Source code for elephant.statistics

# -*- coding: utf-8 -*-
"""
Statistical measures of spike trains (e.g., Fano factor) and functions to
estimate firing rates.

Rate estimation
***************

.. autosummary::
:toctree: _toctree/statistics/

mean_firing_rate
instantaneous_rate
time_histogram
optimal_kernel_bandwidth

Spike interval statistics
*************************

.. autosummary::
:toctree: _toctree/statistics/

isi
cv
cv2
lv
lvr

Statistics across spike trains
******************************

.. autosummary::
:toctree: _toctree/statistics/

fanofactor
complexity_pdf
Complexity

Tutorial
********

:doc:View tutorial <../tutorials/statistics>

Run tutorial interactively:

:target: https://mybinder.org/v2/gh/NeuralEnsemble/elephant/master
?filepath=doc/tutorials/statistics.ipynb

References
----------

.. bibliography:: ../bib/elephant.bib
:labelprefix: st
:keyprefix: statistics-
:style: unsrt

:copyright: Copyright 2014-2020 by the Elephant team, see doc/authors.rst.
"""

from __future__ import division, print_function

import math
import warnings

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

import elephant.conversion as conv
import elephant.kernels as kernels
from elephant.conversion import BinnedSpikeTrain
from elephant.utils import deprecated_alias, check_neo_consistency, \
is_time_quantity, round_binning_errors

# do not import unicode_literals
# (quantities rescale does not work with unicodes)

__all__ = [
"isi",
"mean_firing_rate",
"fanofactor",
"cv",
"cv2",
"lv",
"lvr",
"instantaneous_rate",
"time_histogram",
"complexity_pdf",
"Complexity",
"fftkernel",
"optimal_kernel_bandwidth"
]

