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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?
<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)
def stride_repeat(x, n, axis=0):
'''
Repeat the values in an array in a memory-efficient manner. Array x is
stacked vertically n times.
.. warning::
It is not safe to write to the output array. Multiple
elements may point to the same piece of memory, so
modifying one value may change others.
Parameters
----------
x : 1D array or sequence
Array or sequence containing the data.
n : integer
The number of time to repeat the array.
axis : integer
The axis along which the data will run.
References
----------
`stackoverflow: Repeat NumPy array without replicating data?
<http://stackoverflow.com/a/5568169>`_
'''
if axis not in [0, 1]:
raise ValueError('axis must be 0 or 1')
x = np.asarray(x)
if x.ndim != 1:
raise ValueError('only 1-dimensional arrays can be used')
if n == 1:
if axis == 0:
return np.atleast_2d(x)
else:
return np.atleast_2d(x).T
if n < 1:
raise ValueError('n cannot be less than 1')
# np.lib.stride_tricks.as_strided easily leads to memory corruption for
# non integer shape and strides, i.e. n. See #3845.
n = int(n)
if axis == 0:
shape = (n, x.size)
strides = (0, x.strides[0])
else:
shape = (x.size, n)
strides = (x.strides[0], 0)
return np.lib.stride_tricks.as_strided(x, shape=shape, strides=strides)
def _spectral_helper(x, y=None, NFFT=None, Fs=None, detrend_func=None,
window=None, noverlap=None, pad_to=None,
sides=None, scale_by_freq=None, mode=None):
'''
This is a helper function that implements the commonality between the
psd, csd, spectrogram and complex, magnitude, angle, and phase spectrums.
It is *NOT* meant to be used outside of mlab and may change at any time.
'''
if y is None:
# if y is None use x for y
same_data = True
else:
# The checks for if y is x are so that we can use the same function to
# implement the core of psd(), csd(), and spectrogram() without doing
# extra calculations. We return the unaveraged Pxy, freqs, and t.
same_data = y is x
if Fs is None:
Fs = 2
if noverlap is None:
noverlap = 0
if detrend_func is None:
detrend_func = detrend_none
if window is None:
window = window_hanning
# if NFFT is set to None use the whole signal
if NFFT is None:
NFFT = 256
if mode is None or mode == 'default':
mode = 'psd'
elif mode not in ['psd', 'complex', 'magnitude', 'angle', 'phase']:
raise ValueError("Unknown value for mode %s, must be one of: "
"'default', 'psd', 'complex', "
"'magnitude', 'angle', 'phase'" % mode)
if not same_data and mode != 'psd':
raise ValueError("x and y must be equal if mode is not 'psd'")
# Make sure we're dealing with a numpy array. If y and x were the same
# object to start with, keep them that way
x = np.asarray(x)
if not same_data:
y = np.asarray(y)
if sides is None or sides == 'default':
if np.iscomplexobj(x):
sides = 'twosided'
else:
sides = 'onesided'
elif sides not in ['onesided', 'twosided']:
raise ValueError("Unknown value for sides %s, must be one of: "
"'default', 'onesided', or 'twosided'" % sides)
# zero pad x and y up to NFFT if they are shorter than NFFT
if len(x) < NFFT:
n = len(x)
x = np.resize(x, (NFFT,))
x[n:] = 0
if not same_data and len(y) < NFFT:
n = len(y)
y = np.resize(y, (NFFT,))
y[n:] = 0
if pad_to is None:
pad_to = NFFT
if mode != 'psd':
scale_by_freq = False
elif scale_by_freq is None:
scale_by_freq = True
# For real x, ignore the negative frequencies unless told otherwise
if sides == 'twosided':
numFreqs = pad_to
if pad_to % 2:
freqcenter = (pad_to - 1)//2 + 1
else:
freqcenter = pad_to//2
scaling_factor = 1.
elif sides == 'onesided':
if pad_to % 2:
numFreqs = (pad_to + 1)//2
else:
numFreqs = pad_to//2 + 1
scaling_factor = 2.
