From 99dbf5b1615d9a92e6d55f6b580ef2cdab516aba Mon Sep 17 00:00:00 2001 From: JanNeuendorf <75676159+JanNeuendorf@users.noreply.github.com> Date: Fri, 8 Jan 2021 14:45:24 +0100 Subject: [PATCH 1/8] Add files via upload --- examples/05_correlators.ipynb | 286 ++++++++++++++++++++++++++++++++++ 1 file changed, 286 insertions(+) create mode 100644 examples/05_correlators.ipynb diff --git a/examples/05_correlators.ipynb b/examples/05_correlators.ipynb new file mode 100644 index 00000000..616968e9 --- /dev/null +++ b/examples/05_correlators.ipynb @@ -0,0 +1,286 @@ +{ + "metadata": { + "language_info": { + "codemirror_mode": { + "name": "ipython", + "version": 3 + }, + "file_extension": ".py", + "mimetype": "text/x-python", + "name": "python", + "nbconvert_exporter": "python", + "pygments_lexer": "ipython3", + "version": "3.7.6-final" + }, + "orig_nbformat": 2, + "kernelspec": { + "name": "python3", + "display_name": "Python 3", + "language": "python" + } + }, + "nbformat": 4, + "nbformat_minor": 2, + "cells": [ + { + "source": [ + "## Correlator Example" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "source": [ + "We are often dealing with lists of observables defined at every time slice. For a more convenient analysis, those can be represented using the \"correlators.Corr\" class.\n", + "This is especially useful, if there is not one Obs per time slice, but a whole smearing matrix. We will load an example of such an object." + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 11, + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "64 4\n" + ] + } + ], + "source": [ + "import pyerrors as pe\n", + "import autograd.numpy as np\n", + "P5P5=pe.load_object(\"data/Example_Corr_P5P5.p\")\n", + "print(P5P5.T, P5P5.N)" + ] + }, + { + "source": [ + "What we just printed out, are the only parameters a Corr has. T represents the number of time slices and N the rank of the NxN smearing matrix. \n", + "The content is accessible with P5P5.content and gives a list of np.arrays of obs. There is no formal difference between correlators, which contain a single observable per time slice \n", + "and those, which hold a smearing matrix. \n", + "To initialize a Corr, we only need to pass the content or in the case of N=1, we might pass a list of obs.\n", + "Lets run some code!" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 18, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "output_type": "display_data", + "data": { + "text/plain": "
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\n" + }, + "metadata": { + "needs_background": "light" + } + }, + { + "output_type": "display_data", + "data": { + "text/plain": "
" + }, + "metadata": {} + } + ], + "source": [ + "P5P5.symmetric().projected().show(logscale=True)" + ] + }, + { + "source": [ + "We will go over this line the way it is executed, left to right. \n", + "The correlator has a method called **symmetric()**, which returns a time symmetrized version of itself. \n", + "We did not need to redo the error estimation. While a Corr has a **.gamma_method()** defined, which simply applies the Obs.gamma_method() to every Obs in the correlator, \n", + "this is done automatically, whenever a new corr is initiated. This is a little slow, but convenient. The next method called is **.projected()**. \n", + "This methods *projects* the smearing matrices $G(t)$ with a set of vectors, returning a single value at every $t$. $$v_l^T G(t) v_r$$ Since we did not pass an argument, it defaulted to $v_l=v_r=(1,0,...,0)$, giving us $G(t)[0,0]$ . The method returns another Corr, but this time with N=1. \n", + " The last method **.show()** just allows us to quickly inspect a correlator. \n", + "Let us now look at the GEVP. \n" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 13, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "[-6.26306805e-04 1.64198098e-01 -5.99926044e-01 7.83024479e-01]\n" + ] + } + ], + "source": [ + "eigenvector=P5P5.symmetric().GEVP(t0=3,ts=6)\n", + "print(eigenvector)" + ] + }, + { + "source": [ + "**.GEVP()** Needs two time indices as arguments. It then solves:\n", + "\n", + "$$G(t_s)v=\\lambda G(t_0)v$$\n", + "\n", + "and returns the vector $v$ for the largest eigenvalue. It uses a *Scipy* method, which itself is based on a *LAPACK* method. \n", + "To really see the difference this makes, we can visualize the effective mass, which we also have a method for." + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 14, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "output_type": "display_data", + "data": { + "text/plain": "
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\n" + }, + "metadata": { + "needs_background": "light" + } + }, + { + "output_type": "display_data", + "data": { + "text/plain": "
" + }, + "metadata": {} + } + ], + "source": [ + "P5P5.symmetric().projected().m_eff().show([0,32])\n", + "P5P5.symmetric().projected(eigenvector).m_eff().show([0,32])" + ] + }, + { + "source": [ + "The plateau is definetely larger now. We can use it to extract the pseudoscalar mass. All we need is a plateau range. \n", + "Finding those can be time consuming, if we are dealing with many correlators and we need to zoom in to the plot, to see the range properly.\n", + "We can call a GUI method to help us to visualize the range. Play around with the checkboxes at the bottom of the window to make the program adjust plot to your selected range." + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 15, + "metadata": { + "tags": [] + }, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "The pseudoscalar mass is: Obs[0.64457(66)]\n" + ] + } + ], + "source": [ + "m_eff=P5P5.symmetric().projected(eigenvector).m_eff() # Our lines were getting a little long, so we just assign a new Corr. \n", + "plateau_range=pe.correlators.GUI_range_finder(m_eff)\n", + "m_p=m_eff.plateau(plateau_range)\n", + "print(\"The pseudoscalar mass is: \",m_p)" + ] + }, + { + "source": [ + "For fun you can compare your result with https://arxiv.org/abs/1912.09937v1 Table XV (the first value for $am_{H_{is}}$). \n", + "Before we wrap up, we should look at Corrs and math operations. They can be multiplied by and added to other Corrs (of same N,T), or scaled by an Obs or float. \n", + "Usually the operation is just done for every value at the same time and smearing. \n", + "\n" + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 16, + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "64 4\n" + ] + } + ], + "source": [ + "new_correlator=0.5*P5P5+np.sqrt(P5P5)/np.sin(P5P5**2)+np.arcsinh(P5P5)\n", + "print(new_correlator.T, new_correlator.N)" + ] + }, + { + "source": [ + "This is a senseless but valid expression, which does exactly, what one would expect. It returns a Corr of the same shape as P5P5. \n", + " It is really important, that there is never any confusion about the appropriate time slices. Lets look at *m_eff* once again. \n", + " By default it is calculated as $$ m_{eff}(t)=\\ln(\\frac{Corr[t]}{Corr[t+1]})$$ \n", + " Therefore m_eff is only defined up to the second to last time slice. But the method appends a **None** object, so that m_eff.T=64. \n", + " If we add m_eff to P5P5.projected(), the resulting Corr would still have T=64, with the last item being **None**. \n", + "Another reason for a Corr being **None** at one time slice, is a division by zero or other undefined operation. \n", + "Even if a Corr is partially undefined, math operations still work, as long as T and N are identical.\n", + " " + ], + "cell_type": "markdown", + "metadata": {} + }, + { + "cell_type": "code", + "execution_count": 17, + "metadata": {}, + "outputs": [ + { + "output_type": "stream", + "name": "stdout", + "text": [ + "64\n" + ] + } + ], + "source": [ + "print((m_eff+P5P5.projected()).T)" + ] + }, + { + "cell_type": "code", + "execution_count": null, + "metadata": {}, + "outputs": [], + "source": [] + } + ] +} \ No newline at end of file From d69fde1cc017680b4c953c818b50f031a5362867 Mon Sep 17 00:00:00 2001 From: JanNeuendorf <75676159+JanNeuendorf@users.noreply.github.com> Date: Fri, 8 Jan 2021 14:45:55 +0100 Subject: [PATCH 2/8] Delete pyerrors.py --- pyerrors/pyerrors.py | 1223 ------------------------------------------ 1 file changed, 1223 deletions(-) delete mode 100644 pyerrors/pyerrors.py diff --git a/pyerrors/pyerrors.py b/pyerrors/pyerrors.py deleted file mode 100644 index 6c6a1130..00000000 --- a/pyerrors/pyerrors.py +++ /dev/null @@ -1,1223 +0,0 @@ -#!/usr/bin/env python -# coding: utf-8 - -import pickle -import numpy as np -import autograd.numpy as anp # Thinly-wrapped numpy -from autograd import jacobian -import matplotlib.pyplot as plt -import numdifftools as nd -import scipy.special - - -class Obs: - """Class for a general observable. - - Instances of Obs are the basic objects of a pyerrors error analysis. - They are initialized with a list which contains arrays of samples for - different ensembles/replica and another list of same length which contains - the names of the ensembles/replica. Mathematical operations can be - performed on instances. The result is another instance of Obs. The error of - an instance can be computed with the gamma_method. Also contains additional - methods for output and visualization of the error calculation. - - Attributes - ---------- - e_tag_global -- Integer which determines which part of the name belongs - to the ensemble and which to the replicum. - S_global -- Standard value for S (default 2.0) - S_dict -- Dictionary for S values. If an entry for a given ensemble - exists this overwrites the standard value for that ensemble. - tau_exp_global -- Standard value for tau_exp (default 0.0) - tau_exp_dict -- Dictionary for tau_exp values. If an entry for a given - ensemble exists this overwrites the standard value for that - ensemble. - N_sigma_global -- Standard value for N_sigma (default 1.0) - """ - - e_tag_global = 0 - S_global = 2.0 - S_dict = {} - tau_exp_global = 0.0 - tau_exp_dict = {} - N_sigma_global = 1.0 - - def __init__(self, samples, names): - - if len(samples) != len(names): - raise Exception('Length of samples and names incompatible.') - if len(names) != len(set(names)): - raise Exception('Names are not unique.') - if not all(isinstance(x, str) for x in names): - raise TypeError('All names have to be strings.') - if min(len(x) for x in samples) <= 4: - raise Exception('Samples have to have at least 4 entries.') - - self.names = sorted(names) - self.shape = {} - self.r_values = {} - self.deltas = {} - for name, sample in sorted(zip(names, samples)): - self.shape[name] = np.size(sample) - self.r_values[name] = np.mean(sample) - self.deltas[name] = sample - self.r_values[name] - - self.N = sum(map(np.size, list(self.deltas.values()))) - - self.value = 0 - for name in self.names: - self.value += self.shape[name] * self.r_values[name] - self.value /= self.N - - self.dvalue = 0.0 - self.ddvalue = 0.0 - self.reweighted = 0 - - self.S = {} - self.tau_exp = {} - self.N_sigma = 0 - - self.e_names = {} - self.e_content = {} - - self.e_dvalue = {} - self.e_ddvalue = {} - self.e_tauint = {} - self.e_dtauint = {} - self.e_windowsize = {} - self.e_Q = {} - self.e_rho = {} - self.e_drho = {} - self.e_n_tauint = {} - self.e_n_dtauint = {} - - - def gamma_method(self, **kwargs): - """Calculate the error and related properties of the Obs. - - Keyword arguments - ----------------- - S -- specifies a custom value for the parameter S (default 2.0), can be - a float or an array of floats for different ensembles - tau_exp -- positive value triggers the critical slowing down analysis - (default 0.0), can be a float or an array of floats for - different ensembles - N_sigma -- number of standard deviations from zero until the tail is - attached to the autocorrelation function (default 1) - e_tag -- number of characters which label the ensemble. The remaining - ones label replica (default 0) - fft -- boolean, which determines whether the fft algorithm is used for - the computation of the autocorrelation function (default True) - """ - - if 'e_tag' in kwargs: - e_tag_local = kwargs.get('e_tag') - if not isinstance(e_tag_local, int): - raise TypeError('Error: e_tag is not integer') - else: - e_tag_local = Obs.e_tag_global - - self.e_names = sorted(set([o[:e_tag_local] for o in self.names])) - self.e_content = {} - self.e_dvalue = {} - self.e_ddvalue = {} - self.e_tauint = {} - self.e_dtauint = {} - self.e_windowsize = {} - self.e_n_tauint = {} - self.e_n_dtauint = {} - e_gamma = {} - self.e_rho = {} - self.e_drho = {} - self.dvalue = 0 - self.ddvalue = 0 - - self.S = {} - self.tau_exp = {} - - if kwargs.get('fft') is