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 """ Numerical python functions written for compatability with MATLAB commands with the same names. MATLAB compatible functions --------------------------- :func:cohere Coherence (normalized cross spectral density) :func:csd Cross spectral density using Welch's average periodogram :func:detrend Remove the mean or best fit line from an array :func:find Return the indices where some condition is true; numpy.nonzero is similar but more general. :func:griddata Interpolate irregularly distributed data to a regular grid. :func:prctile Find the percentiles of a sequence :func:prepca Principal Component Analysis :func:psd Power spectral density using Welch's average periodogram :func:rk4 A 4th order runge kutta integrator for 1D or ND systems :func:specgram Spectrogram (spectrum over segments of time) Miscellaneous functions ----------------------- Functions that don't exist in MATLAB, but are useful anyway: :func:cohere_pairs Coherence over all pairs. This is not a MATLAB function, but we compute coherence a lot in my lab, and we compute it for a lot of pairs. This function is optimized to do this efficiently by caching the direct FFTs. :func:rk4 A 4th order Runge-Kutta ODE integrator in case you ever find yourself stranded without scipy (and the far superior scipy.integrate tools) :func:contiguous_regions Return the indices of the regions spanned by some logical mask :func:cross_from_below Return the indices where a 1D array crosses a threshold from below :func:cross_from_above Return the indices where a 1D array crosses a threshold from above :func:complex_spectrum Return the complex-valued frequency spectrum of a signal :func:magnitude_spectrum Return the magnitude of the frequency spectrum of a signal :func:angle_spectrum Return the angle (wrapped phase) of the frequency spectrum of a signal :func:phase_spectrum Return the phase (unwrapped angle) of the frequency spectrum of a signal :func:detrend_mean Remove the mean from a line. :func:demean Remove the mean from a line. This function is the same as :func:detrend_mean except for the default *axis*. :func:detrend_linear Remove the best fit line from a line. :func:detrend_none Return the original line. :func:stride_windows Get all windows in an array in a memory-efficient manner :func:stride_repeat Repeat an array in a memory-efficient manner :func:apply_window Apply a window along a given axis record array helper functions ----------------------------- A collection of helper methods for numpyrecord arrays .. _htmlonly: See :ref:misc-examples-index :func:rec2txt Pretty print a record array :func:rec2csv Store record array in CSV file :func:csv2rec Import record array from CSV file with type inspection :func:rec_append_fields Adds field(s)/array(s) to record array :func:rec_drop_fields Drop fields from record array :func:rec_join Join two record arrays on sequence of fields :func:recs_join A simple join of multiple recarrays using a single column as a key :func:rec_groupby Summarize data by groups (similar to SQL GROUP BY) :func:rec_summarize Helper code to filter rec array fields into new fields For the rec viewer functions(e rec2csv), there are a bunch of Format objects you can pass into the functions that will do things like color negative values red, set percent formatting and scaling, etc. Example usage:: r = csv2rec('somefile.csv', checkrows=0) formatd = dict( weight = FormatFloat(2), change = FormatPercent(2), cost = FormatThousands(2), ) rec2excel(r, 'test.xls', formatd=formatd) rec2csv(r, 'test.csv', formatd=formatd) scroll = rec2gtk(r, formatd=formatd) win = gtk.Window() win.set_size_request(600,800) win.add(scroll) win.show_all() gtk.main() """ from __future__ import (absolute_import, division, print_function, unicode_literals) import six from six.moves import map, xrange, zip import copy import csv import operator import os import warnings import numpy as np import matplotlib.cbook as cbook from matplotlib import docstring from matplotlib.path import Path import math if six.PY3: long = int def logspace(xmin, xmax, N): ''' Return N values logarithmically spaced between xmin and xmax. ''' return np.exp(np.linspace(np.log(xmin), np.log(xmax), N)) def _norm(x): ''' Return sqrt(x dot x). ''' return np.sqrt(np.dot(x, x)) def window_hanning(x): ''' Return x times the hanning window of len(x). See Also -------- :func:window_none :func:window_none is another window algorithm. ''' return np.hanning(len(x))*x def window_none(x): ''' No window function; simply return x. See Also -------- :func:window_hanning :func:window_hanning is another window algorithm. ''' return x 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 cbook.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 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 :func:detrend_mean. 'constant' is the same. 'linear' is the same as :func:detrend_linear. 'none' is the same as :func: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 -------- :func:detrend_mean :func:detrend_mean implements the 'mean' algorithm. :func:detrend_linear :func:detrend_linear implements the 'linear' algorithm. :func:detrend_none :func:detrend_none implements 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, six.string_types): 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) 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 -------- :func:delinear :func:denone :func:delinear and :func:denone are other detrend algorithms. :func:detrend_mean This function is the same as :func:detrend_mean except for the default *axis*. ''' return detrend_mean(x, axis=axis) 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 -------- :func:demean This function is the same as :func:demean except for the default *axis*. :func:detrend_linear :func:detrend_none :func:detrend_linear and :func:detrend_none are other detrend algorithms. :func:detrend :func:detrend is 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) # short-circuit 0-D array. if not x.ndim: return np.array(0., dtype=x.dtype) # short-circuit simple operations if axis == 0 or axis is None or x.ndim <= 1: return x - x.mean(axis) ind = [slice(None)] * x.ndim ind[axis] = np.newaxis return x - x.mean(axis)[ind] 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 -------- :func:denone This function is the same as :func:denone except for the default *axis*, which has no effect. :func:detrend_mean :func:detrend_linear :func:detrend_mean and :func:detrend_linear are other detrend algorithms. :func:detrend :func:detrend is a wrapper around all the detrend algorithms. ''' return x 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 -------- :func:delinear This function is the same as :func:delinear except for the default *axis*. :func:detrend_mean :func:detrend_none :func:detrend_mean and :func:detrend_none are other detrend algorithms. :func:detrend :func:detrend is 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) 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? _ stackoverflow: Using strides for an efficient moving average filter _ ''' 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) 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? _ ''' 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=cbook.dedent(""" 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 :func:window_hanning, :func:window_none, :func:numpy.blackman, :func:numpy.hamming, :func:numpy.bartlett, :func:scipy.signal, :func:scipy.signal.get_window, etc. The default is :func: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=cbook.dedent(""" pad_to : integer 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=cbook.dedent(""" pad_to : integer 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 : integer 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 : {'default', 'constant', 'mean', 'linear', 'none'} or callable 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.pylab module defines :func:~matplotlib.pylab.detrend_none, :func:~matplotlib.pylab.detrend_mean, and :func:~matplotlib.pylab.detrend_linear, but you can use a custom function as well. You can also use a string to choose one of the functions. 