algorithms.spectral¶
Module: algorithms.spectral
¶
Spectral transforms are used in order to estimate the frequencydomain representation of timeseries. Several methods can be used and this module contains implementations of several algorithms for the calculation of spectral transforms.
Functions¶

nitime.algorithms.spectral.
freq_response
(b, a=1.0, n_freqs=1024, sides='onesided')¶ Returns the frequency response of the IIR or FIR filter described by beta and alpha coefficients.
Parameters: b : beta sequence (moving average component)
a : alpha sequence (autoregressive component)
n_freqs : size of frequency grid
sides : {‘onesided’, ‘twosided’}
compute frequencies between [PI,PI), or from [0, PI]
Returns: fgrid, H(e^jw) :
Notes
For a description of the linear constantcoefficient difference equation, see http://en.wikipedia.org/wiki/Ztransform

nitime.algorithms.spectral.
get_spectra
(time_series, method=None)¶ Compute the spectra of an ntuple of time series and all of the pairwise crossspectra.
Parameters: time_series : float array
The timeseries, where time is the last dimension
method : dict, optional
 contains: this_method:’welch’
indicates that
mlab.psd()
will be used in order to calculate the psd/csd, in which case, additional optional inputs (and default values) are:NFFT=64
Fs=2pi
detrend=mlab.detrend_none
window=mlab.window_hanning
n_overlap=0
 this_method:’periodogram_csd’
indicates that
periodogram()
will be used in order to calculate the psd/csd, in which case, additional optional inputs (and default values) are:Skx=None
Sky=None
N=None
sides=’onesided’
normalize=True
Fs=2pi
 this_method:’multi_taper_csd’
indicates that
multi_taper_psd()
used in order to calculate psd/csd, in which case additional optional inputs (and default values) are:BW=0.01
Fs=2pi
sides = ‘onesided’
Returns: f : float array
The central frequencies for the frequency bands for which the spectra are estimated
fxy : float array
A semifilled matrix with the crossspectra of the signals. The csd of signal i and signal j is in f[j][i], but not in f[i][j] (which will be filled with zeros). For i=j fxy[i][j] is the psd of signal i.

nitime.algorithms.spectral.
get_spectra_bi
(x, y, method=None)¶ Computes the spectra of two timeseries and the crossspectrum between them
Parameters: x,y : float arrays
Timeseries data
method : dict, optional
See
get_spectra()
documentation for detailsReturns: f : float array
The central frequencies for the frequency bands for which the spectra are estimated
fxx : float array
The psd of the first signal
fyy : float array
The psd of the second signal
fxy : float array
The crossspectral density of the two signals

nitime.algorithms.spectral.
mtm_cross_spectrum
(tx, ty, weights, sides='twosided')¶ The crossspectrum between two tapered timeseries, derived from a multitaper spectral estimation.
Parameters: tx, ty : ndarray (K, …, N)
The complex DFTs of the tapered sequence
weights : ndarray, or 2tuple or list
Weights can be specified as a length2 list of weights for spectra tx and ty respectively. Alternatively, if tx is ty and this function is computing the spectral density function of a single sequence, the weights can be given as an ndarray of weights for the spectrum. Weights may be
 scalars, if the shape of the array is (K, …, 1)
 vectors, with the shape of the array being the same as tx or ty
sides : str in {‘onesided’, ‘twosided’}
For the symmetric spectra of a real sequence, optionally combine half of the frequencies and scale the duplicate frequencies in the range (0, F_nyquist).
Notes
spectral densities are always computed as
\(S_{xy}^{mt}(f) = \frac{\sum_k [d_k^x(f)s_k^x(f)][d_k^y(f)(s_k^y(f))^{*}]}{[\sum_k d_k^x(f)^2]^{\frac{1}{2}}[\sum_k d_k^y(f)^2]^{\frac{1}{2}}}\)

nitime.algorithms.spectral.
multi_taper_csd
(s, Fs=6.283185307179586, NW=None, BW=None, low_bias=True, adaptive=False, sides='default', NFFT=None)¶ Returns an estimate of the Cross Spectral Density (CSD) function between all (N choose 2) pairs of timeseries in s, using the multitaper method. If the NW product, or the BW and Fs in Hz are not specified by the user, a bandwidth of 4 times the fundamental frequency, corresponding to NW = 4 will be used.
Parameters: s : ndarray
An array of sampled random processes, where the time axis is assumed to be on the last axis. If ndim > 2, the number of time series to compare will still be taken as prod(s.shape[:1])
Fs : float, Sampling rate of the signal
NW : float
The normalized halfbandwidth of the data tapers, indicating a multiple of the fundamental frequency of the DFT (Fs/N). Common choices are n/2, for n >= 4. This parameter is unitless and more MATLAB compatible. As an alternative, set the BW parameter in Hz. See Notes on bandwidth.
BW : float
The samplingrelative bandwidth of the data tapers, in Hz.
adaptive : {True, False}
Use adaptive weighting to combine spectra
low_bias : {True, False}
Rather than use 2NW tapers, only use the tapers that have better than 90% spectral concentration within the bandwidth (still using a maximum of 2NW tapers)
sides : str (optional) [ ‘default’  ‘onesided’  ‘twosided’ ]
This determines which sides of the spectrum to return. For complexvalued inputs, the default is twosided, for realvalued inputs, default is onesided Indicates whether to return a onesided or twosided
Returns: (freqs, csd_est) : ndarrays
The estimatated CSD and the frequency points vector. The CSD{i,j}(f) are returned in a square “matrix” of vectors holding Sij(f). For an input array of (M,N), the output is (M,M,N)
Notes
The bandwidth of the windowing function will determine the number tapers to use. This parameters represents tradeoff between frequency resolution (lower main lobe BW for the taper) and variance reduction (higher BW and number of averaged estimates). Typically, the number of tapers is calculated as 2x the bandwidthtofundamentalfrequency ratio, as these eigenfunctions have the best energy concentration.

