Source code for elephant.signal_processing

# -*- coding: utf-8 -*-
Basic processing procedures for time series (e.g., performing a z-score of a
signal, or filtering a signal).

.. autosummary::
    :toctree: _toctree/signal_processing


:copyright: Copyright 2014-2022 by the Elephant team, see `doc/authors.rst`.
:license: Modified BSD, see LICENSE.txt for details.

from __future__ import division, print_function, unicode_literals

import neo
import numpy as np
import quantities as pq
import scipy.signal

from elephant.utils import deprecated_alias, check_same_units

import warnings

__all__ = [

[docs]def zscore(signal, inplace=True): r""" Apply a z-score operation to one or several `neo.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 a `neo.AnalogSignal` object containing multiple signals is provided, the z-transform is always calculated for each signal individually. If a list of `neo.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 a `neo.AnalogSignal` that contains multiple signals, each signal of the array is treated separately across list elements. Therefore, the number of signals must be identical for each `neo.AnalogSignal` object of the list. Parameters ---------- signal : neo.AnalogSignal or list of neo.AnalogSignal Signals for which to calculate the z-score. inplace : bool, optional If True, the contents of the input `signal` is replaced by the z-transformed signal, if possible, i.e when the signal type is float. If the signal type is not float, an error is raised. If False, a copy of the original `signal` is returned. Default: True Returns ------- signal_ztransformed : neo.AnalogSignal or list of neo.AnalogSignal The output format matches the input format: for each input `neo.AnalogSignal`, a corresponding `neo.AnalogSignal` is returned, containing the z-transformed signal with dimensionless unit. Raises ------ ValueError If `inplace` is True and the type of `signal` is not float. Notes ----- You may supply a list of `neo.AnalogSignal` objects, where each object in the list contains the data of one trial of the experiment, and each signal of the `neo.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 -------- Z-transform a single `neo.AnalogSignal`, containing only a single signal. >>> import neo >>> import numpy as np >>> import quantities as pq >>> from elephant.signal_processing import zscore ... >>> a = neo.AnalogSignal( ... np.array([1, 2, 3, 4, 5, 6]).reshape(-1,1) * pq.mV, ... t_start=0*pq.s, sampling_rate=1000*pq.Hz) >>> zscore(a).as_quantity() [[-1.46385011] [-0.87831007] [-0.29277002] [ 0.29277002] [ 0.87831007] [ 1.46385011]] dimensionless Z-transform a single `neo.AnalogSignal` containing multiple signals. >>> b = neo.AnalogSignal( ... np.transpose([[1, 2, 3, 4, 5, 6], ... [11, 12, 13, 14, 15, 16]]) * pq.mV, ... t_start=0*pq.s, sampling_rate=1000*pq.Hz) >>> zscore(b).as_quantity() [[-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 Z-transform a list of `neo.AnalogSignal`, each one containing more than one signal: >>> c = neo.AnalogSignal( ... np.transpose([[21, 22, 23, 24, 25, 26], ... [31, 32, 33, 34, 35, 36]]) * pq.mV, ... t_start=0*pq.s, sampling_rate=1000*pq.Hz) >>> 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 isinstance(signal, neo.AnalogSignal): signal = [signal] check_same_units(signal, object_type=neo.AnalogSignal) # Calculate mean and standard deviation signal_stacked = np.vstack(signal).magnitude mean = signal_stacked.mean(axis=0) std = signal_stacked.std(axis=0) signal_ztransformed = [] for sig in signal: # Perform inplace operation only if array is of dtype float. # Otherwise, raise an error. if inplace and not np.issubdtype(np.float, sig.dtype): raise ValueError(f"Cannot perform inplace operation as the " f"signal dtype is not float. Source: {}") sig_normalized = sig.magnitude.astype(mean.dtype, copy=not inplace) sig_normalized -= mean # items where std is zero are already zero np.divide(sig_normalized, std, out=sig_normalized, where=std != 0) if inplace: # Replace unit in the original array by dimensionless sig._dimensionality = pq.dimensionless.dimensionality sig_dimless = sig else: # Create new object sig_dimless = sig.duplicate_with_new_data(sig_normalized, units=pq.dimensionless) # todo use flag once is fixed # sig_dimless.array_annotate(**sig.array_annotations) signal_ztransformed.append(sig_dimless) # Return single object, or list of objects if len(signal_ztransformed) == 1: signal_ztransformed = signal_ztransformed[0] return signal_ztransformed