'default', 'constant', and 'mean' call :func:~matplotlib.pylab.detrend_mean. 'linear' calls :func:~matplotlib.pylab.detrend_linear. 'none' calls :func:~matplotlib.pylab.detrend_none. scale_by_freq : boolean, 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. """)) @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. Call signature:: psd(x, NFFT=256, Fs=2, detrend=mlab.detrend_none, window=mlab.window_hanning, noverlap=0, pad_to=None, sides='default', scale_by_freq=None) 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 -------- :func:specgram :func:specgram differs in the default overlap; in not returning the mean of the segment periodograms; and in returning the times of the segments. :func:magnitude_spectrum :func:magnitude_spectrum returns the magnitude spectrum. :func:csd :func: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 @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. Call signature:: csd(x, y, NFFT=256, Fs=2, detrend=mlab.detrend_none, window=mlab.window_hanning, noverlap=0, pad_to=None, sides='default', scale_by_freq=None) 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 -------- :func:psd :func:psd is the 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 @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 -------- :func:magnitude_spectrum :func:magnitude_spectrum returns the absolute value of this function. :func:angle_spectrum :func:angle_spectrum returns the angle of this function. :func:phase_spectrum :func:phase_spectrum returns the phase (unwrapped angle) of this function. :func:specgram :func: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') @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 -------- :func:psd :func:psd returns the power spectral density. :func:complex_spectrum This function returns the absolute value of :func:complex_spectrum. :func:angle_spectrum :func:angle_spectrum returns the angles of the corresponding frequencies. :func:phase_spectrum :func:phase_spectrum returns the phase (unwrapped angle) of the corresponding frequencies. :func:specgram :func:specgram can return the magnitude spectrum of segments within the signal. """ return _single_spectrum_helper(x=x, Fs=Fs, window=window, pad_to=pad_to, sides=sides, mode='magnitude') @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 -------- :func:complex_spectrum This function returns the angle value of :func:complex_spectrum. :func:magnitude_spectrum :func:angle_spectrum returns the magnitudes of the corresponding frequencies. :func:phase_spectrum :func:phase_spectrum returns the unwrapped version of this function. :func:specgram :func:specgram can return the angle spectrum of segments within the signal. """ return _single_spectrum_helper(x=x, Fs=Fs, window=window, pad_to=pad_to, sides=sides, mode='angle') @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 -------- :func:complex_spectrum This function returns the angle value of :func:complex_spectrum. :func:magnitude_spectrum :func:magnitude_spectrum returns the magnitudes of the corresponding frequencies. :func:angle_spectrum :func:angle_spectrum returns the wrapped version of this function. :func:specgram :func:specgram can return the phase spectrum of segments within the signal. """ return _single_spectrum_helper(x=x, Fs=Fs, window=window, pad_to=pad_to, sides=sides, mode='phase') @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: warnings.warn("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*. """ @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 def donothing_callback(*args): pass def cohere_pairs(X, ij, NFFT=256, Fs=2, detrend=detrend_none, window=window_hanning, noverlap=0, preferSpeedOverMemory=True, progressCallback=donothing_callback, returnPxx=False): """ Compute the coherence and phase for all pairs *ij*, in *X*. *X* is a *numSamples* * *numCols* array *ij* is a list of tuples. Each tuple is a pair of indexes into the columns of X for which you want to compute coherence. For example, if *X* has 64 columns, and you want to compute all nonredundant pairs, define *ij* as:: ij = [] for i in range(64): for j in range(i+1,64): ij.append( (i,j) ) *preferSpeedOverMemory* is an optional bool. Defaults to true. If False, limits the caching by only making one, rather than two, complex cache arrays. This is useful if memory becomes critical. Even when *preferSpeedOverMemory* is False, :func:cohere_pairs will still give significant performace gains over calling :func:cohere for each pair, and will use subtantially less memory than if *preferSpeedOverMemory* is True. In my tests with a 43000,64 array over all nonredundant pairs, *preferSpeedOverMemory* = True delivered a 33% performance boost on a 1.7GHZ Athlon with 512MB RAM compared with *preferSpeedOverMemory* = False. But both solutions were more than 10x faster than naively crunching all possible pairs through :func:cohere. Returns ------- Cxy : dictionary of (*i*, *j*) tuples -> coherence vector for that pair. i.e., Cxy[(i,j) = cohere(X[:,i], X[:,j]). Number of dictionary keys is len(ij). Phase : dictionary of phases of the cross spectral density at each frequency for each pair. Keys are (*i*, *j*). freqs : vector of frequencies, equal in length to either the coherence or phase vectors for any (*i*, *j*) key. e.g., to make a coherence Bode plot:: subplot(211) plot( freqs, Cxy[(12,19)]) subplot(212) plot( freqs, Phase[(12,19)]) For a large number of pairs, :func:cohere_pairs can be much more efficient than just calling :func:cohere for each pair, because it caches most of the intensive computations. If :math:N is the number of pairs, this function is :math:O(N) for most of the heavy lifting, whereas calling cohere for each pair is :math:O(N^2). However, because of the caching, it is also more memory intensive, making 2 additional complex arrays with approximately the same number of elements as *X*. See :file:test/cohere_pairs_test.py in the src tree for an example script that shows that this :func:cohere_pairs and :func:cohere give the same results for a given pair. See Also -------- :func:psd For information about the methods used to compute :math:P_{xy}, :math:P_{xx} and :math:P_{yy}. """ numRows, numCols = X.shape # zero pad if X is too short if numRows < NFFT: tmp = X X = np.zeros((NFFT, numCols), X.dtype) X[:numRows, :] = tmp del tmp numRows, numCols = X.shape # get all the columns of X that we are interested in by checking # the ij tuples allColumns = set() for i, j in ij: allColumns.add(i) allColumns.add(j) Ncols = len(allColumns) # for real X, ignore the negative frequencies if np.iscomplexobj(X): numFreqs = NFFT else: numFreqs = NFFT//2+1 # cache the FFT of every windowed, detrended NFFT length segement # of every channel. If preferSpeedOverMemory, cache the conjugate # as well if cbook.iterable(window): if len(window) != NFFT: raise ValueError("The length of the window must be equal to NFFT") windowVals = window else: windowVals = window(np.ones(NFFT, X.dtype)) ind = list(xrange(0, numRows-NFFT+1, NFFT-noverlap)) numSlices = len(ind) FFTSlices = {} FFTConjSlices = {} Pxx = {} slices = range(numSlices) normVal = np.linalg.norm(windowVals)**2 for iCol in allColumns: progressCallback(i/Ncols, 