# -*- coding: utf-8 -*-
'''
Basic processing procedures for analog signals (e.g., performing a z-score of a
signal, or filtering a signal).
:copyright: Copyright 2014-2016 by the Elephant team, see AUTHORS.txt.
:license: Modified BSD, see LICENSE.txt for details.
'''
from __future__ import division, print_function
import numpy as np
import scipy.signal
import quantities as pq
import neo
import numpy.matlib as npm
[docs]def zscore(signal, inplace=True):
'''
Apply a z-score operation to one or several AnalogSignal objects.
The z-score operation subtracts the mean :math:`\\mu` of the signal, and
divides by its standard deviation :math:`\\sigma`:
.. math::
Z(x(t))= \\frac{x(t)-\\mu}{\\sigma}
If an AnalogSignal containing multiple signals is provided, the
z-transform is always calculated for each signal individually.
If a list of AnalogSignal objects is supplied, the mean and standard
deviation are calculated across all objects of the list. Thus, all list
elements are z-transformed by the same values of :math:`\\mu` and
:math:`\\sigma`. For AnalogSignals, each signal of the array is
treated separately across list elements. Therefore, the number of signals
must be identical for each AnalogSignal of the list.
Parameters
----------
signal : neo.AnalogSignal or list of neo.AnalogSignal
Signals for which to calculate the z-score.
inplace : bool
If True, the contents of the input signal(s) is replaced by the
z-transformed signal. Otherwise, a copy of the original
AnalogSignal(s) is returned. Default: True
Returns
-------
neo.AnalogSignal or list of neo.AnalogSignal
The output format matches the input format: for each supplied
AnalogSignal object a corresponding object is returned containing
the z-transformed signal with the unit dimensionless.
Use Case
--------
You may supply a list of AnalogSignal objects, where each object in
the list contains the data of one trial of the experiment, and each signal
of the AnalogSignal corresponds to the recordings from one specific
electrode in a particular trial. In this scenario, you will z-transform the
signal of each electrode separately, but transform all trials of a given
electrode in the same way.
Examples
--------
>>> a = neo.AnalogSignal(
... np.array([1, 2, 3, 4, 5, 6]).reshape(-1,1)*mV,
... t_start=0*s, sampling_rate=1000*Hz)
>>> b = neo.AnalogSignal(
... np.transpose([[1, 2, 3, 4, 5, 6], [11, 12, 13, 14, 15, 16]])*mV,
... t_start=0*s, sampling_rate=1000*Hz)
>>> c = neo.AnalogSignal(
... np.transpose([[21, 22, 23, 24, 25, 26], [31, 32, 33, 34, 35, 36]])*mV,
... t_start=0*s, sampling_rate=1000*Hz)
>>> print zscore(a)
[[-1.46385011]
[-0.87831007]
[-0.29277002]
[ 0.29277002]
[ 0.87831007]
[ 1.46385011]] dimensionless
>>> print zscore(b)
[[-1.46385011 -1.46385011]
[-0.87831007 -0.87831007]
[-0.29277002 -0.29277002]
[ 0.29277002 0.29277002]
[ 0.87831007 0.87831007]
[ 1.46385011 1.46385011]] dimensionless
>>> print zscore([b,c])
[<AnalogSignal(array([[-1.11669108, -1.08361877],
[-1.0672076 , -1.04878252],
[-1.01772411, -1.01394628],
[-0.96824063, -0.97911003],
[-0.91875714, -0.94427378],
[-0.86927366, -0.90943753]]) * dimensionless, [0.0 s, 0.006 s],
sampling rate: 1000.0 Hz)>,
<AnalogSignal(array([[ 0.78170952, 0.84779261],
[ 0.86621866, 0.90728682],
[ 0.9507278 , 0.96678104],
[ 1.03523694, 1.02627526],
[ 1.11974608, 1.08576948],
[ 1.20425521, 1.1452637 ]]) * dimensionless, [0.0 s, 0.006 s],
sampling rate: 1000.0 Hz)>]
'''
# Transform input to a list
if type(signal) is not list:
signal = [signal]
# Calculate mean and standard deviation
m = np.mean(np.concatenate(signal), axis=0)
s = np.std(np.concatenate(signal), axis=0)
if not inplace:
# Create new signal instance
result = []
for sig in signal:
sig_dimless = sig.duplicate_with_new_data(
(sig.magnitude - m.magnitude) / s.magnitude) / sig.units
result.append(sig_dimless)
else:
result = []
# Overwrite signal
for sig in signal:
sig[:] = pq.Quantity(
(sig.magnitude - m.magnitude) / s.magnitude,
units=sig.units)
sig_dimless = sig / sig.units
result.append(sig_dimless)
# Return single object, or list of objects
if len(result) == 1:
return result[0]
else:
return result
[docs]def cross_correlation_function(signal, ch_pairs, env=False, nlags=None):
"""
Computes unbiased estimator of the cross-correlation function.
