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raise ValueError('The keys of inverse covariance matrix are not the same or do not appear in the same order as the x and y values.')
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chol_inv = chol_inv[0]
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else:
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corr = covariance(y_all, correlation=True, **kwargs)
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covdiag = np.diag(1 / np.asarray(dy_f))
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condn = np.linalg.cond(corr)
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if condn > 0.1 / np.finfo(float).eps:
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raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})")
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if condn > 1e13:
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warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning)
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chol = np.linalg.cholesky(corr)
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chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True)
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inverrdiag = np.diag(1 / np.asarray(dy_f))
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chol_inv = invert_corr_cov_cholesky(corr, inverrdiag)
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def general_chisqfunc(p, ivars, pr):
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model = anp.concatenate([anp.array(funcd[key](p, xd[key])).reshape(-1) for key in key_ls])
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@ -350,7 +413,6 @@ def least_squares(x, y, func, priors=None, silent=False, **kwargs):
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fit_result = scipy.optimize.least_squares(chisqfunc_residuals_uncorr, x0, method='lm', ftol=1e-15, gtol=1e-15, xtol=1e-15)
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if kwargs.get('correlated_fit') is True:
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def chisqfunc_residuals(p):
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return general_chisqfunc(p, y_f, p_f)
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@ -1544,6 +1544,92 @@ def covariance(obs, visualize=False, correlation=False, smooth=None, **kwargs):
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return cov
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def invert_corr_cov_cholesky(corr, inverrdiag):
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"""Constructs a lower triangular matrix `chol` via the Cholesky decomposition of the correlation matrix `corr`
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and then returns the inverse covariance matrix `chol_inv` as a lower triangular matrix by solving `chol * x = inverrdiag`.
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Parameters
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----------
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corr : np.ndarray
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correlation matrix
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inverrdiag : np.ndarray
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diagonal matrix, the entries are the inverse errors of the data points considered
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"""
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condn = np.linalg.cond(corr)
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if condn > 0.1 / np.finfo(float).eps:
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raise Exception(f"Cannot invert correlation matrix as its condition number exceeds machine precision ({condn:1.2e})")
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if condn > 1e13:
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warnings.warn("Correlation matrix may be ill-conditioned, condition number: {%1.2e}" % (condn), RuntimeWarning)
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chol = np.linalg.cholesky(corr)
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chol_inv = scipy.linalg.solve_triangular(chol, inverrdiag, lower=True)
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return chol_inv
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def sort_corr(corr, kl, yd):
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""" Reorders a correlation matrix to match the alphabetical order of its underlying y data.
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The ordering of the input correlation matrix `corr` is given by the list of keys `kl`.
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The input dictionary `yd` (with the same keys `kl`) must contain the corresponding y data
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that the correlation matrix is based on.
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This function sorts the list of keys `kl` alphabetically and sorts the matrix `corr`
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according to this alphabetical order such that the sorted matrix `corr_sorted` corresponds
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to the y data `yd` when arranged in an alphabetical order by its keys.
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Parameters
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----------
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corr : np.ndarray
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A square correlation matrix constructed using the order of the y data specified by `kl`.
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The dimensions of `corr` should match the total number of y data points in `yd` combined.
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kl : list of str
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A list of keys that denotes the order in which the y data from `yd` was used to build the
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input correlation matrix `corr`.
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yd : dict of list
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A dictionary where each key corresponds to a unique identifier, and its value is a list of
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y data points. The total number of y data points across all keys must match the dimensions
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of `corr`. The lists in the dictionary can be lists of Obs.
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Returns
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-------
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np.ndarray
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A new, sorted correlation matrix that corresponds to the y data from `yd` when arranged alphabetically by its keys.