'Cacheing FFTs') Slices = np.zeros((numSlices, numFreqs), dtype=np.complex_) for iSlice in slices: thisSlice = X[ind[iSlice]:ind[iSlice]+NFFT, iCol] thisSlice = windowVals*detrend(thisSlice) Slices[iSlice, :] = np.fft.fft(thisSlice)[:numFreqs] FFTSlices[iCol] = Slices if preferSpeedOverMemory: FFTConjSlices[iCol] = np.conj(Slices) Pxx[iCol] = np.divide(np.mean(abs(Slices)**2, axis=0), normVal) del Slices, ind, windowVals # compute the coherences and phases for all pairs using the # cached FFTs Cxy = {} Phase = {} count = 0 N = len(ij) for i, j in ij: count += 1 if count % 10 == 0: progressCallback(count/N, 'Computing coherences') if preferSpeedOverMemory: Pxy = FFTSlices[i] * FFTConjSlices[j] else: Pxy = FFTSlices[i] * np.conj(FFTSlices[j]) if numSlices > 1: Pxy = np.mean(Pxy, axis=0) # Pxy = np.divide(Pxy, normVal) Pxy /= normVal # Cxy[(i,j)] = np.divide(np.absolute(Pxy)**2, Pxx[i]*Pxx[j]) Cxy[i, j] = abs(Pxy)**2 / (Pxx[i]*Pxx[j]) Phase[i, j] = np.arctan2(Pxy.imag, Pxy.real) freqs = Fs/NFFT*np.arange(numFreqs) if returnPxx: return Cxy, Phase, freqs, Pxx else: return Cxy, Phase, freqs def entropy(y, bins): r""" Return the entropy of the data in *y* in units of nat. .. math:: -\sum p_i \ln(p_i) where :math:p_i is the probability of observing *y* in the :math:i^{th} bin of *bins*. *bins* can be a number of bins or a range of bins; see :func:numpy.histogram. Compare *S* with analytic calculation for a Gaussian:: x = mu + sigma * randn(200000) Sanalytic = 0.5 * ( 1.0 + log(2*pi*sigma**2.0) ) """ n, bins = np.histogram(y, bins) n = n.astype(float) n = np.take(n, np.nonzero(n)[0]) # get the positive p = np.divide(n, len(y)) delta = bins[1] - bins[0] S = -1.0 * np.sum(p * np.log(p)) + np.log(delta) return S def normpdf(x, *args): "Return the normal pdf evaluated at *x*; args provides *mu*, *sigma*" mu, sigma = args return 1./(np.sqrt(2*np.pi)*sigma)*np.exp(-0.5 * (1./sigma*(x - mu))**2) def find(condition): "Return the indices where ravel(condition) is true" res, = np.nonzero(np.ravel(condition)) return res def longest_contiguous_ones(x): """ Return the indices of the longest stretch of contiguous ones in *x*, assuming *x* is a vector of zeros and ones. If there are two equally long stretches, pick the first. """ x = np.ravel(x) if len(x) == 0: return np.array([]) ind = (x == 0).nonzero()[0] if len(ind) == 0: return np.arange(len(x)) if len(ind) == len(x): return np.array([]) y = np.zeros((len(x)+2,), x.dtype) y[1:-1] = x dif = np.diff(y) up = (dif == 1).nonzero()[0] dn = (dif == -1).nonzero()[0] i = (dn-up == max(dn - up)).nonzero()[0][0] ind = np.arange(up[i], dn[i]) return ind def longest_ones(x): '''alias for longest_contiguous_ones''' return longest_contiguous_ones(x) class PCA(object): def __init__(self, a, standardize=True): """ compute the SVD of a and store data for PCA. Use project to project the data onto a reduced set of dimensions Parameters ---------- a : np.ndarray A numobservations x numdims array standardize : bool True if input data are to be standardized. If False, only centering will be carried out. Attributes ---------- a A centered unit sigma version of input a. numrows, numcols The dimensions of a. mu A numdims array of means of a. This is the vector that points to the origin of PCA space. sigma A numdims array of standard deviation of a. fracs The proportion of variance of each of the principal components. s The actual eigenvalues of the decomposition. Wt The weight vector for projecting a numdims point or array into PCA space. Y A projected into PCA space. Notes ----- The factor loadings are in the Wt factor, i.e., the factor loadings for the first principal component are given by Wt[0]. This row is also the first eigenvector. """ n, m = a.shape if n < m: raise RuntimeError('we assume data in a is organized with ' 'numrows>numcols') self.numrows, self.numcols = n, m self.mu = a.mean(axis=0) self.sigma = a.std(axis=0) self.standardize = standardize a = self.center(a) self.a = a U, s, Vh = np.linalg.svd(a, full_matrices=False) # Note: .H indicates the conjugate transposed / Hermitian. # The SVD is commonly written as a = U s V.H. # If U is a unitary matrix, it means that it satisfies U.H = inv(U). # The rows of Vh are the eigenvectors of a.H a. # The columns of U are the eigenvectors of a a.H. # For row i in Vh and column i in U, the corresponding eigenvalue is # s[i]**2. self.Wt = Vh # save the transposed coordinates Y = np.dot(Vh, a.T).T self.Y = Y # save the eigenvalues self.s = s**2 # and now the contribution of the individual components vars = self.s/float(len(s)) self.fracs = vars/vars.sum() def project(self, x, minfrac=0.): ''' project x onto the principle axes, dropping any axes where fraction of variance= minfrac if x.ndim == 2: Yreduced = Y[:, mask] else: Yreduced = Y[mask] return Yreduced def center(self, x): ''' center and optionally standardize the data using the mean and sigma from training set a ''' if self.standardize: return (x - self.mu)/self.sigma else: return (x - self.mu) @staticmethod def _get_colinear(): c0 = np.array([ 0.19294738, 0.6202667, 0.45962655, 0.07608613, 0.135818, 0.83580842, 0.07218851, 0.48318321, 0.84472463, 0.18348462, 0.81585306, 0.96923926, 0.12835919, 0.35075355, 0.15807861, 0.837437, 0.10824303, 0.1723387, 0.43926494, 0.83705486]) c1 = np.array([ -1.17705601, -0.513883, -0.26614584, 0.88067144, 1.00474954, -1.1616545, 0.0266109, 0.38227157, 1.80489433, 0.21472396, -1.41920399, -2.08158544, -0.10559009, 1.68999268, 0.34847107, -0.4685737, 1.23980423, -0.14638744, -0.35907697, 0.22442616]) c2 = c0 + 2*c1 c3 = -3*c0 + 4*c1 a = np.array([c3, c0, c1, c2]).T return a def prctile(x, p=(0.0, 25.0, 50.0, 75.0, 100.0)): """ Return the percentiles of *x*. *p* can either be a sequence of percentile values or a scalar. If *p* is a sequence, the ith element of the return sequence is the *p*(i)-th percentile of *x*. If *p* is a scalar, the largest value of *x* less than or equal to the *p* percentage point in the sequence is returned. """ # This implementation derived from scipy.stats.scoreatpercentile def _interpolate(a, b, fraction): """Returns the point at the given fraction between a and b, where 'fraction' must be between 0 and 1. """ return a + (b - a) * fraction per = np.array(p) values = np.sort(x, axis=None) idxs = per / 100 * (values.shape[0] - 1) ai = idxs.astype(int) bi = ai + 1 frac = idxs % 1 # handle cases where attempting to interpolate past last index cond = bi >= len(values) if per.ndim: ai[cond] -= 1 bi[cond] -= 1 frac[cond] += 1 else: if cond: ai -= 1 bi -= 1 frac += 1 return _interpolate(values[ai], values[bi], frac) def prctile_rank(x, p): """ Return the rank for each element in *x*, return the rank 0..len(*p*). e.g., if *p* = (25, 50, 75), the return value will be a len(*x*) array with values in [0,1,2,3] where 0 indicates the value is less than the 25th percentile, 1 indicates the value is >= the 25th and < 50th percentile, ... and 3 indicates the value is above the 75th percentile cutoff. *p* is either an array of percentiles in [0..100] or a scalar which indicates how many quantiles of data you want ranked. """ if not cbook.iterable(p): p = np.arange(100.0/p, 100.0, 100.0/p) else: p = np.asarray(p) if p.max() <= 1 or p.min() < 0 or p.max() > 100: raise ValueError('percentiles should be in range 0..100, not 0..1') ptiles = prctile(x, p) return np.searchsorted(ptiles, x) def center_matrix(M, dim=0): """ Return the matrix *M* with each row having zero mean and unit std. If *dim* = 1 operate on columns instead of rows. (*dim* is opposite to the numpy axis kwarg.) """ M = np.asarray(M, float) if dim: M = (M - M.mean(axis=0)) / M.std(axis=0) else: M = (M - M.mean(axis=1)[:, np.newaxis]) M = M / M.std(axis=1)[:, np.newaxis] return M def rk4(derivs, y0, t): """ Integrate 1D or ND system of ODEs using 4-th order Runge-Kutta. This is a toy implementation which may be useful if you find yourself stranded on a system w/o scipy. Otherwise use :func:scipy.integrate. Parameters ---------- y0 initial state vector t sample times derivs returns the derivative of the system and has the signature dy = derivs(yi, ti) Examples -------- A 2D system:: def derivs6(x,t): d1 = x[0] + 2*x[1] d2 = -3*x[0] + 4*x[1] return (d1, d2) dt = 0.0005 t = arange(0.0, 2.0, dt) y0 = (1,2) yout = rk4(derivs6, y0, t) A 1D system:: alpha = 2 def derivs(x,t): return -alpha*x + exp(-t) y0 = 1 yout = rk4(derivs, y0, t) If you have access to scipy, you should probably be using the scipy.integrate tools rather than this function. """ try: Ny = len(y0) except TypeError: yout = np.zeros((len(t),), float) else: yout = np.zeros((len(t), Ny), float) yout[0] = y0 i = 0 for i in np.arange(len(t)-1): thist = t[i] dt = t[i+1] - thist dt2 = dt/2.0 y0 = yout[i] k1 = np.asarray(derivs(y0, thist)) k2 = np.asarray(derivs(y0 + dt2*k1, thist+dt2)) k3 = np.asarray(derivs(y0 + dt2*k2, thist+dt2)) k4 = np.asarray(derivs(y0 + dt*k3, thist+dt)) yout[i+1] = y0 + dt/6.0*(k1 + 2*k2 + 2*k3 + k4) return yout def bivariate_normal(X, Y, sigmax=1.0, sigmay=1.0, mux=0.0, muy=0.0, sigmaxy=0.0): """ Bivariate Gaussian distribution for equal shape *X*, *Y*. See bivariate normal _ at mathworld. """ Xmu = X-mux Ymu = Y-muy rho = sigmaxy/(sigmax*sigmay) z = Xmu**2/sigmax**2 + Ymu**2/sigmay**2 - 2*rho*Xmu*Ymu/(sigmax*sigmay) denom = 2*np.pi*sigmax*sigmay*np.sqrt(1-rho**2) return np.exp(-z/(2*(1-rho**2))) / denom def get_xyz_where(Z, Cond): """ *Z* and *Cond* are *M* x *N* matrices. *Z* are data and *Cond* is a boolean matrix where some condition is satisfied. Return value is (*x*, *y*, *z*) where *x* and *y* are the indices into *Z* and *z* are the values of *Z* at those indices. *x*, *y*, and *z* are 1D arrays. """ X, Y = np.indices(Z.shape) return X[Cond], Y[Cond], Z[Cond] def get_sparse_matrix(M, N, frac=0.1): """ Return a *M* x *N* sparse matrix with *frac* elements randomly filled. """ data = np.zeros((M, N))*0. for i in range(int(M*N*frac)): x = np.random.randint(0, M-1) y = np.random.randint(0, N-1) data[x, y] = np.random.rand() return data def dist(x, y): """ Return the distance between two points. """ d = x-y return np.sqrt(np.dot(d, d)) def dist_point_to_segment(p, s0, s1): """ Get the distance of a point to a segment. *p*, *s0*, *s1* are *xy* sequences This algorithm from http://geomalgorithms.com/a02-_lines.html """ p = np.asarray(p, float) s0 = np.asarray(s0, float) s1 = np.asarray(s1, float) v = s1 - s0 w = p - s0 c1 = np.dot(w, v) if c1 <= 0: return dist(p, s0) c2 = np.dot(v, v) if c2 <= c1: return dist(p, s1) b = c1 / c2 pb = s0 + b * v return dist(p, pb) def segments_intersect(s1, s2): """ Return *True* if *s1* and *s2* intersect. *s1* and *s2* are defined as:: s1: (x1, y1), (x2, y2) s2: (x3, y3), (x4, y4) """ (x1, y1), (x2, y2) = s1 (x3, y3), (x4, y4) = s2 den = ((y4-y3) * (x2-x1)) - ((x4-x3)*(y2-y1)) n1 = ((x4-x3) * (y1-y3)) - ((y4-y3)*(x1-x3)) n2 = ((x2-x1) * (y1-y3)) - ((y2-y1)*(x1-x3)) if den == 0: # lines parallel return False u1 = n1/den u2 = n2/den return 0.0 <= u1 <= 1.0 and 0.0 <= u2 <= 1.0 def fftsurr(x, detrend=detrend_none, window=window_none): """ Compute an FFT phase randomized surrogate of *x*. """ if cbook.iterable(window): x = window*detrend(x) else: x = window(detrend(x)) z = np.fft.fft(x) a = 2.*np.pi*1j phase = a * np.random.rand(len(x)) z = z*np.exp(phase) return np.fft.ifft(z).real def movavg(x, n): """ Compute the len(*n*) moving average of *x*. """ w = np.empty((n,), dtype=float) w[:] = 1.0/n return np.convolve(x, w, mode='valid') # the following code was written and submitted by Fernando Perez # from the ipython numutils package under a BSD license # begin fperez functions """ A set of convenient utilities for numerical work. Most of this module requires numpy or is meant to be used with it. Copyright (c) 2001-2004, Fernando Perez. All rights reserved. This license was generated from the BSD license template as found in: http://www.opensource.org/licenses/bsd-license.php Redistribution and use in source and binary forms, with or without modification, are permitted provided that the following conditions are met: * Redistributions of source code must retain the above copyright notice, this list of conditions and the following disclaimer. * Redistributions in binary form must reproduce the above copyright notice, this list of conditions and the following disclaimer in the documentation and/or other materials provided with the distribution. * Neither the name of the IPython project nor the names of its contributors may be used to endorse or promote products derived from this software without specific prior written permission. THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT, INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING, BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE POSSIBILITY OF SUCH DAMAGE. """ # ***************************************************************************** # Globals # **************************************************************************** # function definitions exp_safe_MIN = math.log(2.2250738585072014e-308) exp_safe_MAX = 1.7976931348623157e+308 def exp_safe(x): """ Compute exponentials which safely underflow to zero. Slow, but convenient to use. Note that numpy provides proper floating point exception handling with access to the underlying hardware. """ if type(x) is np.ndarray: return np.exp(np.clip(x, exp_safe_MIN, exp_safe_MAX)) else: return math.exp(x) def amap(fn, *args): """ amap(function, sequence[, sequence, ...]) -> array. Works like :func:map, but it returns an array. This is just a convenient shorthand for numpy.array(map(...)). """ return np.array(list(map(fn, *args))) def rms_flat(a): """ Return the root mean square of all the elements of *a*, flattened out. """ return np.sqrt(np.mean(np.abs(a) ** 2)) def l1norm(a): """ Return the *l1* norm of *a*, flattened out. Implemented as a separate function (not a call to :func:norm for speed). """ return np.sum(np.abs(a)) def l2norm(a): """ Return the *l2* norm of *a*, flattened out. Implemented as a separate function (not a call to :func:norm for speed). """ return np.sqrt(np.sum(np.abs(a) ** 2)) def norm_flat(a, p=2): """ norm(a,p=2) -> l-p norm of a.flat Return the l-p norm of *a*, considered as a flat array. This is NOT a true matrix norm, since arrays of arbitrary rank are always flattened. *p* can be a number or the string 'Infinity' to get the L-infinity norm. """ # This function was being masked by a more general norm later in # the file. We may want to simply delete it. if p == 'Infinity': return np.max(np.abs(a)) else: return np.sum(np.abs(a) ** p) ** (1 / p) def frange(xini, xfin=None, delta=None, **kw): """ frange([start,] stop[, step, keywords]) -> array of floats Return a numpy ndarray containing a progression of floats. Similar to :func:numpy.arange, but defaults to a closed interval. frange(x0, x1) returns [x0, x0+1, x0+2, ..., x1]; *start* defaults to 0, and the endpoint *is included*. This behavior is different from that of :func:range and :func:numpy.arange. This is deliberate, since :func:frange will probably be more useful for generating lists of points for function evaluation, and endpoints are often desired in this use. The usual behavior of :func:range can be obtained by setting the keyword *closed* = 0, in this case, :func:frange basically becomes :func:numpy.arange. When *step* is given, it specifies