Calculates the unbiased estimator of the cross-correlation function [1]_
.. math::
R(\\tau) = \\frac{1}{N-|k|} R'(\\tau) \\ ,
where :math:`R'(\\tau) = \\left<x(t)y(t+\\tau)\\right>` in a pairwise
manner, i.e. `signal[ch_pairs[0,0]]` vs `signal2[ch_pairs[0,1]]`,
`signal[ch_pairs[1,0]]` vs `signal2[ch_pairs[1,1]]`, and so on. The
cross-correlation function is obtained by `scipy.signal.fftconvolve`.
Time series in signal are zscored beforehand. Alternatively returns the
Hilbert envelope of :math:`R(\\tau)`, which is useful to determine the
correlation length of oscillatory signals.
Parameters
-----------
signal : neo.AnalogSignal (`nt` x `nch`)
Signal with nt number of samples that contains nch LFP channels
ch_pairs : list (or array with shape `(n,2)`)
list with n channel pairs for which to compute cross-correlation,
each element of list must contain 2 channel indices
env : bool
Return Hilbert envelope of cross-correlation function
Default: False
nlags : int
Defines number of lags for cross-correlation function. Float will be
rounded to nearest integer. Number of samples of output is `2*nlags+1`.
If None, number of samples of output is equal to number of samples of
input signal, namely `nt`
Default: None
Returns
-------
cross_corr : neo.AnalogSgnal (`2*nlag+1` x `n`)
Pairwise cross-correlation functions for channel pairs given by
`ch_pairs`. If `env=True`, the output is the Hilbert envelope of the
pairwise cross-correlation function. This is helpful to compute the
correlation length for oscillating cross-correlation functions
Raises
------
ValueError
If the input signal is not a neo.AnalogSignal.
ValueError
If `ch_pairs` is not a list of channel pair indices with shape `(n,2)`.
KeyError
If keyword `env` is not a boolean.
KeyError
If `nlags` is not an integer or float larger than 0.
Examples
--------
>>> dt = 0.02
>>> N = 2018
>>> f = 0.5
>>> t = np.arange(N)*dt
>>> x = np.zeros((N,2))
>>> x[:,0] = 0.2 * np.sin(2.*np.pi*f*t)
>>> x[:,1] = 5.3 * np.cos(2.*np.pi*f*t)
>>> # Generate neo.AnalogSignals from x
>>> signal = neo.AnalogSignal(x, units='mV', t_start=0.*pq.ms,
>>> sampling_rate=1/dt*pq.Hz, dtype=float)
>>> rho = elephant.signal_processing.cross_correlation_function(
>>> signal, [0,1], nlags=150)
>>> env = elephant.signal_processing.cross_correlation_function(
>>> signal, [0,1], nlags=150, env=True)
>>> plt.plot(rho.times, rho)
>>> plt.plot(env.times, env) # should be equal to one
>>> plt.show()
References
----------
.. [1] Hall & River (2009) "Spectral Analysis of Signals, Spectral Element
Method in Structural Dynamics", Eq. 2.2.3
"""
# Make ch_pairs a 2D array
pairs = np.array(ch_pairs)
if pairs.ndim == 1:
pairs = pairs[:, np.newaxis]
# Check input
if not isinstance(signal, neo.AnalogSignal):
raise ValueError('Input signal is not a neo.AnalogSignal!')
if np.shape(pairs)[1] != 2:
pairs = pairs.T
if np.shape(pairs)[1] != 2:
raise ValueError('ch_pairs is not a list of channel pair indices.'\
'Cannot define pairs for cross-correlation.')
if not isinstance(env, bool):
raise KeyError('env is not a boolean!')
if nlags is not None:
if not isinstance(nlags, (int, float)):
raise KeyError('nlags must be an integer or float larger than 0!')
if nlags <= 0:
raise KeyError('nlags must be an integer or float larger than 0!')