[docs]@deprecated_alias(ch_pairs='channel_pairs', nlags='n_lags', env='hilbert_envelope') def cross_correlation_function(signal, channel_pairs, hilbert_envelope=False, n_lags=None, scaleopt='unbiased'): r""" Computes an estimator of the cross-correlation function :cite:`signal-Stoica2005`. .. 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[channel_pairs[0,0]]` vs `signal[channel_pairs[0,1]]`, `signal[channel_pairs[1,0]]` vs `signal[channel_pairs[1,1]]`, and so on. The input time series are z-scored beforehand. `scaleopt` controls the choice of :math:`R_{xy}(\tau)` normalizer. Alternatively, returns the Hilbert envelope of :math:`R_{xy}(\tau)`, which is useful to determine the correlation length of oscillatory signals. Parameters ---------- signal : (nt, nch) neo.AnalogSignal Signal with `nt` number of samples that contains `nch` LFP channels. channel_pairs : list or (n, 2) np.ndarray List with `n` channel pairs for which to compute cross-correlation. Each element of the list must contain 2 channel indices. If `np.ndarray`, the second axis must have dimension 2. hilbert_envelope : bool, optional If True, returns the Hilbert envelope of cross-correlation function result. Default: False n_lags : int, optional Defines the number of lags for cross-correlation function. If a `float` is passed, it will be rounded to the nearest integer. Number of samples of output is `2*n_lags+1`. If None, the number of samples of the output is equal to the number of samples of the input signal (namely `nt`). Default: None scaleopt : {'none', 'biased', 'unbiased', 'normalized', 'coeff'}, optional Normalization option, equivalent to matlab `xcorr(..., scaleopt)`. Specified as one of the following. * 'none': raw, unscaled cross-correlation .. math:: R_{xy}(\tau) * 'biased': biased estimate of the cross-correlation: .. math:: R_{xy,biased}(\tau) = \frac{1}{N} R_{xy}(\tau) * 'unbiased': unbiased estimate of the cross-correlation: .. math:: R_{xy,unbiased}(\tau) = \frac{1}{N-\tau} R_{xy}(\tau) * 'normalized' or 'coeff': normalizes the sequence so that the autocorrelations at zero lag equal 1: .. math:: R_{xy,coeff}(\tau) = \frac{1}{\sqrt{R_{xx}(0) R_{yy}(0)}} R_{xy}(\tau) Default: 'unbiased' Returns ------- cross_corr : neo.AnalogSignal Shape: `[2*n_lags+1, n]` Pairwise cross-correlation functions for channel pairs given by `channel_pairs`. If `hilbert_envelope` is 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 input `signal` is not a `neo.AnalogSignal`. If `channel_pairs` is not a list of channel pair indices with shape `(n,2)`. If `hilbert_envelope` is not a boolean. If `n_lags` is not a positive integer. If `scaleopt` is not one of the predefined above keywords. Examples -------- >>> import neo >>> import quantities as pq >>> import matplotlib.pyplot as plt >>> from elephant.signal_processing import cross_correlation_function >>> 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 and find cross-correlation >>> signal = neo.AnalogSignal(x, units='mV', t_start=0.*, >>> sampling_rate=1/dt*pq.Hz, dtype=float) >>> rho = cross_correlation_function(signal, [0,1], n_lags=150) >>> env = cross_correlation_function(signal, [0,1], n_lags=150, ... hilbert_envelope=True) ... >>> plt.plot(rho.times, rho) >>> plt.plot(env.times, env) # should be equal to one >>> """ # Make channel_pairs a 2D array pairs = np.asarray(channel_pairs) if pairs.ndim == 1: pairs = np.expand_dims(pairs, axis=0) # Check input if not isinstance(signal, neo.AnalogSignal): raise ValueError('Input signal must be of type neo.AnalogSignal') if pairs.shape[1] != 2: raise ValueError("'channel_pairs' is not a list of channel pair " "indices. Cannot define pairs for cross-correlation.") if not isinstance(hilbert_envelope, bool): raise ValueError("'hilbert_envelope' must be a boolean value") if n_lags is not None: if not isinstance(n_lags, int) or n_lags <= 0: raise ValueError('n_lags must be a non-negative integer') # z-score analog signal