Module reference

Model fitting and Diagnostics

Model fitting classes

class sails.modelfit.VieiraMorfLinearModel

TODO: Description

classmethod fit_model(data, delay_vect)

Estimates the multichannel autoregressive spectral estimators using a Vieira-Morf algorithm. Equations are referenced to Marple, appendix 15.B.

This is the multitrial versions of the algorithm, using the AMVAR method outlined in Ding 2000.

Parameters:

• data (numpy.ndarray) – array of shape [nsignals, nsamples, ntrials]
• delay_vect (numpy.ndarray) – Vector containing evenly spaced delays to be assessed in the model. Delays are represented in samples. Must start with zero.
• assess_order (bool) – Optional True/False value indicating whether additional information about each iteration of the fit should be returned. Defaults to False

Returns:

A populated object containing the fitted forward and backwards coefficients and several other useful variables and methods.

get_residuals(data, forward_parameters=True)

Returns the prediction error from a fitted model. This is a wrapper function for get_residuals()

Parameters:forward_parameters (bool) – If True, use forward parameters, otherwise use backward parameters

class sails.modelfit.OLSLinearModel

TODO: Description

classmethod fit_model(data, delay_vect)

TODO: Description

Model fitting helper functions

sails.modelfit.sliding_window_fit(model_class, data, delay_vect, win_len_samp, win_step_samp, compute_diagnostics=True)

Model diagnostics

class sails.diags.DelayDiagnostics

Class which computes the mutual information as a function of lag from zero lag to the first zero crossing of the autocorrelation function.

classmethod delay_search(data, maxdelay, step, sample_rate, constant_window=True)

Compute MI as a function of lag from zero lag to the first zero crossing of the autocorrelation function

If NafOptions().verbose is True, progress will be reported to stdout.

Parameters:

• data (numpy.ndarray) – array of [signals, samples, trials]
• maxdelay (int) – maximum delay to consider
• step (int) – step to increment the delay by
• sample_rate (float) – sample rate of data
• constant_window (bool) – Flag indicating that the same number of datapoints should be included at each delay. Default is True

class sails.diags.ModelDiagnostics

TODO: Description

sails.modelfit.get_residuals(data, parameters, delay_vect, backwards=False)

TODO: Description

Parameters:

• data – TODO
• parameters – TODO
• delay_vect – TODO
• backwards – TODO

Returns:

TODO

Model Decomposition

class sails.modal.MvarModalDecomposition

Object which calculates the modal decomposition of a fitted model derived from the AbstractLinearModel class.

compute_modal_parameters()

Compute the order 1 or 2 autoregressive coefficients associated with each pole or conjugate pair.

compute_residues()

Compute the residue matrix from the eigenvectors associated with each pole.

excitation(resid_cov)

http://climate-dynamics.org/wp-content/uploads/2015/08/arfit.pdf

classmethod initialise(model, sample_rate=None, normalise=False)

Compute the modal pole-residue decomposition of the A matrix from a fitted MVAR model.

Currently only implemented for a single realisation (A.ndim == 3)

Parameters:

• model – Fitted model object. Must be derived from AbstractLinearModel.
• sample_rate – sample rate of system in Hz
• normalise – Whether to adjust the phase of the eigenvectors

Returns:

Modal decomposition object

modal_freqz()

Compute filter characteristics for each pole

modal_timecourse(data)

Compute the modal time-course from a dataset and a fitted model. This is the tranformation of a set of [nsignals x time] data observations into modal coordinates of shape [nmodes x time].

Parameters:data – TODO

modal_transfer_function(sample_rate, modes=None)

Compute a transfer function matrix based on the excitation period for each mode

Parameters:

• sample_rate – sample rate of system in Hz
• modes – List of mode indices to evaluate over (optional)

mode_indices

Returns a list of tuples of mode indices where each tuple contains the indices of either a pole-pair or an unpaired pole

Returns:list of tuples containing indices into the modes

per_mode_transfer_function(sample_rate, freq_vect)

Extracts the transfer function for each mode by summing the transfer function across pole-pairs where necessary

Parameters:

• sample_rate – sample rate of system in Hz
• freq_vect – Vector of frequencies at which to evaluate function

period_hz

This is an old and deprecated way of accessing peak_frequency

pole_plot(ax=None, normscale=50, plottype='unitcircle')

Plot the poles of the current system.

