# 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 csv
import inspect

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). See Also -------- window_none : Another window algorithm. ''' return np.hanning(len(x))*x
[docs]def window_none(x): ''' No window function; simply return x. See Also -------- window_hanning : Another window algorithm. ''' return x
[docs]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 : integer 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 Specifies the detrend algorithm to use. 'default' is 'mean', which is the same as detrend_mean. 'constant' is the same. '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 : integer The axis along which to do the detrending. See Also -------- 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 isinstance(key, str): raise ValueError("Unknown value for key %s, must be one of: " "'default', 'constant', 'mean', " "'linear', or a function" % key) if not callable(key): raise ValueError("Unknown value for key %s, must be one of: " "'default', 'constant', 'mean', " "'linear', or a function" % key) x = np.asarray(x) if axis is not None and axis+1 > x.ndim: raise ValueError('axis(=%s) out of bounds' % axis) 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)
[docs]@cbook.deprecated("3.1", alternative="detrend_mean") def demean(x, axis=0): ''' Return x minus its mean along the specified axis. Parameters ---------- x : array or sequence Array or sequence containing the data Can have any dimensionality axis : integer The axis along which to take the mean. See numpy.mean for a description of this argument. See Also -------- detrend_mean : Same as demean except for the default *axis*. ''' return detrend_mean(x, axis=axis)
[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 : integer The axis along which to take the mean. See numpy.mean for a description of this argument. See Also -------- 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 : integer This parameter is ignored. It is included for compatibility with detrend_mean See Also -------- 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 : integer The axis along which to take the mean. See numpy.mean for a description of this argument. See Also -------- 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 : integer The number of data points in each window. noverlap : integer The overlap between adjacent windows. Default is 0 (no overlap) axis : integer 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[0], step*x.strides[0]) else: shape = ((x.shape[-1]-noverlap)//step, n) strides = (step*x.strides[0], x.strides[0]) return np.lib.stride_tricks.as_strided(x, shape=shape, strides=strides)
[docs]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 : integer The number of time to repeat the array. axis : integer 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[0]) else: shape = (x.size, n) strides = (x.strides[0], 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, window=None, noverlap=None, pad_to=None, sides=None, scale_by_freq=None, mode=None): ''' This is a helper function that implements the commonality between the psd, csd, spectrogram and complex, magnitude, angle, and phase spectrums. It is *NOT* meant to be used outside of mlab and may change at any time. ''' 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' elif mode not in ['psd', 'complex', 'magnitude', 'angle', 'phase']: raise ValueError("Unknown value for mode %s, must be one of: " "'default', 'psd', 'complex', " "'magnitude', 'angle', 'phase'" % 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' elif sides not in ['onesided', 'twosided']: raise ValueError("Unknown value for sides %s, must be one of: " "'default', 'onesided', or 'twosided'" % 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: pad_to = NFFT 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': numFreqs = pad_to if pad_to % 2: freqcenter = (pad_to - 1)//2 + 1 else: freqcenter = pad_to//2 scaling_factor = 1. elif sides == 'onesided': if pad_to % 2: numFreqs = (pad_to + 1)//2 else: numFreqs = pad_to//2 + 1 scaling_factor = 2. result = stride_windows(x, NFFT, noverlap, axis=0) result = detrend(result, detrend_func, axis=0) result, windowVals = apply_window(result, window, axis=0, return_window=True) 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 = apply_window(resultY, window, axis=0) 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(windowVals).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(windowVals).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(windowVals)**2).sum() else: # In this case, preserve power in the segment, not amplitude result /= np.abs(windowVals).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.concatenate((freqs[freqcenter:], freqs[:freqcenter])) result = np.concatenate((result[freqcenter:, :], result[:freqcenter, :]), 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(x, mode, Fs=None, window=None, pad_to=None, sides=None): ''' This is a helper function that implements the commonality between the complex, magnitude, angle, and phase spectrums. It is *NOT* meant to be used outside of mlab and may change at any time. ''' if mode is None or mode == 'psd' or mode == 'default': raise ValueError('_single_spectrum_helper does not work with %s mode' % mode) if pad_to is None: pad_to = len(x) spec, freqs, _ = _spectral_helper(x=x, y=None, NFFT=len(x), Fs=Fs, detrend_func=detrend_none, window=window, noverlap=0, pad_to=pad_to, sides=sides, scale_by_freq=False, mode=mode) if mode != 'complex': spec = spec.real if spec.ndim == 2 and spec.shape[1] == 1: spec = spec[:, 0] return spec, freqs # Split out these keyword docs so that they can be used elsewhere docstring.interpd.update(Spectral=inspect.cleandoc(""" Fs : scalar The sampling frequency (samples per time unit). It is used to calculate the Fourier frequencies, freqs, in cycles per time unit. The default value is 2. window : callable or ndarray 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. The default is window_hanning. 