and store channel time series in different arrays # Cross-correlation will be calculated between xsig and ysig z_transformed = signal.magnitude - signal.magnitude.mean(axis=0) z_transformed = np.divide(z_transformed, signal.magnitude.std(axis=0), out=z_transformed, where=z_transformed != 0) # transpose (nch, xy, nt) -> (xy, nt, nch) xsig, ysig = np.transpose(z_transformed.T[pairs], (1, 2, 0)) # Define vector of lags tau nt, nch = xsig.shape 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 = scipy.signal.fftconvolve(xsig, ysig[::-1], mode='same', axes=0) if scaleopt == 'biased': xcorr /= nt elif scaleopt == 'unbiased': normalizer = np.expand_dims(nt - np.abs(tau), axis=1) xcorr /= normalizer elif scaleopt in ('normalized', 'coeff'): normalizer = np.sqrt((xsig ** 2).sum(axis=0) * (ysig ** 2).sum(axis=0)) xcorr /= normalizer elif scaleopt != 'none': raise ValueError("Invalid scaleopt mode: '{}'".format(scaleopt)) # Calculate envelope of cross-correlation function with Hilbert transform. # This is useful for transient oscillatory signals. if hilbert_envelope: xcorr = np.abs(scipy.signal.hilbert(xcorr, axis=0)) # Cut off lags outside the desired range if n_lags is not None: tau0 = np.argwhere(tau == 0).item() xcorr = xcorr[tau0 - n_lags: tau0 + n_lags + 1, :] # Return neo.AnalogSignal cross_corr = neo.AnalogSignal(xcorr, units='', t_start=tau[0] * signal.sampling_period, t_stop=tau[-1] * signal.sampling_period, sampling_rate=signal.sampling_rate, dtype=float) return cross_corr
[docs]@deprecated_alias(highpass_freq='highpass_frequency', lowpass_freq='lowpass_frequency', fs='sampling_frequency') def butter(signal, highpass_frequency=None, lowpass_frequency=None, order=4, filter_function='filtfilt', sampling_frequency=1.0, axis=-1): """ Butterworth filtering function for `neo.AnalogSignal`. Filter type is determined according to how values of `highpass_frequency` and `lowpass_frequency` are given (see "Parameters" section for details). Parameters ---------- signal : neo.AnalogSignal or pq.Quantity or np.ndarray Time series data to be filtered. If `pq.Quantity` or `np.ndarray`, the sampling frequency should be given through the keyword argument `fs`. highpass_frequency : pq.Quantity of float, optional High-pass cut-off frequency. If `float`, the given value is taken as frequency in Hz. Default: None lowpass_frequency : pq.Quantity or float, optional Low-pass cut-off frequency. If `float`, the given value is taken as frequency in Hz. Filter type is determined depending on the values of `lowpass_frequency` and `highpass_frequency`: * `highpass_frequency` only (`lowpass_frequency` is None): highpass filter * `lowpass_frequency` only (`highpass_frequency` is None): lowpass filter * `highpass_frequency` < `lowpass_frequency`: bandpass filter * `highpass_frequency` > `lowpass_frequency`: bandstop filter Default: None order : int, optional Order of the Butterworth filter. Default: 4 filter_function : {'filtfilt', 'lfilter', 'sosfiltfilt'}, optional Filtering function to be used. Available filters: * 'filtfilt': `scipy.signal.filtfilt`; * 'lfilter': `scipy.signal.lfilter`; * 'sosfiltfilt': `scipy.signal.sosfiltfilt`. In most applications 'filtfilt' should be used, because it doesn't bring about phase shift due to filtering. For numerically stable filtering, in particular higher order filters, use 'sosfiltfilt' (see Default: 'filtfilt' sampling_frequency : pq.Quantity or float, optional The sampling frequency of the input time series. When given as `float`, its value is taken as frequency in Hz. When `signal` is given as `neo.AnalogSignal`, its attribute is used to specify the sampling frequency and this parameter is ignored. Default: 1.0 axis : int, optional Axis along which filter is applied. Default: last axis (-1) Returns ------- filtered_signal : neo.AnalogSignal or pq.Quantity or np.ndarray Filtered input data. The shape and type is identical to those of the input `signal`. Raises ------ ValueError If `filter_function` is not one of 'lfilter', 'filtfilt', or 'sosfiltfilt'. If both `highpass_frequency` and `lowpass_frequency` are None. Examples -------- >>> import neo >>> import numpy as np >>> import quantities as pq >>> from elephant.signal_processing import