Continuous-time multivariate analysis

Description

Implements continuous-time analogues of several classical techniques of multivariate analysis. The inputs are "fd" (functional data) objects from the fda package.

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

References

Paul, B., Reiss, P. T. and Cui, E. (2023). Continuous-time multivariate analysis. arXiv:2307.09404 [stat.ME]

Continuous-time canonical correlation analysis

Description

A continuous-time version of canonical correlation analysis (CCA).

Usage

cca.ct(fdobj1, fdobj2)

Arguments

Value

A list consisting of

Note

Columns of the output matrix vex2 are flipped as needed to ensure positive correlations.

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

cancor, for classical CCA

Examples

## Not run: 

# CCA relating Canadian daily temperature and precipitation data
require(fda)
data(CanadianWeather)
daybasis <- create.bspline.basis(c(0,365), nbasis=80)
tempfd <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"Temperature.C"], daybasis)$fd
precfd <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"log10precip"], daybasis)$fd
tpcor <- cca.ct(tempfd, precfd)
par(mfrow=1:2)
barplot(tpcor$vex1[,1], horiz=TRUE, las=1, main="Temperature",
            sub="First canonical coefficients vector")
barplot(tpcor$vex2[,1], horiz=TRUE, las=1, main="Log precipitation",
            sub="First canonical coefficients vector")


## End(Not run)

Center a continuous-time multivariate data set

Description

Subtracts the (continuous-time) mean of each of the variables. This is analogous to column-centering an n \times p data matrix.

Usage

center.ct(fdobj)

Arguments

Value

A centered version of the input data.

Author(s)

Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

standardize.ct

Continuous-time correlation or cross-correlation matrix

Description

Computes the correlation matrix of a continuous-time multivariate data set represented as an fd object; or the cross-correlation matrix of two such data sets.

Usage

cor.ct(fdobj1, fdobj2 = fdobj1, common_trend = FALSE)

Arguments

Value

A matrix of (cross-) correlations

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

center.fd, for centering of "fd" objects; inprod.cent

Examples

## Not run: 

# Canadian temperature data

require(fda)
require(corrplot)
data(CanadianWeather)
daybasis <- create.fourier.basis(c(0,365), nbasis=55)
tempfd <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"Temperature.C"], daybasis)$fd

## The following yields a matrix of correlations that are all near 1:
rawcor <- cor.ct(tempfd)
corrplot(rawcor, method = 'square', type = 'lower', tl.col="black", tl.cex = 0.6)
## This occurs due to a strong seasonal trend that is common to all stations
## Removing this common trend leads to a more interesting result:
dtcor  <- cor.ct(tempfd, common_trend = TRUE)
ord <- corrMatOrder(dtcor)
dtcord <- dtcor[ord,ord]
corrplot(dtcord, method = 'square', type = 'lower', tl.col="black", tl.cex = 0.6)


## End(Not run)

Continuous-time covariance or cross-covariance matrix

Description

Computes the covariance matrix of a continuous-time multivariate data set represented as an fd object; or the cross-covariance matrix of two such data sets.

Usage

cov.ct(fdobj1, fdobj2 = fdobj1, common_trend = FALSE)

Arguments

Value

A matrix of (cross-) covariances

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

cor.ct

Examples

# see example for cor.ct, which works similarly

Centered inner product matrix for a basis or pair of bases

Description

Several methods of continous-time multivariate analysis require a matrix of inner products of pairs of centered functions from a basis, such as a B-spline basis, or pairs consisting of one function from each of two bases. This function computes such matrices via 7-point Newton-Cotes integration, which is exact for cubic B-splines. For a Fourier basis with the inner product taken over the entire range, a simple closed form is used instead of integration.

Usage

inprod.cent(basis1, basis2 = basis1, rng = NULL)

Arguments

Value

Matrix of inner products of each pair of centered basis functions.

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

create.bspline.basis from package fda, for the most commonly used basis object type.

Examples

## Not run: 

require(fda)
bbasis6 <- create.bspline.basis(nbasis=6)
inprod.cent(bbasis6)
fbasis7 <- create.fourier.basis(nbasis=7)
inprod.cent(fbasis7)

## End(Not run)

Continuous-time k-means clustering

Description

A continuous-time version of k-means clustering in which each cluster is a time segments or set of time segments.

Usage

kmeans.ct(
  fdobj,
  k,
  common_trend = FALSE,
  init.pts = NULL,
  tol = 0.001,
  max.iter = 100
)

Arguments

Value

Object of class "kmeans.ct", a list consisting of

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

kmeans, plot.kmeans.ct

Examples

## Not run: 

require(fda)
data(CanadianWeather)
daybasis <- create.bspline.basis(c(0,365), nbasis=55)
tempfd <- smooth.basis(day.5, CanadianWeather$dailyAv[,,"Temperature.C"], daybasis)$fd
kmtemp3 <- kmeans.ct(tempfd, 3)
plot(kmtemp3, axes=FALSE)
axesIntervals(); box()
plot(silhouette.ct(kmtemp3), axes=FALSE)
axesIntervals(); box()

## End(Not run)

Continuous-time Fisher's linear discriminant analysis

Description

A continuous-time version of Fisher's LDA, in which segments of the time interval take the place of groups of observations.

Usage

lda.ct(fdobj, partition, part.names = NULL)

Arguments

Details

The means and scaling components of the output are similar to lda, but unlike that function, lda.ct performs only Fisher's LDA and cannot incorporate priors or perform classification.