False: - fft = False - else: - fft = True - - if 'S' in kwargs: - tmp = kwargs.get('S') - if isinstance(tmp, list): - if len(tmp) != len(self.e_names): - raise Exception('Length of S array does not match ensembles.') - for e, e_name in enumerate(self.e_names): - if tmp[e] <= 0: - raise Exception('S has to be larger than 0.') - self.S[e_name] = tmp[e] - else: - if isinstance(tmp, (int, float)): - if tmp <= 0: - raise Exception('S has to be larger than 0.') - for e, e_name in enumerate(self.e_names): - self.S[e_name] = tmp - else: - raise TypeError('S is not in proper format.') - else: - for e, e_name in enumerate(self.e_names): - if e_name in Obs.S_dict: - self.S[e_name] = Obs.S_dict[e_name] - else: - self.S[e_name] = Obs.S_global - - if 'tau_exp' in kwargs: - tmp = kwargs.get('tau_exp') - if isinstance(tmp, list): - if len(tmp) != len(self.e_names): - raise Exception('Length of tau_exp array does not match ensembles.') - for e, e_name in enumerate(self.e_names): - if tmp[e] < 0: - raise Exception('tau_exp smaller than 0.') - self.tau_exp[e_name] = tmp[e] - else: - if isinstance(tmp, (int, float)): - if tmp < 0: - raise Exception('tau_exp smaller than 0.') - for e, e_name in enumerate(self.e_names): - self.tau_exp[e_name] = tmp - else: - raise TypeError('tau_exp is not in proper format.') - else: - for e, e_name in enumerate(self.e_names): - if e_name in Obs.tau_exp_dict: - self.tau_exp[e_name] = Obs.tau_exp_dict[e_name] - else: - self.tau_exp[e_name] = Obs.tau_exp_global - - if 'N_sigma' in kwargs: - self.N_sigma = kwargs.get('N_sigma') - if not isinstance(self.N_sigma, (int, float)): - raise TypeError('N_sigma is not a number.') - else: - self.N_sigma = Obs.N_sigma_global - - if max([len(x) for x in self.names]) <= e_tag_local: - for e, e_name in enumerate(self.e_names): - self.e_content[e_name] = [e_name] - else: - for e, e_name in enumerate(self.e_names): - if len(e_name) < e_tag_local: - self.e_content[e_name] = [e_name] - else: - self.e_content[e_name] = sorted(filter(lambda x: x.startswith(e_name), self.names)) - - for e, e_name in enumerate(self.e_names): - - r_length = [] - for r_name in self.e_content[e_name]: - r_length.append(len(self.deltas[r_name])) - - e_N = np.sum(r_length) - w_max = max(r_length) // 2 - e_gamma[e_name] = np.zeros(w_max) - self.e_rho[e_name] = np.zeros(w_max) - self.e_drho[e_name] = np.zeros(w_max) - - if fft: - for r_name in self.e_content[e_name]: - max_gamma = min(self.shape[r_name], w_max) - # The padding for the fft has to be even - padding = self.shape[r_name] + max_gamma + (self.shape[r_name] + max_gamma) % 2 - e_gamma[e_name][:max_gamma] += np.fft.irfft(np.abs(np.fft.rfft(self.deltas[r_name], padding)) ** 2)[:max_gamma] - else: - for n in range(w_max): - for r_name in self.e_content[e_name]: - if self.shape[r_name] - n >= 0: - e_gamma[e_name][n] += self.deltas[r_name][0:self.shape[r_name] - n].dot(self.deltas[r_name][n:self.shape[r_name]]) - - e_shapes = [] - for r_name in self.e_content[e_name]: - e_shapes.append(self.shape[r_name]) - - div = np.array([]) - mul = np.array([]) - sorted_shapes = sorted(e_shapes) - for i, item in enumerate(sorted_shapes): - if len(div) > w_max: - break - if i == 0: - samples = item - else: - samples = item - sorted_shapes[i - 1] - div = np.append(div, np.repeat(np.sum(sorted_shapes[i:]), samples)) - mul = np.append(mul, np.repeat(len(sorted_shapes) - i, samples)) - div = div - np.arange(len(div)) * mul - - e_gamma[e_name] /= div[:w_max] - - if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny: # Prevent division by zero - self.e_tauint[e_name] = 0.5 - self.e_dtauint[e_name] = 0.0 - self.e_dvalue[e_name] = 0.0 - self.e_ddvalue[e_name] = 0.0 - self.e_windowsize[e_name] = 0 - continue - - self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0] - self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:]))) - # Make sure no entry of tauint is smaller than 0.5 - self.e_n_tauint[e_name][self.e_n_tauint[e_name] < 0.5] = 0.500000000001 - # hep-lat/0306017 eq. (42) - self.e_n_dtauint[e_name] = self.e_n_tauint[e_name] * 2 * np.sqrt(np.abs(np.arange(w_max) - + 0.5 - self.e_n_tauint[e_name]) / e_N) - self.e_n_dtauint[e_name][0] = 0.0 - - - def _compute_drho(i): - tmp = self.e_rho[e_name][i+1:w_max] + np.concatenate([self.e_rho[e_name][i-1::-1], self.e_rho[e_name][1:w_max - 2 * i]]) - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i] - self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N) - - - _compute_drho(1) - if self.tau_exp[e_name] > 0: - # Critical slowing down analysis - for n in range(1, w_max // 2): - _compute_drho(n + 1) - if (self.e_rho[e_name][n] - self.N_sigma * self.e_drho[e_name][n]) < 0 or n >= w_max // 2 - 2: - # Bias correction hep-lat/0306017 eq. (49) included - self.e_tauint[e_name] = self.e_n_tauint[e_name][n] * (1 + (2 * n + 1) / e_N) / (1 + 1 / e_N) + self.tau_exp[e_name] * np.abs(self.e_rho[e_name][n + 1]) - # The absolute makes sure, that the tail contribution is always positive - self.e_dtauint[e_name] = np.sqrt(self.e_n_dtauint[e_name][n] ** 2 + self.tau_exp[e_name] ** 2 * self.e_drho[e_name][n + 1] ** 2) - # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2 - self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) - self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) - self.e_windowsize[e_name] = n - break - else: - # Standard automatic windowing procedure - g_w = self.S[e_name] / np.log((2 * self.e_n_tauint[e_name][1:] + 1) / (2 * self.e_n_tauint[e_name][1:] - 1)) - g_w = np.exp(- np.arange(1, w_max) / g_w) - g_w / np.sqrt(np.arange(1, w_max) * e_N) - for n in range(1, w_max): - if n < w_max // 2 - 2: - _compute_drho(n + 1) - if g_w[n - 1] < 0 or n >= w_max - 1: - self.e_tauint[e_name] = self.e_n_tauint[e_name][n] * (1 + (2 * n + 1) / e_N) / (1 + 1 / e_N) # Bias correction hep-lat/0306017 eq. (49) - self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n] - self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) - self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) - self.e_windowsize[e_name] = n - break - - if len(self.e_content[e_name]) > 1 and self.e_dvalue[e_name] > np.finfo(np.float).eps: - e_mean = 0 - for r_name in self.e_content[e_name]: - e_mean += self.shape[r_name] * self.r_values[r_name] - e_mean /= e_N - xi2 = 0 - for r_name in self.e_content[e_name]: - xi2 += self.shape[r_name] * (self.r_values[r_name] - e_mean) ** 2 - xi2 /= self.e_dvalue[e_name] ** 2 * e_N - self.e_Q[e_name] = 1 - scipy.special.gammainc((len(self.e_content[e_name]) - 1.0) / 2.0, xi2 / 2.0) - else: - self.e_Q[e_name] = None - - self.dvalue += self.e_dvalue[e_name] ** 2 - self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2 - - self.dvalue = np.sqrt(self.dvalue) - if self.dvalue == 0.0: - self.ddvalue = 0.0 - else: - self.ddvalue = np.sqrt(self.ddvalue) / self.dvalue - return 0 - - - def print(self, level=1): - """Print basic properties of the Obs.""" - if level == 0: - print(self) - else: - print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self.dvalue, self.ddvalue, np.abs(self.dvalue / self.value) * 100)) - if len(self.e_names) > 1: - print(' Ensemble errors:') - for e_name in self.e_names: - if len(self.e_names) > 1: - print('', e_name, '\t %3.8e +/- %3.8e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name])) - if self.tau_exp[e_name] > 0: - print(' t_int\t %3.8e +/- %3.8e tau_exp = %3.2f, N_sigma = %1.0i' % (self.e_tauint[e_name], self.e_dtauint[e_name], self.tau_exp[e_name], self.N_sigma)) - else: - print(' t_int\t %3.8e +/- %3.8e S = %3.2f' % (self.e_tauint[e_name], self.e_dtauint[e_name], self.S[e_name])) - if level > 1: - print(self.N, 'samples in', len(self.e_names), 'ensembles:') - for e_name in self.e_names: - print(e_name, ':', self.e_content[e_name]) - - - def plot_tauint(self): - """Plot integrated autocorrelation time for each ensemble.""" - if not self.e_names: - raise Exception('Run the gamma method first.') - for e, e_name in enumerate(self.e_names): - plt.xlabel('W') - plt.ylabel('tauint') - length = int(len(self.e_n_tauint[e_name])) - plt.errorbar(np.arange(length), self.e_n_tauint[e_name][:], yerr=self.e_n_dtauint[e_name][:], linewidth=1, capsize=2) - plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25) - if self.tau_exp[e_name] > 0: - base = self.e_n_tauint[e_name][self.e_windowsize[e_name]] - x_help = np.arange(2 * self.tau_exp[e_name]) - y_help = (x_help + 1) * np.abs(self.e_rho[e_name][self.e_windowsize[e_name]+1]) * (1 - x_help / (2 * (2 * self.tau_exp[e_name] - 1))) + base - x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]) - plt.plot(x_arr, y_help, 'k', linewidth=1) - plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]], - yerr=[self.e_dtauint[e_name]], fmt='k', linewidth=1, capsize=2) - xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5 - plt.title('Tauint ' + e_name + ', tau_exp='+str(np.around(self.tau_exp[e_name], decimals=2))) - else: - xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5) - plt.title('Tauint ' + e_name + ', S='+str(np.around(self.S[e_name], decimals=2))) - plt.xlim(-0.5, xmax) - plt.show() - - - def plot_rho(self): - """Plot normalized autocorrelation function time for each ensemble.""" - if not self.e_names: - raise Exception('Run the gamma method first.') - for e, e_name in enumerate(self.e_names): - plt.xlabel('W') - plt.ylabel('rho') - length = int(len(self.e_drho[e_name])) - plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2) - plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25) - if self.tau_exp[e_name] > 0: - plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]], - [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1) - xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5 - plt.title('Rho ' + e_name + ', tau_exp='+str(np.around(self.tau_exp[e_name], decimals=2))) - else: - xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5) - plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))) - plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1) - plt.xlim(-0.5, xmax) - plt.show() - - - def plot_rep_dist(self): - """Plot replica distribution for each ensemble with more than one replicum.""" - if not self.e_names: - raise Exception('Run the gamma method first.') - for e, e_name in enumerate(self.e_names): - if len(self.e_content[e_name]) == 1: - print('No replica distribution for a single replicum (', e_name, ')') - continue - r_length = [] - sub_r_mean = 0 - for r, r_name in enumerate(self.e_content[e_name]): - r_length.append(len(self.deltas[r_name])) - sub_r_mean += self.shape[r_name] * self.r_values[r_name] - e_N = np.sum(r_length) - sub_r_mean /= e_N - arr = np.zeros(len(self.e_content[e_name])) - for r, r_name in enumerate(self.e_content[e_name]): - arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1)) - plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name])) - plt.title('Replica distribution' + e_name + ' (mean=0, var=1), Q='+str(np.around(self.e_Q[e_name], decimals=2))) - plt.show() - - - def plot_history(self): - """Plot derived Monte Carlo history for each ensemble.""" - if not self.e_names: - raise Exception('Run the gamma method first.') - - for e, e_name in enumerate(self.e_names): - f = plt.figure() - r_length = [] - sub_r_mean = 0 - for r, r_name in enumerate(self.e_content[e_name]): - r_length.append(len(self.deltas[r_name])) - e_N = np.sum(r_length) - x = np.arange(e_N) - tmp = [] - for r, r_name in enumerate(self.e_content[e_name]): - tmp.append(self.deltas[r_name]+self.r_values[r_name]) - y = np.concatenate(tmp, axis=0) - plt.errorbar(x, y, fmt='.', markersize=3) - plt.xlim(-0.5, e_N - 0.5) - plt.title(e_name) - plt.show() - - - def plot_piechart(self): - """Plot piechart which shows the fractional contribution of each - ensemble to the error and returns a dictionary containing the fractions.""" - if not self.e_names: - raise Exception('Run the gamma method first.') - if self.dvalue == 0.0: - raise Exception('Error is 0.0') - labels = self.e_names - sizes = [i ** 2 for i in list(self.e_dvalue.values())] / self.dvalue ** 2 - fig1, ax1 = plt.subplots() - ax1.pie(sizes, labels=labels, startangle=90, normalize=True) - ax1.axis('equal') - plt.show() - - return dict(zip(self.e_names, sizes)) - - def dump(self, name, **kwargs): - """Dump the Obs to a pickle file 'name'. - - Keyword arguments - ----------------- - path -- specifies a custom path for the file (default '.') - """ - if 'path' in kwargs: - file_name = kwargs.get('path') + '/' + name + '.p' - else: - file_name = name + '.p' - with open(file_name, 'wb') as fb: - pickle.dump(self, fb) - - - def __repr__(self): - if self.dvalue == 0.0: - return 'Obs['+str(self.value)+']' - fexp = np.floor(np.log10(self.dvalue)) - if fexp < 0.0: - return 'Obs[{:{form}}({:2.0f})]'.format(self.value, self.dvalue * 10 ** (-fexp + 1), form='.'