# multipletau documentation¶

## General¶

Multipletau provides a multiple-τ algorithm for Python 2.7 and Python 3.x with numpy as its sole dependency.

Multiple-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().

### Obtaining multipletau¶

If you have Python and numpy installed, simply run

pip install multipletau

The source code of multipletau is available at https://github.com/FCS-analysis/multipletau.

### Citing multipletau¶

The multipletau package should be cited like this (replace “x.x.x” with the actual version of multipletau used and “DD Month YYYY” with a matching date).

cite

Paul Müller (2012) Python multiple-tau algorithm (Version x.x.x) [Computer program]. Available at https://pypi.python.org/pypi/multipletau/ (Accessed DD Month YYYY)

You can find out what version you are using by typing (in a Python console):

>>> import multipletau
>>> multipletau.__version__
'0.3.0'


### Usage¶

The package is straightforward to use. Here is a quick example:

>>> import numpy as np
>>> import multipletau
>>> a = np.linspace(2,5,42)
>>> v = np.linspace(1,6,42)
>>> multipletau.correlate(a, v, m=2)
array([[   0.        ,  569.56097561],
[   1.        ,  549.87804878],
[   2.        ,  530.37477692],
[   4.        ,  491.85812017],
[   8.        ,  386.39500297]])


## Methods¶

Summary:

 autocorrelate(a[, m, deltat, normalize, …]) Autocorrelation of a 1-dimensional sequence on a log2-scale. correlate(a, v[, m, deltat, normalize, …]) Cross-correlation of two 1-dimensional sequences on a log2-scale. correlate_numpy(a, v[, deltat, normalize, …]) Convenience function that wraps around numpy.correlate() and returns the correlation in the same format as correlate() does.

### Autocorrelation¶

multipletau.autocorrelate(a, m=16, deltat=1, normalize=False, copy=True, dtype=None, compress='average', ret_sum=False)[source]

Autocorrelation of a 1-dimensional sequence on a log2-scale.

This computes the correlation similar to numpy.correlate() for positive $$k$$ on a base 2 logarithmic scale.

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

$$z_k = \Sigma_n a_n a_{n+k}$$

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 $$M-k$$. 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. compress (str) – strategy for propagating values to the next register ”average” (default): average two measurements when pushing to the next level of the correlator. ”first”: use only the first value when pushing to the next level of the correlator. ”second”: use only the second value when pushing to the next level of the correlator. Using only the first or the second values during propagation completely removes the systematic error at the cost of increasing the statistical error. See https://doi.org/10.1063/1.3491098 for a discussion on the effect of averaging. ret_sum (bool) – return the exact sum $$z_k = \Sigma_n a_n a_{n+k}$$. In addition $$M-k$$ is returned as an array of length N. autocorrelation (ndarray of shape (N,2)) – the lag time (1st column) and the autocorrelation (2nd column). count (ndarray of length N) – only returned if ret_sum is True; the value of $$M-k$$ for each row in autocorrelation.

Notes

Changed in version 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 $$M-k$$ by invoking normalize=True.

For normalizing according to the behavior of numpy.correlate(), use normalize=False.

For complex arrays, this method falls back to the method 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]])


### Cross-correlation¶

multipletau.correlate(a, v, m=16, deltat=1, normalize=False, copy=True, dtype=None, compress='average', ret_sum=False)[source]

Cross-correlation of two 1-dimensional sequences on a log2-scale.

This computes the cross-correlation similar 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. To obtain the correlation for negative lag times swap the input variables a and v.

Parameters: v (a,) – 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 (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. compress (str) – strategy for propagating values to the next register ”average” (default): average two measurements when pushing to the next level of the correlator. ”first”: use only the first value when pushing to the next level of the correlator. ”second”: use only the second value when pushing to the next level of the correlator. Using only the first or the second values during propagation completely removes the systematic error at the cost of increasing the statistical error. See https://doi.org/10.1063/1.3491098 for a discussion on the effect of averaging. ret_sum (bool) – return the exact sum $$z_k = \Sigma_n a_n v_{n+k}$$. In addition $$M-k$$ is returned as an array of length N. cross_correlation (ndarray of shape (N,2)) – the lag time (1st column), the cross-correlation (2nd column). count (ndarray of length N) – only returned if ret_sum is True; the value of $$M-k$$ for each row in autocorrelation.

Notes

Changed in version 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 $$M-k$$ by invoking normalize=True.