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'} Specifies which sides of the spectrum to return. Default gives the default behavior, which returns one-sided for real data and both for complex data. 'onesided' forces the return of a one-sided spectrum, while 'twosided' forces two-sided. """)) docstring.interpd.update(Single_Spectrum=inspect.cleandoc(""" pad_to : int 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). """)) docstring.interpd.update(PSD=inspect.cleandoc(""" pad_to : int 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 The number of data points used in each block for the FFT. A power 2 is most efficient. The default value is 256. 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, optional Specifies 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 : integer 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 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) See Also -------- 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, window=window, noverlap=noverlap, pad_to=pad_to, 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 padded to *NFFT*. Parameters ---------- x, y : 1-D arrays or sequences Arrays or sequences containing the data %(Spectral)s %(PSD)s noverlap : integer 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 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) See Also -------- 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, noverlap=noverlap, pad_to=pad_to, sides=sides, scale_by_freq=scale_by_freq, mode='psd') if Pxy.ndim == 2: if Pxy.shape[1] > 1: Pxy = Pxy.mean(axis=1) else: Pxy = Pxy[:, 0] return Pxy, freqs
[docs]@docstring.dedent_interpd def complex_spectrum(x, Fs=None, window=None, pad_to=None, sides=None): """ Compute the complex-valued frequency spectrum 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)s %(Single_Spectrum)s Returns ------- spectrum : 1-D array The values for the complex spectrum (complex valued) freqs : 1-D array The frequencies corresponding to the elements in *spectrum* See Also -------- magnitude_spectrum Returns the absolute value of this function. angle_spectrum Returns the angle of this function. phase_spectrum Returns the phase (unwrapped angle) of this function. specgram Can return the complex spectrum of segments within the signal. """ return _single_spectrum_helper(x=x, Fs=Fs, window=window, pad_to=pad_to, sides=sides, mode='complex')
[docs]@docstring.dedent_interpd def magnitude_spectrum(x, Fs=None, window=None, pad_to=None, sides=None): """ Compute the magnitude (absolute value) of the frequency spectrum 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)s %(Single_Spectrum)s Returns ------- spectrum : 1-D array The values for the magnitude spectrum (real valued) freqs : 1-D array The frequencies corresponding to the elements in *spectrum* See Also -------- psd Returns the power spectral density. complex_spectrum This function returns the absolute value of complex_spectrum. angle_spectrum Returns the angles of the corresponding frequencies. phase_spectrum Returns the phase (unwrapped angle) of the corresponding frequencies. specgram Can return the complex spectrum of segments within the signal. """ return _single_spectrum_helper(x=x, Fs=Fs, window=window, pad_to=pad_to, sides=sides, mode='magnitude')
[docs]@docstring.dedent_interpd def angle_spectrum(x, Fs=None, window=None, pad_to=None, sides=None): """ Compute the angle of the frequency spectrum (wrapped phase spectrum) 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)s %(Single_Spectrum)s Returns ------- spectrum : 1-D array The values for the angle spectrum in radians (real valued) freqs : 1-D array The frequencies corresponding to the elements in *spectrum* See Also -------- complex_spectrum This function returns the angle value of complex_spectrum. magnitude_spectrum Returns the magnitudes of the corresponding frequencies. phase_spectrum Returns the phase (unwrapped angle) of the corresponding frequencies. specgram Can return the complex spectrum of segments within the signal. """ return _single_spectrum_helper(x=x, Fs=Fs, window=window, pad_to=pad_to, sides=sides, mode='angle')
[docs]@docstring.dedent_interpd def phase_spectrum(x, Fs=None, window=None, pad_to=None, sides=None): """ Compute the phase of the frequency spectrum (unwrapped angle spectrum) 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)s %(Single_Spectrum)s Returns ------- spectrum : 1-D array The values for the phase spectrum in radians (real valued) freqs : 1-D array The frequencies corresponding to the elements in *spectrum* See Also -------- complex_spectrum This function returns the phase value of complex_spectrum. magnitude_spectrum Returns the magnitudes of the corresponding frequencies. angle_spectrum Returns the angle (wrapped phase) of the corresponding frequencies. specgram Can return the complex spectrum of segments within the signal. """ return _single_spectrum_helper(x=x, Fs=Fs, window=window, pad_to=pad_to, sides=sides, mode='phase')
[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, optional What sort of spectrum to use, default is 'psd'. '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*). See Also -------- 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, noverlap=noverlap, pad_to=pad_to, 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
_coh_error = """Coherence is calculated by averaging over *NFFT* length segments. Your signal is too short for your choice of *NFFT*. """
[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): """ 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 : integer 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. See Also -------- :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(_coh_error) 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(object): """ 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 isString = isinstance(bw_method, str) if bw_method is None: pass elif (isString and bw_method == 'scott'): self.covariance_factor = self.scotts_factor elif (isString and bw_method == 'silverman'): self.covariance_factor = self.silverman_factor elif (np.isscalar(bw_method) and not isString): 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