Maintainer: Martin Zaefferer <martin.zaefferer@gmx.de>
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Title: Slow Feature Analysis
Type: Package
Author: Wolfgang Konen <wolfgang.konen@fh-koeln.de>, Martin Zaefferer, Patrick Koch; Bug hunting and testing by Ayodele Fasika, Ashwin Kumar, Prawyn Jebakumar
Description: Slow Feature Analysis (SFA), ported to R based on 'matlab' implementations of SFA: 'SFA toolkit' 1.0 by Pietro Berkes and 'SFA toolkit' 2.8 by Wolfgang Konen.
Version: 1.5
Date: 2022-03-29
Depends: R (≥ 2.0.0)
Imports: stats, MASS, graphics, grDevices
NeedsCompilation: no
Repository: CRAN
RoxygenNote: 7.1.2
Packaged: 2022-03-29 09:40:39 UTC; Martin
Date/Publication: 2022-03-29 10:00:07 UTC

Slow Feature Analysis

Description

Slow Feature Analysis

Details

Package: rSFA
Type: Package
Version: 1.5
Date: 29.03.2022
Maintainer: Martin Zaefferer martin.zaefferer@gmx.de
License: GPL (>= 2)
LazyLoad: yes

Slow Feature Analysis (SFA), ported to R based on the matlab implementations SFA toolkit 1.0 by Pietro Berkes and SFA toolkit 2.8 by Wolfgang Konen.

Author(s)

Wolfgang Konen wolfgang.konen@fh-koeln.de, Martin Zaefferer, Patrick Koch; Bug hunting and testing by Ayodele Fasika, Ashwin Kumar, Prawyn Jebakumar

Add noisy copies for parametric bootstrap

Description

Given training data X with true labels REALCLASS, add new records to X and REALCLASS, which are noisy copies of the training data.

Usage

addNoisyCopies(realclass, x, pars)

Arguments

realclass

true class of training data (can be vector, numerics, integers, factors)

x

a matrix containing the training data

pars

list of parameters:
pars$ncopies: Number of new records to add
pars$ncsort: Defines if training data should be sorted by class. Default is FALSE
pars$ncsigma: The noise in each column of x has the std.dev. pars$ncsigma*(standard deviation of column). Default Value: 0.8
pars$ncmethod: =1: each 'old' record from X in turn is the centroid for a new pattern;
=2: the centroid is the average of all records from the same class, the std.dev. is the same for all classes;
=3: centroid as in '2', the std.dev. is the std.dev. of all records from the same class (*recommended*)

Value

list res
- res contains two list entries: realclass and x (including added copies)

References

sfaPBootstrap

Custom Repeater Function

Description

Faster than customRepmat in matlab package, in certain cases.

Usage

customRep(a, n)

Arguments

a

to be repeated

n

repeat by

Value

Returns the repeated dataset

Custom repmat Function

Description

R version of the matlab function repmat, repeating a by m x n

Usage

customRepmat(a, n, m)

Arguments

a

to be repeated

n

repeater parameter

m

repeater parameter

Value

Returns the repeated dataset

Custom Size Function.

Description

custom R version of matlabs size function. Calls length for vectors, or else calls dim.

Usage

customSize(x, i = NULL)

Arguments

x

object to be checked for size

i

1, 2 or NULL. Defines if both or which size information should be returned.

Value

a vector if i is null, an integer if i is 1 or 2.

Computes the eta value of a signal (slowness)

Description

Computes the eta value of a signal (slowness)

Usage

etaval(x, T = length(x))

Arguments

x

The columns of signal correspond to different input components. Must be normalized (zero mean, unit variance)

T

Time interval

Value

returns the eta value of the signal in a time interval T time units long.

Classifier for SFA demos

Description

Train or apply a Gaussian classifier..