cv = scipy.stats.variation

[docs]def isi(spiketrain, axis=-1): """ Return an array containing the inter-spike intervals of the spike train. Accepts a neo.SpikeTrain, a pq.Quantity array, a np.ndarray, or a list of time spikes. If either a neo.SpikeTrain or pq.Quantity is provided, the return value will be pq.Quantity, otherwise np.ndarray. The units of pq.Quantity will be the same as spiketrain. Visualization of this function is covered in Viziphant: :func:viziphant.statistics.plot_isi_histogram. Parameters ---------- spiketrain : neo.SpikeTrain or pq.Quantity or array-like The spike times. axis : int, optional The axis along which the difference is taken. Default: the last axis Returns ------- intervals : np.ndarray or pq.Quantity The inter-spike intervals of the spiketrain. Warns ----- UserWarning When the input array is not sorted, negative intervals are returned with a warning. Examples -------- >>> from elephant import statistics >>> statistics.isi([0.3, 4.5, 6.7, 9.3]) array([4.2, 2.2, 2.6]) """ if isinstance(spiketrain, neo.SpikeTrain): intervals = np.diff(spiketrain.magnitude, axis=axis) # np.diff makes a copy intervals = pq.Quantity(intervals, units=spiketrain.units, copy=False) else: intervals = np.diff(spiketrain, axis=axis) if (intervals < 0).any(): warnings.warn("ISI evaluated to negative values. " "Please sort the input array.") return intervals
[docs]def mean_firing_rate(spiketrain, t_start=None, t_stop=None, axis=None): """ Return the firing rate of the spike train. The firing rate is calculated as the number of spikes in the spike train in the range [t_start, t_stop] divided by the time interval t_stop - t_start. See the description below for cases when t_start or t_stop is None. Accepts a neo.SpikeTrain, a pq.Quantity array, or a plain np.ndarray. If either a neo.SpikeTrain or pq.Quantity array is provided, the return value will be a pq.Quantity array, otherwise a plain np.ndarray. The units of the pq.Quantity array will be the inverse of the spiketrain. Parameters ---------- spiketrain : neo.SpikeTrain or pq.Quantity or np.ndarray The spike times. t_start : float or pq.Quantity, optional The start time to use for the interval. If None, retrieved from the t_start attribute of spiketrain. If that is not present, default to 0. All spiketrain's spike times below this value are ignored. Default: None t_stop : float or pq.Quantity, optional The stop time to use for the time points. If not specified, retrieved from the t_stop attribute of spiketrain. If that is not present, default to the maximum value of spiketrain. All spiketrain's spike times above this value are ignored. Default: None axis : int, optional The axis over which to do the calculation; has no effect when the input is a neo.SpikeTrain, because a neo.SpikeTrain is always a 1-d vector. If None, do the calculation over the flattened array. Default: None Returns ------- float or pq.Quantity or np.ndarray The firing rate of the spiketrain Raises ------ TypeError If the input spiketrain is a np.ndarray but t_start or t_stop is pq.Quantity. If the input spiketrain is a neo.SpikeTrain or pq.Quantity but t_start or t_stop is not pq.Quantity. ValueError If the input spiketrain is empty. Examples -------- >>> from elephant import statistics >>> statistics.mean_firing_rate([0.3, 4.5, 6.7, 9.3]) 0.4301075268817204 """ if isinstance(spiketrain, neo.SpikeTrain) and t_start is None \ and t_stop is None and axis is None: # a faster approach for a typical use case n_spikes = len(spiketrain) time_interval = spiketrain.t_stop - spiketrain.t_start time_interval = time_interval.rescale(spiketrain.units) rate = n_spikes / time_interval return rate if isinstance(spiketrain, pq.Quantity): # Quantity or neo.SpikeTrain if not is_time_quantity(t_start, allow_none=True): raise TypeError("'t_start' must be a Quantity or None") if not is_time_quantity(t_stop, allow_none=True): raise TypeError("'t_stop' must be a Quantity or None") units = spiketrain.units if t_start is None: t_start = getattr(spiketrain, 't_start', 0 * units) t_start = t_start.rescale(units).magnitude if t_stop is None: t_stop = getattr(spiketrain, 't_stop', np.max(spiketrain, axis=axis)) t_stop = t_stop.rescale(units).magnitude # calculate as a numpy array rates = mean_firing_rate(spiketrain.magnitude, t_start=t_start, t_stop=t_stop, axis=axis) rates = pq.Quantity(rates, units=1. / units) elif isinstance(spiketrain, (np.ndarray, list, tuple)): if isinstance(t_start, pq.Quantity) or isinstance(t_stop, pq.Quantity): raise TypeError("'t_start' and 't_stop' cannot be quantities if " "'spiketrain' is not a Quantity.") spiketrain = np.asarray(spiketrain) if len(spiketrain) == 0: raise ValueError("Empty input spiketrain.") if t_start is None: t_start = 0 if t_stop is None: t_stop = np.max(spiketrain, axis=axis) time_interval = t_stop - t_start if axis and isinstance(t_stop, np.ndarray): t_stop = np.expand_dims(t_stop, axis) rates = np.sum((spiketrain >= t_start) & (spiketrain <= t_stop), axis=axis) / time_interval else: raise TypeError("Invalid input spiketrain type: '{}'. Allowed: " "neo.SpikeTrain, Quantity, ndarray". format(type(spiketrain))) return rates
[docs]def fanofactor(spiketrains, warn_tolerance=0.1 * pq.ms): r""" Evaluates the empirical Fano factor F of the spike counts of a list of neo.SpikeTrain objects. Given the vector v containing the observed spike counts (one per spike train) in the time window [t0, t1], F is defined as: .. math:: F := \frac{var(v)}{mean(v)} The Fano factor is typically computed for spike trains representing the activity of the same neuron over different trials. The higher F, the larger the cross-trial non-stationarity. In theory for a time-stationary Poisson process, F=1. Parameters ---------- spiketrains : list List of neo.SpikeTrain or pq.Quantity or np.ndarray or list of spike times for which to compute the Fano factor of spike counts. warn_tolerance : pq.Quantity In case of a list of input neo.SpikeTrains, if their durations vary by more than warn_tolerence in their absolute values, throw a warning (see Notes). Default: 0.1 ms Returns ------- fano : float The Fano factor of the spike counts of the input spike trains. Returns np.NaN if an empty list is specified, or if all spike trains are empty. Raises ------ TypeError If the input spiketrains are neo.SpikeTrain objects, but warn_tolerance is not a quantity. Notes ----- The check for the equal duration of the input spike trains is performed only if the input is of typeneo.SpikeTrain: if you pass a numpy array, please make sure that they all have the same duration manually. Examples -------- >>> import neo >>> from elephant import statistics >>> spiketrains = [ ... neo.SpikeTrain([0.3, 4.5, 6.7, 9.3], t_stop=10, units='s'), ... neo.SpikeTrain([1.4, 3.3, 8.2], t_stop=10, units='s') ... ] >>> statistics.fanofactor(spiketrains) 0.07142857142857142 """ # Build array of spike counts (one per spike train) spike_counts = np.array([len(st) for st in spiketrains]) # Compute FF if all(count == 0 for count in spike_counts): # empty list of spiketrains reaches this branch, and NaN is returned return np.nan if all(isinstance(st, neo.SpikeTrain) for st in spiketrains): if not is_time_quantity(warn_tolerance): raise TypeError("'warn_tolerance' must be a time quantity.") durations = [(st.t_stop - st.t_start).simplified.item() for st in spiketrains] durations_min = min(durations) durations_max = max(durations) if durations_max - durations_min > warn_tolerance.simplified.item(): warnings.warn("Fano factor calculated for spike trains of " "different duration (minimum: {_min}s, maximum " "{_max}s).".format(_min=durations_min, _max=durations_max)) fano = spike_counts.var() / spike_counts.mean() return fano
