Source code for multipletau.core

#!/usr/bin/python
# -*- coding: utf-8 -*-
"""
A multiple-τ algorithm for Python 2.7 and 3.x.

Copyright (c) 2014 Paul Müller

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"""
from __future__ import division

import numpy as np
import warnings

__all__ = ["autocorrelate", "correlate", "correlate_numpy"]


[docs]def autocorrelate(a, m=16, deltat=1, normalize=False, copy=True, dtype=None): """ Autocorrelation of a 1-dimensional sequence on a log2-scale. This computes the correlation similar to :py:func:`numpy.correlate` for positive :math:`k` on a base 2 logarithmic scale. :func:`numpy.correlate(a, a, mode="full")[len(a)-1:]` :math:`z_k = \Sigma_n a_n a_{n+k}` Parameters ---------- a : array-like input sequence m : even integer defines the number of points on one level, must be an even integer deltat : float distance between bins normalize : bool normalize the result to the square of the average input signal and the factor :math:`M-k`. copy : bool copy input array, set to ``False`` to save memory dtype : object to be converted to a data type object The data type of the returned array and of the accumulator for the multiple-tau computation. Returns ------- autocorrelation : ndarray of shape (N,2) the lag time (1st column) and the autocorrelation (2nd column). Notes ----- .. versionchanged :: 0.1.6 Compute the correlation for zero lag time. The algorithm computes the correlation with the convention of the curve decaying to zero. For experiments like e.g. fluorescence correlation spectroscopy, the signal can be normalized to :math:`M-k` by invoking ``normalize = True``. For normalizing according to the behavior of :py:func:`numpy.correlate`, use ``normalize = False``. For complex arrays, this method falls back to the method :func:`correlate`. Examples -------- >>> from multipletau import autocorrelate >>> autocorrelate(range(42), m=2, dtype=np.float_) array([[ 0.00000000e+00, 2.38210000e+04], [ 1.00000000e+00, 2.29600000e+04], [ 2.00000000e+00, 2.21000000e+04], [ 4.00000000e+00, 2.03775000e+04], [ 8.00000000e+00, 1.50612000e+04]]) """ assert isinstance(copy, bool) assert isinstance(normalize, bool) if dtype is None: dtype = np.dtype(a[0].__class__) else: dtype = np.dtype(dtype) # Complex data if dtype.kind == "c": # run cross-correlation return correlate(a=a, v=a, m=m, deltat=deltat, normalize=normalize, copy=copy, dtype=dtype) elif dtype.kind != "f": warnings.warn("Input dtype is not float; casting to np.float_!") dtype = np.dtype(np.float_) # If copy is false and dtype is the same as the input array, # then this line does not have an effect: trace = np.array(a, dtype=dtype, copy=copy) # Check parameters if m // 2 != m / 2: mold = m m = np.int_((m // 2 + 1) * 2) warnings.warn("Invalid value of m={}. Using m={} instead" .format(mold, m)) else: m = np.int_(m) N = N0 = trace.shape[0] # Find out the length of the correlation function. # The integer k defines how many times we can average over # two neighboring array elements in order to obtain an array of # length just larger than m. k = np.int_(np.floor(np.log2(N / m))) # In the base2 multiple-tau scheme, the length of the correlation # array is (only taking into account values that are computed from # traces that are just larger than m): lenG = m + k * (m // 2) + 1 G = np.zeros((lenG, 2), dtype=dtype) normstat = np.zeros(lenG, dtype=dtype) normnump = np.zeros(lenG, dtype=dtype) traceavg = np.average(trace) # We use the fluctuation of the signal around the mean if normalize: trace -= traceavg assert traceavg != 0, "Cannot normalize: Average of `a` is zero!" # Otherwise the following for-loop will fail: assert N >= 2 * m, "len(a) must be larger than 2m!" # Calculate autocorrelation function for first m+1 bins # Discrete convolution of m elements for n in range(0, m + 1): G[n, 0] = deltat * n # This is the computationally intensive step G[n, 1] = np.sum(trace[:N - n] * trace[n:]) normstat[n] = N - n normnump[n] = N # Now that we calculated the first m elements of G, let us # go on with the next m/2 elements. # Check if len(trace) is even: if