Usage

gaussClassifier(gauss, y, realC, method = "train")

Arguments

gauss

List created by gaussCreate. Contains also the elements:

aligned

=0: do not align the Gaussian classifiers with axes, use full covariance matrix
=1 (default): set the off-diagonals in covariance matrix to 0, i.e. the Gaussian classifier is forced to be aligned with the axes. This is more robust in the case where the data deviate largely from a multivariate normal distribution.

epsD

[defaults to 0.04] replace diagonal elements of COV smaller than epsD with epsD to avoid too small Gaussians

y

K x M matrix where K is the total number of patterns and M is the number of variables used for classification. I.e. each row of y contains the data for one pattern.

realC

1 x K matrix with NCLASS distinct real class labels needed only for method='train'. In case of method="apply" realC is not used and can have any value

method

either "train" (default) or "apply"

Value

list gauss containing

gauss$predC

1 x K matrix: the predicted class

gauss$prob

K x NCLASS matrix: prob(k,n) is the estimated probability that pattern k belongs to class m

See Also

gaussCreate

Create an Gaussian classifier object

Description

Create an Gaussian classifier object

Usage

gaussCreate(nclass, dimY)

Arguments

nclass

number of classes

dimY

dimension

Value

list of defaults for gauss classifier

See Also

gaussClassifier

Load a GAUSS object.

Description

Load a GAUSS object.

Usage

gaussLoad(filename)

Arguments

filename

Load list gauss from this file

Value

list gauss

References

gaussSave

Save a GAUSS object.

Description

Save a GAUSS object.

Usage

gaussSave(gauss, filename)

Arguments

gauss

A list that contains all information about the handled gauss-structure

filename

Save list gauss to this file

References

gaussLoad

Create a new covariance object.

Description

Create a new covariance object.

Usage

lcovCreate(dim)

Arguments

dim

dimension of the covariance object list to be created

Value

list containing the COVMTX, avg and tlen

See Also

lcovUpdate

Fix a covariance object

Description

Computes the definitive covariance matrix and the average of the covariance object referenced by lcov after a series of update operations.

Usage

lcovFix(lcov)

Arguments

lcov

A list that contains all information about the handled covariance-structure

Value

returns the fixed covariance list structure

See Also

lcovCreate lcovPca

Principal Component Analysis on a covariance object

Description

Performs PCA _and_ whitening on the covariance object referenced by lcov. CAUTION: can be numerically instable if covariance matrix is singular, better use LCOV_PCA2 instead /W. Konen/

Usage

lcovPca(lcov, dimRange = NULL)

Arguments

lcov

A list that contains all information about the handled covariance-structure

dimRange

A number or vector for dimensionality reduction:
if it is a number: only the first components 1:dimRange are kept (those with largest eigenvalues)
if it is a range: only the components in the range dimRange[1]..dimRange[2] are kept

Value

returns a list: $W is the whitening matrix, $DW the dewhitening matrix and $D an array containing a list of the eigenvalues. $kvar contains the total variance kept in percent.

Note

lcovFix(lcov) has to be used before this function is applied

See Also

lcovFix lcovPca2

Improved Principal Component Analysis on a covariance object

Description

Performs PCA _and_ whitening on the covariance object referenced by lcov.

Difference to LCOV_PCA: null the rows of W (columns of DW) where the corresponding eigenvalue in D is close to zero (more precisely: if lam/lam_max < EPS = 1e-7). This is numerically stable in the case where the covariance matrix is singular.
- Author: Wolfgang Konen, Cologne Univ., May'2009

Usage

lcovPca2(lcov, dimRange = NULL)

Arguments

lcov

A list that contains all information about the handled covariance-structure

dimRange

A number or vector for dimensionality reduction:
if it is a number: only the first components 1:dimRange are kept (those with largest eigenvalues)
if it is a range: only the components in the range dimRange[1]..dimRange[2] are kept

Value

returns a list: $W is the whitening matrix, $DW the dewhitening matrix and $D an array containing a list of the eigenvalues. $kvar contains the total variance kept in percent.

Note

lcovFix(lcov) has to be used before this function is applied

See Also

lcovFix lcovPca

Transform a covariance object

Description

Computes the definitive covariance matrix and the average of the covariance object referenced by lcov after a series of update operations.

Usage

lcovTransform(lcov, A)

Arguments

lcov

A list that contains all information about the handled covariance-structure to be transformed

A

linear function by which covariance object is to be transformed

Value

returns the fixed covariance list structure

Note

lcovFix(lcov) has to be used before this function is applied

See Also

lcovFix

Update a covariance object

Description

Updates the covariance object referenced by lcov with a new chunk of data DATA.

Usage

lcovUpdate(lcov, DATA)

Arguments

lcov

A list that contains all information about the handled covariance-structure

DATA

Input Data: must be oriented so that each column is a variable and each row a new measurement

Value

list containing the COVMTX, avg and tlen

See Also

lcovCreate

Custom Nonlinear Dimension Calculation

Description

Compute the dimension, used for the customexpansion demo.