def __variation_check(v, with_nan): # ensure the input ia a vector if v.ndim != 1: raise ValueError("The input must be a vector, not a {}-dim matrix.". format(v.ndim)) # ensure we have enough entries if v.size < 2: if with_nan: warnings.warn("The input size is too small. Please provide" "an input with more than 1 entry. Returning NaN" "since the argument with_nan is True") return np.NaN raise ValueError("Input size is too small. Please provide " "an input with more than 1 entry. Set 'with_nan' " "to True to replace the error by a warning.") return None
[docs]@deprecated_alias(v='time_intervals') def cv2(time_intervals, with_nan=False): r""" Calculate the measure of Cv2 for a sequence of time intervals between events :cite:statistics-Holt1996_1806. Given a vector :math:I containing a sequence of intervals, the Cv2 is defined as: .. math:: Cv2 := \frac{1}{N} \sum_{i=1}^{N-1} \frac{2|I_{i+1}-I_i|} {|I_{i+1}+I_i|} The Cv2 is typically computed as a substitute for the classical coefficient of variation (Cv) for sequences of events which include some (relatively slow) rate fluctuation. As with the Cv, Cv2=1 for a sequence of intervals generated by a Poisson process. Parameters ---------- time_intervals : pq.Quantity or np.ndarray or list Vector of consecutive time intervals. with_nan : bool, optional If True, cv2 of a spike train with less than two spikes results in a np.NaN value and a warning is raised. If False, ValueError exception is raised with a spike train with less than two spikes. Default: True Returns ------- float The Cv2 of the inter-spike interval of the input sequence. Raises ------ ValueError If an empty list is specified, or if the sequence has less than two entries and with_nan is False. If a matrix is passed to the function. Only vector inputs are supported. Warns ----- UserWarning If with_nan is True and cv2 is calculated for a sequence with less than two entries, generating a np.NaN. Examples -------- >>> from elephant import statistics >>> statistics.cv2([0.3, 4.5, 6.7, 9.3]) 0.8226190476190478 """ # convert to array, cast to float time_intervals = np.asarray(time_intervals) np_nan = __variation_check(time_intervals, with_nan) if np_nan is not None: return np_nan # calculate Cv2 and return result cv_i = np.diff(time_intervals) / (time_intervals[:-1] + time_intervals[1:]) return 2. * np.mean(np.abs(cv_i))
[docs]@deprecated_alias(v='time_intervals') def lv(time_intervals, with_nan=False): r""" Calculate the measure of local variation Lv for a sequence of time intervals between events :cite:statistics-Shinomoto2003_2823. Given a vector :math:I containing a sequence of intervals, the Lv is defined as: .. math:: Lv := \frac{1}{N} \sum_{i=1}^{N-1} \frac{3(I_i-I_{i+1})^2} {(I_i+I_{i+1})^2} The Lv is typically computed as a substitute for the classical coefficient of variation for sequences of events which include some (relatively slow) rate fluctuation. As with the Cv, Lv=1 for a sequence of intervals generated by a Poisson process. Parameters ---------- time_intervals : pq.Quantity or np.ndarray or list Vector of consecutive time intervals. with_nan : bool, optional If True, the Lv of a spike train with less than two spikes results in a np.NaN value and a warning is raised. If False, a ValueError exception is raised with a spike train with less than two spikes. Default: True Returns ------- float The Lv of the inter-spike interval of the input sequence. Raises ------ ValueError If an empty list is specified, or if the sequence has less than two entries and with_nan is False. If a matrix is passed to the function. Only vector inputs are supported. Warns ----- UserWarning If with_nan is True and the Lv is calculated for a spike train with less than two spikes, generating a np.NaN. Examples -------- >>> from elephant import statistics >>> statistics.lv([0.3, 4.5, 6.7, 9.3]) 0.8306154336734695 """ # convert to array, cast to float time_intervals = np.asarray(time_intervals) np_nan = __variation_check(time_intervals, with_nan) if np_nan is not None: return np_nan cv_i = np.diff(time_intervals) / (time_intervals[:-1] + time_intervals[1:]) return 3. * np.mean(np.power(cv_i, 2))
[docs]def lvr(time_intervals, R=5*pq.ms, with_nan=False): r""" Calculate the measure of revised local variation LvR for a sequence of time intervals between events :cite:statistics-Shinomoto2009_e1000433. Given a vector :math:I containing a sequence of intervals, the LvR is defined as: .. math:: LvR := \frac{3}{N-1} \sum_{i=1}^{N-1} \left(1-\frac{4 I_i I_{i+1}} {(I_i+I_{i+1})^2}\right) \left(1+\frac{4 R}{I_i+I_{i+1}}\right) The LvR is a revised version of the Lv, with enhanced invariance to firing rate fluctuations by introducing a refractoriness constant R. The LvR with R=5ms was shown to outperform other ISI variability measures in spike trains with firing rate fluctuations and sensory stimuli :cite:statistics-Shinomoto2009_e1000433. Parameters ---------- time_intervals : pq.Quantity or np.ndarray or list Vector of consecutive time intervals. Must have time units, if not unit is passed ms are assumed. R : pq.Quantity or int or float Refractoriness constant (R >= 0). If no quantity is passed ms are assumed. Default: 5 ms with_nan : bool, optional If True, LvR of a spike train with less than two spikes results in a np.NaN value and a warning is raised. If False, a ValueError exception is raised with a spike train with less than two spikes. Default: True Returns ------- float The LvR of the inter-spike interval of the input sequence. Raises ------ ValueError If an empty list is specified, or if the sequence has less than two entries and with_nan is False. If a matrix is passed to the function. Only vector inputs are supported. Warns ----- UserWarning If with_nan is True and the lvr is calculated for a spike train with less than two spikes, generating a np.NaN. If R is passed without any units attached milliseconds are assumed. Examples -------- >>> from elephant import statistics >>> statistics.lvr([0.3, 4.5, 6.7, 9.3], R=0.005) 0.833907445980624 """ if isinstance(R, pq.Quantity): R = R.rescale('ms').magnitude else: warnings.warn('No units specified for R, assuming milliseconds (ms)') if R < 0: raise ValueError('R must be >= 0') # check units of intervals if available if isinstance(time_intervals, pq.Quantity): time_intervals = time_intervals.rescale('ms').magnitude else: warnings.warn('No units specified for time_intervals,' ' assuming milliseconds (ms)') # convert to array, cast to float time_intervals = np.asarray(time_intervals) np_nan = __variation_check(time_intervals, with_nan) if np_nan is not None: return np_nan N = len(time_intervals) t = time_intervals[:-1] + time_intervals[1:] frac1 = 4 * time_intervals[:-1] * time_intervals[1:] / t**2 frac2 = 4 * R / t lvr = (3 / (N-1)) * np.sum((1-frac1) * (1+frac2)) return lvr
