# Source code for matplotlib.mlab

"""
Numerical python functions written for compatibility with MATLAB
commands with the same names. Most numerical python functions can be found in
the numpy and scipy libraries. What remains here is code for performing
spectral computations.

Spectral functions
-------------------

cohere
Coherence (normalized cross spectral density)

csd
Cross spectral density using Welch's average periodogram

detrend
Remove the mean or best fit line from an array

psd
Power spectral density using Welch's average periodogram

specgram
Spectrogram (spectrum over segments of time)

complex_spectrum
Return the complex-valued frequency spectrum of a signal

magnitude_spectrum
Return the magnitude of the frequency spectrum of a signal

angle_spectrum
Return the angle (wrapped phase) of the frequency spectrum of a signal

phase_spectrum
Return the phase (unwrapped angle) of the frequency spectrum of a signal

detrend_mean
Remove the mean from a line.

detrend_linear
Remove the best fit line from a line.

detrend_none
Return the original line.

stride_windows
Get all windows in an array in a memory-efficient manner

stride_repeat
Repeat an array in a memory-efficient manner

apply_window
Apply a window along a given axis
"""

import functools
from numbers import Number

import numpy as np

import matplotlib.cbook as cbook
from matplotlib import docstring

[docs]def window_hanning(x):
"""
Return x times the hanning window of len(x).

--------
window_none : Another window algorithm.
"""
return np.hanning(len(x))*x

[docs]def window_none(x):
"""
No window function; simply return x.

--------
window_hanning : Another window algorithm.
"""
return x

[docs]@cbook.deprecated("3.2")
def apply_window(x, window, axis=0, return_window=None):
"""
Apply the given window to the given 1D or 2D array along the given axis.

Parameters
----------
x : 1D or 2D array or sequence
Array or sequence containing the data.

window : function or array.
Either a function to generate a window or an array with length
*x*.shape[*axis*]

axis : int
The axis over which to do the repetition.
Must be 0 or 1.  The default is 0

return_window : bool
If true, also return the 1D values of the window that was applied
"""
x = np.asarray(x)

if x.ndim < 1 or x.ndim > 2:
raise ValueError('only 1D or 2D arrays can be used')
if axis+1 > x.ndim:
raise ValueError('axis(=%s) out of bounds' % axis)

xshape = list(x.shape)
xshapetarg = xshape.pop(axis)

if np.iterable(window):
if len(window) != xshapetarg:
raise ValueError('The len(window) must be the same as the shape '
'of x for the chosen axis')
windowVals = window
else:
windowVals = window(np.ones(xshapetarg, dtype=x.dtype))

if x.ndim == 1:
if return_window:
return windowVals * x, windowVals
else:
return windowVals * x

xshapeother = xshape.pop()

otheraxis = (axis+1) % 2

windowValsRep = stride_repeat(windowVals, xshapeother, axis=otheraxis)

if return_window:
return windowValsRep * x, windowVals
else:
return windowValsRep * x

[docs]def detrend(x, key=None, axis=None):
"""
Return x with its trend removed.

Parameters
----------
x : array or sequence
Array or sequence containing the data.

key : {'default', 'constant', 'mean', 'linear', 'none'} or function
The detrending algorithm to use. 'default', 'mean', and 'constant' are
the same as detrend_mean. 'linear' is the same as detrend_linear.
'none' is the same as detrend_none. The default is 'mean'. See the
corresponding functions for more details regarding the algorithms. Can
also be a function that carries out the detrend operation.

axis : int
The axis along which to do the detrending.