the increment (or decrement). All arguments can be floating point numbers. frange(x0,x1,d) returns [x0,x0+d,x0+2d,...,xfin] where *xfin* <= *x1*. :func:frange can also be called with the keyword *npts*. This sets the number of points the list should contain (and overrides the value *step* might have been given). :func:numpy.arange doesn't offer this option. Examples:: >>> frange(3) array([ 0., 1., 2., 3.]) >>> frange(3,closed=0) array([ 0., 1., 2.]) >>> frange(1,6,2) array([1, 3, 5]) or 1,3,5,7, depending on floating point vagueries >>> frange(1,6.5,npts=5) array([ 1. , 2.375, 3.75 , 5.125, 6.5 ]) """ # defaults kw.setdefault('closed', 1) endpoint = kw['closed'] != 0 # funny logic to allow the *first* argument to be optional (like range()) # This was modified with a simpler version from a similar frange() found # at http://aspn.activestate.com/ASPN/Cookbook/Python/Recipe/66472 if xfin is None: xfin = xini + 0.0 xini = 0.0 if delta is None: delta = 1.0 # compute # of points, spacing and return final list try: npts = kw['npts'] delta = (xfin-xini)/float(npts-endpoint) except KeyError: npts = int(np.round((xfin-xini)/delta)) + endpoint # round finds the nearest, so the endpoint can be up to # delta/2 larger than xfin. return np.arange(npts)*delta+xini # end frange() def identity(n, rank=2, dtype='l', typecode=None): """ Returns the identity matrix of shape (*n*, *n*, ..., *n*) (rank *r*). For ranks higher than 2, this object is simply a multi-index Kronecker delta:: / 1 if i0=i1=...=iR, id[i0,i1,...,iR] = -| \\ 0 otherwise. Optionally a *dtype* (or typecode) may be given (it defaults to 'l'). Since rank defaults to 2, this function behaves in the default case (when only *n* is given) like numpy.identity(n) -- but surprisingly, it is much faster. """ if typecode is not None: dtype = typecode iden = np.zeros((n,)*rank, dtype) for i in range(n): idx = (i,)*rank iden[idx] = 1 return iden def base_repr(number, base=2, padding=0): """ Return the representation of a *number* in any given *base*. """ chars = '0123456789ABCDEFGHIJKLMNOPQRSTUVWXYZ' if number < base: return (padding - 1) * chars[0] + chars[int(number)] max_exponent = int(math.log(number)/math.log(base)) max_power = long(base) ** max_exponent lead_digit = int(number/max_power) return (chars[lead_digit] + base_repr(number - max_power * lead_digit, base, max(padding - 1, max_exponent))) def binary_repr(number, max_length=1025): """ Return the binary representation of the input *number* as a string. This is more efficient than using :func:base_repr with base 2. Increase the value of max_length for very large numbers. Note that on 32-bit machines, 2**1023 is the largest integer power of 2 which can be converted to a Python float. """ # assert number < 2L << max_length shifts = map(operator.rshift, max_length * [number], range(max_length - 1, -1, -1)) digits = list(map(operator.mod, shifts, max_length * [2])) if not digits.count(1): return 0 digits = digits[digits.index(1):] return ''.join(map(repr, digits)).replace('L', '') def log2(x, ln2=math.log(2.0)): """ Return the log(*x*) in base 2. This is a _slow_ function but which is guaranteed to return the correct integer value if the input is an integer exact power of 2. """ try: bin_n = binary_repr(x)[1:] except (AssertionError, TypeError): return math.log(x)/ln2 else: if '1' in bin_n: return math.log(x)/ln2 else: return len(bin_n) def ispower2(n): """ Returns the log base 2 of *n* if *n* is a power of 2, zero otherwise. Note the potential ambiguity if *n* == 1: 2**0 == 1, interpret accordingly. """ bin_n = binary_repr(n)[1:] if '1' in bin_n: return 0 else: return len(bin_n) def isvector(X): """ Like the MATLAB function with the same name, returns *True* if the supplied numpy array or matrix *X* looks like a vector, meaning it has a one non-singleton axis (i.e., it can have multiple axes, but all must have length 1, except for one of them). If you just want to see if the array has 1 axis, use X.ndim == 1. """ return np.prod(X.shape) == np.max(X.shape) # end fperez numutils code # helpers for loading, saving, manipulating and viewing numpy record arrays def safe_isnan(x): ':func:numpy.isnan for arbitrary types' if isinstance(x, six.string_types): return False try: b = np.isnan(x) except NotImplementedError: return False except TypeError: return False else: return b def safe_isinf(x): ':func:numpy.isinf for arbitrary types' if isinstance(x, six.string_types): return False try: b = np.isinf(x) except NotImplementedError: return False except TypeError: return False else: return b def rec_append_fields(rec, names, arrs, dtypes=None): """ Return a new record array with field names populated with data from arrays in *arrs*. If appending a single field, then *names*, *arrs* and *dtypes* do not have to be lists. They can just be the values themselves. """ if (not isinstance(names, six.string_types) and cbook.iterable(names) and len(names) and isinstance(names[0], six.string_types)): if len(names) != len(arrs): raise ValueError("number of arrays do not match number of names") else: # we have only 1 name and 1 array names = [names] arrs = [arrs] arrs = list(map(np.asarray, arrs)) if dtypes is None: dtypes = [a.dtype for a in arrs] elif not cbook.iterable(dtypes): dtypes = [dtypes] if len(arrs) != len(dtypes): if len(dtypes) == 1: dtypes = dtypes * len(arrs) else: raise ValueError("dtypes must be None, a single dtype or a list") old_dtypes = rec.dtype.descr if six.PY2: old_dtypes = [(name.encode('utf-8'), dt) for name, dt in old_dtypes] newdtype = np.dtype(old_dtypes + list(zip(names, dtypes))) newrec = np.recarray(rec.shape, dtype=newdtype) for field in rec.dtype.fields: newrec[field] = rec[field] for name, arr in zip(names, arrs): newrec[name] = arr return newrec def rec_drop_fields(rec, names): """ Return a new numpy record array with fields in *names* dropped. """ names = set(names) newdtype = np.dtype([(name, rec.dtype[name]) for name in rec.dtype.names if name not in names]) newrec = np.recarray(rec.shape, dtype=newdtype) for field in newdtype.names: newrec[field] = rec[field] return newrec def rec_keep_fields(rec, names): """ Return a new numpy record array with only fields listed in names """ if isinstance(names, six.string_types): names = names.split(',') arrays = [] for name in names: arrays.append(rec[name]) return np.rec.fromarrays(arrays, names=names) def rec_groupby(r, groupby, stats): """ *r* is a numpy record array *groupby* is a sequence of record array attribute names that together form the grouping key. e.g., ('date', 'productcode') *stats* is a sequence of (*attr*, *func*, *outname*) tuples which will call x = func(attr) and assign *x* to the record array output with attribute *outname*. For example:: stats = ( ('sales', len, 'numsales'), ('sales', np.mean, 'avgsale') ) Return record array has *dtype* names for each attribute name in the *groupby* argument, with the associated group values, and for each outname name in the *stats* argument, with the associated stat summary output. """ # build a dictionary from groupby keys-> list of indices into r with # those keys rowd = {} for i, row in enumerate(r): key = tuple([row[attr] for attr in groupby]) rowd.setdefault(key, []).append(i) rows = [] # sort the output by groupby keys for key in sorted(rowd): row = list(key) # get the indices for this groupby key ind = rowd[key] thisr = r[ind] # call each stat function for this groupby slice row.extend([func(thisr[attr]) for attr, func, outname in stats]) rows.append(row) # build the output record array with groupby and outname attributes attrs, funcs, outnames = list(zip(*stats)) names = list(groupby) names.extend(outnames) return