Parameters:

• ax – Optional axis on which to plot
• normscale – Scaling factor for points on plot
• plottyle – String defining which plot to make, can be ‘unitcircle’ or ‘eigenspectrum’

Returns:

Reference to axes on which plot was drawn

residue_norm

TODO

transfer_function(sample_rate, freq_vect, modes=None, sum_modes=True)

Compute the transfer function in pole-residue form, splitting the system into modes with separate transfer functions. The full system transfer function is then a linear sum of each modal transfer function.

Parameters:

• sample_rate – sample rate of system in Hz
• freq_vect – Vector of frequencies at which to evaluate function
• modes – List of mode indices to evaluate over (optional)
• sum_modes – Boolean indicating whether to sum modes or return all

Model metric estimatation

class sails.mvar_metrics.FourierMvarMetrics

TODO: Document

classmethod initialise(model, sample_rate, freq_vect, nmodes=None)

Compute some basic values from A across our frequency vector

class sails.mvar_metrics.ModalMvarMetrics

classmethod initialise(model, sample_rate, freq_vect, nmodes=None, sum_modes=True)

Compute some basic values from the pole-residue decomposition of the A matrix

Currently only implemented for a single realisation (A.ndim == 3)

Parameters:

• model – TODO
• sample_rate – TODO
• freq_vect – TODO
• nmodes – TODO

sails.mvar_metrics.modal_transfer_function(evals, evecl, evecr, nchannels, sample_rate=None, freq_vect=None)

Comute the transfer function in pole-residue form, splitting the system into modes with separate transfer functions. The full system transfer function is then a linear sum of each modal transfer function.

Plotting

sails.plotting.root_plot(rts, ax=None, figargs={}, plotargs={})

Plot a set of roots (complex numbers).

Parameters:

• rts – Roots to plot
• ax – Optional Axes on which to place plot.

Returns:

Axes object on which plot was drawn

sails.plotting.plot_vector(metric, x_vect, y_vect=None, x_label=None, y_label=None, title=None, labels=None, line_labels=None, F=None, cmap=<matplotlib.colors.LinearSegmentedColormap object>, triangle=None, diag=False, thresh=None, two_tail_thresh=False, font_size=10, use_latex=False)

Function for plotting frequency domain connectivity at a single time-point.

Params metric:

matrix containing connectivity values [nsignals x signals x frequencies x participants] in which the first dimension refers to source nodes and the second dimension refers to target nodes

Params x_vect:

vector of frequencies to label the x axis

Params y_vect:

y_vect: vector containing the values for the y-axis

Parameters:

• x_label (string [optional]) – label for the x axis
• y_label (string [optional]) – label for the y axis
• title (string [optional]) – title for the figure
• labels (list) – list of node labels for columns and vectors
• line_labels (list) – list of labels for each separate line (participant dimension in metric)
• F (figurehandle [optional]) – handle of existing figure to plot within
• triangle (string [optional]) – string to indicate whether only the ‘upper’ or ‘lower’ triangle of the matrix should be plotted
• diag (bool [optional]) – flag to indicate whether the diagonal elements should be plotted
• thresh (ndarray [optional]) – matrix containing thesholds to be plotted alongside connectivity values [nsignals x nsignals x frequencies]
• two_tailed_thresh (bool [optional]) – flag to indicate whether both signs (+/-) of the threshold should be plotted
• font_size (int [optional]) – override the default font size

sails.plotting.plot_matrix(metric, x_vect, y_vect, x_label=None, y_label=None, z_vect=None, title=None, labels=None, F=None, cmap=<matplotlib.colors.LinearSegmentedColormap object>, font_size=8, use_latex=False, diag=True)

Function for plotting frequency domain connectivity over many time points

Parameters:

• metric (ndarray) – matrix containing connectivity values [nsignals x signals x frequencies x participants] in which the first dimension refers to source nodes and the second dimension refers to target nodes
• x_vect (1d array) – vector of frequencies to label the x axis
• y_vect (1d array [optional]) – vector containing the values for the y-axis
• z_vect (1d array [optional]) – vector containing values for the colour scale
• x_label (string [optional]) – label for the x axis
• y_label (string [optional]) – label for the y axis
• title (string [optional]) – title for the figure
• labels (list) – list of node labels for columns and vectors
• F (figurehandle [optional]) – handle of existing figure to plot within
• cmap (matplotlib colormap [optional]) – matplotlib.cm.<colormapname> to use for colourscale (redundant for plot vector??)
• font_size (int [optional]) – override the default font size

