# -*- coding: utf-8 -*-
"""
The Change point detection algorithm :cite:`cpd-Messer2014_2027` determines if
a spike train `spiketrain` can be considered as a stationary process (constant
firing rate) or not as stationary process (i.e. presence of one or more points
at which the rate increases or decreases). In case of non-stationarity, the
output is a list of detected Change Points (CPs).
Essentially, a set of two-sided windows of width `h`
(`_filter(t, h, spiketrain)`) slides over the spike train within the time
`[h, t_final-h]`. This generates a `_filter_process(time_step, h, spiketrain)`
that assigns at each time `t` the difference between a spike lying in the right
and left windows. If at any time `t` this difference is large 'enough', the
presence of a rate Change Point in a neighborhood of `t` is assumed. A
threshold `test_quantile` for the maximum of the filter_process (max
difference of spike count between the left and right windows) is derived based
on asymptotic considerations. The procedure is repeated for an arbitrary set of
windows with different sizes `h`.
.. autosummary::
:toctree: _toctree/change_point_detection
multiple_filter_test
empirical_parameters
Examples
--------
>>> import quantities as pq
>>> from elephant.change_point_detection import multiple_filter_test
>>> spike_times = [1.1, 1.2, 1.4, 1.6, 1.7, 1.75, 1.8, 1.9, 1.95] * pq.s
>>> change_points = multiple_filter_test(window_sizes=[0.5] * pq.s,
... spiketrain=spike_times, t_final=2.1 * pq.s, alpha=5, n_surrogates=100,
... time_step=0.1 * pq.s) # doctest: +SKIP
[[array(1.5) * s]]
Original code
-------------
Adapted from the published R implementation: :cite:`cpd-Messer2014_2027`
"""
from __future__ import division, print_function, unicode_literals
import numpy as np
import quantities as pq
__all__ = [
"multiple_filter_test",
"empirical_parameters"
]
[docs]
def multiple_filter_test(window_sizes, spiketrain, t_final, alpha,
n_surrogates=1000, test_quantile=None,
test_param=None, time_step=None):
"""
Detects change points.
This function returns the detected change points that corresponds to the
maxima of the *filter processes* - the processes generated by sliding
windows of step `time_step`; at each step the difference between spikes on
the right and left windows is calculated.
Parameters
----------
window_sizes : list of pq.Quantity
list that contains windows sizes
spiketrain : neo.SpikeTrain or pq.Quantity
A spiketrain object to analyze.
t_final : pq.Quantity
The final time of the spike train which is to be analysed
alpha : float
Alpha-quantile in range [0, 100] for the set of maxima of the limit
processes
n_surrogates : int, optional
The number of simulated limit processes.
Default: 1000
test_quantile : float or None, optional
The threshold for the maxima of the filter derivative processes; if any
of these maxima is larger than this value, it is assumed the
presence of a change point (cp) at the time corresponding to that
maximum.
If None, will be set according to the :func:`empirical_parameters`.
Default: None
test_param : (3, num. of windows) np.ndarray or None, optional
first row: list of `h`, second and third rows: empirical means and
variances of the limit process corresponding to `h`. This will be
used to normalize the *filter processes* in order to give to the
every maximum the same impact on the global statistic.
If None, will be set according to the :func:`empirical_parameters`.
Default: None
time_step : pq.Quantity or None, optional
The resolution - the time step at which the windows are slided.
If None, will be set to ``window_size / 20``.
Default: None
Returns
-------
cps : list of list
The change points,
one list for each window size `h`, containing the points detected with
the corresponding `filter_process`. N.B.: only cps whose h-neighborhood
does not include previously detected cps (with smaller window h) are
added to the list.