# z-score analog signal and store channel time series in different arrays
# Cross-correlation will be calculated between xsig and ysig
xsig = np.array([zscore(signal).magnitude[:, pair[0]] \
for pair in pairs]).T
ysig = np.array([zscore(signal).magnitude[:, pair[1]] \
for pair in pairs]).T
# Define vector of lags tau
nt, nch = np.shape(xsig)
tau = (np.arange(nt) - nt//2)
# Calculate cross-correlation by taking Fourier transform of signal,
# multiply in Fourier space, and transform back. Correct for bias due
# to zero-padding
xcorr = np.zeros((nt, nch))
for i in range(nch):
xcorr[:, i] = scipy.signal.fftconvolve(xsig[:, i], ysig[::-1, i],
mode='same')
xcorr = xcorr / npm.repmat((nt-abs(tau)), nch, 1).T
# Calculate envelope of cross-correlation function with Hilbert transform.
# This is useful for transient oscillatory signals.
if env:
for i in range(nch):
xcorr[:, i] = np.abs(scipy.signal.hilbert(xcorr[:, i]))
# Cut off lags outside desired range
if nlags is not None:
nlags = int(np.round(nlags))
tau0 = int(np.argwhere(tau == 0))
xcorr = xcorr[tau0-nlags:tau0+nlags+1, :]
# Return neo.AnalogSignal
cross_corr = neo.AnalogSignal(xcorr,
units='',
t_start=np.min(tau)*signal.sampling_period,
t_stop=np.max(tau)*signal.sampling_period,
sampling_rate=signal.sampling_rate,
dtype=float)
return cross_corr
[docs]def butter(signal, highpass_freq=None, lowpass_freq=None, order=4,
filter_function='filtfilt', fs=1.0, axis=-1):
"""
Butterworth filtering function for neo.AnalogSignal. Filter type is
determined according to how values of `highpass_freq` and `lowpass_freq`
are given (see Parameters section for details).
Parameters
----------
signal : AnalogSignal or Quantity array or NumPy ndarray
Time series data to be filtered. When given as Quantity array or NumPy
ndarray, the sampling frequency should be given through the keyword
argument `fs`.
highpass_freq, lowpass_freq : Quantity or float
High-pass and low-pass cut-off frequencies, respectively. When given as
float, the given value is taken as frequency in Hz.
Filter type is determined depending on values of these arguments:
* highpass_freq only (lowpass_freq = None): highpass filter
* lowpass_freq only (highpass_freq = None): lowpass filter
* highpass_freq < lowpass_freq: bandpass filter
* highpass_freq > lowpass_freq: bandstop filter
order : int
Order of Butterworth filter. Default is 4.
filter_function : string
Filtering function to be used. Either 'filtfilt'
(`scipy.signal.filtfilt()`) or 'lfilter' (`scipy.signal.lfilter()`). In
most applications 'filtfilt' should be used, because it doesn't bring
about phase shift due to filtering. Default is 'filtfilt'.
fs : Quantity or float
The sampling frequency of the input time series. When given as float,
its value is taken as frequency in Hz. When the input is given as neo
AnalogSignal, its attribute is used to specify the sampling
frequency and this parameter is ignored. Default is 1.0.
axis : int
Axis along which filter is applied. Default is -1.
Returns
-------
filtered_signal : AnalogSignal or Quantity array or NumPy ndarray
Filtered input data. The shape and type is identical to those of the
input.
"""
def _design_butterworth_filter(Fs, hpfreq=None, lpfreq=None, order=4):
# set parameters for filter design
Fn = Fs / 2.