Usage

nlDim(n)

Arguments

n

Dimension of input vector

Value

Dimension of expanded vector

See Also

sfa2 nlExpand

Expand a signal in the for Nonlinear Expansion demo

Description

This is an optional expansion function, calculating the expanded data in [x; x^4].

Usage

nlExpand(sfaList, DATA)

Arguments

sfaList

A list that contains all information about the handled sfa-structure

DATA

Input data, each column a different variable

Value

expanded matrix DATA

See Also

sfa2 sfaExpand nlDim

The SFA1 algorithm, linear SFA.

Description

Y = sfa1(X) performs linear Slow Feature Analysis on the input data X and returns the output signals Y ordered by increasing temporal variation, i.e. the first signal Y[,1] is the slowest varying one, Y[,2] the next slowest and so on. The input data have to be organized with each variable in a column and each data (time) point in a row, i.e. X(t,i) is the value of variable nr. i at time t.

Usage

sfa1(x)

Arguments

x

Input data, each column a different variable

Value

list sfaList with all learned information, where sfaList$y contains the outputs

See Also

sfaStep sfa1Create sfaExecute

Create structured list for linear SFA

Description

Create structured list for linear SFA

Usage

sfa1Create(sfaRange, axType = "ORD1", regCt = 0)

Arguments

sfaRange

number of slowly-varying functions to be kept

axType

is the type of derivative approximation to be used, see sfaTimediff

regCt

regularization constant, currently not used

Value

list sfaList contains all arguments passed into sfa1create plus

deg

2

This list will be expanded by other SFA functions with further SFa results

See Also

sfa1 sfaStep sfa2Create

A step in the SFA1 algorithm.

Description

!!! Do not use this function directly, use sfaStep instead !!!

Usage

sfa1Step(sfaList, arg = NULL, step = NULL, method = NULL)

Arguments

sfaList

A list that contains all information about the handled sfa-structure

arg

Input data, each column a different variable

step

Specifies the current SFA step. Must be given in the right sequence: for SFA1 objects: "preprocessing", "sfa"
for SFA2 objects: "preprocessing", "expansion", "sfa" Each time a new step is invoked, the previous one is closed, which might take some time.

method

Method to be used: For sfaList$step="expansion" the choices are "TIMESERIES" or "CLASSIF".
For sfaList$step="sfa" currently no choices.

Value

list sfaList taken from the input, with new information added to this list. Among the new items are:

avg0

mean vector in input space

SF

(sfaRange x sfaRange)-matrix with rows which contain the directions in expanded space with slow signals. The rows are sorted acc. to increasing eigenvalues of time-diff covariance matrix

See Also

sfaStep sfa1Create sfa2Step

The SFA2 algorithm, SFA with degree 2 expansion.

Description

Y = sfa2(X) performs expanded Slow Feature Analysis on the input data X and returns the output signals Y ordered by increasing temporal variation, i.e. the first signal Y[,1] is the slowest varying one, Y[,2] the next slowest varying one and so on. The input data have to be organized with each variable in a column and each data (time) point in a row, i.e. X(t,i) is the value of variable i at time t. By default an expansion to the space of 2nd degree polynomials is done, this can be changed by using different functions for xpDimFun and sfaExpandFun.

Usage

sfa2(
  x,
  method = "SVDSFA",
  ppType = "PCA",
  xpDimFun = xpDim,
  sfaExpandFun = sfaExpand
)

Arguments

x

input data

method

eigenvector calculation method: ="SVDSFA" for singular value decomposition (recommended) or ="GENEIG" for generalized eigenvalues (unstable!). GENEIG is not implemented in the current version, since R lacks an easy option to calculate generalized eigenvalues.

ppType

preprocessing type: ="PCA" (principal component analysis) or ="SFA1" (linear sfa)

xpDimFun

function to calculate dimension of expanded data

sfaExpandFun

function to expand data

Value

list sfaList with all SFA information, among them are

y

a matrix containing the output Y (as described above)

-

all input parameters to sfa2Create

-

all elements of sfaList as specified in sfa2Step

See Also

sfa2Step sfa2Create sfaExecute sfa1

Examples

## prepare input data for simple demo
t=seq.int(from=0,by=0.011,to=2*pi)
x1=sin(t)+cos(11*t)^2
x2=cos(11*t)
x=data.frame(x1,x2)
## perform sfa2 algorithm with data
res = sfa2(x)
## plot slowest varying function of result
plot(t, res$y[,1],type="l",main="output of the slowest varying function")
## see http://www.scholarpedia.org/article/Slow_feature_analysis#The_algorithm
## for detailed description of this example

Create structured list for expanded SFA

Description

'Expanded' SFA means that the input data are expanded into a higher-dimensional space with the function sfaExpandFun. See sfaExpand for the default expansion function.

