Data Structures | Functions

src/nonlinear.h File Reference

STATUS: unstable Functions using nonlinear systems-theory. More...

#include "mathadd.h"
#include "distances.h"
#include "definitions.h"
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Data Structures

struct  TimeDelayReconstruction

Functions

double ** eeg_nonlinear_prediction_error (const EEG *eeg, int embedding_dim, int time_lag, int npredict, double epsilon, double **output, OptArgList *optargs)
double tdelay_attractor_size (TimeDelayReconstruction *p)
int tdelay_estimate_dimension (double Rtol, int m_start, int m_end)
int tdelay_estimate_timelag_autocorr (TimeDelayReconstruction *p)
int tdelay_estimate_timelag_mutual (TimeDelayReconstruction *p, long partitions, long corrlength, double *mutual)
double tdelay_fnn_ratio (TimeDelayReconstruction *p, double Rtol, double Atol)
void tdelay_free (TimeDelayReconstruction *p)
void tdelay_index_i (TimeDelayReconstruction *p, int i, double *x)
double tdelay_index_ij (TimeDelayReconstruction *p, int i, int j)
void tdelay_index_j (TimeDelayReconstruction *p, int j, double *x)
TimeDelayReconstruction * tdelay_init (int m, int tau, double *x, int n)
double tdelay_predict_simple (TimeDelayReconstruction *p, double *sample, int npredict, double epsilon)
 simple nonlinear predicition.
void tdelay_print (FILE *out, TimeDelayReconstruction *p)
double tdelay_simple_nonlinear_prediction_error (TimeDelayReconstruction *reference, double *y, int yn, int npredict, double epsilon)
Array * tdelay_to_array (TimeDelayReconstruction *p)

Detailed Description

STATUS: unstable Functions using nonlinear systems-theory.

In general, the phase space is represented through a two-dimensional DOUBLE-array (a matrix). This can e.g. be all electrodes over time or any tranformation of the raw data.

Time Delay Reconstruction

Time-Delay reconstruction is done as in

\[ \vec{x}_i = \sum_{j=1}^{m}s_{i+(j-1)\tau}\vec{e}_j \]

where$ \tau = \nu \Delta t$ is a multiple of the sampling step.

There is a special structure for time-delay reconstruction to avoid the space-overhead. This structure saves only the 1D original time-series and calculates the other dimensions on the fly.

The disadvantage of this approach is that there need to be special functions to deal with these structs. Some of the functions are implemented both for general phase-spaces (full matrices) and the TimeDelayReconstruction struct. Others are not. You can always convert from a TimeDelayReconstruction-struct to a Matrix using tdelay_to_array() 

 double *data=getdata(100);
 TimeDelayReconstruction *p=tdelay_init( 4, 10, data, 100 );
 Array *space=tdelay_to_array( p );

Functions that are designed to be used with the TimeDelayReconstruction begin with the prefix tdelay_*.

Handling of the struct is done via 

  TimeDelayReconstruction* tdelay_init ( int m, int tau, double *x, int n );
  void        tdelay_free ( TimeDelayReconstruction *p );
  void        tdelay_print( FILE *out, TimeDelayReconstruction *p);

and it can be indexed with 

  double      tdelay_index_ij( TimeDelayReconstruction *p, int i, int j );
  void        tdelay_index_i ( TimeDelayReconstruction *p, int i, double *x);
  void        tdelay_index_j ( TimeDelayReconstruction *p, int j, double *x);

Definition in file nonlinear.h.

Function Documentation

double** eeg_nonlinear_prediction_error ( const EEG *  eeg,
int  embedding_dim,
int  time_lag,
int  npredict,
double  epsilon,
double **  output,
OptArgList *  optargs 
)

Calculate a trial x trial matrix M_ij that contains the prediction error when predicting time-series points in trial j using trial i as the reference.

The function uses tdelay_simple_nonlinear_prediction_error() for prediction.

Parameters:
eeg 
embedding_dim embedding dimensionm for phase space reconstruction
time_lag time-lag in sampling units used for phase space reconstruction
npredict prediction for npredict timesteps ahead
epsilon defines the initial neighbourhood (e.g. 1/4 of the variance in the data points)
allocated memory of eeg->ntrials x eeg->ntrials or ALLOC_IN_FCT
optargs may contain:

  • "channel=int" calculate the trial x trial matrix for this channel, default=0
  • "progress=void*" progressbar callback-function; default=NULL
Returns:
the trial x trial matrix contain the prediction errors; NULL if an error occured

Definition at line 175 of file nonlinear.c.

double tdelay_attractor_size ( TimeDelayReconstruction *  p  ) 

estimate size of the attractor from a scalar time-series using

\[ R = \sqrt{ \frac{1}{N}\sum_{i=1}^{N} (x_i - \langle x\rangle_i)^2 } \]

where

\[ \langle x\rangle_i = \frac{1}{N}\sum_{i=1}^{N} x_i \]

Definition at line 475 of file nonlinear.c.

int tdelay_estimate_dimension ( double  Rtol,
int  m_start,
int  m_end 
)
int tdelay_estimate_timelag_autocorr ( TimeDelayReconstruction *  p  ) 

estimate time-lag based on the first zero of the autorcorrelation function.