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Example
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-------
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>>> import numpy as np
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>>> import pyerrors as pe
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>>> corr = np.array([[1, 0.2, 0.3], [0.2, 1, 0.4], [0.3, 0.4, 1]])
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>>> kl = ['b', 'a']
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>>> yd = {'a': [1, 2], 'b': [3]}
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>>> sorted_corr = pe.obs.sort_corr(corr, kl, yd)
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>>> print(sorted_corr)
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array([[1. , 0.3, 0.4],
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[0.3, 1. , 0.2],
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[0.4, 0.2, 1. ]])
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"""
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kl_sorted = sorted(kl)
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posd = {}
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ofs = 0
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for ki, k in enumerate(kl):
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posd[k] = [i + ofs for i in range(len(yd[k]))]
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ofs += len(posd[k])
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mapping = []
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for k in kl_sorted:
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for i in range(len(yd[k])):
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mapping.append(posd[k][i])
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corr_sorted = np.zeros_like(corr)
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for i in range(corr.shape[0]):
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for j in range(corr.shape[0]):
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corr_sorted[i][j] = corr[mapping[i]][mapping[j]]
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return corr_sorted
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def _smooth_eigenvalues(corr, E):
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"""Eigenvalue smoothing as described in hep-lat/9412087
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@ -152,6 +152,124 @@ def test_alternative_solvers():
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chisquare_values = np.array(chisquare_values)
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assert np.all(np.isclose(chisquare_values, chisquare_values[0]))
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def test_inv_cov_matrix_input_least_squares():
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num_samples = 400
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N = 10
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x = norm.rvs(size=(N, num_samples)) # generate random numbers
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r = np.zeros((N, N))
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for i in range(N):
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for j in range(N):
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r[i, j] = np.exp(-0.8 * np.fabs(i - j)) # element in correlation matrix
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errl = np.sqrt([3.4, 2.5, 3.6, 2.8, 4.2, 4.7, 4.9, 5.1, 3.2, 4.2]) # set y errors
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for i in range(N):
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for j in range(N):
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r[i, j] *= errl[i] * errl[j] # element in covariance matrix
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c = cholesky(r, lower=True)
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y = np.dot(c, x)
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x = np.arange(N)
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x_dict = {}
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y_dict = {}
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for i,item in enumerate(x):
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x_dict[str(item)] = [x[i]]
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for linear in [True, False]:
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data = []
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for i in range(N):
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if linear:
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data.append(pe.Obs([[i + 1 + o for o in y[i]]], ['ens']))
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else:
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data.append(pe.Obs([[np.exp(-(i + 1)) + np.exp(-(i + 1)) * o for o in y[i]]], ['ens']))
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[o.gamma_method() for o in data]
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data_dict = {}
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for i,item in enumerate(x):
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data_dict[str(item)] = [data[i]]
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corr = pe.covariance(data, correlation=True)
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chol = np.linalg.cholesky(corr)
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covdiag = np.diag(1 / np.asarray([o.dvalue for o in data]))
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chol_inv = scipy.linalg.solve_triangular(chol, covdiag, lower=True)
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chol_inv_keys = [""]
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chol_inv_keys_combined_fit = [str(item) for i,item in enumerate(x)]
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if linear:
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def fitf(p, x):
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return p[1] + p[0] * x
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fitf_dict = {}
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for i,item in enumerate(x):
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fitf_dict[str(item)] = fitf
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else:
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def fitf(p, x):
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return p[1] * anp.exp(-p[0] * x)
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fitf_dict = {}
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for i,item in enumerate(x):
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fitf_dict[str(item)] = fitf
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fitpc = pe.least_squares(x, data, fitf, correlated_fit=True)
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fitp_inv_cov = pe.least_squares(x, data, fitf, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,chol_inv_keys])