--------
detrend_mean : Implementation of the 'mean' algorithm.
detrend_linear : Implementation of the 'linear' algorithm.
detrend_none : Implementation of the 'none' algorithm.
"""
if key is None or key in ['constant', 'mean', 'default']:
return detrend(x, key=detrend_mean, axis=axis)
elif key == 'linear':
return detrend(x, key=detrend_linear, axis=axis)
elif key == 'none':
return detrend(x, key=detrend_none, axis=axis)
elif callable(key):
x = np.asarray(x)
if axis is not None and axis + 1 > x.ndim:
raise ValueError(f'axis(={axis}) out of bounds')
if (axis is None and x.ndim == 0) or (not axis and x.ndim == 1):
return key(x)
# try to use the 'axis' argument if the function supports it,
# otherwise use apply_along_axis to do it
try:
return key(x, axis=axis)
except TypeError:
return np.apply_along_axis(key, axis=axis, arr=x)
else:
raise ValueError(
f"Unknown value for key: {key!r}, must be one of: 'default', "
f"'constant', 'mean', 'linear', or a function")

[docs]def detrend_mean(x, axis=None):
"""
Return x minus the mean(x).

Parameters
----------
x : array or sequence
Array or sequence containing the data
Can have any dimensionality

axis : int
The axis along which to take the mean.  See numpy.mean for a
description of this argument.

--------
detrend_linear : Another detrend algorithm.
detrend_none : Another detrend algorithm.
detrend : A wrapper around all the detrend algorithms.
"""
x = np.asarray(x)

if axis is not None and axis+1 > x.ndim:
raise ValueError('axis(=%s) out of bounds' % axis)

return x - x.mean(axis, keepdims=True)

[docs]def detrend_none(x, axis=None):
"""
Return x: no detrending.

Parameters
----------
x : any object
An object containing the data

axis : int
This parameter is ignored.
It is included for compatibility with detrend_mean

--------
detrend_mean : Another detrend algorithm.
detrend_linear : Another detrend algorithm.
detrend : A wrapper around all the detrend algorithms.
"""
return x

[docs]def detrend_linear(y):
"""
Return x minus best fit line; 'linear' detrending.

Parameters
----------
y : 0-D or 1-D array or sequence
Array or sequence containing the data

axis : int
The axis along which to take the mean.  See numpy.mean for a
description of this argument.

--------
detrend_mean : Another detrend algorithm.
detrend_none : Another detrend algorithm.
detrend : A wrapper around all the detrend algorithms.
"""
# This is faster than an algorithm based on linalg.lstsq.
y = np.asarray(y)

if y.ndim > 1:
raise ValueError('y cannot have ndim > 1')

# short-circuit 0-D array.
if not y.ndim:
return np.array(0., dtype=y.dtype)

x = np.arange(y.size, dtype=float)

C = np.cov(x, y, bias=1)
b = C[0, 1]/C[0, 0]

a = y.mean() - b*x.mean()
return y - (b*x + a)

[docs]def stride_windows(x, n, noverlap=None, axis=0):
"""
Get all windows of x with length n as a single array,
using strides to avoid data duplication.

.. warning::

It is not safe to write to the output array.  Multiple
elements may point to the same piece of memory,
so modifying one value may change others.

Parameters
----------
x : 1D array or sequence
Array or sequence containing the data.

n : int
The number of data points in each window.

noverlap : int
The overlap between adjacent windows.
Default is 0 (no overlap)

axis : int
The axis along which the windows will run.

References
----------
stackoverflow: Rolling window for 1D arrays in Numpy?
<http://stackoverflow.com/a/6811241>_
stackoverflow: Using strides for an efficient moving average filter
<http://stackoverflow.com/a/4947453>_
"""
if noverlap is None:
noverlap = 0

if noverlap >= n:
raise ValueError('noverlap must be less than n')
if n < 1:
raise ValueError('n cannot be less than 1')

x = np.asarray(x)

if x.ndim != 1:
raise ValueError('only 1-dimensional arrays can be used')
if n == 1 and noverlap == 0:
if axis == 0:
return x[np.newaxis]
else:
return x[np.newaxis].transpose()
if n > x.size:
raise ValueError('n cannot be greater than the length of x')

# np.lib.stride_tricks.as_strided easily leads to memory corruption for
# non integer shape and strides, i.e. noverlap or n. See #3845.
noverlap = int(noverlap)
n = int(n)

step = n - noverlap
if axis == 0:
shape = (n, (x.shape[-1]-noverlap)//step)
strides = (x.strides, step*x.strides)
else:
shape = ((x.shape[-1]-noverlap)//step, n)
strides = (step*x.strides, x.strides)
return np.lib.stride_tricks.as_strided(x, shape=shape, strides=strides)

[docs]@cbook.deprecated("3.2")
def stride_repeat(x, n, axis=0):
"""
Repeat the values in an array in a memory-efficient manner.  Array x is
stacked vertically n times.