butter >>> noise = neo.AnalogSignal(np.random.normal(size=5000), ... sampling_rate=1000 * pq.Hz, units='mV') >>> filtered_noise = butter(noise, highpass_frequency=250.0 * pq.Hz) >>> filtered_noise AnalogSignal with 1 channels of length 5000; units mV; datatype float64 sampling rate: 1000.0 Hz time: 0.0 s to 5.0 s Let's check that the normal noise power spectrum at zero frequency is close to zero. >>> from elephant.spectral import welch_psd >>> freq, psd = welch_psd(filtered_noise, fs=1000.0) >>> psd.shape (1, 556) >>> freq[0], psd[0, 0] (array(0.) * Hz, array(7.21464674e-08) * mV**2/Hz) """ available_filters = 'lfilter', 'filtfilt', 'sosfiltfilt' if filter_function not in available_filters: raise ValueError("Invalid `filter_function`: {filter_function}. " "Available filters: {available_filters}".format( filter_function=filter_function, available_filters=available_filters)) # design filter if hasattr(signal, 'sampling_rate'): sampling_frequency = signal.sampling_rate.rescale(pq.Hz).magnitude if isinstance(highpass_frequency, pq.quantity.Quantity): highpass_frequency = highpass_frequency.rescale(pq.Hz).magnitude if isinstance(lowpass_frequency, pq.quantity.Quantity): lowpass_frequency = lowpass_frequency.rescale(pq.Hz).magnitude Fn = sampling_frequency / 2. # filter type is determined according to the values of cut-off # frequencies if lowpass_frequency and highpass_frequency: if highpass_frequency < lowpass_frequency: Wn = (highpass_frequency / Fn, lowpass_frequency / Fn) btype = 'bandpass' else: Wn = (lowpass_frequency / Fn, highpass_frequency / Fn) btype = 'bandstop' elif lowpass_frequency: Wn = lowpass_frequency / Fn btype = 'lowpass' elif highpass_frequency: Wn = highpass_frequency / Fn btype = 'highpass' else: raise ValueError( "Either highpass_frequency or lowpass_frequency must be given" ) if filter_function == 'sosfiltfilt': output = 'sos' else: output = 'ba' designed_filter = scipy.signal.butter(order, Wn, btype=btype, output=output) # 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 == 'lfilter': b, a = designed_filter filtered_data = scipy.signal.lfilter(b=b, a=a, x=data, axis=axis) elif filter_function == 'filtfilt': b, a = designed_filter filtered_data = scipy.signal.filtfilt(b=b, a=a, x=data, axis=axis) else: filtered_data = scipy.signal.sosfiltfilt(sos=designed_filter, x=data, axis=axis) if isinstance(signal, neo.AnalogSignal): filtered_data = np.rollaxis(filtered_data, -1, 0) signal_out = signal.duplicate_with_new_data(filtered_data) # todo use flag once is fixed # signal_out.array_annotate(**signal.array_annotations) return signal_out if isinstance(signal, pq.quantity.Quantity): return filtered_data * signal.units return filtered_data
[docs]@deprecated_alias(nco='n_cycles', freq='frequency', fs='sampling_frequency') def wavelet_transform(signal, frequency, n_cycles=6.0, sampling_frequency=1.0, zero_padding=True): r""" Compute the wavelet transform of a given signal with Morlet mother wavelet. The parametrization of the wavelet is based on :cite:`signal-Le2001_83`. Parameters ---------- signal : (Nt, Nch) neo.AnalogSignal or np.ndarray or list Time series data to be wavelet-transformed. When multi-dimensional `np.ndarray` or list is given, the time axis must be the last dimension. If `neo.AnalogSignal`, `Nt` is the number of time points and `Nch` is the number of channels. frequency : float or list of float Center frequency of the Morlet wavelet in Hz. Multiple center frequencies can be given as a list, in which case the function computes the wavelet transforms for all the given frequencies at once. n_cycles : float, optional Size of the mother wavelet (approximate number of oscillation cycles within a wavelet). Corresponds to :math:`nco` in :cite:`signal-Le2001_83`. A larger `n_cycles` value leads to a higher frequency resolution and a lower temporal resolution, and vice versa. Typically used values are in a range of 3–8, but one should be cautious when using a value smaller than ~ 6, in which case the admissibility of the wavelet is not ensured :cite:`signal-Farge1992_395`. Default: 6.0 sampling_frequency : float, optional Sampling rate of the input data in Hz. When `signal` is given as a `neo.AnalogSignal`, the sampling frequency