N % 2 == 1: N -= 1 # Add up every second element trace = (trace[:N:2] + trace[1:N:2]) / 2 N //= 2 # Start iteration for each m/2 values for step in range(1, k + 1): # Get the next m/2 values via correlation of the trace for n in range(1, m // 2 + 1): npmd2 = n + m // 2 idx = m + n + (step - 1) * m // 2 if len(trace[:N - npmd2]) == 0: # This is a shortcut that stops the iteration once the # length of the trace is too small to compute a corre- # lation. The actual length of the correlation function # does not only depend on k - We also must be able to # perform the sum with respect to k for all elements. # For small N, the sum over zero elements would be # computed here. # # One could make this for-loop go up to maxval, where # maxval1 = int(m/2) # maxval2 = int(N-m/2-1) # maxval = min(maxval1, maxval2) # However, we then would also need to find out which # element in G is the last element... G = G[:idx - 1] normstat = normstat[:idx - 1] normnump = normnump[:idx - 1] # Note that this break only breaks out of the current # for loop. However, we are already in the last loop # of the step-for-loop. That is because we calculated # k in advance. break else: G[idx, 0] = deltat * npmd2 * 2**step # This is the computationally intensive step G[idx, 1] = np.sum(trace[:N - npmd2] * trace[npmd2:]) normstat[idx] = N - npmd2 normnump[idx] = N # Check if len(trace) is even: if N % 2 == 1: N -= 1 # Add up every second element trace = (trace[:N:2] + trace[1:N:2]) / 2 N //= 2 if normalize: G[:, 1] /= traceavg**2 * normstat else: G[:, 1] *= N0 / normnump return G
[docs]def correlate(a, v, m=16, deltat=1, normalize=False, copy=True, dtype=None): """ Cross-correlation of two 1-dimensional sequences on a log2-scale. This computes the cross-correlation similar to :py:func:`numpy.correlate` for positive :math:`k` on a base 2 logarithmic scale. :func:`numpy.correlate(a, v, mode="full")[len(a)-1:]` :math:`z_k = \Sigma_n a_n v_{n+k}` Note that only the correlation in the positive direction is computed. To obtain the correlation for negative lag times swap the input variables ``a`` and ``v``. Parameters ---------- a, v : array-like input sequences with equal length m : even integer defines the number of points on one level, must be an even integer deltat : float distance between bins normalize : bool normalize the result to the square of the average input signal and the factor :math:`M-k`. copy : bool copy input array, set to ``False`` to save memory dtype : object to be converted to a data type object The data type of the returned array and of the accumulator for the multiple-tau computation. Returns ------- cross_correlation : ndarray of shape (N,2) the lag time (column 1) and the cross-correlation (column2). Notes ----- .. versionchanged :: 0.1.6 Compute the correlation for zero lag time and correctly normalize the correlation for a complex input sequence `v`. The algorithm computes the correlation with the convention of the curve decaying to zero. For experiments like e.g. fluorescence correlation spectroscopy, the signal can be normalized to :math:`M-k` by invoking ``normalize = True``. For normalizing according to the behavior of :py:func:`numpy.correlate`, use ``normalize = False``. Examples -------- >>> from multipletau import correlate >>> correlate(range(42), range(1,43), m=2, dtype=np.float_) array([[ 0.00000000e+00, 2.46820000e+04], [ 1.00000000e+00, 2.38210000e+04], [ 2.00000000e+00, 2.29600000e+04], [ 4.00000000e+00, 2.12325000e+04], [ 8.00000000e+00, 1.58508000e+04]]) """ assert isinstance(copy, bool) assert isinstance(normalize, bool) # See `autocorrelation` for better documented code. traceavg1 = np.average(v) traceavg2 = np.average(a) if normalize: assert traceavg1 != 0, "Cannot normalize: Average of `v` is zero!" assert traceavg2 != 0, "Cannot normalize: Average of `a` is zero!" if dtype is None: dtype = np.dtype(v[0].__class__) dtype2 = np.dtype(a[0].