nitime.algorithms.spectral.
multi_taper_psd
(s, Fs=6.283185307179586, NW=None, BW=None, adaptive=False, jackknife=True, low_bias=True, sides='default', NFFT=None)¶ Returns an estimate of the PSD function of s using the multitaper method. If the NW product, or the BW and Fs in Hz are not specified by the user, a bandwidth of 4 times the fundamental frequency, corresponding to NW = 4 will be used.
Parameters: s : ndarray
An array of sampled random processes, where the time axis is assumed to be on the last axis
Fs : float
Sampling rate of the signal
NW : float
The normalized halfbandwidth of the data tapers, indicating a multiple of the fundamental frequency of the DFT (Fs/N). Common choices are n/2, for n >= 4. This parameter is unitless and more MATLAB compatible. As an alternative, set the BW parameter in Hz. See Notes on bandwidth.
BW : float
The samplingrelative bandwidth of the data tapers, in Hz.
adaptive : {True/False}
Use an adaptive weighting routine to combine the PSD estimates of different tapers.
jackknife : {True/False}
Use the jackknife method to make an estimate of the PSD variance at each point.
low_bias : {True/False}
Rather than use 2NW tapers, only use the tapers that have better than 90% spectral concentration within the bandwidth (still using a maximum of 2NW tapers)
sides : str (optional) [ ‘default’  ‘onesided’  ‘twosided’ ]
This determines which sides of the spectrum to return. For complexvalued inputs, the default is twosided, for realvalued inputs, default is onesided Indicates whether to return a onesided or twosided
Returns: (freqs, psd_est, var_or_nu) : ndarrays
The first two arrays are the frequency points vector and the estimated PSD. The last returned array differs depending on whether the jackknife was used. It is either
 The jackknife estimated variance of the logpsd, OR
 The degrees of freedom in a chi2 model of how the estimated PSD is distributed about the true logPSD (this is either 2*floor(2*NW), or calculated from adaptive weights)
Notes
The bandwidth of the windowing function will determine the number tapers to use. This parameters represents tradeoff between frequency resolution (lower main lobe BW for the taper) and variance reduction (higher BW and number of averaged estimates). Typically, the number of tapers is calculated as 2x the bandwidthtofundamentalfrequency ratio, as these eigenfunctions have the best energy concentration.

nitime.algorithms.spectral.
periodogram
(s, Fs=6.283185307179586, Sk=None, N=None, sides='default', normalize=True)¶ Takes an Npoint periodogram estimate of the PSD function. The number of points N, or a precomputed FFT Sk may be provided. By default, the PSD function returned is normalized so that the integral of the PSD is equal to the mean squared amplitude (mean energy) of s (see Notes).
Parameters: s : ndarray
Signal(s) for which to estimate the PSD, time dimension in the last axis
Fs : float (optional)
The sampling rate. Defaults to 2*pi
Sk : ndarray (optional)
Precomputed FFT of s
N : int (optional)
Indicates an Npoint FFT where N != s.shape[1]
sides : str (optional) [ ‘default’  ‘onesided’  ‘twosided’ ]
This determines which sides of the spectrum to return. For complexvalued inputs, the default is twosided, for realvalued inputs, default is onesided Indicates whether to return a onesided or twosided
PSD normalize : boolean (optional, default=True) Normalizes the PSD
Returns: (f, psd) : tuple
f: The central frequencies for the frequency bands PSD estimate for each row of s

nitime.algorithms.spectral.
periodogram_csd
(s, Fs=6.283185307179586, Sk=None, NFFT=None, sides='default', normalize=True)¶ Takes an Npoint periodogram estimate of all the cross spectral density functions between rows of s.
The number of points N, or a precomputed FFT Sk may be provided. By default, the CSD function returned is normalized so that the integral of the PSD is equal to the mean squared amplitude (mean energy) of s (see Notes).
 s : ndarray
 Signals for which to estimate the CSD, time dimension in the last axis
 Fs : float (optional)
 The sampling rate. Defaults to 2*pi
 Sk : ndarray (optional)
 Precomputed FFT of rows of s
 NFFT : int (optional)
 Indicates an Npoint FFT where N != s.shape[1]
 sides : str (optional) [ ‘default’  ‘onesided’  ‘twosided’ ]
 This determines which sides of the spectrum to return. For complexvalued inputs, the default is twosided, for realvalued inputs, default is onesided Indicates whether to return a onesided or twosided
 normalize : boolean (optional)
 Normalizes the PSD
Returns: freqs, csd_est : ndarrays
The estimated CSD and the frequency points vector. The CSD{i,j}(f) are returned in a square “matrix” of vectors holding Sij(f). For an input array that is reshaped to (M,N), the output is (M,M,N)