[docs]@deprecated_alias(spiketrain='spiketrains') def instantaneous_rate(spiketrains, sampling_period, kernel='auto', cutoff=5.0, t_start=None, t_stop=None, trim=False, center_kernel=True): """ Estimates instantaneous firing rate by kernel convolution. Visualization of this function is covered in Viziphant: :func:viziphant.statistics.plot_instantaneous_rates_colormesh. Parameters ---------- spiketrains : neo.SpikeTrain or list of neo.SpikeTrain Neo object(s) that contains spike times, the unit of the time stamps, and t_start and t_stop of the spike train. sampling_period : pq.Quantity Time stamp resolution of the spike times. The same resolution will be assumed for the kernel. kernel : 'auto' or Kernel, optional The string 'auto' or callable object of class kernels.Kernel. The kernel is used for convolution with the spike train and its standard deviation determines the time resolution of the instantaneous rate estimation. Currently implemented kernel forms are rectangular, triangular, epanechnikovlike, gaussian, laplacian, exponential, and alpha function. If 'auto', the optimized kernel width for the rate estimation is calculated according to :cite:statistics-Shimazaki2010_171 and with this width a gaussian kernel is constructed. Automatized calculation of the kernel width is not available for other than gaussian kernel shapes. Default: 'auto' cutoff : float, optional This factor determines the cutoff of the probability distribution of the kernel, i.e., the considered width of the kernel in terms of multiples of the standard deviation sigma. Default: 5.0 t_start : pq.Quantity, optional Start time of the interval used to compute the firing rate. If None, t_start is assumed equal to t_start attribute of spiketrain. Default: None t_stop : pq.Quantity, optional End time of the interval used to compute the firing rate (included). If None, t_stop is assumed equal to t_stop attribute of spiketrain. Default: None trim : bool, optional Accounts for the asymmetry of a kernel. If False, the output of the Fast Fourier Transformation being a longer vector than the input vector by the size of the kernel is reduced back to the original size of the considered time interval of the spiketrain using the median of the kernel. False (no trimming) is equivalent to 'same' convolution mode for symmetrical kernels. If True, only the region of the convolved signal is returned, where there is complete overlap between kernel and spike train. This is achieved by reducing the length of the output of the Fast Fourier Transformation by a total of two times the size of the kernel, and t_start and t_stop are adjusted. True (trimming) is equivalent to 'valid' convolution mode for symmetrical kernels. Default: False center_kernel : bool, optional If set to True, the kernel will be translated such that its median is centered on the spike, thus putting equal weight before and after the spike. If False, no adjustment is performed such that the spike sits at the origin of the kernel. Default: True Returns ------- rate : neo.AnalogSignal 2D matrix that contains the rate estimation in unit hertz (Hz) of shape (time, len(spiketrains)) or (time, 1) in case of a single input spiketrain. rate.times contains the time axis of the rate estimate: the unit of this property is the same as the resolution that is given via the argument sampling_period to the function. Raises ------ TypeError If spiketrain is not an instance of neo.SpikeTrain. If sampling_period is not a pq.Quantity. If sampling_period is not larger than zero. If kernel is neither instance of kernels.Kernel nor string 'auto'. If cutoff is neither float nor int. If t_start and t_stop are neither None nor a pq.Quantity. If trim is not bool. ValueError If sampling_period is smaller than zero. If kernel is 'auto' and the function was unable to calculate optimal kernel width for instantaneous rate from input data. Warns ----- UserWarning If cutoff is less than min_cutoff attribute of kernel, the width of the kernel is adjusted to a minimally allowed width. Notes ----- The resulting instantaneous firing rate values smaller than 0, which can happen due to machine precision errors, are clipped to zero. Examples -------- Example 1. Automatic kernel estimation. >>> import neo >>> import quantities as pq >>> from elephant import statistics >>> spiketrain = neo.SpikeTrain([0.3, 4.5, 6.7, 9.3], t_stop=10, units='s') >>> rate = statistics.instantaneous_rate(spiketrain, ... sampling_period=10 * pq.ms, ... kernel='auto') >>> rate AnalogSignal with 1 channels of length 1000; units Hz; datatype float64 annotations: {'t_stop': array(10.) * s, 'kernel': {'type': 'GaussianKernel', 'sigma': '7.273225922958104 s', 'invert': False}} sampling rate: 0.1 1/ms time: 0.0 s to 10.0 s Example 2. Manually set kernel. >>> from elephant import kernels >>> spiketrain = neo.SpikeTrain(, t_stop=1, units='s') >>> kernel = kernels.GaussianKernel(sigma=300 * pq.ms) >>> rate = statistics.instantaneous_rate(spiketrain, ... sampling_period=200 * pq.ms, kernel=kernel, t_start=-1 * pq.s) >>> rate AnalogSignal with 1 channels of length 10; units Hz; datatype float64 annotations: {'t_stop': array(1.) * s, 'kernel': {'type': 'GaussianKernel', 'sigma': '300.0 ms', 'invert': False}} sampling rate: 0.005 1/ms time: -1.0 s to 1.0 s >>> rate.magnitude array([[0.01007419], [0.05842767], [0.22928759], [0.60883028], [1.0938699 ], [1.3298076 ], [1.0938699 ], [0.60883028], [0.22928759], [0.05842767]]) """ def optimal_kernel(st): width_sigma = None if len(st) > 0: width_sigma = optimal_kernel_bandwidth( st.magnitude, times=None, bootstrap=False)['optw'] if width_sigma is None: raise ValueError("Unable to calculate optimal kernel width for " "instantaneous rate from input data.") return kernels.GaussianKernel(width_sigma * st.units) if isinstance(spiketrains, neo.SpikeTrain): if kernel == 'auto': kernel = optimal_kernel(spiketrains) spiketrains = [spiketrains] elif not isinstance(spiketrains, (list, tuple)): raise