.. warning::

It is not safe to write to the output array.  Multiple
elements may point to the same piece of memory, so
modifying one value may change others.

Parameters
----------
x : 1D array or sequence
Array or sequence containing the data.

n : int
The number of time to repeat the array.

axis : int
The axis along which the data will run.

References
----------
stackoverflow: Repeat NumPy array without replicating data?
<http://stackoverflow.com/a/5568169>_
"""
if axis not in [0, 1]:
raise ValueError('axis must be 0 or 1')
x = np.asarray(x)
if x.ndim != 1:
raise ValueError('only 1-dimensional arrays can be used')

if n == 1:
if axis == 0:
return np.atleast_2d(x)
else:
return np.atleast_2d(x).T
if n < 1:
raise ValueError('n cannot be less than 1')

# np.lib.stride_tricks.as_strided easily leads to memory corruption for
# non integer shape and strides, i.e. n. See #3845.
n = int(n)

if axis == 0:
shape = (n, x.size)
strides = (0, x.strides)
else:
shape = (x.size, n)
strides = (x.strides, 0)

return np.lib.stride_tricks.as_strided(x, shape=shape, strides=strides)

def _spectral_helper(x, y=None, NFFT=None, Fs=None, detrend_func=None,
sides=None, scale_by_freq=None, mode=None):
"""
Private helper implementing the common parts between the psd, csd,
spectrogram and complex, magnitude, angle, and phase spectrums.
"""
if y is None:
# if y is None use x for y
same_data = True
else:
# The checks for if y is x are so that we can use the same function to
# implement the core of psd(), csd(), and spectrogram() without doing
# extra calculations.  We return the unaveraged Pxy, freqs, and t.
same_data = y is x

if Fs is None:
Fs = 2
if noverlap is None:
noverlap = 0
if detrend_func is None:
detrend_func = detrend_none
if window is None:
window = window_hanning

# if NFFT is set to None use the whole signal
if NFFT is None:
NFFT = 256

if mode is None or mode == 'default':
mode = 'psd'
cbook._check_in_list(
['default', 'psd', 'complex', 'magnitude', 'angle', 'phase'],
mode=mode)

if not same_data and mode != 'psd':
raise ValueError("x and y must be equal if mode is not 'psd'")

# Make sure we're dealing with a numpy array. If y and x were the same
# object to start with, keep them that way
x = np.asarray(x)
if not same_data:
y = np.asarray(y)

if sides is None or sides == 'default':
if np.iscomplexobj(x):
sides = 'twosided'
else:
sides = 'onesided'
cbook._check_in_list(['default', 'onesided', 'twosided'], sides=sides)

# zero pad x and y up to NFFT if they are shorter than NFFT
if len(x) < NFFT:
n = len(x)
x = np.resize(x, NFFT)
x[n:] = 0

if not same_data and len(y) < NFFT:
n = len(y)
y = np.resize(y, NFFT)
y[n:] = 0

if pad_to is None:

if mode != 'psd':
scale_by_freq = False
elif scale_by_freq is None:
scale_by_freq = True

# For real x, ignore the negative frequencies unless told otherwise
if sides == 'twosided':
if pad_to % 2:
freqcenter = (pad_to - 1)//2 + 1
else:
scaling_factor = 1.
elif sides == 'onesided':
if pad_to % 2:
numFreqs = (pad_to + 1)//2
else:
numFreqs = pad_to//2 + 1
scaling_factor = 2.