Usage

sfa2Create(
  ppRange,
  sfaRange,
  ppType = "SFA1",
  axType = "ORD1",
  regCt = 0,
  opts = NULL,
  xpDimFun = xpDim,
  sfaExpandFun = sfaExpand
)

Arguments

ppRange

umber of dimensions to be kept after preprocessing step - or - a two-number vector with lower and upper dimension number

sfaRange

umber of slowly-varying functions to be kept

ppType

preprocessing type: ="PCA", "PCA2" (principal component analysis) or ="SFA1" (linear sfa)

axType

is the type of derivative approximation to be used, see sfaTimediff

regCt

regularization constant, currently not used

opts

optional list of additional options

xpDimFun

Function to calculate dimension of expanded data

sfaExpandFun

Function to expand data

Value

list sfaList contains all arguments passed into sfa2create plus

xpRange

evaluates to xpDimFun(ppRange)

deg

2

This list will be expanded by other SFA functions with further SFa results

See Also

sfa2 sfaStep sfa1Create

A step in the SFA2 algorithm.

Description

!!! Do not use this function directly, use sfaStep instead !!!

Usage

sfa2Step(sfaList, arg = NULL, step = NULL, method = NULL)

Arguments

sfaList

A list that contains all information about the handled sfa-structure

arg

Input data, each column a different variable

step

Specifies the current SFA step. Must be given in the right sequence: for SFA1 objects: "preprocessing", "sfa"
for SFA2 objects: "preprocessing", "expansion", "sfa" Each time a new step is invoked, the previous one is closed, which might take some time.

method

Method to be used: For sfaList$step="expansion" the choices are "TIMESERIES" or "CLASSIF".
For sfaList$step="sfa" the choices are "SVDSFA" (recommended) or "GENEIG" (unstable). GENEIG is not implemented in the current version, since R lacks the option to calculate generalized eigenvalues easily.

Value

list sfaList taken from the input, with new information added to this list. Among the new items are:

avg0

mean vector in input space

avg1

mean vector in expanded space

W0

(ppRange x ppRange)-matrix, the whitening matrix for the input data

C

covariance matrix of the time-diff of expanded and sphered data

SF

(sfaRange x sfaRange)-matrix with rows which contain the directions in expanded space with slow signals. The rows are sorted acc. to increasing eigenvalues of C

See Also

sfaStep sfa2Create sfa1Step

Backslash operator.

Description

Reproduce what MATLAB's backslash operator can do, using qr() and qr.coef().

Usage

sfaBSh(X, Y)

Arguments

X

X matrix

Y

Y vector

Value

Returns coefficients

Check Condition of a matrix for SFA

Description

Creates warnings with recommendations for different settings, if given matrix is ill-conditioned.

Usage

sfaCheckCondition(matr, datatype)

Arguments

matr

matrix to be checked

datatype

string to identify "input" or "expanded" data

Predict Class for SFA classification

Description

Use a SFA classification model (stored in opts$*Filename), predict & evaluate on new data (xtst,realc_tst).
Author of orig. matlab version: Wolfgang Konen, Jan 2011-Mar 2011.
See also [Berkes05] Pietro Berkes: Pattern recognition with Slow Feature Analysis. Cognitive Sciences EPrint Archive (CogPrint) 4104, http://cogprints.org/4104/ (2005)

Usage

sfaClassPredict(xtst, realcTst, opts)

Arguments

xtst

NTST x IDIM, test input data

realcTst

1 x NTST, test class labels

opts

list with several parameter settings:

gaussdim
...
*Filename

[* = s,g,x] from where to load the models (see sfaClassify)