+str(-int(fexp) + 1) + 'f') - elif fexp == 0.0: - return 'Obs[{:.1f}({:1.1f})]'.format(self.value, self.dvalue) - else: - return 'Obs[{:.0f}({:2.0f})]'.format(self.value, self.dvalue) - - - # Overload comparisons - def __lt__(self, other): - return self.value < other - - def __gt__(self, other): - return self.value > other - - - # Overload math operations - def __add__(self, y): - if isinstance(y, Obs): - return derived_observable(lambda x, **kwargs: x[0] + x[1], [self, y], man_grad=[1, 1]) - else: - if isinstance(y, np.ndarray): - return np.array([self + o for o in y]) - else: - return derived_observable(lambda x, **kwargs: x[0] + y, [self], man_grad=[1]) - - - def __radd__(self, y): - return self + y - - - def __mul__(self, y): - if isinstance(y, Obs): - return derived_observable(lambda x, **kwargs: x[0] * x[1], [self, y], man_grad=[y.value, self.value]) - else: - if isinstance(y, np.ndarray): - return np.array([self * o for o in y]) - else: - return derived_observable(lambda x, **kwargs: x[0] * y, [self], man_grad=[y]) - - - def __rmul__(self, y): - return self * y - - - def __sub__(self, y): - if isinstance(y, Obs): - return derived_observable(lambda x, **kwargs: x[0] - x[1], [self, y], man_grad=[1, -1]) - else: - if isinstance(y, np.ndarray): - return np.array([self - o for o in y]) - else: - return derived_observable(lambda x, **kwargs: x[0] - y, [self], man_grad=[1]) - - - def __rsub__(self, y): - return -1 * (self - y) - - - def __neg__(self): - return -1 * self - - - def __truediv__(self, y): - if isinstance(y, Obs): - return derived_observable(lambda x, **kwargs: x[0] / x[1], [self, y], man_grad=[1 / y.value, - self.value / y.value ** 2]) - else: - if isinstance(y, np.ndarray): - return np.array([self / o for o in y]) - else: - return derived_observable(lambda x, **kwargs: x[0] / y, [self], man_grad=[1 / y]) - - - def __rtruediv__(self, y): - if isinstance(y, Obs): - return derived_observable(lambda x, **kwargs: x[0] / x[1], [y, self], man_grad=[1 / self.value, - y.value / self.value ** 2]) - else: - if isinstance(y, np.ndarray): - return np.array([o / self for o in y]) - else: - return derived_observable(lambda x, **kwargs: y / x[0], [self], man_grad=[-y / self.value ** 2]) - - - def __pow__(self, y): - if isinstance(y, Obs): - return derived_observable(lambda x: x[0] ** x[1], [self, y]) - else: - return derived_observable(lambda x: x[0] ** y, [self]) - - - def __rpow__(self, y): - if isinstance(y, Obs): - return derived_observable(lambda x: x[0] ** x[1], [y, self]) - else: - return derived_observable(lambda x: y ** x[0], [self]) - - - def __abs__(self): - return derived_observable(lambda x: anp.abs(x[0]), [self]) - - - # Overload numpy functions - def sqrt(self): - return derived_observable(lambda x, **kwargs: np.sqrt(x[0]), [self], man_grad=[1 / 2 / np.sqrt(self.value)]) - - - def log(self): - return derived_observable(lambda x, **kwargs: np.log(x[0]), [self], man_grad=[1 / self.value]) - - - def exp(self): - return derived_observable(lambda x, **kwargs: np.exp(x[0]), [self], man_grad=[np.exp(self.value)]) - - - def sin(self): - return derived_observable(lambda x, **kwargs: np.sin(x[0]), [self], man_grad=[np.cos(self.value)]) - - - def cos(self): - return derived_observable(lambda x, **kwargs: np.cos(x[0]), [self], man_grad=[-np.sin(self.value)]) - - - def tan(self): - return derived_observable(lambda x, **kwargs: np.tan(x[0]), [self], man_grad=[1 / np.cos(self.value) ** 2]) - - - def arcsin(self): - return derived_observable(lambda x: anp.arcsin(x[0]), [self]) - - - def arccos(self): - return derived_observable(lambda x: anp.arccos(x[0]), [self]) - - - def arctan(self): - return derived_observable(lambda x: anp.arctan(x[0]), [self]) - - - def sinh(self): - return derived_observable(lambda x, **kwargs: np.sinh(x[0]), [self], man_grad=[np.cosh(self.value)]) - - - def cosh(self): - return derived_observable(lambda x, **kwargs: np.cosh(x[0]), [self], man_grad=[np.sinh(self.value)]) - - - def tanh(self): - return derived_observable(lambda x, **kwargs: np.tanh(x[0]), [self], man_grad=[1 / np.cosh(self.value) ** 2]) - - - def arcsinh(self): - return derived_observable(lambda x: anp.arcsinh(x[0]), [self]) - - - def arccosh(self): - return derived_observable(lambda x: anp.arccosh(x[0]), [self]) - - - def arctanh(self): - return derived_observable(lambda x: anp.arctanh(x[0]), [self]) - - - def sinc(self): - return derived_observable(lambda x: anp.sinc(x[0]), [self]) - - -def derived_observable(func, data, **kwargs): - """Construct a derived Obs according to func(data, **kwargs) using automatic differentiation. - - Parameters - ---------- - func -- arbitrary function of the form func(data, **kwargs). For the - automatic differentiation to work, all numpy functions have to have - the autograd wrapper (use 'import autograd.numpy as anp'). - data -- list of Obs, e.g. [obs1, obs2, obs3]. - - Keyword arguments - ----------------- - num_grad -- if True, numerical derivatives are used instead of autograd - (default False). To control the numerical differentiation the - kwargs of numdifftools.step_generators.MaxStepGenerator - can be used. - man_grad -- manually supply a list or an array which contains the jacobian - of func. Use cautiously, supplying the wrong derivative will - not be intercepted. - bias_correction -- if True, the bias correction specified in - hep-lat/0306017 eq. (19) is performed, not recommended. - (Only applicable for more than 1 replicum) - - Notes - ----- - For simple mathematical operations it can be practical to use anonymous - functions. For the ratio of two observables one can e.g. use - - new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2]) - """ - - data = np.asarray(data) - raveled_data = data.ravel() - - n_obs = len(raveled_data) - new_names = sorted(set([y for x in [o.names for o in raveled_data] for y in x])) - replicas = len(new_names) - - new_shape = {} - for i_data in raveled_data: - for name in new_names: - tmp = i_data.shape.get(name) - if tmp is not None: - if new_shape.get(name) is None: - new_shape[name] = tmp - else: - if new_shape[name] != tmp: - raise Exception('Shapes of ensemble', name, 'do not match.') - - values = np.vectorize(lambda x: x.value)(data) - - new_values = func(values, **kwargs) - - multi = 0 - if isinstance(new_values, np.ndarray): - multi = 1 - - new_r_values = {} - for name in new_names: - tmp_values = np.zeros(n_obs) - for i, item in enumerate(raveled_data): - tmp = item.r_values.get(name) - if tmp is None: - tmp = item.value - tmp_values[i] = tmp - if multi > 0: - tmp_values = np.array(tmp_values).reshape(data.shape) - new_r_values[name] = func(tmp_values, **kwargs) - - if 'man_grad' in kwargs: - deriv = np.asarray(kwargs.get('man_grad')) - if new_values.shape + data.shape != deriv.shape: - raise Exception('Manual derivative does not have correct shape.') - elif kwargs.get('num_grad') is True: - if multi > 0: - raise Exception('Multi mode currently not supported for numerical derivative') - options = { - 'base_step': 0.1, - 'step_ratio': 2.5, - 'num_steps': None, - 'step_nom': None, - 'offset': None, - 'num_extrap': None, - 'use_exact_steps': None, - 'check_num_steps': None, - 'scale': None} - for key in options.keys(): - kwarg = kwargs.get(key) - if kwarg is not None: - options[key] = kwarg - tmp_df = nd.Gradient(func, order=4, **{k:v for k, v in options.items() if v is not None})(values, **kwargs) - if tmp_df.size == 1: - deriv = np.array([tmp_df.real]) - else: - deriv = tmp_df.real - else: - deriv = jacobian(func)(values, **kwargs) - - final_result = np.zeros(new_values.shape, dtype=object) - - for i_val, new_val in np.ndenumerate(new_values): - new_deltas = {} - for j_obs, obs in np.ndenumerate(data): - for name in obs.names: - new_deltas[name] = new_deltas.get(name, 0) + deriv[i_val + j_obs] * obs.deltas[name] - - new_samples = [] - for name in new_names: - new_samples.append(new_deltas[name] + new_r_values[name][i_val]) - - final_result[i_val] = Obs(new_samples, new_names) - - # Bias correction - if replicas > 1 and kwargs.get('bias_correction'): - final_result[i_val].value = (replicas * new_val - final_result[i_val].value) / (replicas - 1) - else: - final_result[i_val].value = new_val - - if multi == 0: - final_result = final_result.item() - - return final_result - - -def reweight(weight, obs, **kwargs): - """Reweight a list of observables.""" - result = [] - for i in range(len(obs)): - if sorted(weight.names) != sorted(obs[i].names): - raise Exception('Error: Ensembles do not fit') - for name in weight.names: - if weight.shape[name] != obs[i].shape[name]: - raise Exception('Error: Shapes of ensemble', name, 'do not fit') - new_samples = [] - for name in sorted(weight.names): - new_samples.append((weight.deltas[name] + weight.r_values[name]) * (obs[i].deltas[name] + obs[i].r_values[name])) - tmp_obs = Obs(new_samples, sorted(weight.names)) - - result.append(derived_observable(lambda x, **kwargs: x[0] / x[1], [tmp_obs, weight], **kwargs)) - result[-1].reweighted = 1 - - return result - - -def correlate(obs_a, obs_b): - """Correlate two observables. - - Keep in mind to only correlate primary observables which have not been reweighted - yet. The reweighting has to be applied after correlating the observables. - Currently only works if ensembles are identical. This is not really necessary. - """ - - if sorted(obs_a.names) != sorted(obs_b.names): - raise Exception('Ensembles do not fit') - for name in obs_a.names: - if obs_a.shape[name] != obs_b.shape[name]: - raise Exception('Shapes of ensemble', name, 'do not fit') - - if obs_a.reweighted == 1: - print('Warning: The first observable is already reweighted.') - if obs_b.reweighted == 1: - print('Warning: The second observable is already reweighted.') - - new_samples = [] - for name in sorted(obs_a.names): - new_samples.append((obs_a.deltas[name] + obs_a.r_values[name]) * (obs_b.deltas[name] + obs_b.r_values[name])) - - return Obs(new_samples, sorted(obs_a.names)) - - -def covariance(obs1, obs2, correlation=False, **kwargs): - """Calculates the covariance of two observables. - - covariance(obs, obs) is equal to obs.dvalue ** 2 - The gamma method has to be applied first to both observables. - - If abs(covariance(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance - is constrained to the maximum value in order to make sure that covariance - matrices are positive semidefinite. - - Keyword arguments - ----------------- - correlation -- if true the correlation instead of the covariance is - returned (default False) - """ - - for name in sorted(set(obs1.names + obs2.names)): - if (obs1.shape.get(name) != obs2.shape.get(name)) and (obs1.shape.get(name) is not None) and (obs2.shape.get(name) is not None): - raise Exception('Shapes of ensemble', name, 'do not fit') - - if obs1.e_names == {} or obs2.e_names == {}: - raise Exception('The gamma method has to be applied to both Obs first.') - - dvalue = 0 - - for e_name in obs1.e_names: - - if e_name not in obs2.e_names: - continue - - gamma = 0 - r_length = [] - for r_name in obs1.e_content[e_name]: - if r_name not in obs2.e_content[e_name]: - continue - - r_length.append(len(obs1.deltas[r_name])) - - gamma += np.sum(obs1.deltas[r_name] * obs2.deltas[r_name]) - - e_N = np.sum(r_length) - - tau_combined = (obs1.e_tauint[e_name] + obs2.e_tauint[e_name]) / 2 - dvalue += gamma / e_N * (1 + 1 / e_N) / e_N * 2 * tau_combined - - if np.abs(dvalue / obs1.dvalue / obs2.dvalue) > 1.0: - dvalue = np.sign(dvalue) * obs1.dvalue * obs2.dvalue - - if correlation: - dvalue = dvalue / obs1.dvalue / obs2.dvalue - - return dvalue - - -def covariance2(obs1, obs2, correlation=False, **kwargs): - """Alternative implementation of the covariance of two observables. - - covariance(obs, obs) is equal to obs.dvalue ** 2 - The gamma method has to be applied first to both observables. - - If abs(covariance(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance - is constrained to the maximum value in order to make sure that covariance - matrices are positive semidefinite. - - Keyword arguments - ----------------- - correlation -- if true the correlation instead of the covariance is - returned (default False) - """ - - for name in sorted(set(obs1.names + obs2.names)): - if (obs1.shape.get(name) != obs2.shape.get(name)) and (obs1.shape.get(name) is not None) and (obs2.shape.get(name) is not None): - raise Exception('Shapes of ensemble', name, 'do not fit') - - if