# - filter type is determined according to the values of cut-off
# frequencies
if lpfreq and hpfreq:
if hpfreq < lpfreq:
Wn = (hpfreq / Fn, lpfreq / Fn)
btype = 'bandpass'
else:
Wn = (lpfreq / Fn, hpfreq / Fn)
btype = 'bandstop'
elif lpfreq:
Wn = lpfreq / Fn
btype = 'lowpass'
elif hpfreq:
Wn = hpfreq / Fn
btype = 'highpass'
else:
raise ValueError(
"Either highpass_freq or lowpass_freq must be given"
)
# return filter coefficients
return scipy.signal.butter(order, Wn, btype=btype)
# design filter
Fs = signal.sampling_rate.rescale(pq.Hz).magnitude \
if hasattr(signal, 'sampling_rate') else fs
Fh = highpass_freq.rescale(pq.Hz).magnitude \
if isinstance(highpass_freq, pq.quantity.Quantity) else highpass_freq
Fl = lowpass_freq.rescale(pq.Hz).magnitude \
if isinstance(lowpass_freq, pq.quantity.Quantity) else lowpass_freq
b, a = _design_butterworth_filter(Fs, Fh, Fl, order)
# When the input is AnalogSignal, the axis for time index (i.e. the
# first axis) needs to be rolled to the last
data = np.asarray(signal)
if isinstance(signal, neo.AnalogSignal):
data = np.rollaxis(data, 0, len(data.shape))
# apply filter
if filter_function is 'lfilter':
filtered_data = scipy.signal.lfilter(b, a, data, axis=axis)
elif filter_function is 'filtfilt':
filtered_data = scipy.signal.filtfilt(b, a, data, axis=axis)
else:
raise ValueError(
"filter_func must to be either 'filtfilt' or 'lfilter'"
)
if isinstance(signal, neo.AnalogSignal):
return signal.duplicate_with_new_data(np.rollaxis(filtered_data, -1, 0))
elif isinstance(signal, pq.quantity.Quantity):
return filtered_data * signal.units
else:
return filtered_data
[docs]def hilbert(signal, N='nextpow'):
'''
Apply a Hilbert transform to an AnalogSignal object in order to obtain its
(complex) analytic signal.
The time series of the instantaneous angle and amplitude can be obtained as
the angle (np.angle) and absolute value (np.abs) of the complex analytic
signal, respectively.
By default, the function will zero-pad the signal to a length corresponding
to the next higher power of 2. This will provide higher computational
efficiency at the expense of memory. In addition, this circumvents a
situation where for some specific choices of the length of the input,
scipy.signal.hilbert() will not terminate.
Parameters
-----------
signal : neo.AnalogSignal
Signal(s) to transform
N : string or int
Defines whether the signal is zero-padded.
'none': no padding
'nextpow': zero-pad to the next length that is a power of 2
int: directly specify the length to zero-pad to (indicates the
number of Fourier components, see parameter N of
scipy.signal.hilbert()).
Default: 'nextpow'.
Returns
-------
neo.AnalogSignal
Contains the complex analytic signal(s) corresponding to the input
signals. The unit of the analytic signal is dimensionless.
Example
-------
Create a sine signal at 5 Hz with increasing amplitude and calculate the
instantaneous phases
>>> t = np.arange(0, 5000) * ms
>>> f = 5. * Hz
>>> a = neo.AnalogSignal(
... np.array(
... (1 + t.magnitude / t[-1].magnitude) * np.sin(
... 2. * np.pi * f * t.rescale(s))).reshape((-1,1))*mV,
... t_start=0*s, sampling_rate=1000*Hz)
>>> analytic_signal = hilbert(a, N='nextpow')
>>> angles = np.angle(analytic_signal)
>>> amplitudes = np.abs(analytic_signal)
>>> print angles
[[-1.57079633]
[-1.51334228]
[-1.46047675]
...,
[-1.73112977]
[-1.68211683]
[-1.62879501]]
>>> plt.plot(t,angles)
'''
# Length of input signals
n_org = signal.shape[0]
# Right-pad signal to desired length using the signal itself
if type(N) == int:
# User defined padding
n = N
elif N == 'nextpow':
# To speed up calculation of the Hilbert transform, make sure we change
# the signal to be of a length that is a power of two. Failure to do so
# results in computations of certain signal lengths to not finish (or
# finish in absurd time). This might be a bug in scipy (0.16), e.g.,
# the following code will not terminate for this value of k:
#
# import numpy
# import scipy.signal
# k=679346
# t = np.arange(0, k) / 1000.
# a = (1 + t / t[-1]) * np.sin(2 * np.pi * 5 * t)
# analytic_signal = scipy.signal.hilbert(a)
#
# For this reason, nextpow is the default setting for now.
n = 2 ** (int(np.log2(n_org - 1)) + 1)
elif N == 'none':
# No padding
n = n_org
else:
raise ValueError("'{}' is an unknown N.".format(N))
output = signal.duplicate_with_new_data(
scipy.signal.hilbert(signal.magnitude, N=n, axis=0)[:n_org])
return output / output.units