np.rec.fromrecords(rows, names=names) def rec_summarize(r, summaryfuncs): """ *r* is a numpy record array *summaryfuncs* is a list of (*attr*, *func*, *outname*) tuples which will apply *func* to the array *r*[attr] and assign the output to a new attribute name *outname*. The returned record array is identical to *r*, with extra arrays for each element in *summaryfuncs*. """ names = list(r.dtype.names) arrays = [r[name] for name in names] for attr, func, outname in summaryfuncs: names.append(outname) arrays.append(np.asarray(func(r[attr]))) return np.rec.fromarrays(arrays, names=names) def rec_join(key, r1, r2, jointype='inner', defaults=None, r1postfix='1', r2postfix='2'): """ Join record arrays *r1* and *r2* on *key*; *key* is a tuple of field names -- if *key* is a string it is assumed to be a single attribute name. If *r1* and *r2* have equal values on all the keys in the *key* tuple, then their fields will be merged into a new record array containing the intersection of the fields of *r1* and *r2*. *r1* (also *r2*) must not have any duplicate keys. The *jointype* keyword can be 'inner', 'outer', 'leftouter'. To do a rightouter join just reverse *r1* and *r2*. The *defaults* keyword is a dictionary filled with {column_name:default_value} pairs. The keywords *r1postfix* and *r2postfix* are postfixed to column names (other than keys) that are both in *r1* and *r2*. """ if isinstance(key, six.string_types): key = (key, ) for name in key: if name not in r1.dtype.names: raise ValueError('r1 does not have key field %s' % name) if name not in r2.dtype.names: raise ValueError('r2 does not have key field %s' % name) def makekey(row): return tuple([row[name] for name in key]) r1d = {makekey(row): i for i, row in enumerate(r1)} r2d = {makekey(row): i for i, row in enumerate(r2)} r1keys = set(r1d) r2keys = set(r2d) common_keys = r1keys & r2keys r1ind = np.array([r1d[k] for k in common_keys]) r2ind = np.array([r2d[k] for k in common_keys]) common_len = len(common_keys) left_len = right_len = 0 if jointype == "outer" or jointype == "leftouter": left_keys = r1keys.difference(r2keys) left_ind = np.array([r1d[k] for k in left_keys]) left_len = len(left_ind) if jointype == "outer": right_keys = r2keys.difference(r1keys) right_ind = np.array([r2d[k] for k in right_keys]) right_len = len(right_ind) def key_desc(name): ''' if name is a string key, use the larger size of r1 or r2 before merging ''' dt1 = r1.dtype[name] if dt1.type != np.string_: return (name, dt1.descr[0][1]) dt2 = r2.dtype[name] if dt1 != dt2: raise ValueError("The '{}' fields in arrays 'r1' and 'r2' must " "have the same dtype".format(name)) if dt1.num > dt2.num: return (name, dt1.descr[0][1]) else: return (name, dt2.descr[0][1]) keydesc = [key_desc(name) for name in key] def mapped_r1field(name): """ The column name in *newrec* that corresponds to the column in *r1*. """ if name in key or name not in r2.dtype.names: return name else: return name + r1postfix def mapped_r2field(name): """ The column name in *newrec* that corresponds to the column in *r2*. """ if name in key or name not in r1.dtype.names: return name else: return name + r2postfix r1desc = [(mapped_r1field(desc[0]), desc[1]) for desc in r1.dtype.descr if desc[0] not in key] r2desc = [(mapped_r2field(desc[0]), desc[1]) for desc in r2.dtype.descr if desc[0] not in key] all_dtypes = keydesc + r1desc + r2desc if six.PY2: all_dtypes = [(name.encode('utf-8'), dt) for name, dt in all_dtypes] newdtype = np.dtype(all_dtypes) newrec = np.recarray((common_len + left_len + right_len,), dtype=newdtype) if defaults is not None: for thiskey in defaults: if thiskey not in newdtype.names: warnings.warn('rec_join defaults key="%s" not in new dtype ' 'names "%s"' % (thiskey, newdtype.names)) for name in newdtype.names: dt = newdtype[name] if dt.kind in ('f', 'i'): newrec[name] = 0 if jointype != 'inner' and defaults is not None: # fill in the defaults enmasse newrec_fields = list(newrec.dtype.fields) for k, v in six.iteritems(defaults): if k in newrec_fields: newrec[k] = v for field in r1.dtype.names: newfield = mapped_r1field(field) if common_len: newrec[newfield][:common_len] = r1[field][r1ind] if (jointype == "outer" or jointype == "leftouter") and left_len: newrec[newfield][common_len:(common_len+left_len)] = ( r1[field][left_ind] ) for field in r2.dtype.names: newfield = mapped_r2field(field) if field not in key and common_len: newrec[newfield][:common_len] = r2[field][r2ind] if jointype == "outer" and right_len: newrec[newfield][-right_len:] = r2[field][right_ind] newrec.sort(order=key) return newrec def recs_join(key, name, recs, jointype='outer', missing=0., postfixes=None): """ Join a sequence of record arrays on single column key. This function only joins a single column of the multiple record arrays *key* is the column name that acts as a key *name* is the name of the column that we want to join *recs* is a list of record arrays to join *jointype* is a string 'inner' or 'outer' *missing* is what any missing field is replaced by *postfixes* if not None, a len recs sequence of postfixes returns a record array with columns [rowkey, name0, name1, ... namen-1]. or if postfixes [PF0, PF1, ..., PFN-1] are supplied, [rowkey, namePF0, namePF1, ... namePFN-1]. Example:: r = recs_join("date", "close", recs=[r0, r1], missing=0.) """ results = [] aligned_iters = cbook.align_iterators(operator.attrgetter(key), *[iter(r) for r in recs]) def extract(r): if r is None: return missing else: return r[name] if jointype == "outer": for rowkey, row in aligned_iters: results.append([rowkey] + list(map(extract, row))) elif jointype == "inner": for rowkey, row in aligned_iters: if None not in row: # throw out any Nones results.append([rowkey] + list(map(extract, row))) if postfixes is None: postfixes = ['%d' % i for i in range(len(recs))] names = ",".join([key] + ["%s%s" % (name, postfix) for postfix in postfixes]) return np.rec.fromrecords(results, names=names) 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"""~!@#\$%^&*()-=+~\|]}[{';: /?.>,<""") 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, six.string_types): 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 approriate 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 # a series of classes for describing the format intentions of various rec views class FormatObj(object): def tostr(self, x): return self.toval(x) def toval(self, x): return str(x) def fromstr(self, s): return s def __hash__(self): """ override the hash function of any of the formatters, so that we don't create duplicate excel format styles """ return hash(self.__class__) class FormatString(FormatObj): def tostr(self, x): val = repr(x) return val[1:-1] class FormatFormatStr(FormatObj): def __init__(self, fmt): self.fmt = fmt def tostr(self, x): if x is None: return 'None' return self.fmt % self.toval(x) class FormatFloat(FormatFormatStr): def __init__(self, precision=4, scale=1.): FormatFormatStr.__init__(self, '%%1.%df' % precision) self.precision = precision self.scale = scale def __hash__(self): return hash((self.__class__, self.precision, self.scale)) def toval(self, x): if x is not None: x = x * self.scale return x def fromstr(self, s): return float(s)/self.scale class FormatInt(FormatObj): def tostr(self, x): return '%d' % int(x) def toval(self, x): return int(x) def fromstr(self, s): return int(s) class FormatBool(FormatObj): def toval(self, x): return str(x) def fromstr(self, s): return bool(s) class FormatPercent(FormatFloat): def __init__(self, precision=4): FormatFloat.__init__(self, precision, scale=100.) class FormatThousands(FormatFloat): def __init__(self, precision=4): FormatFloat.