Simulation

class sails.simulate.AbstractSigGen(extraargs=None)

Abstract base class for signal generation

generate_signal(meg_reader)

Generate a custom signal for embedding in a dataset

Parameters:meg_reader (naf.meg.abstractdata.AbstractMEGReader) – MEG data for which to generate signal Return type:numpy.ndarray Returns:numpy array of shape (samples). Note that samples must be equal to the value of meg_reader.num_samples

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.AbstractLagSigGen(extraargs=None)

Generic class for making lagged signals

generate_basesignal(nsamples)

Function defining the starting signal which will be lagged throughout the network

generate_noise(nsamples)

Function defining the additive noise added at each lag

generate_signal(sample_rate=None, num_samples=None, noise_dB=None)

Method for generating a set of time courses based on either the structure of a meg_reader object or passed parameters and the self.generate_parameters method defined in this class

Returns a timeseries of dimension [samples x signals]

get_lags()

Method returning the lagging of the signal through the system. Should return a [nsignals x nsignals x nlags] matrix in which the first dimension refers to the source node and the second dimension the target node. Each non-zero value indicates that the signal in the first dimension will be added into the signal in the second dimension at the specified weight.

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.AbstractMVARSigGen(extraargs=None)

Generic class for making MVAR signals

check_stability()

Checks the stablility of the parameter matrix by the magnitude of the largest eigenvalue of the first lag in the parameter matrix. Lutkepohl 2005 ch 1 for more detail

generate_model()

Returns a LinearModel containing containing the true model parameters.

generate_parameters()

Method to generate the parameters of the MVAR system. Should return a [nsignals x nsignals x nlags] parameter matrix in which the first dimension refers to the source node and the second dimension the target node

generate_signal(sample_rate=None, num_samples=None, noise_dB=None, num_realisations=1)

Method for generating a set of time courses based on either the structure of a meg_reader object or passed parameters and the self.generate_parameters method defined in this class

Returns a timeseries of dimension [samples x signals]

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.Baccala2001_fig1(extraargs=None)

https://doi.org/10.1007/PL00007990

generate_parameters()

Method to generate the parameters of the MVAR system. Should return a [nsignals x nsignals x nlags] parameter matrix in which the first dimension refers to the source node and the second dimension the target node

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.Baccala2001_fig2(extraargs=None)

https://doi.org/10.1007/PL00007990

generate_parameters()

Method to generate the parameters of the MVAR system. Should return a [nsignals x nsignals x nlags] parameter matrix in which the first dimension refers to the source node and the second dimension the target node

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.Baccala2001_fig3(extraargs=None)

https://doi.org/10.1007/PL00007990

generate_parameters()

Method to generate the parameters of the MVAR system. Should return a [nsignals x nsignals x nlags] parameter matrix in which the first dimension refers to the source node and the second dimension the target node

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.Baccala2001_fig4(extraargs=None)

https://doi.org/10.1007/PL00007990

generate_parameters()

Method to generate the parameters of the MVAR system. Should return a [nsignals x nsignals x nlags] parameter matrix in which the first dimension refers to the source node and the second dimension the target node

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.Baccala2001_fig5(extraargs=None)

https://doi.org/10.1007/PL00007990

generate_parameters()

Method to generate the parameters of the MVAR system. Should return a [nsignals x nsignals x nlags] parameter matrix in which the first dimension refers to the source node and the second dimension the target node

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.Fasoula2013_eqn26(extraargs=None)

https://doi.org/10.1016/j.jneumeth.2013.02.021

generate_parameters()

Method to generate the parameters of the MVAR system. Should return a [nsignals x nsignals x nlags] parameter matrix in which the first dimension refers to the source node and the second dimension the target node

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.Schelter2006_fig1(extraargs=None)

https://doi.org/10.1016/j.jneumeth.2005.09.001

generate_parameters()

Method to generate the parameters of the MVAR system. Should return a [nsignals x nsignals x nlags] parameter matrix in which the first dimension refers to the source node and the second dimension the target node

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.PascualMarqui2014_fig3(extraargs=None)

https://dx.doi.org/10.3389/fnhum.2014.00448

generate_parameters()

Method to generate the parameters of the MVAR system. Should return a [nsignals x nsignals x nlags] parameter matrix in which the first dimension refers to the source node and the second dimension the target node

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.Korzeniewska2003Lag(extraargs=None)

Generate realisations of the time-lagged network defined in Korzeniewska 2003 figure 1.

https://doi.org/10.1016/S0165-0270(03)00052-9

generate_basesignal(nsamples)

Generate oscillatory signal to be propogated through the netowrk. This creates a single resonance at one-half of the Nyqvist frequency via direct pole placement.

