#!/usr/bin/python # -*- coding: utf-8 -*- """ A multiple-τ algorithm for Python 2.7 and 3.x. Copyright (c) 2014 Paul Müller Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: 1. Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. 2. Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. 3. Neither the name of multipletau nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL INFRAE OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. """ from __future__ import division import numpy as np import warnings __all__ = ["autocorrelate", "correlate", "correlate_numpy"] 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 according to :py:func:`numpy.correlate` for positive :math:`k` on a base 2 logarithmic scale. numpy.correlate(a, a, mode="full")[len(a)-1:] :math:`z_k = \Sigma_n a_n a_{n+k}` Note that only the correlation in the positive direction is computed. Parameters ---------- a : array_like input sequence of real numbers 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 `M-k`. copy : bool copy input array, set to False to save memory dtype : dtype, optional The type of the returned array and of the accumulator in which the elements are summed. By default, the dtype of `a` is used. Returns ------- autocorrelation : ndarray Nx2 array containing lag time and autocorrelation Notes ----- 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 `M-k` by invoking: normalize = True For emulating the numpy.correlate behavior on a logarithmic scale (default behavior) use: normalize = False Examples -------- >>> from numpy import dtype >>> from multipletau import autocorrelate >>> autocorrelate(range(42), m=2, dtype=dtype(float)) array([[ 1.00000000e+00, 2.29600000e+04], [ 2.00000000e+00, 2.21000000e+04], [ 4.00000000e+00, 2.03775000e+04], [ 8.00000000e+00, 1.50612000e+04]]) """ traceavg = np.average(a) if normalize and traceavg == 0: raise ZeroDivisionError("Normalization not possible. "+ "The average of the input *binned_array* is zero.") trace = np.array(a, dtype=dtype, copy=copy) dtype = trace.dtype if dtype.kind in ["b","i","u"]: warnings.warn("Converting input data type ({}) to float.". format(dtype)) dtype = np.dtype(float) trace = np.array(a, dtype=dtype, copy=copy) # Complex data if dtype.kind == "c": raise NotImplementedError( "Please use `multipletau.correlate` for complex data.") # Check parameters if np.around(m/2) != m/2: mold = 1*m m = int((np.around(m/2)+1) * 2) warnings.warn("Invalid value of m={}. Using m={} instead" .format(mold,m)) else: m = int(m) N = N0 = len(trace) # 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 = 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 = np.int(np.floor(m + k*m/2)) 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: trace -= traceavg if N < 2*m: # Otherwise the following for-loop will fail: raise ValueError("len(binned_array) must be larger than 2m.") ## Calculate autocorrelation function for first m bins # Discrete convolution of m elements for n in range(1,m+1): G[n-1,0] = deltat * n # This is the computationally intensive step G[n-1,1] = np.sum(trace[:N-n]*trace[n:], dtype=dtype) normstat[n-1] = N-n normnump[n-1] = 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+1:2])/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,int(m/2)+1): idx = int(m + n - 1 + (step-1)*m/2) if len(trace[:N-(n+m/2)]) == 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 repect 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 * (n+m/2) * 2**step # This is the computationally intensive step G[idx,1] = np.sum(trace[:N-(n+m/2)]*trace[(n+m/2):], dtype=dtype) normstat[idx] = N-(n+m/2) 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+1:2])/2 N /= 2 if normalize: G[:,1] /= traceavg**2 * normstat else: G[:,1] *= N0/normnump return G 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 according to :py:func:`numpy.correlate` for positive :math:`k` on a base 2 logarithmic scale. 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. 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 `M-k`. copy : bool copy input array, set to False to save memory dtype : dtype, optional The type of the returned array and of the accumulator in which the elements are summed. By default, the dtype of `a` is used. Returns ------- crosscorrelation : ndarray Nx2 array containing lag time and cross-correlation Notes ----- 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 `M-k` by invoking: normalize = True For emulating the numpy.correlate behavior on a logarithmic scale (default behavior) use: normalize = False Examples -------- >>> from numpy import dtype >>> from multipletau import correlate >>> correlate(range(42), range(1,43), m=2, dtype=dtype(float)) array([[ 1.00000000e+00, 2.38210000e+04], [ 2.00000000e+00, 2.29600000e+04], [ 4.00000000e+00, 2.12325000e+04], [ 8.00000000e+00, 1.58508000e+04]]) """ ## See `autocorrelation` for better documented code. traceavg1 = np.average(v) traceavg2 = np.average(a) if normalize and traceavg1*traceavg2 == 0: raise ZeroDivisionError("Normalization not possible. "+ "The average of the input *binned_array* is zero.") trace1 = np.array(v, dtype=dtype, copy=copy) dtype = trace1.dtype if dtype.kind in ["b","i","u"]: warnings.warn( "Converting input data type ({}) to float.".format(dtype)) dtype = np.dtype(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) # Complex data if dtype.kind == "c": trace1.imag *= -1 # Check parameters if np.around(m/2) != m/2: mold = 1*m m = int((np.around(m/2)+1) * 2) warnings.warn("Invalid value of m={}. Using m={} instead" .format(mold,m)) else: m = int(m) if len(a) != len(v): raise ValueError("Input arrays must be of equal length.") N = N0 = len(trace1) # 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 = 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 = np.int(np.floor(m + k*m/2)) 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 -= traceavg1 trace2 -= traceavg2 if N < 2*m: # Otherwise the following for-loop will fail: raise ValueError("len(binned_array) must be larger than 2m.") # Calculate autocorrelation function for first m bins for n in range(1,m+1): G[n-1,0] = deltat * n G[n-1,1] = np.sum(trace1[:N-n]*trace2[n:]) normstat[n-1] = N-n normnump[n-1] = 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+1:2])/2 trace2 = (trace2[:N:2]+trace2[1:N+1:2])/2 N /= 2 for step in range(1,k+1): # Get the next m/2 values of the trace for n in range(1,int(m/2)+1): idx = int(m + n - 1 + (step-1)*m/2) if len(trace1[:N-(n+m/2)]) == 0: # Abort G = G[:idx-1] normstat = normstat[:idx-1] normnump = normnump[:idx-1] break else: G[idx,0] = deltat * (n+m/2) * 2**step G[idx,1] = np.sum(trace1[:N-(n+m/2)]*trace2[(n+m/2):]) normstat[idx] = N-(n+m/2) 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+1:2])/2 trace2 = (trace2[:N:2]+trace2[1:N+1:2])/2 N /= 2 if normalize: G[:,1] /= traceavg1*traceavg2 * normstat else: G[:,1] *= N0/normnump return G def correlate_numpy(a, v, deltat=1, normalize=False, dtype=None, copy=True): """ Convenience function that wraps around numpy.correlate and returns the data as multipletau.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 (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 : dtype, optional The type of the returned array and of the accumulator in which the elements are summed. By default, the dtype of `a` is used. Returns ------- crosscorrelation : ndarray Nx2 array containing lag time and cross-correlation """ avg = np.average(a) vvg = np.average(v) if dtype is None: dtype = a.dtype if len(a) != len(v): raise ValueError("Arrays must be of same length.") ab = np.array(a, dtype=dtype, copy=copy) vb = np.array(v, dtype=dtype, copy=copy) Gd = np.correlate(ab-avg, vb-vvg, 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)) G[:,1] = Gd G[:,0] = np.arange(len(Gd))*deltat return G