if not np.iterable(window):
window = window(np.ones(NFFT, x.dtype))
if len(window) != NFFT:
raise ValueError(
"The window length must match the data's first dimension")

result = stride_windows(x, NFFT, noverlap, axis=0)
result = detrend(result, detrend_func, axis=0)
result = result * window.reshape((-1, 1))
result = np.fft.fft(result, n=pad_to, axis=0)[:numFreqs, :]
freqs = np.fft.fftfreq(pad_to, 1/Fs)[:numFreqs]

if not same_data:
# if same_data is False, mode must be 'psd'
resultY = stride_windows(y, NFFT, noverlap)
resultY = detrend(resultY, detrend_func, axis=0)
resultY = resultY * window.reshape((-1, 1))
resultY = np.fft.fft(resultY, n=pad_to, axis=0)[:numFreqs, :]
result = np.conj(result) * resultY
elif mode == 'psd':
result = np.conj(result) * result
elif mode == 'magnitude':
result = np.abs(result) / np.abs(window).sum()
elif mode == 'angle' or mode == 'phase':
# we unwrap the phase later to handle the onesided vs. twosided case
result = np.angle(result)
elif mode == 'complex':
result /= np.abs(window).sum()

if mode == 'psd':

# Also include scaling factors for one-sided densities and dividing by
# the sampling frequency, if desired. Scale everything, except the DC
# component and the NFFT/2 component:

# if we have a even number of frequencies, don't scale NFFT/2
if not NFFT % 2:
slc = slice(1, -1, None)
# if we have an odd number, just don't scale DC
else:
slc = slice(1, None, None)

result[slc] *= scaling_factor

# MATLAB divides by the sampling frequency so that density function
# has units of dB/Hz and can be integrated by the plotted frequency
# values. Perform the same scaling here.
if scale_by_freq:
result /= Fs
# Scale the spectrum by the norm of the window to compensate for
# windowing loss; see Bendat & Piersol Sec 11.5.2.
result /= (np.abs(window)**2).sum()
else:
# In this case, preserve power in the segment, not amplitude
result /= np.abs(window).sum()**2

t = np.arange(NFFT/2, len(x) - NFFT/2 + 1, NFFT - noverlap)/Fs

if sides == 'twosided':
# center the frequency range at zero
freqs = np.roll(freqs, -freqcenter, axis=0)
result = np.roll(result, -freqcenter, axis=0)
elif not pad_to % 2:
# get the last value correctly, it is negative otherwise
freqs[-1] *= -1

# we unwrap the phase here to handle the onesided vs. twosided case
if mode == 'phase':
result = np.unwrap(result, axis=0)

return result, freqs, t

def _single_spectrum_helper(
mode, x, Fs=None, window=None, pad_to=None, sides=None):
"""
Private helper implementing the commonality between the complex, magnitude,
angle, and phase spectrums.
"""
cbook._check_in_list(['complex', 'magnitude', 'angle', 'phase'], mode=mode)

if pad_to is None:

spec, freqs, _ = _spectral_helper(x=x, y=None, NFFT=len(x), Fs=Fs,
detrend_func=detrend_none, window=window,
sides=sides,
scale_by_freq=False,
mode=mode)
if mode != 'complex':
spec = spec.real

if spec.ndim == 2 and spec.shape == 1:
spec = spec[:, 0]

return spec, freqs

# Split out these keyword docs so that they can be used elsewhere
docstring.interpd.update(
Spectral="""\
Fs : float, default: 2
The sampling frequency (samples per time unit).  It is used to calculate
the Fourier frequencies, *freqs*, in cycles per time unit.

window : callable or ndarray, default: .window_hanning
A function or a vector of length *NFFT*.  To create window vectors see
.window_hanning, .window_none, numpy.blackman, numpy.hamming,
numpy.bartlett, scipy.signal, scipy.signal.get_window, etc.  If a
function is passed as the argument, it must take a data segment as an
argument and return the windowed version of the segment.

