# Source code for multipletau.core

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

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"]

[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
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