TypeError( "'spiketrains' must be a list of neo.SpikeTrain's or a single " "neo.SpikeTrain. Found: '{}'".format(type(spiketrains))) if not is_time_quantity(sampling_period): raise TypeError( "The 'sampling_period' must be a time Quantity. \n" "Found: {}".format(type(sampling_period))) if sampling_period.magnitude < 0: raise ValueError("The 'sampling_period' ({}) must be non-negative.". format(sampling_period)) if not (isinstance(kernel, kernels.Kernel) or kernel == 'auto'): raise TypeError( "'kernel' must be either instance of class elephant.kernels.Kernel" " or the string 'auto'. Found: %s, value %s" % (type(kernel), str(kernel))) if not isinstance(cutoff, (float, int)): raise TypeError("'cutoff' must be float or integer") if not is_time_quantity(t_start, allow_none=True): raise TypeError("'t_start' must be a time Quantity") if not is_time_quantity(t_stop, allow_none=True): raise TypeError("'t_stop' must be a time Quantity") if not isinstance(trim, bool): raise TypeError("'trim' must be bool") check_neo_consistency(spiketrains, object_type=neo.SpikeTrain, t_start=t_start, t_stop=t_stop) if kernel == 'auto': if len(spiketrains) == 1: kernel = optimal_kernel(spiketrains) else: raise ValueError("Cannot estimate a kernel for a list of spike " "trains. Please provide a kernel explicitly " "rather than 'auto'.") if t_start is None: t_start = spiketrains.t_start if t_stop is None: t_stop = spiketrains.t_stop units = pq.CompoundUnit( "{}*s".format(sampling_period.rescale('s').item())) t_start = t_start.rescale(spiketrains.units) t_stop = t_stop.rescale(spiketrains.units) n_bins = int(((t_stop - t_start) / sampling_period).simplified) + 1 time_vectors = np.zeros((len(spiketrains), n_bins), dtype=np.float64) hist_range_end = t_stop + sampling_period.rescale(spiketrains.units) hist_range = (t_start.item(), hist_range_end.item()) for i, st in enumerate(spiketrains): time_vectors[i], _ = np.histogram(st.magnitude, bins=n_bins, range=hist_range) if cutoff < kernel.min_cutoff: cutoff = kernel.min_cutoff warnings.warn("The width of the kernel was adjusted to a minimally " "allowed width.") # An odd number of points correctly resolves the median index and the # fact that the peak of an instantaneous rate should be centered at t=0 # for symmetric kernels applied on a single spike at t=0. # See issue https://github.com/NeuralEnsemble/elephant/issues/360 n_half = math.ceil(cutoff * ( kernel.sigma / sampling_period).simplified.item()) cutoff_sigma = cutoff * kernel.sigma.rescale(units).magnitude if center_kernel: # t_arr must be centered at the kernel median. # Not centering on the kernel median leads to underestimating the # instantaneous rate in cases when sampling_period >> kernel.sigma. median = kernel.icdf(0.5).rescale(units).item() else: median = 0 t_arr = np.linspace(-cutoff_sigma + median, stop=cutoff_sigma + median, num=2 * n_half + 1, endpoint=True) * units if center_kernel: # keep the full convolve range and do the trimming afterwards; # trimming is performed according to the kernel median index fft_mode = 'full' elif trim: # no median index trimming is involved fft_mode = 'valid' else: # no median index trimming is involved fft_mode = 'same' time_vectors = time_vectors.T # make it (time, units) kernel_arr = np.expand_dims(kernel(t_arr).rescale(pq.Hz).magnitude, axis=1) rate = scipy.signal.fftconvolve(time_vectors, kernel_arr, mode=fft_mode) # the convolution of non-negative vectors is non-negative rate = np.clip(rate, a_min=0, a_max=None, out=rate) if center_kernel: # account for the kernel asymmetry median_id = kernel.median_index(t_arr) # the size of kernel() output matches the input size, len(t_arr) kernel_array_size = len(t_arr) if not trim: rate = rate[median_id: -kernel_array_size + median_id] else: rate = rate[2 * median_id: -2 * (kernel_array_size - median_id)] t_start = t_start + median_id * units t_stop = t_stop - (kernel_array_size - median_id) * units else: # FIXME: don't shrink the output array # (to be consistent with center_kernel=True) # n points have n-1 intervals; # instantaneous rate is a list of intervals; # hence, the last element is excluded rate = rate[:-1] kernel_annotation = dict(type=type(kernel).__name__, sigma=str(kernel.sigma), invert=kernel.invert) rate = neo.AnalogSignal(signal=rate, sampling_period=sampling_period, units=pq.Hz, t_start=t_start, t_stop=t_stop, kernel=kernel_annotation) return rate
[docs]@deprecated_alias(binsize='bin_size') def time_histogram(spiketrains, bin_size, t_start=None, t_stop=None, output='counts', binary=False): """ Time Histogram of a list of neo.SpikeTrain objects. Visualization of this function is covered in Viziphant: :func:viziphant.statistics.plot_time_histogram. Parameters ---------- spiketrains : list of neo.SpikeTrain neo.SpikeTrains with a common time axis (same t_start and t_stop) bin_size : pq.Quantity Width of the histogram's time bins. t_start : pq.Quantity, optional Start time of the histogram. Only events in spiketrains falling between t_start and t_stop (both included) are considered in the histogram. If None, the maximum t_start of all neo.SpikeTrains is used as t_start. Default: None t_stop : pq.Quantity, optional Stop time of the histogram. Only events in spiketrains falling between t_start and t_stop (both included) are considered in the histogram. If None, the minimum t_stop of all neo.SpikeTrains is used as t_stop. Default: None output : {'counts', 'mean', 'rate'}, optional Normalization of the histogram. Can be one of: * 'counts': spike counts at each bin (as integer numbers) * 'mean': mean spike counts per spike train * 'rate': mean spike rate per spike train. Like 'mean', but the counts are additionally normalized by the bin width. Default: 'counts' binary : bool, optional If True, indicates whether all neo.SpikeTrain objects should first be binned to a binary representation (using the conversion.BinnedSpikeTrain class) and the calculation of the histogram is based on this representation. Note that the output is not binary, but a histogram of the converted, binary representation. Default: False Returns ------- neo.AnalogSignal A neo.AnalogSignal object containing the histogram values. neo.AnalogSignal[j] is the histogram computed between t_start + j * bin_size and t_start + (j + 1) * bin_size. Raises ------ ValueError If output is not 'counts', 'mean' or 'rate'. Warns ----- UserWarning If t_start is None and the objects in spiketrains have different t_start values. If t_stop is None and the objects in spiketrains have different t_stop values. See also -------- elephant.conversion.BinnedSpikeTrain Examples -------- >>> import neo >>> import quantities as pq >>> from elephant import statistics >>> spiketrains = [ ... neo.SpikeTrain([0.3, 4.5, 6.7, 9.3], t_stop=10, units='s'), ... neo.SpikeTrain([0.7, 4.3, 8.2], t_stop=10, units='s') ... ] >>> hist = statistics.time_histogram(spiketrains, bin_size=1 * pq.s) >>> hist AnalogSignal with 1 channels of length 10; units dimensionless; datatype int64 annotations: {'normalization': 'counts'} sampling rate: 1.0 1/s time: 0.0 s to 10.0 s >>> hist.magnitude.flatten() array([2, 0, 0, 0, 2, 0, 1, 0, 1, 1]) """ # Bin the spike trains and sum across columns bs = BinnedSpikeTrain(spiketrains, t_start=t_start, t_stop=t_stop, bin_size=bin_size) if binary: bs = bs.binarize(copy=False) bin_hist = bs.get_num_of_spikes(axis=0) # Flatten array bin_hist = np.ravel(bin_hist) # Renormalise the histogram if output == 'counts': # Raw bin_hist = pq.Quantity(bin_hist, units=pq.dimensionless, copy=False) elif output == 'mean': # Divide by number of input spike trains bin_hist = pq.Quantity(bin_hist / len(spiketrains), units=pq.dimensionless, copy=False) elif output == 'rate': # Divide by number of input spike trains and bin width bin_hist = bin_hist / (len(spiketrains) * bin_size) else: raise ValueError(f'Parameter output ({output}) is not valid.') return neo.AnalogSignal(signal=np.expand_dims(bin_hist, axis=1), sampling_period=bin_size, units=bin_hist.units, t_start=bs.t_start, normalization=output, copy=False)
[docs]@deprecated_alias(binsize='bin_size') def complexity_pdf(spiketrains, bin_size): """ Complexity Distribution of a list of neo.SpikeTrain objects :cite:statistics-Gruen2007_96. Deprecated in favor of :meth:Complexity.pdf. Probability density computed from the complexity histogram which is the histogram of the entries of the population histogram of clipped (binary) spike trains computed with a bin width of bin_size. It provides for each complexity (== number of active neurons per bin) the number of occurrences. The normalization of that histogram to 1 is the probability density. Parameters ---------- spiketrains : list of neo.SpikeTrain Spike trains with a common time axis (same t_start and t_stop) bin_size : pq.Quantity Width of the histogram's time bins. Returns ------- complexity_distribution : neo.AnalogSignal A neo.AnalogSignal object containing the histogram values. neo.AnalogSignal[j] is the histogram computed between t_start + j * bin_size and t_start + (j + 1) * bin_size. See also -------- elephant.conversion.BinnedSpikeTrain """ warnings.warn("'complexity_pdf' is deprecated in favor of the Complexity " "class which has a 'pdf' method", DeprecationWarning) complexity = Complexity(spiketrains, bin_size=bin_size) return complexity.pdf()
[docs]class Complexity(object): """ Class for complexity distribution (i.e. number of synchronous spikes found) :cite:statistics-Gruen2007_96 of a list of neo.SpikeTrain objects. Complexity is calculated by counting the number of spikes (i.e. non-empty bins) that occur separated by spread - 1 or less empty bins, within and across spike trains in the spiketrains list. Implementation (without spread) is based on the cited above paper. Parameters ---------- spiketrains : list of neo.SpikeTrain Spike trains with a common time axis (same t_start and t_stop) sampling_rate : pq.Quantity or None, optional Sampling rate of the spike trains with units of 1/time. Used to shift the epoch edges in order to avoid rounding errors. If None using the epoch to slice spike trains may introduce rounding errors. Default: None bin_size : pq.Quantity or None, optional Width of the histogram's time bins with units of time. The user must specify the bin_size or the sampling_rate. * If None and the sampling_rate is available 1/sampling_rate is used. * If both are given then bin_size is used. Default: None binary : bool, optional * If True then the time histograms will be binary. * If False the total number of synchronous spikes is counted in the time histogram. Default: True spread : int, optional Number of bins in which to check for synchronous spikes. Spikes that occur separated by spread - 1 or less empty bins are considered synchronous. * spread = 0 corresponds to a bincount accross spike trains. * spread = 1 corresponds to counting consecutive spikes. * spread = 2 corresponds to counting consecutive spikes and spikes separated by exactly 1 empty bin. * spread = n corresponds to counting spikes separated by exactly or less than n - 1 empty bins. Default: 0 tolerance : float or None, optional Tolerance for rounding errors in the binning process and in the input data. If None possible binning errors are not accounted for. Default: 1e-8 Attributes ---------- epoch : neo.Epoch An epoch object containing complexity values, left edges and durations of all intervals with at least one spike. * epoch.array_annotations['complexity'] contains the complexity values per spike. * epoch.times contains the left edges. * epoch.durations contains the durations. time_histogram : neo.Analogsignal A neo.AnalogSignal object containing the histogram values. neo.AnalogSignal[j] is the histogram computed between t_start + j * binsize and t_start + (j + 1) * binsize. * If binary = True : Number of neurons that spiked in each bin, regardless of the number of spikes. * If binary = False : Number of neurons and spikes per neurons in each bin. complexity_histogram : np.ndarray The number of occurrences of events of different complexities. complexity_hist[i] corresponds to the number of events of complexity i for i > 0. Raises ------ ValueError When t_stop is smaller than t_start. When both sampling_rate and bin_size are not specified. When spread is not a positive integer. When spiketrains is an empty list. When t_start is not the same for all spiketrains When t_stop is not the same for all spiketrains TypeError When spiketrains is not a list. When the elements in spiketrains are not instances of neo.SpikeTrain Warns ----- UserWarning If no sampling rate is supplied which may lead to rounding errors when using the epoch to slice spike trains. Notes ----- * Note that with most common parameter combinations spike times can end up on bin edges. This makes the binning susceptible to rounding errors which is accounted for by moving spikes which are within tolerance of the next bin edge into the following bin. This can be adjusted using the tolerance parameter and turned off by setting tolerance=None. See also -------- elephant.conversion.BinnedSpikeTrain elephant.spike_train_synchrony.Synchrotool Examples -------- >>> import neo >>> import quantities as pq >>> from elephant.statistics import Complexity >>> sampling_rate = 1/pq.ms >>> st1 = neo.SpikeTrain([1, 4, 6] * pq.ms, t_stop=10.0 * pq.ms) >>> st2 = neo.SpikeTrain([1, 5, 8] * pq.ms, t_stop=10.0 * pq.ms) >>> sts = [st1, st2] >>> # spread = 0, a simple bincount >>> cpx = Complexity(sts, sampling_rate=sampling_rate) Complexity calculated at sampling rate precision >>> print(cpx.complexity_histogram) [5 4 1] >>> print(cpx.time_histogram.flatten()) [0 2 0 0 1 1 1 0 1 0] dimensionless >>> print(cpx.time_histogram.times) [0. 