"""
if test_quantile is None and test_param is None:
test_quantile, test_param = empirical_parameters(window_sizes, t_final,
alpha, n_surrogates,
time_step)
elif test_quantile is None:
test_quantile = empirical_parameters(window_sizes, t_final, alpha,
n_surrogates, time_step)[0]
elif test_param is None:
test_param = empirical_parameters(window_sizes, t_final, alpha,
n_surrogates, time_step)[1]
# List of lists of detected change points (CPs), to be returned
cps = []
for i, h in enumerate(window_sizes):
# automatic setting of time_step
dt_temp = h / 20 if time_step is None else time_step
# filter_process for window of size h
t, differences = _filter_process(dt_temp, h, spiketrain, t_final,
test_param)
time_index = np.arange(len(differences))
# Point detected with window h
cps_window = []
while np.max(differences) > test_quantile:
cp_index = np.argmax(differences)
# from index to time
cp = cp_index * dt_temp + h
# before repeating the procedure, the h-neighbourgs of detected CP
# are discarded, because rate changes into it are alrady explained
mask_fore = time_index > cp_index - int((h / dt_temp).simplified)
mask_back = time_index < cp_index + int((h / dt_temp).simplified)
differences[mask_fore & mask_back] = 0
# check if the neighbourhood of detected cp does not contain cps
# detected with other windows
neighbourhood_free = True
# iterate on lists of cps detected with smaller window
for j in range(i):
# iterate on CPs detected with the j-th smallest window
for c_pre in cps[j]:
if c_pre - h < cp < c_pre + h:
neighbourhood_free = False
break
# if none of the previously detected CPs falls in the h-
# neighbourhood
if neighbourhood_free:
# add the current CP to the list
cps_window.append(cp)
# add the present list to the grand list
cps.append(cps_window)
return cps
def _brownian_motion(t_in, t_fin, x_in, time_step):
"""
Generate a Brownian Motion.
Parameters
----------
t_in : quantities,
initial time
t_fin : quantities,
final time
x_in : float,
initial point of the process: _brownian_motio(0) = x_in
time_step : quantities,
resolution, time step at which brownian increments are summed
Returns
-------
Brownian motion on [t_in, t_fin], with resolution time_step and initial
state x_in
"""
u = 1 * pq.s
try:
t_in_sec = t_in.rescale(u).magnitude
except ValueError:
raise ValueError("t_in must be a time quantity")
try:
t_fin_sec = t_fin.rescale(u).magnitude
except ValueError:
raise ValueError("t_fin must be a time quantity")
try:
dt_sec = time_step.rescale(u).magnitude
except ValueError:
raise ValueError("dt must be a time quantity")
x = np.random.normal(0, np.sqrt(dt_sec),
size=int((t_fin_sec - t_in_sec) / dt_sec))
s = np.cumsum(x)
return s + x_in
def _limit_processes(window_sizes, t_final, time_step):
"""
Generate the limit processes (depending only on t_final and h), one for
each window size `h` in H. The distribution of maxima of these processes
is used to derive threshold `test_quantile` and parameters `test_param`.
Parameters
----------
window_sizes : list of quantities
set of windows' size
t_final : quantity object
end of limit process
time_step : quantity object
resolution, time step at which the windows are slided
Returns
-------
limit_processes : list of numpy array
each entries contains the limit processes for each h,
evaluated in [h,T-h] with steps time_step
"""
limit_processes = []
u = 1 * pq.s
try:
window_sizes_sec = window_sizes.rescale(u).magnitude
except ValueError:
raise ValueError("window_sizes must be a list of times")
try:
dt_sec = time_step.rescale(u).magnitude
except ValueError:
raise ValueError("time_step must be a time quantity")
w = _brownian_motion(0 * u, t_final, 0, time_step)
for h in window_sizes_sec:
# BM on [h,T-h], shifted in time t-->t+h
brownian_right = w[int(2 * h / dt_sec):]
# BM on [h,T-h], shifted in time t-->t-h
brownian_left = w[:int(-2 * h / dt_sec)]
# BM on [h,T-h]
brownian_center = w[int(h / dt_sec):int(-h / dt_sec)]
modul = np.abs(brownian_right + brownian_left - 2 * brownian_center)
limit_process_h = modul / (np.sqrt(2 * h))
limit_processes.append(limit_process_h)
return limit_processes
[docs]
def empirical_parameters(window_sizes, t_final, alpha, n_surrogates=1000,
time_step=None):
r"""
This function generates the threshold and the null parameters.
The filter processes (`h`) have been proved to converge (for `t_final`,
:math:`h \to \infty`) to a continuous functional of a Brownian motion
('limit_process'). Using a MonteCarlo technique, maxima of
these limit_processes are collected.