obs1.e_names == {} or obs2.e_names == {}: - raise Exception('The gamma method has to be applied to both Obs first.') - - dvalue = 0 - e_gamma = {} - e_dvalue = {} - e_n_tauint = {} - e_rho = {} - - for e_name in obs1.e_names: - - if e_name not in obs2.e_names: - continue - - r_length = [] - for r_name in obs1.e_content[e_name]: - r_length.append(len(obs1.deltas[r_name])) - - e_N = np.sum(r_length) - w_max = max(r_length) // 2 - e_gamma[e_name] = np.zeros(w_max) - - for r_name in obs1.e_content[e_name]: - if r_name not in obs2.e_content[e_name]: - continue - max_gamma = min(obs1.shape[r_name], w_max) - # The padding for the fft has to be even - padding = obs1.shape[r_name] + max_gamma + (obs1.shape[r_name] + max_gamma) % 2 - e_gamma[e_name][:max_gamma] += (np.fft.irfft(np.fft.rfft(obs1.deltas[r_name], padding) * np.conjugate(np.fft.rfft(obs2.deltas[r_name], padding)))[:max_gamma] - + np.fft.irfft(np.fft.rfft(obs2.deltas[r_name], padding) * np.conjugate(np.fft.rfft(obs1.deltas[r_name], padding)))[:max_gamma]) / 2.0 - - if np.all(e_gamma[e_name]) == 0.0: - continue - - e_shapes = [] - for r_name in obs1.e_content[e_name]: - e_shapes.append(obs1.shape[r_name]) - - div = np.array([]) - mul = np.array([]) - sorted_shapes = sorted(e_shapes) - for i, item in enumerate(sorted_shapes): - if len(div) > w_max: - break - if i == 0: - samples = item - else: - samples = item - sorted_shapes[i - 1] - div = np.append(div, np.repeat(np.sum(sorted_shapes[i:]), samples)) - mul = np.append(mul, np.repeat(len(sorted_shapes) - i, samples)) - div = div - np.arange(len(div)) * mul - - e_gamma[e_name] /= div[:w_max] - - e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0] - e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], e_rho[e_name][1:]))) - # Make sure no entry of tauint is smaller than 0.5 - e_n_tauint[e_name][e_n_tauint[e_name] < 0.5] = 0.500000000001 - - - window = max(obs1.e_windowsize[e_name], obs2.e_windowsize[e_name]) - # Bias correction hep-lat/0306017 eq. (49) - e_dvalue[e_name] = 2 * (e_n_tauint[e_name][window] + obs1.tau_exp[e_name] * np.abs(e_rho[e_name][window + 1])) * (1 + (2 * window + 1) / e_N) * e_gamma[e_name][0] / e_N - - dvalue += e_dvalue[e_name] - - if np.abs(dvalue / obs1.dvalue / obs2.dvalue) > 1.0: - dvalue = np.sign(dvalue) * obs1.dvalue * obs2.dvalue - - if correlation: - dvalue = dvalue / obs1.dvalue / obs2.dvalue - - return dvalue - - -def covariance3(obs1, obs2, correlation=False, **kwargs): - """Another alternative implementation of the covariance of two observables. - - covariance2(obs, obs) is equal to obs.dvalue ** 2 - Currently only works if ensembles are identical. - The gamma method has to be applied first to both observables. - - If abs(covariance2(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance - is constrained to the maximum value in order to make sure that covariance - matrices are positive semidefinite. - - Keyword arguments - ----------------- - correlation -- if true the correlation instead of the covariance is - returned (default False) - plot -- if true, the integrated autocorrelation time for each ensemble is - plotted. - """ - - for name in sorted(set(obs1.names + obs2.names)): - if (obs1.shape.get(name) != obs2.shape.get(name)) and (obs1.shape.get(name) is not None) and (obs2.shape.get(name) is not None): - raise Exception('Shapes of ensemble', name, 'do not fit') - - if obs1.e_names == {} or obs2.e_names == {}: - raise Exception('The gamma method has to be applied to both Obs first.') - - tau_exp = [] - S = [] - for e_name in sorted(set(obs1.e_names + obs2.e_names)): - t_1 = obs1.tau_exp.get(e_name) - t_2 = obs2.tau_exp.get(e_name) - if t_1 is None: - t_1 = 0 - if t_2 is None: - t_2 = 0 - tau_exp.append(max(t_1, t_2)) - S_1 = obs1.S.get(e_name) - S_2 = obs2.S.get(e_name) - if S_1 is None: - S_1 = Obs.S_global - if S_2 is None: - S_2 = Obs.S_global - S.append(max(S_1, S_2)) - - check_obs = obs1 + obs2 - check_obs.gamma_method(tau_exp=tau_exp, S=S) - - if kwargs.get('plot'): - check_obs.plot_tauint() - check_obs.plot_rho() - - cov = (check_obs.dvalue ** 2 - obs1.dvalue ** 2 - obs2.dvalue ** 2) / 2 - - if np.abs(cov / obs1.dvalue / obs2.dvalue) > 1.0: - cov = np.sign(cov) * obs1.dvalue * obs2.dvalue - - if correlation: - cov = cov / obs1.dvalue / obs2.dvalue - - return cov - - -def use_time_reversal_symmetry(data1, data2, **kwargs): - """Combine two correlation functions (lists of Obs) according to time reversal symmetry - - Keyword arguments - ----------------- - minus -- if True, multiply the second correlation function by a minus sign. - """ - if kwargs.get('minus'): - sign = -1 - else: - sign = 1 - - result = [] - T = int(len(data1)) - for i in range(T): - result.append(derived_observable(lambda x, **kwargs: (x[0] + sign * x[1]) / 2, [data1[i], data2[T - i - 1]], **kwargs)) - - return result - - -def pseudo_Obs(value, dvalue, name, samples=1000): - """Generate a pseudo Obs with given value, dvalue and name - - The standard number of samples is a 1000. This can be adjusted. - """ - if dvalue <= 0.0: - return Obs([np.zeros(samples) + value], [name]) - else: - for _ in range(100): - deltas = [np.random.normal(0.0, dvalue * np.sqrt(samples), samples)] - deltas -= np.mean(deltas) - deltas *= dvalue / np.sqrt((np.var(deltas) / samples)) / np.sqrt(1 + 3 / samples) - deltas += value - res = Obs(deltas, [name]) - res.gamma_method(S=2, tau_exp=0) - if abs(res.dvalue - dvalue) < 1e-10 * dvalue: - break - - res.value = float(value) - - return res - - -def dump_object(obj, name, **kwargs): - """Dump object into pickle file. - - Keyword arguments - ----------------- - path -- specifies a custom path for the file (default '.') - """ - if 'path' in kwargs: - file_name = kwargs.get('path') + '/' + name + '.p' - else: - file_name = name + '.p' - with open(file_name, 'wb') as fb: - pickle.dump(obj, fb) - - -def load_object(path): - """Load object from pickle file. """ - with open(path, 'rb') as file: - return pickle.load(file) - - -def plot_corrs(observables, **kwargs): - """Plot lists of Obs. - - Parameters - ---------- - observables -- list of lists of Obs, where the nth entry is considered to be the - correlation function - at x0=n e.g. [[f_A_0,f_A_1],[f_P_0,f_P_1]] or [f_A,f_P], where f_A and f_P are lists of Obs. - - Keyword arguments - ----------------- - xrange -- list of two values, determining the range of the x-axis e.g. [4, 8] - yrange -- list of two values, determining the range of the y-axis e.g. [0.2, 1.1] - prange -- list of two values, visualizing the width of the plateau e.g. [10, 15] - reference -- float valued variable which is shown as horizontal line for reference - plateau -- Obs which is shown as horizontal line with errorbar for reference - shift -- shift x by given value - label -- list of labels, has to have the same length as observables - exp -- plot exponential from fitting routine - """ - - if 'shift' in kwargs: - shift = kwargs.get('shift') - else: - shift = 0 - - if 'label' in kwargs: - label = kwargs.get('label') - if len(label) != len(observables): - raise Exception('label has to be a list with exactly one entry per entry of observables.') - else: - label = [] - for j in range(len(observables)): - label.append(str(j + 1)) - - - f = plt.figure() - for j in range(len(observables)): - T = len(observables[j]) - - x = np.arange(T) + shift - y = np.zeros(T) - y_err = np.zeros(T) - - for i in range(T): - y[i] = observables[j][i].value - y_err[i] = observables[j][i].dvalue - - plt.errorbar(x, y, yerr=y_err, ls='none', fmt='o', capsize=3, - markersize=5, lw=1, label=label[j]) - - if kwargs.get('logscale'): - plt.yscale('log') - - if 'xrange' in kwargs: - xrange = kwargs.get('xrange') - plt.xlim(xrange[0], xrange[1]) - visible_y = y[int(xrange[0] + 0.5):int(xrange[1] + 0.5)] - visible_y_err = y_err[int(xrange[0] + 0.5):int(xrange[1] + 0.5)] - y_start = np.min(visible_y - visible_y_err) - y_stop = np.max(visible_y + visible_y_err) - span = y_stop - y_start - if np.isfinite(y_start) and np.isfinite(y_stop): - plt.ylim(y_start - 0.1 * span, y_stop + 0.1 * span) - - if 'yrange' in kwargs: - yrange = kwargs.get('yrange') - plt.ylim(yrange[0], yrange[1]) - - if 'reference' in kwargs: - y_value = kwargs.get('reference') - plt.axhline(y=y_value, linewidth=2, color='k', alpha=0.25) - - if 'prange' in kwargs: - prange = kwargs.get('prange') - plt.axvline(x=prange[0] - 0.5, ls='--', c='k', lw=1, alpha=0.5) - plt.axvline(x=prange[1] + 0.5, ls='--', c='k', lw=1, alpha=0.5) - - if 'plateau' in kwargs: - plateau = kwargs.get('plateau') - if isinstance(plateau, Obs): - plt.axhline(y=plateau.value, linewidth=2, color='k', alpha=0.6, label='Plateau') - plt.axhspan(plateau.value - plateau.dvalue, plateau.value + plateau.dvalue, alpha=0.25, color='k') - elif isinstance(plateau, list): - for i in range(len(plateau)): - plt.axhline(y=plateau[i].value, linewidth=2, color='C' + str(i), alpha=0.6, label='Plateau' + str(i + 1)) - plt.axhspan(plateau[i].value - plateau[i].dvalue, plateau[i].value + plateau[i].dvalue, - color='C' + str(i), alpha=0.25) - else: - raise Exception('Improper input for plateau.') - - if kwargs.get('exp'): - fit_result = kwargs.get('exp') - y_fit = fit_result[1].value * np.exp(-fit_result[0].value * x) - plt.plot(x, y_fit, color='k') - if not (fit_result[0].e_names == {} and fit_result[1].e_names == {}): - y_fit_err = np.sqrt((y_fit * fit_result[0].dvalue) ** 2 + 2 * covariance(fit_result[0], fit_result[1])* y_fit * - np.exp(-fit_result[0].value * x) + (np.exp(-fit_result[0].value * x) * fit_result[1].dvalue) ** 2) - plt.fill_between(x, y_fit + y_fit_err, y_fit - y_fit_err, color='k', alpha=0.1) - - plt.xlabel('$x_0/a$') - - if 'ylabel' in kwargs: - plt.ylabel(kwargs.get('ylabel')) - - if 'save' in kwargs: - lgd = plt.legend(loc=0) - else: - lgd = plt.legend(bbox_to_anchor=(1.04, 1), loc='upper left') - plt.show() - - if 'save' in kwargs: - save = kwargs.get('save') - if not isinstance(save, str): - raise Exception('save has to be a string.') - f.savefig(save + '.pdf', bbox_extra_artists=(lgd,), bbox_inches='tight') - - -def merge_obs(list_of_obs): - """Combine all observables in list_of_obs into one new observable - - It is not possible to combine obs which are based on the same replicum - """ - replist = [item for obs in list_of_obs for item in obs.names] - if (len(replist) == len(set(replist))) is False: - raise Exception('list_of_obs contains duplicate replica: %s' %(str(replist))) - new_dict = {} - for o in list_of_obs: - new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0) - for key in set(o.deltas) | set(o.r_values)}) - - return Obs(list(new_dict.values()), list(new_dict.keys())) From 7087804e712f51978ca4aa44e97957117f631bdd Mon Sep 17 00:00:00 2001 From: JanNeuendorf <75676159+JanNeuendorf@users.noreply.github.com> Date: Fri, 8 Jan 2021 14:46:37 +0100 Subject: [PATCH 3/8] Add files via upload --- pyerrors/correlators.py | 652 +++++++++++++++++++++ pyerrors/pyerrors.py | 1233 +++++++++++++++++++++++++++++++++++++++ 2 files changed, 1885 insertions(+) create mode 100644 pyerrors/correlators.py create mode 100644 pyerrors/pyerrors.py diff --git a/pyerrors/correlators.py b/pyerrors/correlators.py new file mode 100644 index 00000000..89845ac9 --- /dev/null +++ b/pyerrors/correlators.py @@ -0,0 +1,652 @@ +import autograd.numpy as np +from .pyerrors import * +from .fits import standard_fit +from matplotlib import pyplot as plt +from matplotlib.ticker import NullFormatter # useful for `logit` scale +from matplotlib.backends.backend_tkagg import FigureCanvasTkAgg +import PySimpleGUI as sg +import matplotlib + +class Corr: + """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 inconvinient + to iterate over all timeslices for every operation. This is especially true, when dealing with smearing matrices. + + The correlator can have two types of content: An Obs at every timeslice OR a GEVP + smearing matrix at every timeslice. Other dependency (eg. spacial) are not supported. + """ + + def __init__(self, data_input,padding_front=0,padding_back=0): + #All data_input should be a list of things at different timeslices. This needs to be verified + + if not isinstance(data_input,list): + raise TypeError('Corr__init__ expects a list of timeslices.') + # data_input can have multiple shapes. The simplest one is a list of Obs. + #We check, if this is the case + if all([isinstance(item,Obs) for item in data_input]): + self.content=[np.asarray([item]) for item in data_input] + #Wrapping the Obs in an array ensures that the data structure is consistent with smearing matrices. + self.N=1 # number of smearings + + #data_input in the form [np.array(Obs,NxN)] + elif all([isinstance(item,np.ndarray) or item==None for item in data_input]) and any([isinstance(item,np.ndarray)for item in data_input]): + self.content= data_input + + noNull=[a for a in self.content if not (a is None)] #To check if the matrices are correct for all undefined elements + self.N= noNull[0].shape[0] + # The checks are now identical to the case above + if self.N>1 and noNull[0].shape[0]!