sides : {'default', 'onesided', 'twosided'}, optional
Which sides of the spectrum to return. 'default' is one-sided for real
data and two-sided for complex data. 'onesided' forces the return of a
one-sided spectrum, while 'twosided' forces two-sided.""",

Single_Spectrum="""\
pad_to : int, optional
The number of points to which the data segment is padded when performing
the FFT.  While not increasing the actual resolution of the spectrum (the
minimum distance between resolvable peaks), this can give more points in
the plot, allowing for more detail. This corresponds to the *n* parameter
in the call to fft().  The default is None, which sets *pad_to* equal to
the length of the input signal (i.e. no padding).""",

PSD="""\
pad_to : int, optional
The number of points to which the data segment is padded when performing
the FFT.  This can be different from *NFFT*, which specifies the number
of data points used.  While not increasing the actual resolution of the
spectrum (the minimum distance between resolvable peaks), this can give
more points in the plot, allowing for more detail. This corresponds to
the *n* parameter in the call to fft(). The default is None, which sets
*pad_to* equal to *NFFT*

NFFT : int, default: 256
The number of data points used in each block for the FFT.  A power 2 is
most efficient.  This should *NOT* be used to get zero padding, or the
scaling of the result will be incorrect; use *pad_to* for this instead.

detrend : {'none', 'mean', 'linear'} or callable, default 'none'
The function applied to each segment before fft-ing, designed to remove
the mean or linear trend.  Unlike in MATLAB, where the *detrend* parameter
is a vector, in Matplotlib is it a function.  The :mod:~matplotlib.mlab
module defines .detrend_none, .detrend_mean, and .detrend_linear,
but you can use a custom function as well.  You can also use a string to
choose one of the functions: 'none' calls .detrend_none. 'mean' calls
.detrend_mean. 'linear' calls .detrend_linear.

scale_by_freq : bool, default: True
Whether the resulting density values should be scaled by the scaling
frequency, which gives density in units of Hz^-1.  This allows for
integration over the returned frequency values.  The default is True for
MATLAB compatibility.""")

[docs]@docstring.dedent_interpd
def psd(x, NFFT=None, Fs=None, detrend=None, window=None,
noverlap=None, pad_to=None, sides=None, scale_by_freq=None):
r"""
Compute the power spectral density.

The power spectral density :math:P_{xx} by Welch's average
periodogram method.  The vector *x* is divided into *NFFT* length
segments.  Each segment is detrended by function *detrend* and
windowed by function *window*.  *noverlap* gives the length of
the overlap between segments.  The :math:|\mathrm{fft}(i)|^2
of each segment :math:i are averaged to compute :math:P_{xx}.

If len(*x*) < *NFFT*, it will be zero padded to *NFFT*.

Parameters
----------
x : 1-D array or sequence
Array or sequence containing the data

%(Spectral)s

%(PSD)s

noverlap : int
The number of points of overlap between segments.
The default value is 0 (no overlap).

Returns
-------
Pxx : 1-D array
The values for the power spectrum :math:P_{xx} (real valued)

freqs : 1-D array
The frequencies corresponding to the elements in *Pxx*

References
----------
Bendat & Piersol -- Random Data: Analysis and Measurement Procedures, John
Wiley & Sons (1986)

--------
specgram
specgram differs in the default overlap; in not returning the mean of
the segment periodograms; and in returning the times of the segments.

magnitude_spectrum : returns the magnitude spectrum.

csd : returns the spectral density between two signals.
"""
Pxx, freqs = csd(x=x, y=None, NFFT=NFFT, Fs=Fs, detrend=detrend,
sides=sides, scale_by_freq=scale_by_freq)
return Pxx.real, freqs

[docs]@docstring.dedent_interpd
def csd(x, y, NFFT=None, Fs=None, detrend=None, window=None,
noverlap=None, pad_to=None, sides=None, scale_by_freq=None):
"""
Compute the cross-spectral density.