The threshold is defined as the alpha quantile of this set of maxima.
Namely:
test_quantile := alpha quantile of
:math:`{\max_{h \in \text{window\_sizes}} \max_{t \in [h, t_{final}-h]}
\text{limit\_process}_h(t)}`
Parameters
----------
window_sizes : list of pq.Quantity
list that contains windows sizes
t_final : pq.Quantity
The final time of the spike train which is to be analysed
alpha : float
Alpha-quantile in range [0, 100] for the set of maxima of the limit
processes
n_surrogates : int, optional
The number of simulated limit processes.
Default: 1000
time_step : pq.Quantity or None, optional
The resolution - the time step at which the windows are slided.
If None, will be set to ``window_size / 20``.
Default: None
Returns
-------
test_quantile : float
The threshold for the maxima of the filter derivative processes; if any
of these maxima is larger than this value, it is assumed the
presence of a change point (cp) at the time corresponding to that
maximum.
test_param : (3, num. of windows) np.ndarray
first row: list of `h`, second and third rows: empirical means and
variances of the limit process corresponding to `h`. This will be
used to normalize the *filter processes* in order to give to the
every maximum the same impact on the global statistic.
Examples
--------
>>> import quantities as pq
>>> from elephant.change_point_detection import empirical_parameters
>>> test_quantile, test_param = empirical_parameters(
... window_sizes=[0.5] * pq.s, t_final=2.1 * pq.s, alpha=5,
... n_surrogates=100, time_step=0.1 * pq.s)
>>> test_quantile # doctest: +SKIP
1.8133759165692873
>>> test_param # doctest: +SKIP
array([[0.5 ],
[1.74482974],
[0.24290945]])
"""
# try:
# window_sizes_sec = window_sizes.rescale(u)
# except ValueError:
# raise ValueError("H must be a list of times")
# window_sizes_mag = window_sizes_sec.magnitude
# try:
# t_final_sec = t_final.rescale(u)
# except ValueError:
# raise ValueError("T must be a time quantity")
# t_final_mag = t_final_sec.magnitude
if not isinstance(window_sizes, pq.Quantity):
raise ValueError("window_sizes must be a list of time quantities")
if not isinstance(t_final, pq.Quantity):
raise ValueError("t_final must be a time quantity")
if not isinstance(n_surrogates, int):
raise TypeError("n_surrogates must be an integer")
if not (isinstance(time_step, pq.Quantity) or (time_step is None)):
raise ValueError("time_step must be a time quantity")
if t_final <= 0:
raise ValueError("t_final needs to be strictly positive")
if alpha * (100.0 - alpha) < 0:
raise ValueError("alpha needs to be in (0,100)")
if np.min(window_sizes) <= 0:
raise ValueError("window size needs to be strictly positive")
if np.max(window_sizes) >= t_final / 2:
raise ValueError("window size too large")
if time_step is not None:
for h in window_sizes:
if int(h.rescale('us')) % int(time_step.rescale('us')) != 0:
raise ValueError(
"Every window size h must be a multiple of time_step")
# Generate a matrix M*: n X m where n = n_surrogates is the number of
# simulated limit processes and m is the number of chosen window sizes.
# Elements are: M*(i,h) = max(t in T)[`limit_process_h`(t)],
# for each h in H and surrogate i
maxima_matrix = []
for i in range(n_surrogates):
# mh_star = []
simu = _limit_processes(window_sizes, t_final, time_step)
# for i, h in enumerate(window_sizes_mag):
# # max over time of the limit process generated with window h
# m_h = np.max(simu[i])
# mh_star.append(m_h)
# max over time of the limit process generated with window h
mh_star = [np.max(x) for x in simu]
maxima_matrix.append(mh_star)
maxima_matrix = np.asanyarray(maxima_matrix)
# these parameters will be used to normalize both the limit_processes (H0)
# and the filter_processes
null_mean = maxima_matrix.mean(axis=0)
null_var = maxima_matrix.var(axis=0)
# matrix normalization by mean and variance of the limit process, in order
# to give, for every h, the same impact on the global maximum
matrix_normalized = (maxima_matrix - null_mean) / np.sqrt(null_var)
great_maxs = np.max(matrix_normalized, axis=1)
test_quantile = np.percentile(great_maxs, 100.0 - alpha)
null_parameters = [window_sizes, null_mean, null_var]
test_param = np.asanyarray(null_parameters)
return test_quantile, test_param
def _filter(t_center, window, spiketrain):
"""
This function calculates the difference of spike counts in the left and
right side of a window of size h centered in t and normalized by its
variance. The variance of this count can be expressed as a combination of
mean and var of the I.S.I. lying inside the window.