Value

list res containing

res$errtst

1 x 2 matrix: error rate with / w/o SFA on test set

res$ytst

output from SFA when applied to test data

res$predT

predictions with SFA + GaussClassifier on test set

res$predX

predictions w/o SFA (only GaussClassifier) on test set (only if opts.xFilename exists)

See Also

sfaClassify sfaExecute

Predict Class for SFA classification

Description

Create a SFA classification mode, predict & evaluate on new data (xtst,realc_tst).
Author of orig. matlab version: Wolfgang Konen, May 2009 - Jan 2010
See also [Berkes05] Pietro Berkes: Pattern recognition with Slow Feature Analysis. Cognitive Sciences EPrint Archive (CogPrint) 4104, http://cogprints.org/4104/ (2005)

Usage

sfaClassify(x, realclass, xtst = 0, realcTst = 0, opts)

Arguments

x

NREC x IDIM, training input data

realclass

1 x NREC, training class labels

xtst

NTST x IDIM, test input data

realcTst

1 x NTST, test class labels

opts

list with several parameter settings:

gaussdim
...
*Filename

[* = s,g,x] from where to load the models (see sfaClassify)

Value

list res containing

res$errtrn

1 x 2 matrix: error rate with / w/o SFA on training set

res$errtst

1 x 2 matrix: error rate with / w/o SFA on test set

res$y

output from SFA when applied to training data

res$ytst

output from SFA when applied to test data

res$predT

predictions with SFA + GaussClassifier on test set

res$predX

predictions w/o SFA (only GaussClassifier) on test set (only if opts.xFilename exists)

See Also

sfaClassPredict sfaExecute

Execute learned function for input data

Description

After completion of the learning phase (step="sfa") this function can be used to apply the learned function to the input data.
The execution is completed in 4 steps:
1. projection on the input principal components (dimensionality reduction)
2. expansion (if necessary)
3. projection on the whitened (expanded) space
4. projection on the slow functions

Usage

sfaExecute(sfaList, DATA, prj = NULL, ncomp = NULL)

Arguments

sfaList

A list that contains all information about the handled sfa-structure

DATA

Input data, each column a different variable

prj

If not NULL, the preprocessing step 1 is skipped for SFA2

ncomp

number of learned functions to be used

Value

matrix DATA containing the calculated output

See Also

sfa2 sfa1 sfaStep

Degree 2 Expansion

Description

Expand a signal in the space of polynomials of degree 2. This is the default expansion function used by rSFA.

Usage

sfaExpand(sfaList, DATA)

Arguments

sfaList

A list that contains all information about the handled sfa-structure

DATA

Input data, each column a different variable

Value

expanded matrix DATA

See Also

sfa2 nlExpand xpDim

Return a SFA function as a quadratic form.

Description

sfaGetHf returns function number NR in the sfa object referenced by HDL in the form of a quadratic form
q(x) = 1/2*x'*H*x + f'*x + c
Of course, this only works if a quadratic expansion was used during training. The quadratic form can lie in different spaces, i.e. it can receive as input preprocessed or non-preprocessed vectors. This is specified by setting the argument WHERE. The quadratic form lies - in the preprocessed space for WHERE==0 (e.g. the whitened space if the preprocessing type is PCA) - in the PCA space (i.e. projected on the principal components but not whitened, works only if PCA was used for preprocessing) for WHERE==1 - in the input, mean-free space for WHERE==2 - in the input space for WHERE==3 In general you will need to set WHERE to 2 or 3, but working in the preprocessed spaces can often drastically improve the speed of analysis.

Usage

sfaGetHf(sfaList, nr, where)

Arguments

sfaList

A list that contains all information about the handled sfa-structure

nr

function number

where

WHERE parameter

Value

list res
- res contains: res$H res$f res$c

See Also

sfa2Create

Helper Function of SFA.

Description

Helper Function of SFA.

Usage

sfaGetIntRange(range)

Arguments

range

Range

Value

numeric vector int

Load a SFA object.

Description

Load a SFA object.