Note: the original base-signal in the paper is a realisation from an AR model. I couldn’t find the parameters for this mode, so use pole placement here. The spectral profile will be different to the paper, but the connectivity patterns are the same.

get_lags()

Method returning the lagging of the signal through the system. Should return a [nsignals x nsignals x nlags] matrix in which the first dimension refers to the source node and the second dimension the target node. Each non-zero value indicates that the signal in the first dimension will be added into the signal in the second dimension at the specified weight.

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

class sails.simulate.SailsTutorialExample(extraargs=None)

generate_basesignal(f, r, sample_rate, num_samples)

Generate a simple signal by pole placement

generate_signal(f1, f2, sample_rate, num_samples, noise_dB=None, num_realisations=1, magnitude_noise=0.02)

Method for generating a set of time courses based on either the structure of a meg_reader object or passed parameters and the self.generate_parameters method defined in this class

Returns a timeseries of dimension [samples x signals x realisations]

get_num_sources()

Return the number of sources which this class provides.

If not override, returns 1

Support routines

Abstract Classes

class sails.modelfit.AbstractLinearModel

TODO: Description

companion

Parameter matrix in companion form

compute_diagnostics(data, compute_pc=False)

Compute several diagnostic metrics from a fitted MVAR model and a dataset.

Return type:diags.ModelDiagnostics Returns:ModelDiagnostics object containing LL, AIC, BIC, stability ratio, durbin-watson, R squared and percent consistency measures

get_residuals(data)

Returns the prediction error from a fitted model. This is a wrapper function for get_residuals()

Parameters:data – TODO

nsignals

Number of signals in fitted model

order

Order of fitted model

class sails.mvar_metrics.AbstractMVARMetrics

TODO: Description

PSD

Power Spectral Density matrix

S

Spectral Matrix of shape [nsignals, nsignals, order, epochs]

coherency

TODO: Document

d_directed_transfer_function

TODO: Document

directed_transfer_function

TODO: Document

ff_directed_transfer_function

TODO: Document

geweke_granger_causality

TODO: Document

imaginary_coherence

TODO: Document

inv_S

TODO: Document

isolated_effective_coherence

TODO: Document

magnitude_squared_coherence

TODO: Document

partial_coherence

TODO: Document

partial_directed_coherence

TODO: Document

phase_coherence

TODO: Document

Statistical Functions

sails.mvar_metrics.sdf_spectrum(A, sigma, sample_rate, freq_vect)

Estimate of the Spectral Density as found on wikipedia and Quirk & Liu 1983

This assumes that the spectral representation of A is invertable

sails.mvar_metrics.psd_spectrum(A, sigma, sample_rate, freq_vect)

Estimate the PSD representation of a set of MVAR coefficients as stated in Penny 2000 Signal processing course array.ps 7.41

This assumes that the spectral representation of A is invertable

TODO: This doesn’t behave as expected for univariate?, sdf_spectrum recommended

Parameters:

• A – TODO
• sigma – TODO
• sample_rate – TODO
• freq_vect – TODO

Returns:

TODO

sails.mvar_metrics.ar_spectrum(A, sigma, sample_rate, freq_vect)

Estimate the spectral representation of a set of MVAR coefficients as suggested by Baccala & Sameshima 2001, the equation without a number just below equation 13.

Parameters:

• A – TODO
• sigma – TODO
• sample_rate – TODO
• freq_vect – TODO

Returns:

TODO

sails.mvar_metrics.transfer_function(Af)

Function for computing the transfer function of a system

Parameters:Af – Frequency domain version of parameters (can be calculated using ar_spectrum function) [nchannels x nchannels x nfreqs x nepochs] Returns:numpy array of dimensions [nsignals x nsignals x frequency x epochs]

sails.mvar_metrics.spectral_matrix(H, noise_cov)

Function for computing the spectral matrix, the matrix of spectra and cross-spectra.

