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
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). 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]@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 : 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 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]@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]@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 : 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' 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: 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. 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.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. ''' cbook._check_in_list(['complex', 'magnitude', 'angle', 'phase'], 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
[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 : 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( "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
def _csv2rec(fname, comments='#', skiprows=0, checkrows=0, delimiter=',', converterd=None, names=None, missing='', missingd=None, use_mrecords=False, dayfirst=False, yearfirst=False): """ Load data from comma/space/tab delimited file in *fname* into a numpy record array and return the record array. If *names* is *None*, a header row is required to automatically assign the recarray names. The headers will be lower cased, spaces will be converted to underscores, and illegal attribute name characters removed. If *names* is not *None*, it is a sequence of names to use for the column names. In this case, it is assumed there is no header row. - *fname*: can be a filename or a file handle. Support for gzipped files is automatic, if the filename ends in '.gz' - *comments*: the character used to indicate the start of a comment in the file, or *None* to switch off the removal of comments - *skiprows*: is the number of rows from the top to skip - *checkrows*: is the number of rows to check to validate the column data type. When set to zero all rows are validated. - *converterd*: if not *None*, is a dictionary mapping column number or munged column name to a converter function. - *names*: if not None, is a list of header names. In this case, no header will be read from the file - *missingd* is a dictionary mapping munged column names to field values which signify that the field does not contain actual data and should be masked, e.g., '0000-00-00' or 'unused' - *missing*: a string whose value signals a missing field regardless of the column it appears in - *use_mrecords*: if True, return an mrecords.fromrecords record array if any of the data are missing - *dayfirst*: default is False so that MM-DD-YY has precedence over DD-MM-YY. See http://labix.org/python-dateutil#head-b95ce2094d189a89f80f5ae52a05b4ab7b41af47 for further information. - *yearfirst*: default is False so that MM-DD-YY has precedence over YY-MM-DD. See http://labix.org/python-dateutil#head-b95ce2094d189a89f80f5ae52a05b4ab7b41af47 for further information. If no rows are found, *None* is returned """ if converterd is None: converterd = dict() if missingd is None: missingd = {} import dateutil.parser import datetime fh = cbook.to_filehandle(fname) delimiter = str(delimiter) class FH: """ For space-delimited files, we want different behavior than comma or tab. Generally, we want multiple spaces to be treated as a single separator, whereas with comma and tab we want multiple commas to return multiple (empty) fields. The join/strip trick below effects this. """ def __init__(self, fh): self.fh = fh def close(self): self.fh.close() def seek(self, arg): self.fh.seek(arg) def fix(self, s): return ' '.join(s.split()) def __next__(self): return self.fix(next(self.fh)) def __iter__(self): for line in self.fh: yield self.fix(line) if delimiter == ' ': fh = FH(fh) reader = csv.reader(fh, delimiter=delimiter) def process_skiprows(reader): if skiprows: for i, row in enumerate(reader): if i >= (skiprows-1): break return fh, reader process_skiprows(reader) def ismissing(name, val): "Should the value val in column name be masked?" return val == missing or val == missingd.get(name) or val == '' def with_default_value(func, default): def newfunc(name, val): if ismissing(name, val): return default else: return func(val) return newfunc def mybool(x): if x == 'True': return True elif x == 'False': return False else: raise ValueError('invalid bool') dateparser = dateutil.parser.parse def mydateparser(x): # try and return a datetime object d = dateparser(x, dayfirst=dayfirst, yearfirst=yearfirst) return d mydateparser = with_default_value(mydateparser, datetime.datetime(1, 1, 1)) myfloat = with_default_value(float, np.nan) myint = with_default_value(int, -1) mystr = with_default_value(str, '') mybool = with_default_value(mybool, None) def mydate(x): # try and return a date object d = dateparser(x, dayfirst=dayfirst, yearfirst=yearfirst) if d.hour > 0 or d.minute > 0 or d.second > 0: raise ValueError('not a date') return d.date() mydate = with_default_value(mydate, datetime.date(1, 1, 1)) def get_func(name, item, func): # promote functions in this order funcs = [mybool, myint, myfloat, mydate, mydateparser, mystr] for func in funcs[funcs.index(func):]: try: func(name, item) except Exception: continue return func raise ValueError('Could not find a working conversion function') # map column names that clash with builtins -- TODO - extend this list itemd = { 'return': 'return_', 'file': 'file_', 'print': 'print_', } def get_converters(reader, comments): converters = None i = 0 for row in reader: if (len(row) and comments is not None and row[0].startswith(comments)): continue if i == 0: converters = [mybool]*len(row) if checkrows and i > checkrows: break i += 1 for j, (name, item) in enumerate(zip(names, row)): func = converterd.get(j) if func is None: func = converterd.get(name) if func is None: func = converters[j] if len(item.strip()): func = get_func(name, item, func) else: # how should we handle custom converters and defaults? func = with_default_value(func, None) converters[j] = func return converters # Get header and remove invalid characters needheader = names is None if needheader: for row in reader: if (len(row) and comments is not None and row[0].startswith(comments)): continue headers = row break # remove these chars delete = set(r"""[email protected]#$%^&*()-=+~\|}[]{';: /?.>,<""") delete.add('"') names = [] seen = dict() for i, item in enumerate(headers): item = item.strip().lower().replace(' ', '_') item = ''.join([c for c in item if c not in delete]) if not len(item): item = 'column%d' % i item = itemd.get(item, item) cnt = seen.get(item, 0) if cnt > 0: names.append(item + '_%d' % cnt) else: names.append(item) seen[item] = cnt+1 else: if isinstance(names, str): names = [n.strip() for n in names.split(',')] # get the converter functions by inspecting checkrows converters = get_converters(reader, comments) if converters is None: raise ValueError('Could not find any valid data in CSV file') # reset the reader and start over fh.seek(0) reader = csv.reader(fh, delimiter=delimiter) process_skiprows(reader) if needheader: while True: # skip past any comments and consume one line of column header row = next(reader) if (len(row) and comments is not None and row[0].startswith(comments)): continue break # iterate over the remaining rows and convert the data to date # objects, ints, or floats as appropriate rows = [] rowmasks = [] for i, row in enumerate(reader): if not len(row): continue if comments is not None and row[0].startswith(comments): continue # Ensure that the row returned always has the same nr of elements row.extend([''] * (len(converters) - len(row))) rows.append([func(name, val) for func, name, val in zip(converters, names, row)]) rowmasks.append([ismissing(name, val) for name, val in zip(names, row)]) fh.close() if not len(rows): return None if use_mrecords and np.any(rowmasks): r = np.ma.mrecords.fromrecords(rows, names=names, mask=rowmasks) else: r = np.rec.fromrecords(rows, names=names) return r
[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 ------- values : (# 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