__init__(self, precision, scale=1e-3) class FormatMillions(FormatFloat): def __init__(self, precision=4): FormatFloat.__init__(self, precision, scale=1e-6) class FormatDate(FormatObj): def __init__(self, fmt): self.fmt = fmt def __hash__(self): return hash((self.__class__, self.fmt)) def toval(self, x): if x is None: return 'None' return x.strftime(self.fmt) def fromstr(self, x): import dateutil.parser return dateutil.parser.parse(x).date() class FormatDatetime(FormatDate): def __init__(self, fmt='%Y-%m-%d %H:%M:%S'): FormatDate.__init__(self, fmt) def fromstr(self, x): import dateutil.parser return dateutil.parser.parse(x) defaultformatd = { np.bool_: FormatBool(), np.int16: FormatInt(), np.int32: FormatInt(), np.int64: FormatInt(), np.float32: FormatFloat(), np.float64: FormatFloat(), np.object_: FormatObj(), np.string_: FormatString(), } def get_formatd(r, formatd=None): 'build a formatd guaranteed to have a key for every dtype name' if formatd is None: formatd = dict() for i, name in enumerate(r.dtype.names): dt = r.dtype[name] format = formatd.get(name) if format is None: format = defaultformatd.get(dt.type, FormatObj()) formatd[name] = format return formatd def csvformat_factory(format): format = copy.deepcopy(format) if isinstance(format, FormatFloat): format.scale = 1. # override scaling for storage format.fmt = '%r' return format def rec2txt(r, header=None, padding=3, precision=3, fields=None): """ Returns a textual representation of a record array. Parameters ---------- r: numpy recarray header: list column headers padding: space between each column precision: number of decimal places to use for floats. Set to an integer to apply to all floats. Set to a list of integers to apply precision individually. Precision for non-floats is simply ignored. fields : list If not None, a list of field names to print. fields can be a list of strings like ['field1', 'field2'] or a single comma separated string like 'field1,field2' Examples -------- For precision=[0,2,3], the output is :: ID Price Return ABC 12.54 0.234 XYZ 6.32 -0.076 """ if fields is not None: r = rec_keep_fields(r, fields) if cbook.is_numlike(precision): precision = [precision]*len(r.dtype) def get_type(item, atype=int): tdict = {None: int, int: float, float: str} try: atype(str(item)) except: return get_type(item, tdict[atype]) return atype def get_justify(colname, column, precision): ntype = column.dtype if np.issubdtype(ntype, np.character): fixed_width = int(ntype.str[2:]) length = max(len(colname), fixed_width) return 0, length+padding, "%s" # left justify if np.issubdtype(ntype, np.integer): length = max(len(colname), np.max(list(map(len, list(map(str, column)))))) return 1, length+padding, "%d" # right justify if np.issubdtype(ntype, np.floating): fmt = "%." + str(precision) + "f" length = max( len(colname), np.max(list(map(len, list(map(lambda x: fmt % x, column))))) ) return 1, length+padding, fmt # right justify return (0, max(len(colname), np.max(list(map(len, list(map(str, column))))))+padding, "%s") if header is None: header = r.dtype.names justify_pad_prec = [get_justify(header[i], r.__getitem__(colname), precision[i]) for i, colname in enumerate(r.dtype.names)] justify_pad_prec_spacer = [] for i in range(len(justify_pad_prec)): just, pad, prec = justify_pad_prec[i] if i == 0: justify_pad_prec_spacer.append((just, pad, prec, 0)) else: pjust, ppad, pprec = justify_pad_prec[i-1] if pjust == 0 and just == 1: justify_pad_prec_spacer.append((just, pad-padding, prec, 0)) elif pjust == 1 and just == 0: justify_pad_prec_spacer.append((just, pad, prec, padding)) else: justify_pad_prec_spacer.append((just, pad, prec, 0)) def format(item, just_pad_prec_spacer): just, pad, prec, spacer = just_pad_prec_spacer if just == 0: return spacer*' ' + str(item).ljust(pad) else: if get_type(item) == float: item = (prec % float(item)) elif get_type(item) == int: item = (prec % int(item)) return item.rjust(pad) textl = [] textl.append(''.join([format(colitem, justify_pad_prec_spacer[j]) for j, colitem in enumerate(header)])) for i, row in enumerate(r): textl.append(''.join([format(colitem, justify_pad_prec_spacer[j]) for j, colitem in enumerate(row)])) if i == 0: textl[0] = textl[0].rstrip() text = os.linesep.join(textl) return text def rec2csv(r, fname, delimiter=',', formatd=None, missing='', missingd=None, withheader=True): """ Save the data from numpy recarray *r* into a comma-/space-/tab-delimited file. The record array dtype names will be used for column headers. *fname*: can be a filename or a file handle. Support for gzipped files is automatic, if the filename ends in '.gz' *withheader*: if withheader is False, do not write the attribute names in the first row for formatd type FormatFloat, we override the precision to store full precision floats in the CSV file See Also -------- :func:csv2rec For information about *missing* and *missingd*, which can be used to fill in masked values into your CSV file. """ delimiter = str(delimiter) if missingd is None: missingd = dict() def with_mask(func): def newfunc(val, mask, mval): if mask: return mval else: return func(val) return newfunc if r.ndim != 1: raise ValueError('rec2csv only operates on 1 dimensional recarrays') formatd = get_formatd(r, formatd) funcs = [] for i, name in enumerate(r.dtype.names): funcs.append(with_mask(csvformat_factory(formatd[name]).tostr)) fh, opened = cbook.to_filehandle(fname, 'wb', return_opened=True) writer = csv.writer(fh, delimiter=delimiter) header = r.dtype.names if withheader: writer.writerow(header) # Our list of specials for missing values mvals = [] for name in header: mvals.append(missingd.get(name, missing)) ismasked = False if len(r): row = r[0] ismasked = hasattr(row, '_fieldmask') for row in r: if ismasked: row, rowmask = row.item(), row._fieldmask.item() else: rowmask = [False] * len(row) writer.writerow([func(val, mask, mval) for func, val, mask, mval in zip(funcs, row, rowmask, mvals)]) if opened: fh.close() def griddata(x, y, z, xi, yi, interp='nn'): """Interpolates from a nonuniformly spaced grid to some other grid. Fits a surface of the form z = f(x, y) to the data in the (usually) nonuniformly spaced vectors (x, y, z), then interpolates this surface at the points specified by (xi, yi) to produce zi. Parameters ---------- x, y, z : 1d array_like Coordinates of grid points to interpolate from. xi, yi : 1d or 2d array_like Coordinates of grid points to interpolate to. interp : string key from {'nn', 'linear'} Interpolation algorithm, either 'nn' for natural neighbor, or 'linear' for linear interpolation. Returns ------- 2d float array Array of values interpolated at (xi, yi) points. Array will be masked is any of (xi, yi) are outside the convex hull of (x, y). Notes ----- If interp is 'nn' (the default), uses natural neighbor interpolation based on Delaunay triangulation. This option is only available if the mpl_toolkits.natgrid module is installed. This can be downloaded from https://github.com/matplotlib/natgrid. The (xi, yi) grid must be regular and monotonically increasing in this case. If interp is 'linear', linear interpolation is used via matplotlib.tri.LinearTriInterpolator. Instead of using griddata, more flexible functionality and other interpolation options are available using a matplotlib.tri.Triangulation and a matplotlib.tri.TriInterpolator. """ # Check input arguments. x = np.asanyarray(x, dtype=np.float64) y = np.asanyarray(y, dtype=np.float64) z = np.asanyarray(z, dtype=np.float64) if x.shape != y.shape or x.shape != z.shape or x.ndim != 1: raise ValueError("x, y and z must be equal-length 1-D arrays") xi = np.asanyarray(xi, dtype=np.float64) yi = np.asanyarray(yi, dtype=np.float64) if