The cross spectral density :math:P_{xy} by Welch's average
periodogram method.  The vectors *x* and *y* are divided into
*NFFT* length segments.  Each segment is detrended by function
*detrend* and windowed by function *window*.  *noverlap* gives
the length of the overlap between segments.  The product of
the direct FFTs of *x* and *y* are averaged over each segment
to compute :math:P_{xy}, with a scaling to correct for power
loss due to windowing.

If len(*x*) < *NFFT* or len(*y*) < *NFFT*, they will be zero

Parameters
----------
x, y : 1-D arrays or sequences
Arrays or sequences containing the data

%(Spectral)s

%(PSD)s

noverlap : int
The number of points of overlap between segments.
The default value is 0 (no overlap).

Returns
-------
Pxy : 1-D array
The values for the cross spectrum :math:P_{xy} before scaling (real
valued)

freqs : 1-D array
The frequencies corresponding to the elements in *Pxy*

References
----------
Bendat & Piersol -- Random Data: Analysis and Measurement Procedures, John
Wiley & Sons (1986)

--------
psd : equivalent to setting y = x.
"""
if NFFT is None:
NFFT = 256
Pxy, freqs, _ = _spectral_helper(x=x, y=y, NFFT=NFFT, Fs=Fs,
detrend_func=detrend, window=window,
sides=sides, scale_by_freq=scale_by_freq,
mode='psd')

if Pxy.ndim == 2:
if Pxy.shape > 1:
Pxy = Pxy.mean(axis=1)
else:
Pxy = Pxy[:, 0]
return Pxy, freqs

_single_spectrum_docs = """\
Compute the {quantity} of *x*.
Data is padded to a length of *pad_to* and the windowing function *window* is
applied to the signal.

Parameters
----------
x : 1-D array or sequence
Array or sequence containing the data

{Spectral}

{Single_Spectrum}

Returns
-------
spectrum : 1-D array
The {quantity}.
freqs : 1-D array
The frequencies corresponding to the elements in *spectrum*.

--------
psd
Returns the power spectral density.
complex_spectrum
Returns the complex-valued frequency spectrum.
magnitude_spectrum
Returns the absolute value of the complex_spectrum.
angle_spectrum
Returns the angle of the complex_spectrum.
phase_spectrum
Returns the phase (unwrapped angle) of the complex_spectrum.
specgram
Can return the complex spectrum of segments within the signal.
"""

complex_spectrum = functools.partial(_single_spectrum_helper, "complex")
complex_spectrum.__doc__ = _single_spectrum_docs.format(
quantity="complex-valued frequency spectrum",
**docstring.interpd.params)
magnitude_spectrum = functools.partial(_single_spectrum_helper, "magnitude")
magnitude_spectrum.__doc__ = _single_spectrum_docs.format(
quantity="magnitude (absolute value) of the frequency spectrum",
**docstring.interpd.params)
angle_spectrum = functools.partial(_single_spectrum_helper, "angle")
angle_spectrum.__doc__ = _single_spectrum_docs.format(
quantity="angle of the frequency spectrum (wrapped phase spectrum)",
**docstring.interpd.params)
phase_spectrum = functools.partial(_single_spectrum_helper, "phase")
phase_spectrum.__doc__ = _single_spectrum_docs.format(
quantity="phase of the frequency spectrum (unwrapped phase spectrum)",
**docstring.interpd.params)

[docs]@docstring.dedent_interpd
def specgram(x, NFFT=None, Fs=None, detrend=None, window=None,
noverlap=None, pad_to=None, sides=None, scale_by_freq=None,
mode=None):
"""
Compute a spectrogram.

Compute and plot a spectrogram of data in x.  Data are split into
NFFT length segments and the spectrum of each section is
computed.  The windowing function window is applied to each
segment, and the amount of overlap of each segment is
specified with noverlap.

Parameters
----------
x : array-like
1-D array or sequence.