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fitp_inv_cov_combined_fit = pe.least_squares(x_dict, data_dict, fitf_dict, correlated_fit = True, inv_chol_cov_matrix = [chol_inv,chol_inv_keys_combined_fit])
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for i in range(2):
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diff_inv_cov = fitp_inv_cov[i] - fitpc[i]
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diff_inv_cov.gamma_method()
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assert(diff_inv_cov.is_zero(atol=0.0))
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diff_inv_cov_combined_fit = fitp_inv_cov_combined_fit[i] - fitpc[i]
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diff_inv_cov_combined_fit.gamma_method()
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assert(diff_inv_cov_combined_fit.is_zero(atol=1e-12))
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def test_least_squares_invalid_inv_cov_matrix_input():
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xvals = []
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yvals = []
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err = 0.1
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def func_valid(a,x):
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return a[0] + a[1] * x
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for x in range(1, 8, 2):
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xvals.append(x)
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yvals.append(pe.pseudo_Obs(x + np.random.normal(0.0, err), err, 'test1') + pe.pseudo_Obs(0, err / 100, 'test2', samples=87))
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[o.gamma_method() for o in yvals]
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#dictionaries for a combined fit
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xvals_dict = { }
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yvals_dict = { }
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for i,item in enumerate(np.arange(1, 8, 2)):
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xvals_dict[str(item)] = [xvals[i]]
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yvals_dict[str(item)] = [yvals[i]]
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chol_inv_keys_combined_fit = ['1', '3', '5', '7']
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chol_inv_keys_combined_fit_invalid = ['2', '7', '100', '8']
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func_dict_valid = {"1": func_valid,"3": func_valid,"5": func_valid,"7": func_valid}
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corr_valid = pe.covariance(yvals, correlation = True)
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chol = np.linalg.cholesky(corr_valid)
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covdiag = np.diag(1 / np.asarray([o.dvalue for o in yvals]))
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chol_inv_valid = scipy.linalg.solve_triangular(chol, covdiag, lower=True)
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chol_inv_keys = [""]
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pe.least_squares(xvals, yvals,func_valid, correlated_fit = True, inv_chol_cov_matrix = [chol_inv_valid,chol_inv_keys])
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pe.least_squares(xvals_dict, yvals_dict,func_dict_valid, correlated_fit = True, inv_chol_cov_matrix = [chol_inv_valid,chol_inv_keys_combined_fit])
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chol_inv_invalid_shape1 = np.zeros((len(yvals),len(yvals)-1))
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chol_inv_invalid_shape2 = np.zeros((len(yvals)+2,len(yvals)))
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# for an uncombined fit
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with pytest.raises(TypeError):
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pe.least_squares(xvals, yvals, func_valid, correlated_fit = True, inv_chol_cov_matrix = [chol_inv_invalid_shape1,chol_inv_keys])
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with pytest.raises(TypeError):
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pe.least_squares(xvals, yvals, func_valid,correlated_fit = True, inv_chol_cov_matrix = [chol_inv_invalid_shape2,chol_inv_keys])
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with pytest.raises(ValueError):
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pe.least_squares(xvals, yvals, func_valid,correlated_fit = True, inv_chol_cov_matrix = [chol_inv_valid,chol_inv_keys_combined_fit_invalid])
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#repeat for a combined fit
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with pytest.raises(TypeError):
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pe.least_squares(xvals_dict, yvals_dict,func_dict_valid, correlated_fit = True, inv_chol_cov_matrix = [chol_inv_invalid_shape1,chol_inv_keys_combined_fit])
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with pytest.raises(TypeError):
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pe.least_squares(xvals_dict, yvals_dict,func_dict_valid, correlated_fit = True, inv_chol_cov_matrix = [chol_inv_invalid_shape2,chol_inv_keys_combined_fit])
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with pytest.raises(ValueError):
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pe.least_squares(xvals_dict, yvals_dict,func_dict_valid, correlated_fit = True, inv_chol_cov_matrix = [chol_inv_valid,chol_inv_keys_combined_fit_invalid])
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def test_correlated_fit():
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num_samples = 400
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@ -1063,6 +1063,27 @@ def test_covariance_reorder_non_overlapping_data():
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assert np.isclose(corr1[0, 1], corr2[0, 1], atol=1e-14)
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def test_sort_corr():
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xd = {
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'b': [1, 2, 3],
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||||
'a': [2.2, 4.4],
|
||||
'c': [3.7, 5.1]
|
||||
}
|
||||
|
||||
yd = {k : pe.cov_Obs(xd[k], [.2 * o for o in xd[k]], k) for k in xd}
|
||||
key_orig = list(yd.keys())
|
||||
y_all = np.concatenate([np.array(yd[key]) for key in key_orig])
|
||||
[o.gm() for o in y_all]
|
||||
cov = pe.covariance(y_all)
|
||||
|
||||
key_ls = key_sorted = sorted(key_orig)
|
||||
y_sorted = np.concatenate([np.array(yd[key]) for key in key_sorted])
|
||||
[o.gm() for o in y_sorted]
|
||||
cov_sorted = pe.covariance(y_sorted)
|
||||
retcov = pe.obs.sort_corr(cov, key_orig, yd)
|
||||
assert np.sum(retcov - cov_sorted) == 0
|
||||
|
||||
|
||||
def test_empty_obs():
|
||||
o = pe.Obs([np.random.rand(100)], ['test'])
|
||||
q = o + pe.Obs([], [], means=[])
|
||||
|
|
|
|||
Loading…
Add table
Reference in a new issue