1. 2. 3. 4. 5. 6. 7. 8. 9.] ms >>> # spread = 1, consecutive spikes >>> cpx = Complexity(sts, sampling_rate=sampling_rate, spread=1) Complexity calculated at sampling rate precision >>> print(cpx.complexity_histogram) [5 4 1] >>> print(cpx.time_histogram.flatten()) [0 2 0 0 3 3 3 0 1 0] dimensionless >>> # spread = 2, consecutive spikes and separated by 1 empty bin >>> cpx = Complexity(sts, sampling_rate=sampling_rate, spread=2) Complexity calculated at sampling rate precision >>> print(cpx.complexity_histogram) [4 0 1 0 1] >>> print(cpx.time_histogram.flatten()) [0 2 0 0 4 4 4 4 4 0] dimensionless >>> pdf = cpx.pdf() >>> pdf AnalogSignal with 1 channels of length 3; units dimensionless; datatype float64 sampling rate: 1.0 dimensionless time: 0.0 dimensionless to 3.0 dimensionless >>> pdf.magnitude array([[0.5], [0.4], [0.1]]) """ def __init__(self, spiketrains, sampling_rate=None, bin_size=None, binary=True, spread=0, tolerance=1e-8): check_neo_consistency(spiketrains, object_type=neo.SpikeTrain) if bin_size is None and sampling_rate is None: raise ValueError('No bin_size or sampling_rate was specified!') if spread < 0: raise ValueError('Spread must be >=0') self.input_spiketrains = spiketrains self.t_start = spiketrains.t_start self.t_stop = spiketrains.t_stop self.sampling_rate = sampling_rate self.bin_size = bin_size self.binary = binary self.spread = spread self.tolerance = tolerance if bin_size is None and sampling_rate is not None: self.bin_size = 1 / self.sampling_rate if spread == 0: self.time_histogram, self.complexity_histogram = \ self._histogram_no_spread() self.epoch = self._epoch_no_spread() else: self.epoch = self._epoch_with_spread() self.time_histogram, self.complexity_histogram = \ self._histogram_with_spread()
[docs] def pdf(self): """ Probability density computed from the complexity histogram. Returns ------- pdf : neo.AnalogSignal A neo.AnalogSignal object containing the pdf values. neo.AnalogSignal[j] is the histogram computed between t_start + j * binsize and t_start + (j + 1) * binsize. """ norm_hist = self.complexity_histogram / self.complexity_histogram.sum() # Convert the Complexity pdf to an neo.AnalogSignal pdf = neo.AnalogSignal( np.expand_dims(norm_hist, axis=1), units=pq.dimensionless, t_start=0 * pq.dimensionless, sampling_period=1 * pq.dimensionless) return pdf
def _histogram_no_spread(self): """ Calculate the complexity histogram and time histogram for spread = 0 """ # Computing the population histogram with parameter binary=True to # clip the spike trains before summing time_hist = time_histogram(self.input_spiketrains, self.bin_size, binary=self.binary) # Computing the histogram of the entries of pophist complexity_hist = np.histogram( time_hist.magnitude, bins=range(0, len(self.input_spiketrains) + 2)) return time_hist, complexity_hist def _histogram_with_spread(self): """ Calculate the complexity histogram and time histogram for spread > 0 """ complexity_hist = np.bincount( self.epoch.array_annotations['complexity']) num_bins = (self.t_stop - self.t_start).rescale( self.bin_size.units).item() / self.bin_size.item() num_bins = round_binning_errors(num_bins, tolerance=self.tolerance) time_hist = np.zeros(num_bins, dtype=int) start_bins = (self.epoch.times - self.t_start).rescale( self.bin_size.units).magnitude / self.bin_size.item() stop_bins = (self.epoch.times + self.epoch.durations - self.t_start ).rescale(self.bin_size.units ).magnitude / self.bin_size.item() if self.sampling_rate is not None: shift = (.5 / self.sampling_rate / self.bin_size).simplified.item() # account for the first bin not being shifted in the epoch creation # if the shift would move it past t_start if self.epoch.times == self.t_start: start_bins[1:] += shift else: start_bins += shift stop_bins += shift start_bins = round_binning_errors(start_bins, tolerance=self.tolerance) stop_bins = round_binning_errors(stop_bins, tolerance=self.tolerance) for idx, (start, stop) in enumerate(zip(start_bins, stop_bins)): time_hist[start:stop] = \ self.epoch.array_annotations['complexity'][idx] time_hist = neo.AnalogSignal( signal=np.expand_dims(time_hist, axis=1), sampling_period=self.bin_size, units=pq.dimensionless, t_start=self.t_start) empty_bins = (self.t_stop - self.t_start - self.epoch.durations.sum()) empty_bins = empty_bins.rescale(self.bin_size.units ).magnitude / self.bin_size.item() empty_bins = round_binning_errors(empty_bins, tolerance=self.tolerance) complexity_hist = empty_bins return time_hist, complexity_hist def _epoch_no_spread(self): """ Get an epoch object of the complexity distribution with spread = 0 """ left_edges = self.time_histogram.times durations = self.bin_size * np.ones(self.time_histogram.shape) if self.sampling_rate: # ensure that spikes are not on the bin edges bin_shift = .5 / self.sampling_rate left_edges -= bin_shift # Ensure that an epoch does not start before the minimum t_start. # Note: all spike trains share the same t_start and t_stop. if left_edges < self.t_start: left_edges = self.t_start durations -= bin_shift else: warnings.warn('No sampling rate specified. ' 'Note that using the complexity epoch to get ' 'precise spike times can lead to rounding errors.') complexity = self.time_histogram.magnitude.flatten() complexity = complexity.astype(np.uint16) epoch = neo.Epoch(left_edges, durations=durations, array_annotations={'complexity': complexity}) return epoch def _epoch_with_spread(self): """ Get an epoch object of the complexity distribution with spread > 0 """ bst = conv.BinnedSpikeTrain(self.input_spiketrains, binsize=self.bin_size, tolerance=self.tolerance) if self.binary: bst = bst.binarize(copy=False) bincount = bst.get_num_of_spikes(axis=0) nonzero_indices = np.nonzero(bincount) left_diff = np.diff(nonzero_indices, prepend=-self.spread - 1) right_diff = np.diff(nonzero_indices, append=len(bincount) + self.spread + 1) # standalone bins (no merging required) single_bin_indices = np.logical_and(left_diff > self.spread, right_diff > self.spread) single_bins = nonzero_indices[single_bin_indices] # bins separated by fewer than spread bins form clusters # that have to be merged cluster_start_indices = np.logical_and(left_diff > self.spread, right_diff <= self.spread) cluster_starts = nonzero_indices[cluster_start_indices] cluster_stop_indices = np.logical_and(left_diff <= self.spread, right_diff > self.spread) cluster_stops = nonzero_indices[cluster_stop_indices] + 1 single_bin_complexities = bincount[single_bins] cluster_complexities = [bincount[start:stop].sum() for start, stop in zip(cluster_starts, cluster_stops)] # merge standalone bins and clusters and sort them combined_starts = np.concatenate((single_bins, cluster_starts)) combined_stops = np.concatenate((single_bins + 1, cluster_stops)) combined_complexities = np.concatenate((single_bin_complexities, cluster_complexities)) sorting = np.argsort(combined_starts, kind='mergesort') left_edges = bst.bin_edges[combined_starts[sorting]] right_edges = bst.bin_edges[combined_stops[sorting]] complexities = combined_complexities[sorting].astype(np.uint16) if self.sampling_rate: # ensure that spikes are not on the bin edges bin_shift = .5 / self.sampling_rate left_edges -= bin_shift right_edges -= bin_shift else: warnings.warn('No sampling rate specified. ' 'Note that using the complexity epoch to get ' 'precise spike times can lead to rounding errors.') # Ensure that an epoch does not start before the minimum t_start. # Note: all spike trains share the same t_start and t_stop. left_edges = max(self.t_start, left_edges) complexity_epoch = neo.Epoch(times=left_edges, durations=right_edges - left_edges, array_annotations={'complexity': complexities}) return complexity_epoch