=noNull[0].shape[1]: + raise Exception("Smearing matrices are not NxN") + if (not all([item.shape==noNull[0].shape for item in noNull])): + raise Exception("Items in data_input are not of identical shape."+str(noNull)) + + else: # In case its a list of something else. + raise Exception ("data_input contains item of wrong type") + + #We now apply some padding to our list. In case that our list represents a correlator of length T but is not defined at every value. + #An undefined timeslice is represented by the None object + self.content=[None]*padding_front+self.content+[None]*padding_back + self.T=len(self.content) #for convenience: will be used a lot + + + self.gamma_method() + + + def gamma_method(self): + for item in self.content: + if not(item is None): + if self.N==1: + item[0].gamma_method() + else: + for i in range(self.N): + for j in range(self.N): + item[i,j].gamma_method() + + + + + #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 projected(self,vector_l=None,vector_r=None): + if self.N==1: + raise Exception("Trying to project a Corr, that already has N=1.") + #This Exception is in no way necessary. One could just return self + #But there is no scenario, where a user would want that to happen and the error message might be more informative. + + self.gamma_method() + + if vector_l is None: + vector_l,vector_r=np.asarray( [1.]+(self.N-1)*[0.]),np.asarray( [1.]+(self.N-1)*[0.]) + elif(vector_r is None): + vector_r=vector_l + + if not vector_l.shape==vector_r.shape==(self.N,): + raise Exception("Vectors are of wrong shape!") + + #We always normalize before projecting! But we only raise a warning, when it is clear, they where not meant to be normalized. + if (not(0.951: + transposed=[None if (G is None) else G.T for G in self.content] + return 0.5*(Corr(transposed)+self) + if self.N==1: + raise Exception("Trying to symmetrize a smearing matrix, that already has N=1.") + + + #We also include a simple GEVP method based on Scipy.linalg + def GEVP(self,t0,ts): + if (self.content[t0] is None) or (self.content[ts] is None): + raise Exception("Corr not defined at t0/ts") + G0, Gt=np.empty([self.N,self.N],dtype="double"),np.empty([self.N,self.N],dtype="double") + for i in range(self.N): + for j in range(self.N): + G0[i,j]=self.content[t0][i,j].value + Gt[i,j]=self.content[ts][i,j].value + + sp_val,sp_vec=scipy.linalg.eig(Gt,G0) + sp_vec=sp_vec[:,np.argmax(sp_val)] #we only want the eigenvector belonging to the biggest eigenvalue. + sp_vec=sp_vec/np.sqrt(sp_vec@sp_vec) + return sp_vec + + def deriv(self,symmetric=False): #Defaults to forward derivative f'(t)=f(t+1)-f(t) + if not symmetric: + newcontent=[] + for t in range(self.T-1): + if (self.content[t] is None) or (self.content[t+1] is None): + newcontent.append(None) + else: + newcontent.append(self.content[t+1]-self.content[t]) + if(all([x is None for x in newcontent])): + raise Exception("Derivative is undefined at all timeslices") + return Corr(newcontent, padding_back=1) + if symmetric: + newcontent=[] + for t in range(1,self.T-1): + if (self.content[t-1] is None) or (self.content[t+1] is None): + newcontent.append(None) + else: + newcontent.append(0.5*(self.content[t+1]-self.content[t-1])) + if(all([x is None for x in newcontent])): + raise Exception("Derivative is undefined at all timeslices") + return Corr(newcontent, padding_back=1,padding_front=1) + + #effective mass at every timeslice + def m_eff(self, periodic=False): + if self.N!=1: + raise Exception("Correlator must be projected before getting m_eff") + if not periodic: + newcontent=[] + for t in range(self.T-1): + if (self.content[t] is None) or (self.content[t+1] is None): + newcontent.append(None) + else: + newcontent.append(self.content[t]/self.content[t+1]) + if(all([x is None for x in newcontent])): + raise Exception("m_eff is undefined at all timeslices") + + return np.log(Corr(newcontent,padding_back=1)) + + else: #This is usually not very stable. One could default back to periodic=False. + newcontent=[] + for t in range(1,self.T-1): + if (self.content[t] is None) or (self.content[t+1] is None)or (self.content[t-1] is None): + newcontent.append(None) + else: + newcontent.append((self.content[t+1]+self.content[t-1])/(2*self.content[t])) + if(all([x is None for x in newcontent])): + raise Exception("m_eff is undefined at all timeslices") + return np.arccosh(Corr(newcontent,padding_back=1,padding_front=1)) + + + #We want to apply a pe.standard_fit directly to the Corr using an arbitrary function and range. + def fit(self,function,fitrange=None): + if self.N!=1: + raise Exception("Correlator must be projected before fitting") + + if fitrange is None: + fitrange=[0,self.T] + + xs = [x for x in range(fitrange[0],fitrange[1]) if not self.content[x] is None] + ys = [self.content[x][0] for x in range(fitrange[0],fitrange[1]) if not self.content[x] is None] + result = standard_fit(xs, ys,function,silent=(True)) + [item.gamma_method() for item in result if isinstance(item,Obs)] + return result + + #we want to quickly get a plateau + def plateau(self,plateau_range,method="fit"): + if self.N!=1: + raise Exception("Correlator must be projected before getting a plateau") + if(all([self.content[t] is None for t in range(plateau_range[0],plateau_range[1])])): + raise Exception("plateau is undefined at all timeslices in plateaurange") + if method=="fit": + def const_func(a,t): + return a[0]+a[1]*0 # At some point pe.standard fit had an issue with single parameter fits. Being careful does not hurt + return self.fit(const_func,plateau_range)[0] + elif method in ["avg","average","mean"]: + returnvalue= np.mean([item[0] for item in self.content if not item is None]) + returnvalue.gamma_method() + return returnvalue + + else: + raise Exception("Unsupported plateau method: "+method) + + #quick and dirty plotting function to view Correlator inside Jupyter + #If one would not want to import pyplot, this could easily be replaced by a call to pe.plot_corrs + #This might be a bit more flexible later + def show(self,xrange=None,logscale=False): + if self.N!=1: + raise Exception("Correlator must be projected before plotting") + if xrange is None: + xrange=[0,self.T] + + x,y,y_err=self.plottable() + plt.errorbar(x,y,y_err,fmt="o-") + if logscale: + plt.yscale("log") + else: + # we generate ylim instead of using autoscaling. + y_min=min([ (x[0].value-x[0].dvalue) for x in self.content[xrange[0]:xrange[1]] if(not x is None)]) + y_max=max([ (x[0].value+x[0].dvalue) for x in self.content[xrange[0]:xrange[1]] if(not x is None)]) + plt.ylim([y_min-0.1*(y_max-y_min),y_max+0.1*(y_max-y_min)]) + + plt.xlabel(r"$n_t$ [a]") + plt.xlim(xrange) + plt.title("Quickplot") + plt.grid() + + plt.show() + plt.clf() + return + + def dump(self,filename): + dump_object(self,filename) + return + + + + + def __repr__(self): + return("Corr[T="+str(self.T)+" , N="+str(self.N)+" , content="+str(self.content)+"]") + def __str__(self): + return ("Corr[T="+str(self.T)+" , N="+str(self.N)+" , content="+str(self.content)+"]") + + #We define the basic operations, that can be performed with correlators. + #While */+- get defined here, they only work for Corr*Obs and not Obs*Corr. + #This is because Obs*Corr checks Obs.__mul__ first and does not catch an exception. + #One could try and tell Obs to check if the y in __mul__ is a Corr and + + def __add__(self, y): + if isinstance(y, Corr): + if ((self.N!=y.N) or (self.T!=y.T) ): + raise Exception("Addition of Corrs with different shape") + newcontent=[] + for t in range(self.T): + if (self.content[t] is None) or (y.content[t] is None): + newcontent.append(None) + else: + newcontent.append(self.content[t]+y.content[t]) + return Corr(newcontent) + + elif isinstance(y, Obs) or isinstance(y, int) or isinstance(y,float): + newcontent=[] + for t in range(self.T): + if (self.content[t] is None): + newcontent.append(None) + else: + newcontent.append(self.content[t]+y) + return Corr(newcontent) + else: + raise TypeError("Corr + wrong type") + + def __mul__(self,y): + if isinstance(y,Corr): + if not((self.N==1 or y.N==1 or self.N==y.N) and self.T==y.T): + raise Exception("Multiplication of Corr object requires N=N or N=1 and T=T") + newcontent=[] + for t in range(self.T): + if (self.content[t] is None) or (y.content[t] is None): + newcontent.append(None) + else: + newcontent.append(self.content[t]*y.content[t]) + return Corr(newcontent) + + elif isinstance(y, Obs) or isinstance(y, int) or isinstance(y,float): + newcontent=[] + for t in range(self.T): + if (self.content[t] is None): + newcontent.append(None) + else: + newcontent.append(self.content[t]*y) + return Corr(newcontent) + else: + raise TypeError("Corr * wrong type") + + def __truediv__(self,y): + if isinstance(y,Corr): + if not((self.N==1 or y.N==1 or self.N==y.N) and self.T==y.T): + raise Exception("Multiplication of Corr object requires N=N or N=1 and T=T") + newcontent=[] + for t in range(self.T): + if (self.content[t] is None) or (y.content[t] is None): + newcontent.append(None) + else: + newcontent.append(self.content[t]/y.content[t]) + #Here we set the entire timeslice to undefined, if one of the smearings has encountered an division by zero. + #While this might throw away perfectly good values in other smearings, we will never have to check, if all values in our matrix are defined + for t in range(self.T): + if newcontent[t] is None: + continue + if np.isnan(np.sum(newcontent[t]).value): + newcontent[t]=None + + if all([item is None for item in newcontent]): + raise Exception("Division returns completely undefined correlator") + + + + return Corr(newcontent) + + elif isinstance(y, Obs): + if y.value==0: + raise Exception("Division by Zero will return undefined correlator") + newcontent=[] + for t in range(self.T): + if (self.content[t] is None): + newcontent.append(None) + else: + newcontent.append(self.content[t]/y) + return Corr(newcontent) + + elif isinstance(y, int) or isinstance(y,float): + if y==0: + raise Exception("Division by Zero will return undefined correlator") + newcontent=[] + for t in range(self.T): + if (self.content[t] is None): + newcontent.append(None) + else: + newcontent.append(self.content[t]/y) + return Corr(newcontent) + else: + raise TypeError("Corr / wrong type") + + def __neg__(self): + newcontent=[None if (item is None) else -1.