__class__) if dtype != dtype2: if dtype.kind == "c" or dtype2.kind == "c": # The user might try to combine complex64 and float128. warnings.warn( "Input dtypes not equal; casting to np.complex_!") dtype = np.dtype(np.complex_) else: warnings.warn("Input dtypes not equal; casting to np.float_!") dtype = np.dtype(np.float_) else: dtype = np.dtype(dtype) if dtype.kind not in ["c", "f"]: warnings.warn("Input dtype is not float; casting to np.float_!") dtype = np.dtype(np.float_) trace1 = np.array(v, dtype=dtype, copy=copy) # Prevent traces from overwriting each other if a is v: # Force copying trace 2 copy = True trace2 = np.array(a, dtype=dtype, copy=copy) assert trace1.shape[0] == trace2.shape[0], "`a`,`v` must have same length!" # Complex data if dtype.kind == "c": np.conjugate(trace1, out=trace1) # Check parameters if m // 2 != m / 2: mold = m m = np.int_(m // 2 + 1) * 2 warnings.warn("Invalid value of m={}. Using m={} instead" .format(mold, m)) else: m = np.int_(m) N = N0 = trace1.shape[0] # Find out the length of the correlation function. # The integer k defines how many times we can average over # two neighboring array elements in order to obtain an array of # length just larger than m. k = np.int_(np.floor(np.log2(N / m))) # In the base2 multiple-tau scheme, the length of the correlation # array is (only taking into account values that are computed from # traces that are just larger than m): lenG = m + k * m // 2 + 1 G = np.zeros((lenG, 2), dtype=dtype) normstat = np.zeros(lenG, dtype=dtype) normnump = np.zeros(lenG, dtype=dtype) # We use the fluctuation of the signal around the mean if normalize: trace1 -= np.conj(traceavg1) trace2 -= traceavg2 # Otherwise the following for-loop will fail: assert N >= 2 * m, "len(a) must be larger than 2m!" # Calculate autocorrelation function for first m+1 bins for n in range(0, m + 1): G[n, 0] = deltat * n G[n, 1] = np.sum(trace1[:N - n] * trace2[n:]) normstat[n] = N - n normnump[n] = N # Check if len(trace) is even: if N % 2 == 1: N -= 1 # Add up every second element trace1 = (trace1[:N:2] + trace1[1:N:2]) / 2 trace2 = (trace2[:N:2] + trace2[1:N:2]) / 2 N //= 2 for step in range(1, k + 1): # Get the next m/2 values of the trace for n in range(1, m // 2 + 1): npmd2 = (n + m // 2) idx = m + n + (step - 1) * m // 2 if len(trace1[:N - npmd2]) == 0: # Abort G = G[:idx - 1] normstat = normstat[:idx - 1] normnump = normnump[:idx - 1] break else: G[idx, 0] = deltat * npmd2 * 2**step G[idx, 1] = np.sum( trace1[:N - npmd2] * trace2[npmd2:]) normstat[idx] = N - npmd2 normnump[idx] = N # Check if len(trace) is even: if N % 2 == 1: N -= 1 # Add up every second element trace1 = (trace1[:N:2] + trace1[1:N:2]) / 2 trace2 = (trace2[:N:2] + trace2[1:N:2]) / 2 N //= 2 if normalize: G[:, 1] /= traceavg1 * traceavg2 * normstat else: G[:, 1] *= N0 / normnump return G
[docs]def correlate_numpy(a, v, deltat=1, normalize=False, dtype=None, copy=True): """ Convenience function that wraps around :py:func:`numpy.correlate` and returns the correlation in the same format as :func:`correlate` does. Parameters ---------- a, v : array-like input sequences deltat : float distance between bins normalize : bool normalize the result to the square of the average input signal and the factor :math:`M-k`. The resulting curve follows the convention of decaying to zero for large lag times. copy : bool copy input array, set to ``False`` to save memory dtype : object to be converted to a data type object The data type of the returned array. Returns ------- cross_correlation : ndarray of shape (N,2) the lag time (column 1) and the cross-correlation (column 2). Notes ----- .. versionchanged :: 0.1.6 Removed false normalization when `normalize==False`. """ ab = np.array(a, dtype=dtype, copy=copy) vb = np.array(v, dtype=dtype, copy=copy) assert ab.shape[0] == vb.shape[0], "`a`,`v` must have same length!" avg = np.average(ab) vvg = np.average(vb) if normalize: ab -= avg vb -= vvg assert avg != 0, "Cannot normalize: Average of `a` is zero!" assert vvg != 0, "Cannot normalize: Average of `v` is zero!" Gd = np.correlate(ab, vb, mode="full")[len(ab) - 1:] if normalize: N = len(Gd) m = N - np.arange(N) Gd /= m * avg * vvg G = np.zeros((len(Gd), 2), dtype=dtype) G[:, 1] = Gd G[:, 0] = np.arange(len(Gd)) * deltat return G