multipletau reference

Provides a multiple-τ algorithm for Python 2.7 and Python 3.x and requires the package numpy.

Multipe-τ correlation is computed on a logarithmic scale (less data points are computed) and is thus much faster than conventional correlation on a linear scale such as numpy.correlate().

Recommended literature:

• Klaus Schaetzel and Rainer Peters; Noise on multiple-tau photon correlation data. Proc. SPIE 1430, Photon Correlation Spectroscopy: Multicomponent Systems, 109 (June 1, 1991); http://doi.org/10.1117/12.44160
• Thorsten Wohland, Rudolf Rigler, and Horst Vogel; The Standard Deviation in Fluorescence Correlation Spectroscopy. Biophysical Journal, 80 (June 1, 2001); http://dx.doi.org/10.1016/S0006-3495(01)76264-9

The source code of multipletau is available at https://github.com/paulmueller/multipletau.

multipletau.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 numpy.correlate() for positive \(k\) on a base 2 logarithmic scale.

numpy.correlate(a, a, mode=”full”)[len(a)-1:]

\(z_k = \Sigma_n a_n a_{n+k}\)

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]])

multipletau.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 numpy.correlate() for positive \(k\) on a base 2 logarithmic scale.

numpy.correlate(a, v, mode=”full”)[len(a)-1:]

\(z_k = \Sigma_n a_n v_{n+k}\)

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]])

multipletau.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

testing multipletau

This module is used for testing the multiple-tau algorithm. Executing this module as a package by invoking

python -m test

creates plots that illustrate the difference between multipletau.correlate() and numpy.correlate().

test.noise_exponential(N, tau=20, variance=1, deltat=1)

Generate exponentially correlated noise.

Parameters:

N : integer

Total number of samples

tau : float

Correlation time of the exponential in deltat

variance : float

Variance of the noise

deltat : float

Bin size of output array, defines the time scale of tau

Returns:

a : ndarray

Exponentially correlated noise.

test.noise_cross_exponential(N, tau=20, variance=1, deltat=1)

Generate exponentially cross-correlated noise.

Parameters:

N : integer

Total number of samples

tau : float

Correlation time of the exponential in deltat

variance : float

Variance of the noise

deltat : float

Bin size of output array, defines the time scale of tau

Returns:

a, randarray : ndarrays

Array a has positive exponential correlation to the ‘truly’ random array randarray.