*item for item in self.content] + return Corr(newcontent) + + def __sub__(self,y): + return self +(-y) + + def __pow__(self, y): + if isinstance(y, Obs) or isinstance(y,int) or isinstance(y,float): + newcontent=[None if (item is None) else item**y for item in self.content] + return Corr(newcontent) + else: + raise TypeError("type of exponent not supported") + + #The numpy functions: + def sqrt(self): + return self**0.5 + + def log(self): + newcontent=[None if (item is None) else np.log(item) for item in self.content] + return Corr(newcontent) + + def exp(self): + newcontent=[None if (item is None) else np.exp(item) for item in self.content] + return Corr(newcontent) + + def sin(self): + newcontent=[None if (item is None) else np.sin(item) for item in self.content] + return Corr(newcontent) + + def cos(self): + newcontent=[None if (item is None) else np.cos(item) for item in self.content] + return Corr(newcontent) + + def tan(self): + newcontent=[None if (item is None) else np.tan(item) for item in self.content] + + for t in range(self.T): + if newcontent[t] is None: + continue + if np.isnan(np.sum(newcontent[t]).value): + newcontent[t]=None + if all([item is None for item in newcontent]): + raise Exception("Operation returns completely undefined correlator") + + return Corr(newcontent) + + def sinh(self): + newcontent=[None if (item is None) else np.sinh(item) for item in self.content] + return Corr(newcontent) + + def cosh(self): + newcontent=[None if (item is None) else np.cosh(item) for item in self.content] + return Corr(newcontent) + + def tanh(self): + newcontent=[None if (item is None) else np.tanh(item) for item in self.content] + for t in range(self.T): + if newcontent[t] is None: + continue + if np.isnan(np.sum(newcontent[t]).value): + newcontent[t]=None + if all([item is None for item in newcontent]): + raise Exception("Operation returns completely undefined correlator") + return Corr(newcontent) + + def arcsin(self): + newcontent=[None if (item is None) else np.arcsin(item) for item in self.content] + for t in range(self.T): + if newcontent[t] is None: + continue + if np.isnan(np.sum(newcontent[t]).value): + newcontent[t]=None + if all([item is None for item in newcontent]): + raise Exception("Operation returns completely undefined correlator") + return Corr(newcontent) + + def arccos(self): + newcontent=[None if (item is None) else np.arccos(item) for item in self.content] + for t in range(self.T): + if newcontent[t] is None: + continue + if np.isnan(np.sum(newcontent[t]).value): + newcontent[t]=None + if all([item is None for item in newcontent]): + raise Exception("Operation returns completely undefined correlator") + return Corr(newcontent) + + def arctan(self): + newcontent=[None if (item is None) else np.arctan(item) for item in self.content] + for t in range(self.T): + if newcontent[t] is None: + continue + if np.isnan(np.sum(newcontent[t]).value): + newcontent[t]=None + if all([item is None for item in newcontent]): + raise Exception("Operation returns completely undefined correlator") + return Corr(newcontent) + + def arcsinh(self): + newcontent=[None if (item is None) else np.arcsinh(item) for item in self.content] + for t in range(self.T): + if newcontent[t] is None: + continue + if np.isnan(np.sum(newcontent[t]).value): + newcontent[t]=None + if all([item is None for item in newcontent]): + raise Exception("Operation returns completely undefined correlator") + return Corr(newcontent) + + def arccosh(self): + newcontent=[None if (item is None) else np.arccosh(item) for item in self.content] + for t in range(self.T): + if newcontent[t] is None: + continue + if np.isnan(np.sum(newcontent[t]).value): + newcontent[t]=None + if all([item is None for item in newcontent]): + raise Exception("Operation returns completely undefined correlator") + return Corr(newcontent) + + def arctanh(self): + newcontent=[None if (item is None) else np.arctanh(item) for item in self.content] + for t in range(self.T): + if newcontent[t] is None: + continue + if np.isnan(np.sum(newcontent[t]).value): + newcontent[t]=None + if all([item is None for item in newcontent]): + raise Exception("Operation returns completely undefined correlator") + return Corr(newcontent) + + + #right hand side operations (require tweak in main module to work) + def __rsub__(self,y): + return -self+y + def __rmul__(self, y): + return self * y + def __radd__(self,y): + return self + y + + + +#One of the most common tasks is to select a range for a plateau or a fit. This is best done visually. +def GUI_range_finder(corr, current_range=None): + T=corr.T + if corr.N!=1: + raise Exception("The Corr needs to be projected to select a range.") + #We need to define few helper functions for the Gui + def get_figure(corr,values): + fig = matplotlib.figure.Figure(figsize=(7, 4), dpi=100) + fig.clf() + x,y,err=corr.plottable() + ax=fig.add_subplot(111,label="main")#.plot(t, 2 * np.sin(2 * np.pi * t)) + end=int(max(values["range_start"],values["range_end"])) + start=int(min(values["range_start"],values["range_end"])) + db=[0.1,0.2,0.8] + ax.errorbar(x,y,err, fmt="-o",color=[0.4,0.6,0.8]) + ax.errorbar(x[start:end],y[start:end],err[start:end], fmt="-o",color=db) + offset=int(0.3*(end-start)) + xrange=[max(min(start-1,int(start-offset)),0),min(max(int(end+offset),end+1),T-1)] + ax.grid() + if values["Plateau"]: + plateau=corr.plateau([start,end]) + ax.hlines(plateau.value,0,T+1,lw=plateau.dvalue,color="red",alpha=0.5) + ax.hlines(plateau.value,0,T+1,lw=1,color="red") + ax.set_title(r"Current Plateau="+str(plateau)[4:-1]) + if(values["Crop X"]): + ax.set_xlim(xrange) + ax.set_xticks([x for x in ax.get_xticks() if (x-int(x)==0) and (0<=x= 0: + e_gamma[e_name][n] += self.deltas[r_name][0:self.shape[r_name] - n].dot(self.deltas[r_name][n:self.shape[r_name]]) + + e_shapes = [] + for r_name in self.e_content[e_name]: + e_shapes.append(self.shape[r_name]) + + div = np.array([]) + mul = np.array([]) + sorted_shapes = sorted(e_shapes) + for i, item in enumerate(sorted_shapes): + if len(div) > w_max: + break + if i == 0: + samples = item + else: + samples = item - sorted_shapes[i - 1] + div = np.append(div, np.repeat(np.sum(sorted_shapes[i:]), samples)) + mul = np.append(mul, np.repeat(len(sorted_shapes) - i, samples)) + div = div - np.arange(len(div)) * mul + + e_gamma[e_name] /= div[:w_max] + + if np.abs(e_gamma[e_name][0]) < 10 * np.finfo(float).tiny: # Prevent division by zero + self.e_tauint[e_name] = 0.5 + self.e_dtauint[e_name] = 0.0 + self.e_dvalue[e_name] = 0.0 + self.e_ddvalue[e_name] = 0.0 + self.e_windowsize[e_name] = 0 + continue + + self.e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0] + self.e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], self.e_rho[e_name][1:]))) + # Make sure no entry of tauint is smaller than 0.5 + self.e_n_tauint[e_name][self.e_n_tauint[e_name] < 0.5] = 0.500000000001 + # hep-lat/0306017 eq. (42) + self.e_n_dtauint[e_name] = self.e_n_tauint[e_name] * 2 * np.sqrt(np.abs(np.arange(w_max) + + 0.5 - self.e_n_tauint[e_name]) / e_N) + self.e_n_dtauint[e_name][0] = 0.0 + + + def _compute_drho(i): + tmp = self.e_rho[e_name][i+1:w_max] + np.concatenate([self.e_rho[e_name][i-1::-1], self.e_rho[e_name][1:w_max - 2 * i]]) - 2 * self.e_rho[e_name][i] * self.e_rho[e_name][1:w_max - i] + self.e_drho[e_name][i] = np.sqrt(np.sum(tmp ** 2) / e_N) + + + _compute_drho(1) + if self.tau_exp[e_name] > 0: + # Critical slowing down analysis + for n in range(1, w_max // 2): + _compute_drho(n + 1) + if (self.e_rho[e_name][n] - self.N_sigma * self.e_drho[e_name][n]) < 0 or n >= w_max // 2 - 2: + # Bias correction hep-lat/0306017 eq. (49) included + self.e_tauint[e_name] = self.e_n_tauint[e_name][n] * (1 + (2 * n + 1) / e_N) / (1 + 1 / e_N) + self.tau_exp[e_name] * np.abs(self.e_rho[e_name][n + 1]) + # The absolute makes sure, that the tail contribution is always positive + self.e_dtauint[e_name] = np.sqrt(self.e_n_dtauint[e_name][n] ** 2 + self.tau_exp[e_name] ** 2 * self.e_drho[e_name][n + 1] ** 2) + # Error of tau_exp neglected so far, missing term: self.e_rho[e_name][n + 1] ** 2 * d_tau_exp ** 2 + self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) + self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) + self.e_windowsize[e_name] = n + break + else: + # Standard automatic windowing procedure + g_w = self.S[e_name] / np.log((2 * self.e_n_tauint[e_name][1:] + 1) / (2 * self.e_n_tauint[e_name][1:] - 1)) + g_w = np.exp(- np.arange(1, w_max) / g_w) - g_w / np.sqrt(np.arange(1, w_max) * e_N) + for n in range(1, w_max): + if n < w_max // 2 - 2: + _compute_drho(n + 1) + if g_w[n - 1] < 0 or n >= w_max - 1: + self.e_tauint[e_name] = self.e_n_tauint[e_name][n] * (1 + (2 * n + 1) / e_N) / (1 + 1 / e_N) # Bias correction hep-lat/0306017 eq. (49) + self.e_dtauint[e_name] = self.e_n_dtauint[e_name][n] + self.e_dvalue[e_name] = np.sqrt(2 * self.e_tauint[e_name] * e_gamma[e_name][0] * (1 + 1 / e_N) / e_N) + self.e_ddvalue[e_name] = self.e_dvalue[e_name] * np.sqrt((n + 0.5) / e_N) + self.e_windowsize[e_name] = n + break + + if len(self.e_content[e_name]) > 1 and self.e_dvalue[e_name] > np.finfo(np.float).eps: + e_mean = 0 + for r_name in self.e_content[e_name]: + e_mean += self.shape[r_name] * self.r_values[r_name] + e_mean /= e_N + xi2 = 0 + for r_name in self.e_content[e_name]: + xi2 += self.shape[r_name] * (self.r_values[r_name] - e_mean) ** 2 + xi2 /= self.e_dvalue[e_name] ** 2 * e_N + self.e_Q[e_name] = 1 - scipy.special.gammainc((len(self.e_content[e_name]) - 1.0) / 2.0, xi2 / 2.0) + else: + self.e_Q[e_name] = None + + self.dvalue += self.e_dvalue[e_name] ** 2 + self.ddvalue += (self.e_dvalue[e_name] * self.e_ddvalue[e_name]) ** 2 + + self.dvalue = np.sqrt(self.dvalue) + if self.dvalue == 0.0: + self.ddvalue = 0.0 + else: + self.ddvalue = np.sqrt(self.ddvalue) / self.dvalue + return 0 + + + def print(self, level=1): + """Print basic properties of the Obs.""" + if level == 0: + print(self) + else: + print('Result\t %3.8e +/- %3.8e +/- %3.8e (%3.3f%%)' % (self.value, self.dvalue, self.ddvalue, np.abs(self.dvalue / self.value) * 100)) + if len(self.e_names) > 1: + print(' Ensemble errors:') + for e_name in self.e_names: + if len(self.e_names) > 1: + print('', e_name, '\t %3.8e +/- %3.8e' % (self.e_dvalue[e_name], self.e_ddvalue[e_name])) + if self.tau_exp[e_name] > 0: + print(' t_int\t %3.8e +/- %3.8e tau_exp = %3.2f, N_sigma = %1.0i' % (self.e_tauint[e_name], self.e_dtauint[e_name], self.tau_exp[e_name], self.N_sigma)) + else: + print(' t_int\t %3.8e +/- %3.8e S = %3.2f' % (self.e_tauint[e_name], self.e_dtauint[e_name], self.S[e_name])) + if level > 1: + print(self.N, 'samples in', len(self.e_names), 'ensembles:') + for e_name in self.e_names: + print(e_name, ':', self.e_content[e_name]) + + + def plot_tauint(self): + """Plot integrated autocorrelation time for each ensemble.""" + if not self.e_names: + raise Exception('Run the gamma method first.') + for e, e_name in enumerate(self.e_names): + plt.xlabel('W') + plt.ylabel('tauint') + length = int(len(self.e_n_tauint[e_name])) + plt.errorbar(np.arange(length), self.e_n_tauint[e_name][:], yerr=self.e_n_dtauint[e_name][:], linewidth=1, capsize=2) + plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25) + if self.tau_exp[e_name] > 0: + base = self.e_n_tauint[e_name][self.e_windowsize[e_name]] + x_help = np.arange(2 * self.tau_exp[e_name]) + y_help = (x_help + 1) * np.abs(self.e_rho[e_name][self.e_windowsize[e_name]+1]) * (1 - x_help / (2 * (2 * self.tau_exp[e_name] - 1))) + base + x_arr = np.arange(self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]) + plt.plot(x_arr, y_help, 'k', linewidth=1) + plt.errorbar([self.e_windowsize[e_name] + 2 * self.tau_exp[e_name]], [self.e_tauint[e_name]], + yerr=[self.e_dtauint[e_name]], fmt='k', linewidth=1, capsize=2) + xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5 + plt.title('Tauint ' + e_name + ', tau_exp='+str(np.around(self.tau_exp[e_name], decimals=2))) + else: + xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5) + plt.title('Tauint ' + e_name + ', S='+str(np.around(self.S[e_name], decimals=2))) + plt.xlim(-0.5, xmax) + plt.show() + + + def plot_rho(self): + """Plot normalized autocorrelation function time for each ensemble.""" + if not self.e_names: + raise Exception('Run the gamma method first.') + for e, e_name in enumerate(self.e_names): + plt.xlabel('W') + plt.ylabel('rho') + length = int(len(self.e_drho[e_name])) + plt.errorbar(np.arange(length), self.e_rho[e_name][:length], yerr=self.e_drho[e_name][:], linewidth=1, capsize=2) + plt.axvline(x=self.e_windowsize[e_name], color='r', alpha=0.25) + if self.tau_exp[e_name] > 0: + plt.plot([self.e_windowsize[e_name] + 1, self.e_windowsize[e_name] + 1 + 2 * self.tau_exp[e_name]], + [self.e_rho[e_name][self.e_windowsize[e_name] + 1], 0], 'k-', lw=1) + xmax = self.e_windowsize[e_name] + 2 * self.tau_exp[e_name] + 1.5 + plt.title('Rho ' + e_name + ', tau_exp='+str(np.around(self.tau_exp[e_name], decimals=2))) + else: + xmax = max(10.5, 2 * self.e_windowsize[e_name] - 0.5) + plt.title('Rho ' + e_name + ', S=' + str(np.around(self.S[e_name], decimals=2))) + plt.plot([-0.5, xmax], [0, 0], 'k--', lw=1) + plt.xlim(-0.5, xmax) + plt.show() + + + def plot_rep_dist(self): + """Plot replica distribution for each ensemble with more than one replicum.""" + if not self.e_names: + raise Exception('Run the gamma method first.') + for e, e_name in enumerate(self.e_names): + if len(self.e_content[e_name]) == 1: + print('No replica distribution for a single replicum (', e_name, ')') + continue + r_length = [] + sub_r_mean = 0 + for r, r_name in enumerate(self.e_content[e_name]): + r_length.append(len(self.deltas[r_name])) + sub_r_mean += self.shape[r_name] * self.r_values[r_name] + e_N = np.sum(r_length) + sub_r_mean /= e_N + arr = np.zeros(len(self.e_content[e_name])) + for r, r_name in enumerate(self.e_content[e_name]): + arr[r] = (self.r_values[r_name] - sub_r_mean) / (self.e_dvalue[e_name] * np.sqrt(e_N / self.shape[r_name] - 1)) + plt.hist(arr, rwidth=0.8, bins=len(self.e_content[e_name])) + plt.title('Replica distribution' + e_name + ' (mean=0, var=1), Q='+str(np.around(self.e_Q[e_name], decimals=2))) + plt.show() + + + def plot_history(self): + """Plot derived Monte Carlo history for each ensemble.""" + if not self.e_names: + raise Exception('Run the gamma method first.') + + for e, e_name in enumerate(self.e_names): + f = plt.figure() + r_length = [] + sub_r_mean = 0 + for r, r_name in enumerate(self.e_content[e_name]): + r_length.append(len(self.deltas[r_name])) + e_N = np.sum(r_length) + x = np.arange(e_N) + tmp = [] + for r, r_name in enumerate(self.e_content[e_name]): + tmp.append(self.deltas[r_name]+self.r_values[r_name]) + y = np.concatenate(tmp, axis=0) + plt.errorbar(x, y, fmt='.', markersize=3) + plt.xlim(-0.5, e_N - 0.5) + plt.title(e_name) + plt.show() + + + def plot_piechart(self): + """Plot piechart which shows the fractional contribution of each + ensemble to the error and returns a dictionary containing the fractions.""" + if not self.e_names: + raise Exception('Run the gamma method first.') + if self.dvalue == 0.0: + raise Exception('Error is 0.0') + labels = self.e_names + sizes = [i ** 2 for i in list(self.e_dvalue.values())] / self.dvalue ** 2 + fig1, ax1 = plt.subplots() + ax1.pie(sizes, labels=labels, startangle=90, normalize=True) + ax1.axis('equal') + plt.show() + + return dict(zip(self.e_names, sizes)) + + def dump(self, name, **kwargs): + """Dump the Obs to a pickle file 'name'. + + Keyword arguments + ----------------- + path -- specifies a custom path for the file (default '.') + """ + if 'path' in kwargs: + file_name = kwargs.get('path') + '/' + name + '.p' + else: + file_name = name + '.p' + with open(file_name, 'wb') as fb: + pickle.dump(self, fb) + + + def __repr__(self): + if self.dvalue == 0.0: + return 'Obs['+str(self.value)+']' + fexp = np.floor(np.log10(self.dvalue)) + if fexp < 0.0: + return 'Obs[{:{form}}({:2.0f})]'.format(self.value, self.dvalue * 10 ** (-fexp + 1), form='.'