%(Spectral)s

%(PSD)s

noverlap : int, optional
The number of points of overlap between blocks.  The default
value is 128.
mode : str, default: 'psd'
What sort of spectrum to use:
'psd'
Returns the power spectral density.
'complex'
Returns the complex-valued frequency spectrum.
'magnitude'
Returns the magnitude spectrum.
'angle'
Returns the phase spectrum without unwrapping.
'phase'
Returns the phase spectrum with unwrapping.

Returns
-------
spectrum : array-like
2-D array, columns are the periodograms of successive segments.

freqs : array-like
1-D array, frequencies corresponding to the rows in *spectrum*.

t : array-like
1-D array, the times corresponding to midpoints of segments
(i.e the columns in *spectrum*).

--------
psd : differs in the overlap and in the return values.
complex_spectrum : similar, but with complex valued frequencies.
magnitude_spectrum : similar single segment when mode is 'magnitude'.
angle_spectrum : similar to single segment when mode is 'angle'.
phase_spectrum : similar to single segment when mode is 'phase'.

Notes
-----
detrend and scale_by_freq only apply when *mode* is set to 'psd'.

"""
if noverlap is None:
noverlap = 128  # default in _spectral_helper() is noverlap = 0
if NFFT is None:
NFFT = 256  # same default as in _spectral_helper()
if len(x) <= NFFT:
cbook._warn_external("Only one segment is calculated since parameter "
"NFFT (=%d) >= signal length (=%d)." %
(NFFT, len(x)))

spec, freqs, t = _spectral_helper(x=x, y=None, NFFT=NFFT, Fs=Fs,
detrend_func=detrend, window=window,
sides=sides,
scale_by_freq=scale_by_freq,
mode=mode)

if mode != 'complex':
spec = spec.real  # Needed since helper implements generically

return spec, freqs, t

[docs]@docstring.dedent_interpd
def cohere(x, y, NFFT=256, Fs=2, detrend=detrend_none, window=window_hanning,
noverlap=0, pad_to=None, sides='default', scale_by_freq=None):
r"""
The coherence between *x* and *y*.  Coherence is the normalized
cross spectral density:

.. math::

C_{xy} = \frac{|P_{xy}|^2}{P_{xx}P_{yy}}

Parameters
----------
x, y
Array or sequence containing the data

%(Spectral)s

%(PSD)s

noverlap : int
The number of points of overlap between blocks.  The default value
is 0 (no overlap).

Returns
-------
The return value is the tuple (*Cxy*, *f*), where *f* are the
frequencies of the coherence vector. For cohere, scaling the
individual densities by the sampling frequency has no effect,
since the factors cancel out.

--------
:func:psd, :func:csd :
For information about the methods used to compute :math:P_{xy},
:math:P_{xx} and :math:P_{yy}.
"""
if len(x) < 2 * NFFT:
raise ValueError(
"Coherence is calculated by averaging over *NFFT* length "
"segments.  Your signal is too short for your choice of *NFFT*.")
Pxx, f = psd(x, NFFT, Fs, detrend, window, noverlap, pad_to, sides,
scale_by_freq)
Pyy, f = psd(y, NFFT, Fs, detrend, window, noverlap, pad_to, sides,
scale_by_freq)
Pxy, f = csd(x, y, NFFT, Fs, detrend, window, noverlap, pad_to, sides,
scale_by_freq)
Cxy = np.abs(Pxy) ** 2 / (Pxx * Pyy)
return Cxy, f

[docs]class GaussianKDE:
"""
Representation of a kernel-density estimate using Gaussian kernels.

Parameters
----------
dataset : array-like
Datapoints to estimate from. In case of univariate data this is a 1-D
array, otherwise a 2-D array with shape (# of dims, # of data).

bw_method : str, scalar or callable, optional
The method used to calculate the estimator bandwidth.  This can be
'scott', 'silverman', a scalar constant or a callable.  If a
scalar, this will be used directly as kde.factor.  If a
callable, it should take a GaussianKDE instance as only
parameter and return a scalar. If None (default), 'scott' is used.