result = stride_windows(x, NFFT, noverlap, axis=0)
result = detrend(result, detrend_func, axis=0)
result, windowVals = apply_window(result, window, axis=0,
return_window=True)
result = np.fft.fft(result, n=pad_to, axis=0)[:numFreqs, :]
freqs = np.fft.fftfreq(pad_to, 1/Fs)[:numFreqs]
if not same_data:
# if same_data is False, mode must be 'psd'
resultY = stride_windows(y, NFFT, noverlap)
resultY = detrend(resultY, detrend_func, axis=0)
resultY = apply_window(resultY, window, axis=0)
resultY = np.fft.fft(resultY, n=pad_to, axis=0)[:numFreqs, :]
result = np.conj(result) * resultY
elif mode == 'psd':
result = np.conj(result) * result
elif mode == 'magnitude':
result = np.abs(result) / np.abs(windowVals).sum()
elif mode == 'angle' or mode == 'phase':
# we unwrap the phase later to handle the onesided vs. twosided case
result = np.angle(result)
elif mode == 'complex':
result /= np.abs(windowVals).sum()
if mode == 'psd':
# Also include scaling factors for one-sided densities and dividing by
# the sampling frequency, if desired. Scale everything, except the DC
# component and the NFFT/2 component:
# if we have a even number of frequencies, don't scale NFFT/2
if not NFFT % 2:
slc = slice(1, -1, None)
# if we have an odd number, just don't scale DC
else:
slc = slice(1, None, None)
result[slc] *= scaling_factor
# MATLAB divides by the sampling frequency so that density function
# has units of dB/Hz and can be integrated by the plotted frequency
# values. Perform the same scaling here.
if scale_by_freq:
result /= Fs
# Scale the spectrum by the norm of the window to compensate for
# windowing loss; see Bendat & Piersol Sec 11.5.2.
result /= (np.abs(windowVals)**2).sum()
else:
# In this case, preserve power in the segment, not amplitude
result /= np.abs(windowVals).sum()**2
t = np.arange(NFFT/2, len(x) - NFFT/2 + 1, NFFT - noverlap)/Fs
if sides == 'twosided':
# center the frequency range at zero
freqs = np.concatenate((freqs[freqcenter:], freqs[:freqcenter]))
result = np.concatenate((result[freqcenter:, :],
result[:freqcenter, :]), 0)
elif not pad_to % 2:
# get the last value correctly, it is negative otherwise
freqs[-1] *= -1
# we unwrap the phase here to handle the onesided vs. twosided case
if mode == 'phase':
result = np.unwrap(result, axis=0)
return result, freqs, t
def _single_spectrum_helper(x, mode, Fs=None, window=None, pad_to=None,
sides=None):
'''
This is a helper function that implements the commonality between the
complex, magnitude, angle, and phase spectrums.
It is *NOT* meant to be used outside of mlab and may change at any time.
'''
if mode is None or mode == 'psd' or mode == 'default':
raise ValueError('_single_spectrum_helper does not work with %s mode'
% mode)
if pad_to is None:
pad_to = len(x)
spec, freqs, _ = _spectral_helper(x=x, y=None, NFFT=len(x), Fs=Fs,
detrend_func=detrend_none, window=window,
noverlap=0, pad_to=pad_to,
sides=sides,
scale_by_freq=False,
mode=mode)
if mode != 'complex':
spec = spec.real
if spec.ndim == 2 and spec.shape[1] == 1:
spec = spec[:, 0]
return spec, freqs
# Split out these keyword docs so that they can be used elsewhere
docstring.interpd.update(Spectral=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
'''
x = np.asarray(x)
if x.shape[-1] != self.numcols:
raise ValueError('Expected an array with dims[-1]==%d' %
self.numcols)
Y = np.dot(self.Wt, self.center(x).T).T
mask = self.fracs >= 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
<http://mathworld.wolfram.com/BivariateNormalDistribution.html>`_
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. <Fernando.Perez@colorado.edu>
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.