is taken from its attribute and this parameter is ignored. Default: 1.0 zero_padding : bool, optional Specifies whether the data length is extended to the least power of 2 greater than the original length, by padding zeros to the tail, for speeding up the computation. If True, the extended part is cut out from the final result before returned, so that the output has the same length as the input. Default: True Returns ------- signal_wt : np.ndarray Wavelet transform of the input data. When `frequency` was given as a list, the way how the wavelet transforms for different frequencies are returned depends on the input type: * when the input was a `neo.AnalogSignal`, the returned array has shape (`Nt`, `Nch`, `Nf`), where `Nf` = `len(freq)`, such that the last dimension indexes the frequencies; * when the input was a `np.ndarray` or list of shape (`a`, `b`, ..., `c`, `Nt`), the returned array has a shape (`a`, `b`, ..., `c`, `Nf`, `Nt`), such that the second last dimension indexes the frequencies. To summarize, `signal_wt.ndim` = `signal.ndim` + 1, with the additional dimension in the last axis (for `neo.AnalogSignal` input) or the second last axis (`np.ndarray` or list input) indexing the frequencies. Raises ------ ValueError If `frequency` (or one of the values in `frequency` when it is a list) is greater than the half of `sampling_frequency`. If `n_cycles` is not positive. Notes ----- `n_cycles` is related to the wavelet number :math:`w` as :math:`w \sim 2 \pi \frac{n_{\text{cycles}}}{6}` as defined in :cite:`signal-Le2001_83`. Examples -------- >>> import neo >>> import numpy as np >>> import quantities as pq >>> from elephant.signal_processing import wavelet_transform >>> noise = neo.AnalogSignal(np.random.normal(size=7), ... sampling_rate=11 * pq.Hz, units='mV') The wavelet frequency must be less than the half of the sampling rate; picking at 5 Hz. >>> wavelet_transform(noise, frequency=5) array([[-1.00890049+3.003473j ], [-1.43664254-2.8389273j ], [ 3.02499511+0.96534578j], [-2.79543976+1.4581079j ], [ 0.94387304-2.98159518j], [ 1.41476471+2.77389985j], [-2.95996766-0.9872236j ]]) """ def _morlet_wavelet_ft(freq, n_cycles, fs, n): # Generate the Fourier transform of Morlet wavelet as defined # in Le van Quyen et al. J Neurosci Meth 111:83-98 (2001). sigma = n_cycles / (6. * freq) freqs = np.fft.fftfreq(n, 1.0 / fs) heaviside = np.array(freqs > 0., dtype=np.float) ft_real = np.sqrt(2 * np.pi * freq) * sigma * np.exp( -2 * (np.pi * sigma * (freqs - freq)) ** 2) * heaviside * fs ft_imag = np.zeros_like(ft_real) return ft_real + 1.0j * ft_imag data = np.asarray(signal) # When the input is AnalogSignal, the axis for time index (i.e. the # first axis) needs to be rolled to the last if isinstance(signal, neo.AnalogSignal): data = np.rollaxis(data, 0, data.ndim) # When the input is AnalogSignal, use its attribute to specify the # sampling frequency if hasattr(signal, 'sampling_rate'): sampling_frequency = signal.sampling_rate if isinstance(sampling_frequency, pq.quantity.Quantity): sampling_frequency = sampling_frequency.rescale('Hz').magnitude if isinstance(frequency, (list, tuple, np.ndarray)): freqs = np.asarray(frequency) else: freqs = np.array([frequency, ]) if isinstance(freqs[0], pq.quantity.Quantity): freqs = [f.rescale('Hz').magnitude for f in freqs] # check whether the given central frequencies are less than the # Nyquist frequency of the signal if np.any(freqs >= sampling_frequency / 2): raise ValueError("'frequency' elements must be less than the half of " "the 'sampling_frequency' ({}) Hz" .format(sampling_frequency)) # check if n_cycles is positive if n_cycles <= 0: raise ValueError("`n_cycles` must be positive") n_orig = data.shape[-1] if zero_padding: n = 2 ** (int(np.log2(n_orig)) + 1) else: n = n_orig # generate Morlet wavelets (in the frequency domain) wavelet_fts = np.empty([len(freqs), n], dtype=np.complex) for i, f in enumerate(freqs): wavelet_fts[i] = _morlet_wavelet_ft(f, n_cycles, sampling_frequency, n) # perform wavelet transform by convoluting the signal with the wavelets if data.ndim == 1: data = np.expand_dims(data, 0) data = np.expand_dims(data, data.ndim - 1) data = np.fft.ifft(np.fft.fft(data, n) * wavelet_fts) signal_wt = data[..., 0:n_orig] # reshape the result array according to the input if isinstance(signal, neo.AnalogSignal): signal_wt = np.rollaxis(signal_wt, -1) if not isinstance(frequency, (list, tuple, np.ndarray)): signal_wt = signal_wt[..., 0] else: if signal.ndim == 1: signal_wt = signal_wt[0] if not isinstance(frequency, (list, tuple, np.ndarray)): signal_wt = signal_wt[..., 0, :] return signal_wt