Parameters:

• H – The transfer matrix [nchannels x nchannels x nfreqs x nepochs]
• noise_cov – The noise covariance matrix of the system [nchannels x nchannels x nepochs]

Returns:

numpy array of dimensions [nsignals x nsignals x frequency x epochs]

sails.mvar_metrics.coherency(S)

Method for computing the Coherency. This is the complex form of coherence, from which metrics such as magnitude squared coherence can be derived

Params S:The spectral matrix: 3D or 4D [nchannels x nchannels x nfreqs x nepochs] Returns:numpy array of dimensions [nsignals x nsignals x frequency x epochs]

sails.mvar_metrics.partial_coherence(inv_S)

Method for computing the Partial Coherence.

Params inv_S:The inverse spectral matrix [nchannels x nchannels x nfreqs x nepochs] Returns:numpy array of dimensions [nsignals x nsignals x frequency x epochs]

sails.mvar_metrics.partial_directed_coherence(Af)

Function to estimate the partial directed coherence from a set of multivariate parameters.

Parameters:Af – Frequency domain version of parameters (can be calculated using ar_spectrum function) Returns:PDC : ndarray containing the PDC estimates from the parameters of dimensions [nchannels x nchannels x order]

sails.mvar_metrics.isolated_effective_coherence(Af, noise_cov)

Function for estimating the Isolated Effective Coherence as defined in Pascual-Marqui et al 2014

Parameters:

• Af – Frequency domain version of parameters (can be calculated using ar_spectrum function) [nchannels x nchannels x nfreqs x nepochs]
• noise_cov – The noise covariance matrix of the system [nchannels x nchannels x nepochs]

Returns:

iec: numpy array of dimensions [nsignals x nsignals x frequency x epochs]

sails.mvar_metrics.directed_transfer_function(H)

Method for computing the Directed Transfer Function as defined in Kaminski & Blinowska 1991

Parameters:H – The transfer matrix [nchannels x nchannels x nfreqs x nepochs] Returns:numpy array of dimensions [nsignals x nsignals x frequency x epochs]

sails.modal.adjust_phase(x)

Method for normalising eigenvector matrices to meet the conditions in eqn 21 of Neumaier & Schneider 2001.

Parameters:x – Vector of complex numbers to adjust the phase of Returns:Vector of numbers with adjusted phase

sails.stats.find_cov_multitrial(a, b, ddof=1)

Estimate the average covariance between two datasets across many trials

Parameters:

• a (ndarray) – a multivariate time series of shape nsignals x nsamples x ntrials
• b (ndarray) – a multivariate time series of shape nsignals x nsamples x ntrials
• ddof (None orint) – if None, no normalisation will be performed. If an integer, covariance will be normalised by (nsamples - ddof)

Return type:

ndarray

Returns:

A covariance matrix between a and b of shape nsignals x nsignals

sails.stats.find_cov(a, b, ddof=1)

Estimate the covariance between two datasets.

Parameters:

• a (ndarray) – a multivariate time series of shape nsignals x nsamples
• b (ndarray) – a multivariate time series of shape nsignals x nsamples
• ddof (int) – if None, no normalisation will be performed. If an integer, covariance will be normalised by (nsamples - ddof)

Return type:

ndarray

Returns:

A covariance matrix between a and b of shape nsignals x nsignals

sails.stats.mutual_information(A, B, r=None, nbins=21, base=<ufunc 'log2'>)

Calculate the mutual information between two signals.

Parameters:

• A (ndarray) – First signal, of shape [channels x samples]
• B (ndarray) – Second signal, of shape [channels x samples]
• r (tuple (float, float)) – Range of the random variables used for bin calculations. If not set, takes the minimum and maximum values from the combination of A and B
• nbins (int) – Number of bins to compute the frequencies within
• base (function) – Base to use for MI calculation - defaults to np.log2

Return type:

ndarray

Returns:

the mutual information within each signal passed

sails.stats.durbin_watson(residuals, step=1)

Test for autocorrelation in the residuals of a model.

The result is a value between 0 and 4. A value of 0 indicates a positive autocorrelaiton, 2 indicates no correlation and a value of 4 indicates a negative autocorrelation.

If many trials are found the statistic is estimated for each trial separately.

Parameters:

• residuals (ndarray) – Residuals of shape [channels, samples, ntrials] If only two dimensions, will be treated as [channels, samples] If only one dimension, will be treated as [samples]
• step (int) – step-size to use through residuals when calculating DW. Defaults to 1.

Return type:

ndarray

Returns:

durbin-watson index

sails.stats.percent_consistency(data, residuals)

Estimate how well a model retains the correlational structure of a dataset.