Attributes
----------
dataset : ndarray
The dataset with which gaussian_kde was initialized.

dim : int
Number of dimensions.

num_dp : int
Number of datapoints.

factor : float
The bandwidth factor, obtained from kde.covariance_factor, with which
the covariance matrix is multiplied.

covariance : ndarray
The covariance matrix of *dataset*, scaled by the calculated bandwidth
(kde.factor).

inv_cov : ndarray
The inverse of *covariance*.

Methods
-------
kde.evaluate(points) : ndarray
Evaluate the estimated pdf on a provided set of points.

kde(points) : ndarray
Same as kde.evaluate(points)

"""

# This implementation with minor modification was too good to pass up.
# from scipy: https://github.com/scipy/scipy/blob/master/scipy/stats/kde.py

def __init__(self, dataset, bw_method=None):
self.dataset = np.atleast_2d(dataset)
if not np.array(self.dataset).size > 1:
raise ValueError("dataset input should have multiple elements.")

self.dim, self.num_dp = np.array(self.dataset).shape

if bw_method is None:
pass
elif cbook._str_equal(bw_method, 'scott'):
self.covariance_factor = self.scotts_factor
elif cbook._str_equal(bw_method, 'silverman'):
self.covariance_factor = self.silverman_factor
elif isinstance(bw_method, Number):
self._bw_method = 'use constant'
self.covariance_factor = lambda: bw_method
elif callable(bw_method):
self._bw_method = bw_method
self.covariance_factor = lambda: self._bw_method(self)
else:
raise ValueError("bw_method should be 'scott', 'silverman', a "
"scalar or a callable")

# Computes the covariance matrix for each Gaussian kernel using
# covariance_factor().

self.factor = self.covariance_factor()
# Cache covariance and inverse covariance of the data
if not hasattr(self, '_data_inv_cov'):
self.data_covariance = np.atleast_2d(
np.cov(
self.dataset,
rowvar=1,
bias=False))
self.data_inv_cov = np.linalg.inv(self.data_covariance)

self.covariance = self.data_covariance * self.factor ** 2
self.inv_cov = self.data_inv_cov / self.factor ** 2
self.norm_factor = (np.sqrt(np.linalg.det(2 * np.pi * self.covariance))
* self.num_dp)

[docs]    def scotts_factor(self):
return np.power(self.num_dp, -1. / (self.dim + 4))

[docs]    def silverman_factor(self):
return np.power(
self.num_dp * (self.dim + 2.0) / 4.0, -1. / (self.dim + 4))

#  Default method to calculate bandwidth, can be overwritten by subclass
covariance_factor = scotts_factor

[docs]    def evaluate(self, points):
"""
Evaluate the estimated pdf on a set of points.

Parameters
----------
points : (# of dimensions, # of points)-array
Alternatively, a (# of dimensions,) vector can be passed in and
treated as a single point.

Returns
-------
(# of points,)-array
The values at each point.

Raises
------
ValueError : if the dimensionality of the input points is different
than the dimensionality of the KDE.

"""
points = np.atleast_2d(points)

dim, num_m = np.array(points).shape
if dim != self.dim:
raise ValueError("points have dimension {}, dataset has dimension "
"{}".format(dim, self.dim))

result = np.zeros(num_m)

if num_m >= self.num_dp:
# there are more points than data, so loop over data
for i in range(self.num_dp):
diff = self.dataset[:, i, np.newaxis] - points
tdiff = np.dot(self.inv_cov, diff)
energy = np.sum(diff * tdiff, axis=0) / 2.0
result = result + np.exp(-energy)
else:
# loop over points
for i in range(num_m):
diff = self.dataset - points[:, i, np.newaxis]
tdiff = np.dot(self.inv_cov, diff)
energy = np.sum(diff * tdiff, axis=0) / 2.0
result[i] = np.sum(np.exp(-energy), axis=0)

result = result / self.norm_factor

return result

__call__ = evaluate