[docs]@deprecated_alias(N='padding') def hilbert(signal, padding='nextpow'): """ Apply a Hilbert transform to a `neo.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` function) and absolute value (`np.abs` function) 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` function will not terminate. Parameters ---------- signal : neo.AnalogSignal Signal(s) to transform. padding : int, {'none', 'nextpow'}, or None, optional Defines whether the signal is zero-padded. The `padding` argument corresponds to `N` in `scipy.signal.hilbert(signal, N=padding)` function. If 'none' or None, no padding. If 'nextpow', zero-pad to the next length that is a power of 2. If it is an `int`, directly specify the length to zero-pad to (indicates the number of Fourier components). Default: 'nextpow' Returns ------- neo.AnalogSignal Contains the complex analytic signal(s) corresponding to the input `signal`. The unit of the returned `neo.AnalogSignal` is dimensionless. Raises ------ ValueError: If `padding` is not an integer or neither 'nextpow' nor 'none' (None). Examples -------- Create a sine signal at 5 Hz with increasing amplitude and calculate the instantaneous phases: >>> import neo >>> import numpy as np >>> import quantities as pq >>> import matplotlib.pyplot as plt >>> from elephant.signal_processing import hilbert >>> t = np.arange(0, 5000) * >>> f = 5. * pq.Hz >>> a = neo.AnalogSignal( ... np.array( ... (1 + t.magnitude / t[-1].magnitude) * np.sin( ... 2. * np.pi * f * t.rescale(pq.s))).reshape( ... (-1,1)) * pq.mV, ... t_start=0*pq.s, ... sampling_rate=1000*pq.Hz) ... >>> analytic_signal = hilbert(a, padding='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 isinstance(padding, int): # User defined padding n = padding elif padding == '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 padding == 'none' or padding is None: # No padding n = n_org else: raise ValueError("Invalid padding '{}'.".format(padding)) output = signal.duplicate_with_new_data( scipy.signal.hilbert(signal.magnitude, N=n, axis=0)[:n_org]) # todo use flag once is fixed # output.array_annotate(**signal.array_annotations) return output / output.units
[docs]def rauc(signal, baseline=None, bin_duration=None, t_start=None, t_stop=None): """ Calculate the rectified area under the curve (RAUC) for a `neo.AnalogSignal`. The signal is optionally divided into bins with duration `bin_duration`, and the rectified signal (absolute value) is integrated within each bin to find the area under the curve. The mean or median of the signal or an arbitrary baseline may optionally be subtracted before rectification. Parameters ---------- signal : neo.AnalogSignal The signal to integrate. If `signal` contains more than one channel, each is integrated separately. baseline : pq.Quantity or {'mean', 'median'}, optional A factor to subtract from the signal before rectification. If 'mean', the mean value of the entire `signal` is subtracted on a channel-by-channel basis. If 'median', the median value of the entire `signal` is subtracted on a channel-by-channel basis. Default: None bin_duration : pq.Quantity, optional The length of time that each integration should span. If None, there will be only one bin spanning the entire signal duration. If `bin_duration` does not divide evenly into the signal duration, the end of the signal is padded with zeros to accomodate the final, overextending bin. Default: None t_start : pq.Quantity, optional Time to start the algorithm. If None, starts at the beginning of `signal`. Default: None t_stop : pq.Quantity, optional Time to end the algorithm. If None, ends at the last time of `signal`. The signal is cropped using `signal.time_slice(t_start, t_stop)` after baseline removal. Useful if you want the RAUC for a short section of the signal but want the mean or median calculation (`baseline`='mean' or `baseline`='median') to use the entire signal for better baseline estimation. Default: None Returns ------- pq.Quantity or neo.AnalogSignal If the number of bins is 1, the returned object is a scalar or vector `pq.Quantity` containing a single RAUC value for each channel. Otherwise, the returned object is a `neo.AnalogSignal` containing the RAUC(s) for each bin stored as a sample, with times corresponding to the center of each bin. The output signal will have the same number of channels as the input signal. Raises ------ ValueError If `signal` is not `neo.AnalogSignal`. If `bin_duration` is not None or `pq.Quantity`. If `baseline` is not None, 'mean', 'median', or `pq.Quantity`. See Also -------- neo.AnalogSignal.time_slice : how `t_start` and `t_stop` are used Examples -------- >>> import neo >>> import numpy as np >>> import quantities as pq >>> from elephant.signal_processing import rauc >>> signal = neo.AnalogSignal(np.arange(10), sampling_rate=20 * pq.Hz, ... units='mV') >>> rauc(signal) array(2.025) * mV/Hz """ if not isinstance(signal, neo.AnalogSignal): raise ValueError('Input signal is not a neo.AnalogSignal!') if baseline is None: pass elif baseline == 'mean': # subtract mean from each channel signal = signal - signal.mean(axis=0) elif baseline == 'median': # subtract median from each channel signal = signal - np.median(signal.as_quantity(), axis=0) elif isinstance(baseline, pq.Quantity): # subtract arbitrary baseline signal = signal - baseline else: raise ValueError("baseline must be either None, 'mean', 'median', or " "a Quantity. Got {}".format(baseline)) # slice the signal after subtracting baseline signal = signal.time_slice(t_start, t_stop) if bin_duration is not None: # from bin duration, determine samples per bin and number of bins if isinstance(bin_duration, pq.Quantity): samples_per_bin = int( np.round( bin_duration.rescale('s') / signal.sampling_period.rescale('s'))) n_bins = int(np.ceil(signal.shape[0] / samples_per_bin)) else: raise ValueError("bin_duration must be a Quantity. Got {}".format( bin_duration)) else: # all samples in one bin samples_per_bin = signal.shape[0] n_bins = 1 # store the actual bin duration bin_duration = samples_per_bin * signal.sampling_period # reshape into equal size bins, padding the end with zeros if necessary n_channels = signal.shape[1] sig_binned = signal.as_quantity().copy() sig_binned.resize(n_bins * samples_per_bin, n_channels, refcheck=False) sig_binned = sig_binned.reshape(n_bins, samples_per_bin, n_channels) # rectify and integrate over each bin rauc = np.trapz(np.abs(sig_binned), dx=signal.sampling_period, axis=1) if n_bins == 1: # return a single value for each channel return rauc.squeeze() # return an AnalogSignal with times corresponding to center of each bin t_start = signal.t_start.rescale(bin_duration.units) + bin_duration / 2 rauc_sig = neo.AnalogSignal(rauc, t_start=t_start, sampling_period=bin_duration) return rauc_sig
[docs]def derivative(signal): """ Calculate the derivative of a `neo.AnalogSignal`. Parameters ---------- signal : neo.AnalogSignal The signal to differentiate. If `signal` contains more than one channel, each is differentiated separately. Returns ------- derivative_sig : neo.AnalogSignal The returned object is a `neo.AnalogSignal` containing the differences between each successive sample value of the input signal divided by the sampling period. Times are centered between the successive samples of the input. The output signal will have the same number of channels as the input signal. Raises ------ TypeError If `signal` is not a `neo.AnalogSignal`. Examples -------- >>> import neo >>> import numpy as np >>> import quantities as pq >>> from elephant.signal_processing import derivative >>> signal = neo.AnalogSignal([0, 3, 4, 11, -1], sampling_rate=1 * pq.Hz, ... units='mV') >>> print(derivative(signal)) [[ 3.] [ 1.] [ 7.] [-12.]] mV*Hz """ if not isinstance(signal, neo.AnalogSignal): raise TypeError('Input signal is not a neo.AnalogSignal!') derivative_sig = neo.AnalogSignal( np.diff(signal.as_quantity(), axis=0) / signal.sampling_period, t_start=signal.t_start + signal.sampling_period / 2, sampling_period=signal.sampling_period) return derivative_sig