This uses the percent consistency measure outlined in Ding et al 2000. All auto and cross correlations within a multivariate data set are computed and compared to the auto and cross correlations of the data as predicted by a model (in this case the data - model residuals, as these are easily obtained).

Parameters:

• data (numpy.ndarray) – An array containing the observed data, of shape [nsignals x nsamples x ntrials]
• residuals – An array containing the remaining variance of data after model fitting, of shape [nsignals x nsamples x ntrials]

Rtype float: Returns:

A value between 0 and 100 indicating the percentage of the autocorrelation in the data that is retained in the model.

sails.stats.rsquared_from_residuals(data, residuals, per_signal=False)

Estimate the variance explained by a model from the original data and the model residuals

If data and residuals are 2D, they will be treated as [channels, nsamples]

If data and residuals are 1D, they will be treated as [nsamples]

Parameters:

• data (ndarray) – Array containing the original data, of shape [channels, nsamples, ntrials]
• residuals (ndarray) – Array containing the model residuals, of shape [channels, nsamples, ntrials]
• per_signal (bool) – Boolean indicating whether the variance explained should be computed separately per channel

Return type:

float

Returns:

Value between 0 and 1 indicating the proportion of variance explained by the model.

sails.stats.mahalanobis(X, Y=None, sigma=None)

Estimate the mahalanobis distance of each point in an array across the samples dimension.

Parameters:

• X (ndarray) – Array containing data of points: shape [channels x samples]
• Y (ndarray or None) – Optional Array containing data of second points: shape [channels x samples]. If not provided, the origin will be used as the point for comparison.
• sigma (ndarray or None) – A precomputed covariance matrix to use in the calculation, of shape [channels x channels]. If not provided, sigma will be calculated from X

Return type:

ndarray

Returns:

The mahalanobis distance for each sample, of shape [samples]

sails.stats.profile_likelihood(X)

Find the ‘elbow’ point in a curve.

This function uses the automatic dimensionality selection method from Zhu et al 2006 to estimate the elbow or inflection point in a given curve.

Method can be found in Mu Zhu, Ali Ghodsi. Computational Statistics & Data Analysis 51 (2006) 918 - 930

Parameters:X (ndarray) – 1d array containing the curve to be analysed, of shape [samples] Return type:ndarray Returns:The log likelihood that each point on the curve separates the distributions to the left and right, of shape [samples]

sails.stats.stability_index(A)

Compute the stability index as defined in Lutkepohl 2006 pg 11.

This is a proxy for the stationarity assumption of MVAR models, if the magnitude of the largest principal component of the model parameters is less than 1, the model is stable and thus, stationary.

Parameters:a (numpy.ndarray) – The MVAR parameter matrix Return type:float Returns:The stability index

sails.stats.stability_ratio(A)

Compute the stability ratio as a measure of stationarity

A stronger test for stationarity than the stability index. Computes the ratio between the largest eigenvalue of the parameters and all the others. If the largest parameter is larger than all other parmeters (indicated by a value < 1) we have a super-stable system.

Parameters:a (numpy.ndarray) – The MVAR parameter matrix Rtype float: Returns:The stability ratio

sails.modelvalidation.approx_model_selection_criteria(A, EF, method='ss')

Estimate Akaike’s Information Criterion and Bayes Information Criterion from a model fit using an approximation of the loglikelihood.

Two approximations are available:

‘ss’: The sum square of the residuals. This approximation is outlined in Burnham & Anderson 2004 and Chatfield 2003. ‘det’: The determinant of the residual covariance matrix.

The sum squares method is used as default.

Parameters:

• A – Array containing the parameters of a fitted model, of shape [channels x channels x maxorder]
• EF – Array containing the residuals of the fitted model, of shape [channels x samples x trials]
• method – string indicating which method should be used to approximate the loglikelihood (‘ss’ or ‘det’)

Returns:

tuple of (aLL, AIC, BIC) aLL: Approximation of the loglikelihood (determined by the method defined above) AIC: Estimate of Akaike’s Information Criterion BIC: Estimate of the Bayesian Information Criterion

Tutorial helper functions

sails.tutorial_utils.generate_pink_roots(power, order=24)

Generates the poles of a system with a pink spectrum.

Parameters:

• power – Power of pole
• order – Order of system to generate

Returns:

Roots of the system (length of order)

sails.tutorial_utils.model_from_roots(roots)

Sets up a univariate model based on the roots of a polynomial

sails.tutorial_utils.find_example_path()

Returns the path to the example data directory