Usage

sfaLoad(filename)

Arguments

filename

Load list sfaList from this file name

Value

list sfaList

References

sfaSave

Perform non-linear regression

Description

Given the data in arg, expand them nonlinearly in the same way as it was done in the SFA-object sfaList (expanded dimension M) and search the vector RCOEF of M constant coefficients, such that the sum of squared residuals between a given function in time FUNC and the function
R(t) = (v(t) - v0)' * RCOEF, t=1,...,T,
is minimal

Usage

sfaNlRegress(sfaList, arg, func)

Arguments

sfaList

A list that contains all information about the handled sfa-structure

arg

Input data, each column a different variable

func

(T x 1) the function to be fitted nonlinearly

Value

returns a list res with elements

res$R

(T x 1) the function fitted by NL-regression

res$rcoef

(M x 1) the coefficients for the NL-expanded dimensions

Parametric Bootstrap

Description

If training set too small, augment it with parametric bootstrap

Usage

sfaPBootstrap(realclass, x, sfaList)

Arguments

realclass

true class of training data (can be vector, numerics, integers, factors)

x

matrix containing the training data

sfaList

list with several parameter settings, e.g. as created by sfa2Create
sfaList$xpDimFun (=xpDim by default) calculated dimension of expaned SFA space
sfaList$deg degree of expansion (should not be 1, not implemented)
sfaList$ppRange ppRange for SFA algorithm
sfaList$nclass number of unique classes
sfaList$doPB do (1) or do no (0) param. bootstrap.

Value

a list list containing:

x

training set extended to minimu number of recors1.5*(xpdim+nclass), if necessary

realclass

training class labels, extended analogously

See Also

addNoisyCopies

Preprocessing for SFA classification

Description

Helper function for sfaClassify

Usage

sfaPreproc(sfaList, x, opts)

Arguments

sfaList

A list that contains all information about the handled sfa-structure

x

Input data, each column a different variable

opts

list

Value

preprocessed data

See Also

sfaClassPredict sfaClassify

Save a SFA object.

Description

Save a SFA object.

Usage

sfaSave(sfaList, filename)

Arguments

sfaList

A list that contains all information about the handled sfa-structure

filename

Save list sfaList to this file

References

sfaLoad

Update a step of the SFA algorithm.

Description

sfaStep() updates the current step of the SFA algorithm. Depending on sfaList$deg it calls either sfa1Step or sfa2Step to do the main work. See further documentation there

Usage

sfaStep(sfaList, arg, step = NULL, method = NULL)

Arguments

sfaList

A list that contains all information about the handled sfa-structure

arg

Input data, each column a different variable

step

Specifies the current SFA step. Must be given in the right sequence: for SFA1 objects: "preprocessing", "sfa"
for SFA2 objects: "preprocessing", "expansion", "sfa" Each time a new step is invoked, the previous one is closed, which might take some time.

method

Method to be used: For sfaList$step="expansion" the choices are "TIMESERIES" or "CLASSIF".
For sfaList$step="sfa" (sfa2Step only) the choices are "SVDSFA" (recommended) or "GENEIG" (unstable).

Value

list sfaList taken from the input, with new information added to this list. See sfa1Step or sfa2Step for details.

See Also

sfa1Step sfa2Step sfa1Create sfa2Create sfaExecute

Examples

   ## Suppose you have divided your training data into two chunks,
   ## DATA1 and DATA2. Let the number of input dimensions be N. To apply
   ## SFA on them write:
   ## Not run:  
   sfaList = sfa2Create(N,xpDim(N))
   sfaList = sfaStep(sfaList, DATA1, "preprocessing")
   sfaList = sfaStep(sfaList, DATA2)
   sfaList = sfaStep(sfaList, DATA1, "expansion")
   sfaList = sfaStep(sfaList, DATA2)
   sfaList = sfaStep(sfaList, NULL, "sfa")
   output1 = sfaExecute(sfaList, DATA1)
   output2 = sfaExecute(sfaList, DATA2)
   
## End(Not run)

Calculates the first derivative of signal data

Description

Calculates the first derivative of signal data

Usage

sfaTimediff(DATA, axType = "ORD1")

Arguments

DATA

The matrix of signals for which the derivative is calculated (one column per signal)

axType

Type of interpolation: "ORD1" (default) first order, "SCD" second ,"TRD" third, "ORD3a" cubic polynom

Value

matrix DATA
- DATA contains the derivative signals, with the same structure as the input data.

Note

setting axType to invalid values will lead to first order interpolation.

Degree 2 Dimension Calculation

Description

Compute the dimension of a vector expanded in the space of polynomials of 2nd degree.

Usage

xpDim(n)

Arguments

n

Dimension of input vector

Value

Dimension of expanded vector

See Also

sfa2 sfaExpand