+str(-int(fexp) + 1) + 'f') + elif fexp == 0.0: + return 'Obs[{:.1f}({:1.1f})]'.format(self.value, self.dvalue) + else: + return 'Obs[{:.0f}({:2.0f})]'.format(self.value, self.dvalue) + + + # Overload comparisons + def __lt__(self, other): + return self.value < other + + def __gt__(self, other): + return self.value > other + + + # Overload math operations + def __add__(self, y): + if isinstance(y, Obs): + return derived_observable(lambda x, **kwargs: x[0] + x[1], [self, y], man_grad=[1, 1]) + else: + if isinstance(y, np.ndarray): + return np.array([self + o for o in y]) + elif(y.__class__.__name__=="Corr"): + return NotImplemented + else: + return derived_observable(lambda x, **kwargs: x[0] + y, [self], man_grad=[1]) + def __radd__(self, y): + return self + y + + + def __mul__(self, y): + if isinstance(y, Obs): + return derived_observable(lambda x, **kwargs: x[0] * x[1], [self, y], man_grad=[y.value, self.value]) + else: + if isinstance(y, np.ndarray): + return np.array([self * o for o in y]) + elif(y.__class__.__name__=="Corr"): + return NotImplemented + + else: + return derived_observable(lambda x, **kwargs: x[0] * y, [self], man_grad=[y]) + + def __rmul__(self, y): + return self * y + + + def __sub__(self, y): + if isinstance(y, Obs): + return derived_observable(lambda x, **kwargs: x[0] - x[1], [self, y], man_grad=[1, -1]) + else: + if isinstance(y, np.ndarray): + return np.array([self - o for o in y]) + + elif(y.__class__.__name__=="Corr"): + return NotImplemented + + else: + return derived_observable(lambda x, **kwargs: x[0] - y, [self], man_grad=[1]) + + + def __rsub__(self, y): + return -1 * (self - y) + + + def __neg__(self): + return -1 * self + + + def __truediv__(self, y): + if isinstance(y, Obs): + return derived_observable(lambda x, **kwargs: x[0] / x[1], [self, y], man_grad=[1 / y.value, - self.value / y.value ** 2]) + else: + if isinstance(y, np.ndarray): + return np.array([self / o for o in y]) + + elif(y.__class__.__name__=="Corr"): + return NotImplemented + + else: + return derived_observable(lambda x, **kwargs: x[0] / y, [self], man_grad=[1 / y]) + + + def __rtruediv__(self, y): + if isinstance(y, Obs): + return derived_observable(lambda x, **kwargs: x[0] / x[1], [y, self], man_grad=[1 / self.value, - y.value / self.value ** 2]) + else: + if isinstance(y, np.ndarray): + return np.array([o / self for o in y]) + else: + return derived_observable(lambda x, **kwargs: y / x[0], [self], man_grad=[-y / self.value ** 2]) + + + def __pow__(self, y): + if isinstance(y, Obs): + return derived_observable(lambda x: x[0] ** x[1], [self, y]) + else: + return derived_observable(lambda x: x[0] ** y, [self]) + + + def __rpow__(self, y): + if isinstance(y, Obs): + return derived_observable(lambda x: x[0] ** x[1], [y, self]) + else: + return derived_observable(lambda x: y ** x[0], [self]) + + + def __abs__(self): + return derived_observable(lambda x: anp.abs(x[0]), [self]) + + + # Overload numpy functions + def sqrt(self): + return derived_observable(lambda x, **kwargs: np.sqrt(x[0]), [self], man_grad=[1 / 2 / np.sqrt(self.value)]) + + + def log(self): + return derived_observable(lambda x, **kwargs: np.log(x[0]), [self], man_grad=[1 / self.value]) + + + def exp(self): + return derived_observable(lambda x, **kwargs: np.exp(x[0]), [self], man_grad=[np.exp(self.value)]) + + + def sin(self): + return derived_observable(lambda x, **kwargs: np.sin(x[0]), [self], man_grad=[np.cos(self.value)]) + + + def cos(self): + return derived_observable(lambda x, **kwargs: np.cos(x[0]), [self], man_grad=[-np.sin(self.value)]) + + + def tan(self): + return derived_observable(lambda x, **kwargs: np.tan(x[0]), [self], man_grad=[1 / np.cos(self.value) ** 2]) + + + def arcsin(self): + return derived_observable(lambda x: anp.arcsin(x[0]), [self]) + + + def arccos(self): + return derived_observable(lambda x: anp.arccos(x[0]), [self]) + + + def arctan(self): + return derived_observable(lambda x: anp.arctan(x[0]), [self]) + + + def sinh(self): + return derived_observable(lambda x, **kwargs: np.sinh(x[0]), [self], man_grad=[np.cosh(self.value)]) + + + def cosh(self): + return derived_observable(lambda x, **kwargs: np.cosh(x[0]), [self], man_grad=[np.sinh(self.value)]) + + + def tanh(self): + return derived_observable(lambda x, **kwargs: np.tanh(x[0]), [self], man_grad=[1 / np.cosh(self.value) ** 2]) + + + def arcsinh(self): + return derived_observable(lambda x: anp.arcsinh(x[0]), [self]) + + + def arccosh(self): + return derived_observable(lambda x: anp.arccosh(x[0]), [self]) + + + def arctanh(self): + return derived_observable(lambda x: anp.arctanh(x[0]), [self]) + + + def sinc(self): + return derived_observable(lambda x: anp.sinc(x[0]), [self]) + + +def derived_observable(func, data, **kwargs): + """Construct a derived Obs according to func(data, **kwargs) using automatic differentiation. + + Parameters + ---------- + func -- arbitrary function of the form func(data, **kwargs). For the + automatic differentiation to work, all numpy functions have to have + the autograd wrapper (use 'import autograd.numpy as anp'). + data -- list of Obs, e.g. [obs1, obs2, obs3]. + + Keyword arguments + ----------------- + num_grad -- if True, numerical derivatives are used instead of autograd + (default False). To control the numerical differentiation the + kwargs of numdifftools.step_generators.MaxStepGenerator + can be used. + man_grad -- manually supply a list or an array which contains the jacobian + of func. Use cautiously, supplying the wrong derivative will + not be intercepted. + bias_correction -- if True, the bias correction specified in + hep-lat/0306017 eq. (19) is performed, not recommended. + (Only applicable for more than 1 replicum) + + Notes + ----- + For simple mathematical operations it can be practical to use anonymous + functions. For the ratio of two observables one can e.g. use + + new_obs = derived_observable(lambda x: x[0] / x[1], [obs1, obs2]) + """ + + data = np.asarray(data) + raveled_data = data.ravel() + + n_obs = len(raveled_data) + new_names = sorted(set([y for x in [o.names for o in raveled_data] for y in x])) + replicas = len(new_names) + + new_shape = {} + for i_data in raveled_data: + for name in new_names: + tmp = i_data.shape.get(name) + if tmp is not None: + if new_shape.get(name) is None: + new_shape[name] = tmp + else: + if new_shape[name] != tmp: + raise Exception('Shapes of ensemble', name, 'do not match.') + + values = np.vectorize(lambda x: x.value)(data) + + new_values = func(values, **kwargs) + + multi = 0 + if isinstance(new_values, np.ndarray): + multi = 1 + + new_r_values = {} + for name in new_names: + tmp_values = np.zeros(n_obs) + for i, item in enumerate(raveled_data): + tmp = item.r_values.get(name) + if tmp is None: + tmp = item.value + tmp_values[i] = tmp + if multi > 0: + tmp_values = np.array(tmp_values).reshape(data.shape) + new_r_values[name] = func(tmp_values, **kwargs) + + if 'man_grad' in kwargs: + deriv = np.asarray(kwargs.get('man_grad')) + if new_values.shape + data.shape != deriv.shape: + raise Exception('Manual derivative does not have correct shape.') + elif kwargs.get('num_grad') is True: + if multi > 0: + raise Exception('Multi mode currently not supported for numerical derivative') + options = { + 'base_step': 0.1, + 'step_ratio': 2.5, + 'num_steps': None, + 'step_nom': None, + 'offset': None, + 'num_extrap': None, + 'use_exact_steps': None, + 'check_num_steps': None, + 'scale': None} + for key in options.keys(): + kwarg = kwargs.get(key) + if kwarg is not None: + options[key] = kwarg + tmp_df = nd.Gradient(func, order=4, **{k:v for k, v in options.items() if v is not None})(values, **kwargs) + if tmp_df.size == 1: + deriv = np.array([tmp_df.real]) + else: + deriv = tmp_df.real + else: + deriv = jacobian(func)(values, **kwargs) + + final_result = np.zeros(new_values.shape, dtype=object) + + for i_val, new_val in np.ndenumerate(new_values): + new_deltas = {} + for j_obs, obs in np.ndenumerate(data): + for name in obs.names: + new_deltas[name] = new_deltas.get(name, 0) + deriv[i_val + j_obs] * obs.deltas[name] + + new_samples = [] + for name in new_names: + new_samples.append(new_deltas[name] + new_r_values[name][i_val]) + + final_result[i_val] = Obs(new_samples, new_names) + + # Bias correction + if replicas > 1 and kwargs.get('bias_correction'): + final_result[i_val].value = (replicas * new_val - final_result[i_val].value) / (replicas - 1) + else: + final_result[i_val].value = new_val + + if multi == 0: + final_result = final_result.item() + + return final_result + + +def reweight(weight, obs, **kwargs): + """Reweight a list of observables.""" + result = [] + for i in range(len(obs)): + if sorted(weight.names) != sorted(obs[i].names): + raise Exception('Error: Ensembles do not fit') + for name in weight.names: + if weight.shape[name] != obs[i].shape[name]: + raise Exception('Error: Shapes of ensemble', name, 'do not fit') + new_samples = [] + for name in sorted(weight.names): + new_samples.append((weight.deltas[name] + weight.r_values[name]) * (obs[i].deltas[name] + obs[i].r_values[name])) + tmp_obs = Obs(new_samples, sorted(weight.names)) + + result.append(derived_observable(lambda x, **kwargs: x[0] / x[1], [tmp_obs, weight], **kwargs)) + result[-1].reweighted = 1 + + return result + + +def correlate(obs_a, obs_b): + """Correlate two observables. + + Keep in mind to only correlate primary observables which have not been reweighted + yet. The reweighting has to be applied after correlating the observables. + Currently only works if ensembles are identical. This is not really necessary. + """ + + if sorted(obs_a.names) != sorted(obs_b.names): + raise Exception('Ensembles do not fit') + for name in obs_a.names: + if obs_a.shape[name] != obs_b.shape[name]: + raise Exception('Shapes of ensemble', name, 'do not fit') + + if obs_a.reweighted == 1: + print('Warning: The first observable is already reweighted.') + if obs_b.reweighted == 1: + print('Warning: The second observable is already reweighted.') + + new_samples = [] + for name in sorted(obs_a.names): + new_samples.append((obs_a.deltas[name] + obs_a.r_values[name]) * (obs_b.deltas[name] + obs_b.r_values[name])) + + return Obs(new_samples, sorted(obs_a.names)) + + +def covariance(obs1, obs2, correlation=False, **kwargs): + """Calculates the covariance of two observables. + + covariance(obs, obs) is equal to obs.dvalue ** 2 + The gamma method has to be applied first to both observables. + + If abs(covariance(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance + is constrained to the maximum value in order to make sure that covariance + matrices are positive semidefinite. + + Keyword arguments + ----------------- + correlation -- if true the correlation instead of the covariance is + returned (default False) + """ + + for name in sorted(set(obs1.names + obs2.names)): + if (obs1.shape.get(name) != obs2.shape.get(name)) and (obs1.shape.get(name) is not None) and (obs2.shape.get(name) is not None): + raise Exception('Shapes of