Definition at line 331 of file nonlinear.c.

int tdelay_estimate_timelag_mutual ( TimeDelayReconstruction *  p,
long  partitions,
long  corrlength,
double *  mutual 
)

estimate time-lag based on the first local minimum of the mutual information.

This function is "inspired" (stolen and adapted) from TISEAN 3.1, http://www.mpipks-dresden.mpg.de/~tisean

Ref: R. Hegger, H. Kantz, and T. Schreiber, Practical implementation of nonlinear time series methods: The TISEAN package, CHAOS 9, 413 (1999)

Parameters:
p the phase-space rep of the signal
partitions number of partitions to use (if <0, 16 is used as default)
corrlength maximum corrlength (if <0, 20 is used as default)
mutual if != NULL, the array is filled with the mutual information for each time-lag (corrlength-long)
Returns:
the minimum of mutual

Definition at line 246 of file nonlinear.c.

double tdelay_fnn_ratio ( TimeDelayReconstruction *  p,
double  Rtol,
double  Atol 
)

Estimate number of false-nearest-neighbours as proposed in

KENNEL et al. DETERMINING EMBEDDING DIMENSION FOR PHASE-SPACE RECONSTRUCTION USING A GEOMETRICAL CONSTRUCTION. Physical Review A (1992) vol. 45 (6) pp. 3403-3411

by applying two criteria.

(1) if

\[ \frac{ |x_{i+m\tau} - x^{r}_{i+m\tau} |} {} > R_{tol} \]

then state i is

Definition at line 386 of file nonlinear.c.

void tdelay_free ( TimeDelayReconstruction *  p  ) 

Definition at line 572 of file nonlinear.c.

void tdelay_index_i ( TimeDelayReconstruction *  p,
int  i,
double *  x 
)

Get an element from phase-space reconstruction:

\[ \vec{x}_i = \sum_{j=1}^{m}s_{i+(j-1)\tau}\vec{e}_j \]

Parameters:
p is the phase-space rep. of the data
i as in the formula above
x output (m long vector containing the final vector)

Definition at line 523 of file nonlinear.c.

double tdelay_index_ij ( TimeDelayReconstruction *  p,
int  i,
int  j 
)

Get an element from phase-space reconstruction:

\[ \vec{x}_i = \sum_{j=1}^{m}s_{i+(j-1)\tau}\vec{e}_j \]

Parameters:
p is the phase-space rep. of the data
i,j as in the formula above

Definition at line 500 of file nonlinear.c.

void tdelay_index_j ( TimeDelayReconstruction *  p,
int  j,
double *  x 
)

Get a dimension from phase-space reconstruction:

\[ \vec{x}_i = \sum_{j=1}^{m}s_{i+(j-1)\tau}\vec{e}_j \]

Parameters:
p is the phase-space rep. of the data
j as in the formula above
x output (n long vector containing the final vector)

Definition at line 546 of file nonlinear.c.

TimeDelayReconstruction* tdelay_init ( int  m,
int  tau,
double *  x,
int  n 
)

Definition at line 562 of file nonlinear.c.

double tdelay_predict_simple ( TimeDelayReconstruction *  p,
double *  sample,
int  npredict,
double  epsilon 
)

simple nonlinear predicition.

Using the Simple-Prediction algorithm from

Kantz & Schreiber 1997 Nonlinear Time-Series Analysis. Cambridge University Press

this function calculates the current+npredict's sample of the time series from which sample has been taken based on the time-series in p. It's basically an average over all points that are npredict time-steps after those points in p that are closer to sample than epsilon.

It's also described here:

R. Hegger, H. Kantz, and T. Schreiber, Practical implementation of nonlinear time series methods: The TISEAN package, CHAOS 9, 413 (1999)

Parameters:
p prediction time-series
sample predict time-points after this sample (must be of same dimension as p->m)
npredict predict n time-steps
epsilon epsilon-ball placed around all points in p to look for sample
Returns:
the predicted sample in the time-series

Definition at line 93 of file nonlinear.c.

void tdelay_print ( FILE *  out,
TimeDelayReconstruction *  p 
)

Definition at line 576 of file nonlinear.c.

double tdelay_simple_nonlinear_prediction_error ( TimeDelayReconstruction *  reference,
double *  y,
int  yn,
int  npredict,
double  epsilon 
)

Calculates the root mean square prediction error averaged over all samples in y. I.e. the function predicts the npredict'th following sample for each point in y and sums up the RMSE for all these predictions. The result is divided by yn.

\[ result = \frac{1}{yn}\sum_{i=1}^{i=yn} RMSE( predict(y,npredict), y[i+npredict] ) \]

with $ RMSE(r,d) = \sqrt{ \frac{1}{\#r} \sum{ (r-d)^2)}} $ For prediction, the function uses tdelay_predict_simple().

Parameters:
reference is the reference signal used for prediction
y signal to predict for
yn length of y
npredict prediction for npredict timesteps ahead
epsilon defines the initial neighbourhood
Returns:
root mean square prediction error cumulated (see above)

Definition at line 138 of file nonlinear.c.

Array* tdelay_to_array ( TimeDelayReconstruction *  p  ) 
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