Value

Object of class "lda.ct", a list consisting of

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

plot.lda.ct; lda, for the classical version

Examples

## see end of example in ?pca.ct

Compute means of basis functions

Description

Given a basis object as defined in the fda package (see basisfd), this function simply computes the vector of means of the basis functions. Used internally.

Usage

meanbasis(basis, rng = NULL)

Arguments

Value

Vector of means of the basis functions

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

Examples

require(fda)
bbasis6 <- create.bspline.basis(nbasis=6)
meanbasis(bbasis6)
meanbasis(bbasis6, c(.3,.6))
fbasis11 <- create.fourier.basis(nbasis=11)
meanbasis(fbasis11)

Continuous-time principal component analysis

Description

A continuous-time version of principal component analysis.

Usage

pca.ct(fdobj, cor = FALSE, common_trend = FALSE)

Arguments

Value

Returns a list including:

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

cov.ct; princomp, for the classical version

Examples

## Not run: 

# Data for one session from a classic EEG data set
require(fda)
require(eegkit)
data(eegdata)
data(eegcoord)
longdat <- subset(eegdata, subject=="co2a0000369" & trial==0)
widedat <- reshape(longdat, direction="wide", drop=c("subject","group","condition","trial"),
                 v.names="voltage",idvar="channel")

# Convert time series for 64 channels to a functional data object
bsb <- create.bspline.basis(c(0,255),nbasis=30)
fdo <- Data2fd(argvals=0:255, y=t(as.matrix(widedat[,-1])), basisobj=bsb)
plot(fdo)

# Now do PCA and display first loadings for 3 PC's,
# along with percent variance explained by each
pcc <- pca.ct(fdo)
pve <- 100*pcc$var/sum(pcc$var)
par(mfrow=c(1,3))
cidx <- match(widedat[,1],rownames(eegcoord))
eegspace(eegcoord[cidx,4:5],pcc$loadings[,1], colorlab="PC1 loadings",
         main=paste0(round(pve[1],0), "%"), mar=c(17,3,12,2), cex.main=2)
eegspace(eegcoord[cidx,4:5],pcc$loadings[,2], colorlab="PC2 loadings",
         main=paste0(round(pve[2],0), "%"), mar=c(17,3,12,2), cex.main=2)
eegspace(eegcoord[cidx,4:5],pcc$loadings[,3], colorlab="PC3 loadings",
         main=paste0(round(pve[3],0), "%"), mar=c(17,3,12,2), cex.main=2)

# Linear discriminant analysis: discriminating among the 1st, 2nd and 3rd portions
#  of the time interval
ld <- lda.ct(fdo, c(85,170))
plot(ld)
eegspace(eegcoord[cidx,4:5],ld$scaling[,1], colorlab="LD1 coefficients",
         mar=c(17,3,12,2), cex.main=2)
eegspace(eegcoord[cidx,4:5],ld$scaling[,2], colorlab="LD2 coefficients",
         mar=c(17,3,12,2), cex.main=2)


## End(Not run)

Plot a kmeans.ct object

Description

Plots a continuous-time k-means clustering object generated by a call to kmeans.ct.

Usage

## S3 method for class 'kmeans.ct'
plot(
  x,
  plottype = "functions",
  mark.transitions = TRUE,
  col = NULL,
  lty = NULL,
  xlab = "Time",
  ylab = NULL,
  legend = TRUE,
  ncol.legend = 1,
  cex.legend = 1,
  ...
)

Arguments

Value

None; a plot is generated.

Author(s)

Philip Tzvi Reiss <reiss@stat.haifa.ac.il> and Biplab Paul <paul.biplab497@gmail.com>

See Also

kmeans.ct, which includes an example

Plot an lda.ct object

Description

Plots the Fisher's linear discriminant functions generated by a call to lda.ct.

Usage

## S3 method for class 'lda.ct'
plot(x, ylab = "Discriminants", xlab = "Time", which = NULL, col = NULL, ...)

Arguments

Value

None; a plot is generated.

Author(s)

Biplab Paul <paul.biplab497@gmail.com> and Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

lda.ct

Examples

## see the example at the end of ?pca.ct

Plot a silhouette.ct object

Description

Plots the silhouette index, generated by a call to silhouette.ct, for a continuous-time k-means clustering object.

Usage

## S3 method for class 'silhouette.ct'
plot(x, mark.transitions = TRUE, xlab = "Time", ylab = "Silhouette", ...)

Arguments

Value

None; a plot is generated.

Author(s)

Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

kmeans.ct, which includes an example; silhouette.ct

Silhouettes for continuous-time k-means clustering

Description

Computes the silhouette index, at a grid of time points, for a continuous-time k-means clustering object produced by kmeans.ct.

Usage

silhouette.ct(kmobj, ngrid = 5000)

Arguments

Value

Object of class "silhouette.ct", a list consisting of

Note

An error is issued if the grid of time points contains one or more of the cluster transition points. This should not ordinarily occur, but if it does, it can be remedied by modifying ngrid.

Author(s)

Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

kmeans.ct, which includes an example; plot.silhouette.ct

Center and scale a continuous-time multivariate data set

Description

Subtracts the (continuous-time) mean and divides by the (continuous-time) standard deviation of each of the variables. This is the continuous-time analogue of taking an n \times p data matrix, subtracting the mean of each column, and dividing by the standard deviation of each column, as is done by scale(..., center=TRUE, scale=TRUE).

Usage

standardize.ct(fdobj)

Arguments

Value

A standardized (centered and scaled) version of the input data.

Author(s)

Philip Tzvi Reiss <reiss@stat.haifa.ac.il>

See Also

center.ct