ensemble', name, 'do not fit') + + if obs1.e_names == {} or obs2.e_names == {}: + raise Exception('The gamma method has to be applied to both Obs first.') + + dvalue = 0 + + for e_name in obs1.e_names: + + if e_name not in obs2.e_names: + continue + + gamma = 0 + r_length = [] + for r_name in obs1.e_content[e_name]: + if r_name not in obs2.e_content[e_name]: + continue + + r_length.append(len(obs1.deltas[r_name])) + + gamma += np.sum(obs1.deltas[r_name] * obs2.deltas[r_name]) + + e_N = np.sum(r_length) + + tau_combined = (obs1.e_tauint[e_name] + obs2.e_tauint[e_name]) / 2 + dvalue += gamma / e_N * (1 + 1 / e_N) / e_N * 2 * tau_combined + + if np.abs(dvalue / obs1.dvalue / obs2.dvalue) > 1.0: + dvalue = np.sign(dvalue) * obs1.dvalue * obs2.dvalue + + if correlation: + dvalue = dvalue / obs1.dvalue / obs2.dvalue + + return dvalue + + +def covariance2(obs1, obs2, correlation=False, **kwargs): + """Alternative implementation of the covariance of two observables. + + covariance(obs, obs) is equal to obs.dvalue ** 2 + The gamma method has to be applied first to both observables. + + If abs(covariance(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance + is constrained to the maximum value in order to make sure that covariance + matrices are positive semidefinite. + + Keyword arguments + ----------------- + correlation -- if true the correlation instead of the covariance is + returned (default False) + """ + + for name in sorted(set(obs1.names + obs2.names)): + if (obs1.shape.get(name) != obs2.shape.get(name)) and (obs1.shape.get(name) is not None) and (obs2.shape.get(name) is not None): + raise Exception('Shapes of ensemble', name, 'do not fit') + + if obs1.e_names == {} or obs2.e_names == {}: + raise Exception('The gamma method has to be applied to both Obs first.') + + dvalue = 0 + e_gamma = {} + e_dvalue = {} + e_n_tauint = {} + e_rho = {} + + for e_name in obs1.e_names: + + if e_name not in obs2.e_names: + continue + + r_length = [] + for r_name in obs1.e_content[e_name]: + r_length.append(len(obs1.deltas[r_name])) + + e_N = np.sum(r_length) + w_max = max(r_length) // 2 + e_gamma[e_name] = np.zeros(w_max) + + for r_name in obs1.e_content[e_name]: + if r_name not in obs2.e_content[e_name]: + continue + max_gamma = min(obs1.shape[r_name], w_max) + # The padding for the fft has to be even + padding = obs1.shape[r_name] + max_gamma + (obs1.shape[r_name] + max_gamma) % 2 + e_gamma[e_name][:max_gamma] += (np.fft.irfft(np.fft.rfft(obs1.deltas[r_name], padding) * np.conjugate(np.fft.rfft(obs2.deltas[r_name], padding)))[:max_gamma] + + np.fft.irfft(np.fft.rfft(obs2.deltas[r_name], padding) * np.conjugate(np.fft.rfft(obs1.deltas[r_name], padding)))[:max_gamma]) / 2.0 + + if np.all(e_gamma[e_name]) == 0.0: + continue + + e_shapes = [] + for r_name in obs1.e_content[e_name]: + e_shapes.append(obs1.shape[r_name]) + + div = np.array([]) + mul = np.array([]) + sorted_shapes = sorted(e_shapes) + for i, item in enumerate(sorted_shapes): + if len(div) > w_max: + break + if i == 0: + samples = item + else: + samples = item - sorted_shapes[i - 1] + div = np.append(div, np.repeat(np.sum(sorted_shapes[i:]), samples)) + mul = np.append(mul, np.repeat(len(sorted_shapes) - i, samples)) + div = div - np.arange(len(div)) * mul + + e_gamma[e_name] /= div[:w_max] + + e_rho[e_name] = e_gamma[e_name][:w_max] / e_gamma[e_name][0] + e_n_tauint[e_name] = np.cumsum(np.concatenate(([0.5], e_rho[e_name][1:]))) + # Make sure no entry of tauint is smaller than 0.5 + e_n_tauint[e_name][e_n_tauint[e_name] < 0.5] = 0.500000000001 + + + window = max(obs1.e_windowsize[e_name], obs2.e_windowsize[e_name]) + # Bias correction hep-lat/0306017 eq. (49) + e_dvalue[e_name] = 2 * (e_n_tauint[e_name][window] + obs1.tau_exp[e_name] * np.abs(e_rho[e_name][window + 1])) * (1 + (2 * window + 1) / e_N) * e_gamma[e_name][0] / e_N + + dvalue += e_dvalue[e_name] + + if np.abs(dvalue / obs1.dvalue / obs2.dvalue) > 1.0: + dvalue = np.sign(dvalue) * obs1.dvalue * obs2.dvalue + + if correlation: + dvalue = dvalue / obs1.dvalue / obs2.dvalue + + return dvalue + + +def covariance3(obs1, obs2, correlation=False, **kwargs): + """Another alternative implementation of the covariance of two observables. + + covariance2(obs, obs) is equal to obs.dvalue ** 2 + Currently only works if ensembles are identical. + The gamma method has to be applied first to both observables. + + If abs(covariance2(obs1, obs2)) > obs1.dvalue * obs2.dvalue, the covariance + is constrained to the maximum value in order to make sure that covariance + matrices are positive semidefinite. + + Keyword arguments + ----------------- + correlation -- if true the correlation instead of the covariance is + returned (default False) + plot -- if true, the integrated autocorrelation time for each ensemble is + plotted. + """ + + for name in sorted(set(obs1.names + obs2.names)): + if (obs1.shape.get(name) != obs2.shape.get(name)) and (obs1.shape.get(name) is not None) and (obs2.shape.get(name) is not None): + raise Exception('Shapes of ensemble', name, 'do not fit') + + if obs1.e_names == {} or obs2.e_names == {}: + raise Exception('The gamma method has to be applied to both Obs first.') + + tau_exp = [] + S = [] + for e_name in sorted(set(obs1.e_names + obs2.e_names)): + t_1 = obs1.tau_exp.get(e_name) + t_2 = obs2.tau_exp.get(e_name) + if t_1 is None: + t_1 = 0 + if t_2 is None: + t_2 = 0 + tau_exp.append(max(t_1, t_2)) + S_1 = obs1.S.get(e_name) + S_2 = obs2.S.get(e_name) + if S_1 is None: + S_1 = Obs.S_global + if S_2 is None: + S_2 = Obs.S_global + S.append(max(S_1, S_2)) + + check_obs = obs1 + obs2 + check_obs.gamma_method(tau_exp=tau_exp, S=S) + + if kwargs.get('plot'): + check_obs.plot_tauint() + check_obs.plot_rho() + + cov = (check_obs.dvalue ** 2 - obs1.dvalue ** 2 - obs2.dvalue ** 2) / 2 + + if np.abs(cov / obs1.dvalue / obs2.dvalue) > 1.0: + cov = np.sign(cov) * obs1.dvalue * obs2.dvalue + + if correlation: + cov = cov / obs1.dvalue / obs2.dvalue + + return cov + + +def use_time_reversal_symmetry(data1, data2, **kwargs): + """Combine two correlation functions (lists of Obs) according to time reversal symmetry + + Keyword arguments + ----------------- + minus -- if True, multiply the second correlation function by a minus sign. + """ + if kwargs.get('minus'): + sign = -1 + else: + sign = 1 + + result = [] + T = int(len(data1)) + for i in range(T): + result.append(derived_observable(lambda x, **kwargs: (x[0] + sign * x[1]) / 2, [data1[i], data2[T - i - 1]], **kwargs)) + + return result + + +def pseudo_Obs(value, dvalue, name, samples=1000): + """Generate a pseudo Obs with given value, dvalue and name + + The standard number of samples is a 1000. This can be adjusted. + """ + if dvalue <= 0.0: + return Obs([np.zeros(samples) + value], [name]) + else: + for _ in range(100): + deltas = [np.random.normal(0.0, dvalue * np.sqrt(samples), samples)] + deltas -= np.mean(deltas) + deltas *= dvalue / np.sqrt((np.var(deltas) / samples)) / np.sqrt(1 + 3 / samples) + deltas += value + res = Obs(deltas, [name]) + res.gamma_method(S=2, tau_exp=0) + if abs(res.dvalue - dvalue) < 1e-10 * dvalue: + break + + res.value = float(value) + + return res + + +def dump_object(obj, name, **kwargs): + """Dump object into pickle file. + + Keyword arguments + ----------------- + path -- specifies a custom path for the file (default '.') + """ + if 'path' in kwargs: + file_name = kwargs.get('path') + '/' + name + '.p' + else: + file_name = name + '.p' + with open(file_name, 'wb') as fb: + pickle.dump(obj, fb) + + +def load_object(path): + """Load object from pickle file. """ + with open(path, 'rb') as file: + return pickle.load(file) + + +def plot_corrs(observables, **kwargs): + """Plot lists of Obs. + + Parameters + ---------- + observables -- list of lists of Obs, where the nth entry is considered to be the + correlation function + at x0=n e.g. [[f_A_0,f_A_1],[f_P_0,f_P_1]] or [f_A,f_P], where f_A and f_P are lists of Obs. + + Keyword arguments + ----------------- + xrange -- list of two values, determining the range of the x-axis e.g. [4, 8] + yrange -- list of two values, determining the range of the y-axis e.g. [0.2, 1.1] + prange -- list of two values, visualizing the width of the plateau e.g. [10, 15] + reference -- float valued variable which is shown as horizontal line for reference + plateau -- Obs which is shown as horizontal line with errorbar for reference + shift -- shift x by given value + label -- list of labels, has to have the same length as observables + exp -- plot exponential from fitting routine + """ + + if 'shift' in kwargs: + shift = kwargs.get('shift') + else: + shift = 0 + + if 'label' in kwargs: + label = kwargs.get('label') + if len(label) != len(observables): + raise Exception('label has to be a list with exactly one entry per entry of observables.') + else: + label = [] + for j in range(len(observables)): + label.append(str(j + 1)) + + + f = plt.figure() + for j in range(len(observables)): + T = len(observables[j]) + + x = np.arange(T) + shift + y = np.zeros(T) + y_err = np.zeros(T) + + for i in range(T): + y[i] = observables[j][i].value + y_err[i] = observables[j][i].dvalue + + plt.errorbar(x, y, yerr=y_err, ls='none', fmt='o', capsize=3, + markersize=5, lw=1, label=label[j]) + + if kwargs.get('logscale'): + plt.yscale('log') + + if 'xrange' in kwargs: + xrange = kwargs.get('xrange') + plt.xlim(xrange[0], xrange[1]) + visible_y = y[int(xrange[0] + 0.5):int(xrange[1] + 0.5)] + visible_y_err = y_err[int(xrange[0] + 0.5):int(xrange[1] + 0.5)] + y_start = np.min(visible_y - visible_y_err) + y_stop = np.max(visible_y + visible_y_err) + span = y_stop - y_start + if np.isfinite(y_start) and np.isfinite(y_stop): + plt.ylim(y_start - 0.1 * span, y_stop + 0.1 * span) + + if 'yrange' in kwargs: + yrange = kwargs.get('yrange') + plt.ylim(yrange[0], yrange[1]) + + if 'reference' in kwargs: + y_value = kwargs.get('reference') + plt.axhline(y=y_value, linewidth=2, color='k', alpha=0.25) + + if 'prange' in kwargs: + prange = kwargs.get('prange') + plt.axvline(x=prange[0] - 0.5, ls='--', c='k', lw=1, alpha=0.5) + plt.axvline(x=prange[1] + 0.5, ls='--', c='k', lw=1, alpha=0.5) + + if 'plateau' in kwargs: + plateau = kwargs.get('plateau') + if isinstance(plateau, Obs): + plt.axhline(y=plateau.value, linewidth=2, color='k', alpha=0.6, label='Plateau') + plt.axhspan(plateau.value - plateau.dvalue, plateau.value + plateau.dvalue, alpha=0.25, color='k') + elif isinstance(plateau, list): + for i in range(len(plateau)): + plt.axhline(y=plateau[i].value, linewidth=2, color='C' + str(i), alpha=0.6, label='Plateau' + str(i + 1)) + plt.axhspan(plateau[i].value - plateau[i].dvalue, plateau[i].value + plateau[i].dvalue, + color='C' + str(i), alpha=0.25) + else: + raise Exception('Improper input for plateau.') + + if kwargs.get('exp'): + fit_result = kwargs.get('exp') + y_fit = fit_result[1].value * np.exp(-fit_result[0].value * x) + plt.plot(x, y_fit, color='k') + if not (fit_result[0].e_names == {} and fit_result[1].e_names == {}): + y_fit_err = np.sqrt((y_fit * fit_result[0].dvalue) ** 2 + 2 * covariance(fit_result[0], fit_result[1])* y_fit * + np.exp(-fit_result[0].value * x) + (np.exp(-fit_result[0].value * x) * fit_result[1].dvalue) ** 2) + plt.fill_between(x, y_fit + y_fit_err, y_fit - y_fit_err, color='k', alpha=0.1) + + plt.xlabel('$x_0/a$') + + if 'ylabel' in kwargs: + plt.ylabel(kwargs.get('ylabel')) + + if 'save' in kwargs: + lgd = plt.legend(loc=0) + else: + lgd = plt.legend(bbox_to_anchor=(1.04, 1), loc='upper left') + plt.show() + + if 'save' in kwargs: + save = kwargs.get('save') + if not isinstance(save, str): + raise Exception('save has to be a string.') + f.savefig(save + '.pdf', bbox_extra_artists=(lgd,), bbox_inches='tight') + + +def merge_obs(list_of_obs): + """Combine all observables in list_of_obs into one new observable + + It is not possible to combine obs which are based on the same replicum + """ + replist = [item for obs in list_of_obs for item in obs.names] + if (len(replist) == len(set(replist))) is False: + raise Exception('list_of_obs contains duplicate replica: %s' %(str(replist))) + new_dict = {} + for o in list_of_obs: + new_dict.update({key: o.deltas.get(key, 0) + o.r_values.get(key, 0) + for key in set(o.deltas) | set(o.r_values)}) + + return Obs(list(new_dict.values()), list(new_dict.keys())) From 28eba5edbf4343551d03a3f6fb50c72b28bcf4e9 Mon Sep 17 00:00:00 2001 From: JanNeuendorf <75676159+JanNeuendorf@users.noreply.github.com> Date: Fri, 8 Jan 2021 14:47:19 +0100 Subject: [PATCH 4/8] Add files via upload --- 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