For normalizing according to the behavior of 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]])


### Cross-correlation (NumPy)¶

multipletau.correlate_numpy(a, v, deltat=1, normalize=False, dtype=None, copy=True)[source]

Convenience function that wraps around numpy.correlate() and returns the correlation in the same format as correlate() does.

Parameters: v (a,) – 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 (object to be converted to a data type object) – The data type of the returned array. cross_correlation – the lag time (column 1) and the cross-correlation (column 2). ndarray of shape (N,2)

Notes

Changed in version 0.1.6: Removed false normalization when normalize==False.

### Constants¶

multipletau.core.ZERO_CUTOFF = 1e-15

Defines the cutoff when the absolute mean trace divided by the median of the absolute value of the trace is treated as zero.

# Examples¶

## Comparison of correlation methods¶

This example illustrates the differences between the multipletau correlation methods (multipletau.autocorrelate(), multipletau.correlate()) and numpy.correlate().

This example requires noise_generator.py to be present in the current working directory. compare_correlation_methods.py

  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 from matplotlib import pylab as plt import numpy as np from multipletau import autocorrelate, correlate, correlate_numpy from noise_generator import noise_exponential, noise_cross_exponential # starting parameters N = np.int(np.pi * 1e3) countrate = 250. * 1e-3 # in Hz taudiff = 55. # in us deltat = 2e-6 # time discretization [s] normalize = True # time factor taudiff *= deltat # create noise for autocorrelation data = noise_exponential(N, taudiff, deltat=deltat) data -= np.average(data) if normalize: data += countrate # perform autocorrelation (multipletau) gac_mt = autocorrelate(data, deltat=deltat, normalize=normalize) # numpy.correlate for comparison gac_np = correlate_numpy(data, data, deltat=deltat, normalize=normalize) # calculate model curve for autocorrelation x = gac_np[:, 0] amp = np.correlate(data - np.average(data), data - np.average(data), mode="valid") if normalize: amp /= len(data) * countrate**2 y = amp * np.exp(-x / taudiff) # create noise for cross-correlation a, v = noise_cross_exponential(N, taudiff, deltat=deltat) a -= np.average(a) v -= np.average(v) if normalize: a += countrate v += countrate gcc_forw_mt = correlate(a, v, deltat=deltat, normalize=normalize) # forward gcc_back_mt = correlate(v, a, deltat=deltat, normalize=normalize) # backward # numpy.correlate for comparison gcc_forw_np = correlate_numpy(a, v, deltat=deltat, normalize=normalize) # calculate the model curve for cross-correlation xcc = gac_np[:, 0] ampcc = np.correlate(a - np.average(a), v - np.average(v), mode="valid") if normalize: ampcc /= len(a) * countrate**2 ycc = ampcc * np.exp(-xcc / taudiff) # plotting fig = plt.figure(figsize=(8, 5)) fig.canvas.set_window_title('comparing multipletau') # autocorrelation ax1 = fig.add_subplot(211) ax1.plot(gac_np[:, 0], gac_np[:, 1], "-", color="gray", label="correlate (numpy)") ax1.plot(x, y, "g-", label="input model") ax1.plot(gac_mt[:, 0], gac_mt[:, 1], "-", color="#B60000", label="autocorrelate") ax1.legend(loc=0, fontsize='small') ax1.set_xlabel("lag channel") ax1.set_ylabel("autocorrelation") ax1.set_xscale('log') ax1.set_xlim(x.min(), x.max()) ax1.set_ylim(-y.max()*.2, y.max()*1.1) # cross-correlation ax2 = fig.add_subplot(212) ax2.plot(gcc_forw_np[:, 0], gcc_forw_np[:, 1], "-", color="gray", label="forward (numpy)") ax2.plot(xcc, ycc, "g-", label="input model") ax2.plot(gcc_forw_mt[:, 0], gcc_forw_mt[:, 1], "-", color="#B60000", label="forward") ax2.plot(gcc_back_mt[:, 0], gcc_back_mt[:, 1], "-", color="#5D00B6", label="backward") ax2.set_xlabel("lag channel") ax2.set_ylabel("cross-correlation") ax2.legend(loc=0, fontsize='small') ax2.set_xscale('log') ax2.set_xlim(x.min(), x.max()) ax2.set_ylim(-ycc.max()*.2, ycc.max()*1.1) plt.tight_layout() plt.show()