xi.ndim != yi.ndim: raise ValueError("xi and yi must be arrays with the same number of " "dimensions (1 or 2)") if xi.ndim == 2 and xi.shape != yi.shape: raise ValueError("if xi and yi are 2D arrays, they must have the same " "shape") if xi.ndim == 1: xi, yi = np.meshgrid(xi, yi) if interp == 'nn': use_nn_interpolation = True elif interp == 'linear': use_nn_interpolation = False else: raise ValueError("interp keyword must be one of 'linear' (for linear " "interpolation) or 'nn' (for natural neighbor " "interpolation). Default is 'nn'.") # Remove masked points. mask = np.ma.getmask(z) if mask is not np.ma.nomask: x = x.compress(~mask) y = y.compress(~mask) z = z.compressed() if use_nn_interpolation: try: from mpl_toolkits.natgrid import _natgrid except ImportError: raise RuntimeError( "To use interp='nn' (Natural Neighbor interpolation) in " "griddata, natgrid must be installed. Either install it " "from http://github.com/matplotlib/natgrid or use " "interp='linear' instead.") if xi.ndim == 2: # natgrid expects 1D xi and yi arrays. xi = xi[0, :] yi = yi[:, 0] # Override default natgrid internal parameters. _natgrid.seti(b'ext', 0) _natgrid.setr(b'nul', np.nan) if np.min(np.diff(xi)) < 0 or np.min(np.diff(yi)) < 0: raise ValueError("Output grid defined by xi,yi must be monotone " "increasing") # Allocate array for output (buffer will be overwritten by natgridd) zi = np.empty((yi.shape[0], xi.shape[0]), np.float64) # Natgrid requires each array to be contiguous rather than e.g. a view # that is a non-contiguous slice of another array. Use numpy.require # to deal with this, which will copy if necessary. x = np.require(x, requirements=['C']) y = np.require(y, requirements=['C']) z = np.require(z, requirements=['C']) xi = np.require(xi, requirements=['C']) yi = np.require(yi, requirements=['C']) _natgrid.natgridd(x, y, z, xi, yi, zi) # Mask points on grid outside convex hull of input data. if np.any(np.isnan(zi)): zi = np.ma.masked_where(np.isnan(zi), zi) return zi else: # Linear interpolation performed using a matplotlib.tri.Triangulation # and a matplotlib.tri.LinearTriInterpolator. from .tri import Triangulation, LinearTriInterpolator triang = Triangulation(x, y) interpolator = LinearTriInterpolator(triang, z) return interpolator(xi, yi) ################################################## # Linear interpolation algorithms ################################################## @cbook.deprecated("2.2", alternative="np.interp") def less_simple_linear_interpolation(x, y, xi, extrap=False): """ This function provides simple (but somewhat less so than :func:cbook.simple_linear_interpolation) linear interpolation. :func:simple_linear_interpolation will give a list of point between a start and an end, while this does true linear interpolation at an arbitrary set of points. This is very inefficient linear interpolation meant to be used only for a small number of points in relatively non-intensive use cases. For real linear interpolation, use scipy. """ x = np.asarray(x) y = np.asarray(y) xi = np.atleast_1d(xi) s = list(y.shape) s[0] = len(xi) yi = np.tile(np.nan, s) for ii, xx in enumerate(xi): bb = x == xx if np.any(bb): jj, = np.nonzero(bb) yi[ii] = y[jj[0]] elif xx < x[0]: if extrap: yi[ii] = y[0] elif xx > x[-1]: if extrap: yi[ii] = y[-1] else: jj, = np.nonzero(x < xx) jj = max(jj) yi[ii] = y[jj] + (xx-x[jj])/(x[jj+1]-x[jj]) * (y[jj+1]-y[jj]) return yi def slopes(x, y): """ :func:slopes calculates the slope *y*'(*x*) The slope is estimated using the slope obtained from that of a parabola through any three consecutive points. This method should be superior to that described in the appendix of A CONSISTENTLY WELL BEHAVED METHOD OF INTERPOLATION by Russel W. Stineman (Creative Computing July 1980) in at least one aspect: Circles for interpolation demand a known aspect ratio between *x*- and *y*-values. For many functions, however, the abscissa are given in different dimensions, so an aspect ratio is completely arbitrary. The parabola method gives very similar results to the circle method for most regular cases but behaves much better in special cases. Norbert Nemec, Institute of Theoretical Physics, University or Regensburg, April 2006 Norbert.Nemec at physik.uni-regensburg.de (inspired by a original implementation by Halldor Bjornsson, Icelandic Meteorological Office, March 2006 halldor at vedur.is) """ # Cast key variables as float. x = np.asarray(x, float) y = np.asarray(y, float) yp = np.zeros(y.shape, float) dx = x[1:] - x[:-1] dy = y[1:] - y[:-1] dydx = dy/dx yp[1:-1] = (dydx[:-1] * dx[1:] + dydx[1:] * dx[:-1])/(dx[1:] + dx[:-1]) yp[0] = 2.0 * dy[0]/dx[0] - yp[1] yp[-1] = 2.0 * dy[-1]/dx[-1] - yp[-2] return yp def stineman_interp(xi, x, y, yp=None): """ Given data vectors *x* and *y*, the slope vector *yp* and a new abscissa vector *xi*, the function :func:stineman_interp uses Stineman interpolation to calculate a vector *yi* corresponding to *xi*. Here's an example that generates a coarse sine curve, then interpolates over a finer abscissa:: x = linspace(0,2*pi,20); y = sin(x); yp = cos(x) xi = linspace(0,2*pi,40); yi = stineman_interp(xi,x,y,yp); plot(x,y,'o',xi,yi) The interpolation method is described in the article A CONSISTENTLY WELL BEHAVED METHOD OF INTERPOLATION by Russell W. Stineman. The article appeared in the July 1980 issue of Creative Computing with a note from the editor stating that while they were: not an academic journal but once in a while something serious and original comes in adding that this was "apparently a real solution" to a well known problem. For *yp* = *None*, the routine automatically determines the slopes using the :func:slopes routine. *x* is assumed to be sorted in increasing order. For values xi[j] < x[0] or xi[j] > x[-1], the routine tries an extrapolation. The relevance of the data obtained from this, of course, is questionable... Original implementation by Halldor Bjornsson, Icelandic Meteorolocial Office, March 2006 halldor at vedur.is Completely reworked and optimized for Python by Norbert Nemec, Institute of Theoretical Physics, University or Regensburg, April 2006 Norbert.Nemec at physik.uni-regensburg.de """ # Cast key variables as float. x = np.asarray(x, float) y = np.asarray(y, float) if x.shape != y.shape: raise ValueError("'x' and 'y' must be of same shape") if yp is None: yp = slopes(x, y) else: yp = np.asarray(yp, float) xi = np.asarray(xi, float) yi = np.zeros(xi.shape, float) # calculate linear slopes dx = x[1:] - x[:-1] dy = y[1:] - y[:-1] s = dy/dx # note length of s is N-1 so last element is #N-2 # find the segment each xi is in # this line actually is the key to the efficiency of this implementation idx = np.searchsorted(x[1:-1], xi) # now we have generally: x[idx[j]] <= xi[j] <= x[idx[j]+1] # except at the boundaries, where it may be that xi[j] < x[0] or # xi[j] > x[-1] # the y-values that would come out from a linear interpolation: sidx = s.take(idx) xidx = x.take(idx) yidx = y.take(idx) xidxp1 = x.take(idx+1) yo = yidx + sidx * (xi - xidx) # the difference that comes when using the slopes given in yp # using the yp slope of the left point dy1 = (yp.take(idx) - sidx) * (xi - xidx) # using the yp slope of the right point dy2 = (yp.take(idx+1)-sidx) * (xi - xidxp1) dy1dy2 = dy1*dy2 # The following is optimized for Python. The solution actually # does more calculations than necessary but exploiting the power # of numpy, this is far more efficient than coding a loop by hand # in Python yi = yo + dy1dy2 * np.choose(np.array(np.sign(dy1dy2), np.int32)+1, ((2*xi-xidx-xidxp1)/((dy1-dy2)*(xidxp1-xidx)), 0.0, 1/(dy1+dy2),)) return yi 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.