Parameters
----------
t_center : quantity
time on which the window is centered
window : quantity
window's size
spiketrain : list, numpy array or SpikeTrain
spike train to analyze
Returns
-------
difference : float,
difference of spike count normalized by its variance
"""
u = 1 * pq.s
try:
t_sec = t_center.rescale(u).magnitude
except AttributeError:
raise ValueError("t must be a quantities object")
# tm = t_sec.magnitude
try:
h_sec = window.rescale(u).magnitude
except AttributeError:
raise ValueError("h must be a time quantity")
# hm = h_sec.magnitude
try:
spk_sec = spiketrain.rescale(u).magnitude
except AttributeError:
raise ValueError(
"spiketrain must be a list (array) of times or a neo spiketrain")
# cut spike-train on the right
train_right = spk_sec[(t_sec < spk_sec) & (spk_sec < t_sec + h_sec)]
# cut spike-train on the left
train_left = spk_sec[(t_sec - h_sec < spk_sec) & (spk_sec < t_sec)]
# spike count in the right side
count_right = train_right.size
# spike count in the left side
count_left = train_left.size
# form spikes to I.S.I
isi_right = np.diff(train_right)
isi_left = np.diff(train_left)
if isi_right.size == 0:
mu_ri = 0
sigma_ri = 0
else:
# mean of I.S.I inside the window
mu_ri = np.mean(isi_right)
# var of I.S.I inside the window
sigma_ri = np.var(isi_right)
if isi_left.size == 0:
mu_le = 0
sigma_le = 0
else:
mu_le = np.mean(isi_left)
sigma_le = np.var(isi_left)
if (sigma_le > 0) & (sigma_ri > 0):
s_quad = (sigma_ri / mu_ri**3) * h_sec + (sigma_le / mu_le**3) * h_sec
else:
s_quad = 0
if s_quad == 0:
difference = 0
else:
difference = (count_right - count_left) / np.sqrt(s_quad)
return difference
def _filter_process(time_step, h, spk, t_final, test_param):
"""
Given a spike train `spk` and a window size `h`, this function generates
the `filter derivative process` by evaluating the function `_filter`
in steps of `time_step`.
Parameters
----------
h : quantity object
window's size
t_final : quantity,
time on which the window is centered
spk : list, array or SpikeTrain
spike train to analyze
time_step : quantity object, time step at which the windows are slided
resolution
test_param : matrix, the means of the first row list of `h`,
the second row Empirical and the third row variances of
the limit processes `Lh` are used to normalize the number
of elements inside the windows
Returns
-------
time_domain : numpy array
time domain of the `filter derivative process`
filter_process : array,
values of the `filter derivative process`
"""
u = 1 * pq.s
try:
h_sec = h.rescale(u).magnitude
except AttributeError:
raise ValueError("h must be a time quantity")
try:
t_final_sec = t_final.rescale(u).magnitude
except AttributeError:
raise ValueError("t_final must be a time quanity")
try:
dt_sec = time_step.rescale(u).magnitude
except AttributeError:
raise ValueError("time_step must be a time quantity")
# domain of the process
time_domain = np.arange(h_sec, t_final_sec - h_sec, dt_sec)
filter_trajectrory = []
# taken from the function used to generate the threshold
emp_mean_h = test_param[1][test_param[0] == h]
emp_var_h = test_param[2][test_param[0] == h]
for t in time_domain:
filter_trajectrory.append(_filter(t * u, h, spk))
filter_trajectrory = np.asanyarray(filter_trajectrory)
# ordered normalization to give each process the same impact on the max
filter_process = (
np.abs(filter_trajectrory) - emp_mean_h) / np.sqrt(emp_var_h)
return time_domain, filter_process