def nextpow2(x): """ Return the smallest integral power of 2 that is equal or larger than x. """ log2_n = math.ceil(math.log2(x)) n = 2 ** log2_n return n def fftkernel(x, w): """ Applies the Gauss kernel smoother to an input signal using FFT algorithm. Parameters ---------- x : np.ndarray Vector with sample signal. w : float Kernel bandwidth (the standard deviation) in unit of the sampling resolution of x. Returns ------- y : np.ndarray The smoothed signal. Notes ----- 1. MAY 5/23, 2012 Author Hideaki Shimazaki RIKEN Brain Science Insitute http://2000.jukuin.keio.ac.jp/shimazaki 2. Ported to Python: Subhasis Ray, NCBS. Tue Jun 10 10:42:38 IST 2014 """ L = len(x) Lmax = L + 3 * w n = nextpow2(Lmax) X = np.fft.fft(x, n) f = np.arange(0, n, 1.0) / n f = np.concatenate((-f[:int(n / 2)], f[int(n / 2):0:-1])) K = np.exp(-0.5 * (w * 2 * np.pi * f) ** 2) y = np.fft.ifft(X * K, n) y = y[:L].copy() return y def logexp(x): if x < 1e2: y = np.log(1 + np.exp(x)) else: y = x return y def ilogexp(x): if x < 1e2: y = np.log(np.exp(x) - 1) else: y = x return y def cost_function(x, N, w, dt): """ Computes the cost function for sskernel. Cn(w) = sum_{i,j} int k(x - x_i) k(x - x_j) dx - 2 sum_{i~=j} k(x_i - x_j) """ yh = np.abs(fftkernel(x, w / dt)) # density # formula for density C = np.sum(yh ** 2) * dt - 2 * np.sum(yh * x) * \ dt + 2 / np.sqrt(2 * np.pi) / w / N C = C * N * N # formula for rate # C = dt*sum( yh.^2 - 2*yh.*y_hist + 2/sqrt(2*pi)/w*y_hist ) return C, yh
[docs]@deprecated_alias(tin='times', w='bandwidth') def optimal_kernel_bandwidth(spiketimes, times=None, bandwidth=None, bootstrap=False): """ Calculates optimal fixed kernel bandwidth :cite:statistics-Shimazaki2010_171, given as the standard deviation sigma. Original matlab code (sskernel.m) http://2000.jukuin.keio.ac.jp/shimazaki/res/kernel.html has been ported to Python by Subhasis Ray, NCBS. Parameters ---------- spiketimes : np.ndarray Sequence of spike times (sorted to be ascending). times : np.ndarray or None, optional Time points at which the kernel bandwidth is to be estimated. If None, spiketimes is used. Default: None bandwidth : np.ndarray or None, optional Vector of kernel bandwidths (standard deviation sigma). If specified, optimal bandwidth is selected from this. If None, bandwidth is obtained through a golden-section search on a log-exp scale. Default: None bootstrap : bool, optional If True, calculates the 95% confidence interval using Bootstrap. Default: False Returns ------- dict 'y' : np.ndarray Estimated density. 't' : np.ndarray Points at which estimation was computed. 'optw' : float Optimal kernel bandwidth given as standard deviation sigma 'w' : np.ndarray Kernel bandwidths examined (standard deviation sigma). 'C' : np.ndarray Cost functions of bandwidth. 'confb95' : tuple of np.ndarray Bootstrap 95% confidence interval: (lower level, upper level). If bootstrap is False, confb95 is None. 'yb' : np.ndarray Bootstrap samples. If bootstrap is False, yb is None. If no optimal kernel could be found, all entries of the dictionary are set to None. """ if times is None: time = np.max(spiketimes) - np.min(spiketimes) isi = np.diff(spiketimes) isi = isi[isi > 0].copy() dt = np.min(isi) times = np.linspace(np.min(spiketimes), np.max(spiketimes), min(int(time / dt + 0.5), 1000)) # The 1000 seems somewhat arbitrary t = times else: time = np.max(times) - np.min(times) spiketimes = spiketimes[(spiketimes >= np.min(times)) & (spiketimes <= np.max(times))].copy() isi = np.diff(spiketimes) isi = isi[isi > 0].copy() dt = np.min(isi) if dt > np.min(np.diff(times)): t = np.linspace(np.min(times), np.max(times), min(int(time / dt + 0.5), 1000)) else: t = times dt = np.min(np.diff(times)) yhist, bins = np.histogram(spiketimes, np.r_[t - dt / 2, t[-1] + dt / 2]) N = np.sum(yhist) yhist = yhist / (N * dt) # density optw = None y = None if bandwidth is not None: C = np.zeros(len(bandwidth)) Cmin = np.inf for k, w_ in enumerate(bandwidth): C[k], yh = cost_function(yhist, N, w_, dt) if C[k] < Cmin: Cmin = C[k] optw = w_ y = yh else: # Golden section search on a log-exp scale wmin = 2 * dt wmax = max(spiketimes) - min(spiketimes) imax = 20 # max iterations bandwidth = np.zeros(imax) C = np.zeros(imax) tolerance = 1e-5 phi = 0.5 * (np.sqrt(5) + 1) # The Golden ratio a = ilogexp(wmin) b = ilogexp(wmax) c1 = (phi - 1) * a + (2 - phi) * b c2 = (2 - phi) * a + (phi - 1) * b f1, y1 = cost_function(yhist, N, logexp(c1), dt) f2, y2 = cost_function(yhist, N, logexp(c2), dt) k = 0 while (np.abs(b - a) > (tolerance * (np.abs(c1) + np.abs(c2)))) \ and (k < imax): if f1 < f2: b = c2 c2 = c1 c1 = (phi - 1) * a + (2 - phi) * b f2 = f1 f1, y1 = cost_function(yhist, N, logexp(c1), dt) bandwidth[k] = logexp(c1) C[k] = f1 optw = logexp(c1) y = y1 / (np.sum(y1 * dt)) else: a = c1 c1 = c2 c2 = (2 - phi) * a + (phi - 1) * b f1 = f2 f2, y2 = cost_function(yhist, N, logexp(c2), dt) bandwidth[k] = logexp(c2) C[k] = f2 optw = logexp(c2) y = y2 / np.sum(y2 * dt) k = k + 1 # Bootstrap confidence intervals confb95 = None yb = None # If bootstrap is requested, and an optimal kernel was found if bootstrap and optw: nbs = 1000 yb = np.zeros((nbs, len(times))) for ii in range(nbs): idx = np.floor(np.random.rand(N) * N).astype(int) xb = spiketimes[idx] y_histb, bins = np.histogram( xb, np.r_[t - dt / 2, t[-1] + dt / 2]) / dt / N yb_buf = fftkernel(y_histb, optw / dt).real yb_buf = yb_buf / np.sum(yb_buf * dt) yb[ii, :] = np.interp(times, t, yb_buf) ybsort = np.sort(yb, axis=0) y95b = ybsort[np.floor(0.05 * nbs).astype(int), :] y95u = ybsort[np.floor(0.95 * nbs).astype(int), :] confb95 = (y95b, y95u) # Only perform interpolation if y could be calculated if y is not None: y = np.interp(times, t, y) return {'y': y, 't': times, 'optw': optw, 'w': bandwidth, 'C': C, 'confb95': confb95, 'yb': yb}
def sskernel(*args, **kwargs): warnings.warn("'sskernel' function is deprecated; " "use 'optimal_kernel_bandwidth'", DeprecationWarning) return optimal_kernel_bandwidth(*args, **kwargs)