Package 'ctmva'

Continuous-time canonical correlation analysis

Description

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

Usage

cca.ct(fdobj1, fdobj2)

Arguments

fdobj1, fdobj2

a pair of continuous-time multivariate data sets, of class "fd"

Value

A list consisting of

vex1, vex2	matrices defining the canonical variates. The first columns of each give the coefficients defining the first pair of canonical variates; and so on.
cor	canonical correlations, i.e., correlations between the pairs of canonical variates

Note

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

Author(s)

Biplab Paul <[email protected]> and Philip Tzvi Reiss <[email protected]>

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)

---

Curve correlation

Description

Inputs raw data representing two curves, applies penalized B-spline smoothing to the two curves, and computes the curve correlation between them via a call to cor.ct.

Usage

ccor(y, time, curve = NULL, k = 15, min.overlap = 0, min.n = 8)

Arguments

y	either a vector or a two-column matrix, without missing values; see Details
time	a vector of time points
curve	curve indicator; see Details
k	number of B-spline basis functions
min.overlap	minimum overlap of the two curves' time ranges
min.n	minimum number of observations per curve

Details

If y is a two-column matrix, the two curves are observed at the time points given by time; in this case length(time) must equal nrow(y), and curve is ignored. If y is a vector, it must have the same length as both time and curve. In this case y contains the observations on both curves, while elements of time and curve identify the observation time and the curve being observed, respectively.

Value

A list with components

y, time	the supplied y and time
mod1, mod2	models for the two curves, outputted by gam
fd1, fd2	functional data objects (see fd) for the two curves
estcor	estimated curve correlation

Author(s)

Philip Tzvi Reiss <[email protected]>, Noemi Foa, Dror Arbiv and Biplab Paul <[email protected]>

See Also

cor.ct, b.spline

Examples

# see example for ccor_posim

---

Bootstrap confidence interval estimation for curve correlation

Description

Inputs raw data representing two curves, and computes a bootstrap confidence interval for the curve correlation between them.

Usage

ccor_boot(
  y,
  time,
  curve = NULL,
  k = 15,
  ndraw = 299,
  conf = 0.95,
  min.overlap = 0,
  min.n = 8
)

Arguments

y, time, curve, k, min.overlap, min.n	see ccor
ndraw	number of bootstrap samples
conf	confidence level

Value

A list with components

cor	curve correlation (for the full data)
boot.cor	curve correlations for the ndraw bootstrap samples
ci	confidence interval for the curve correlation

Author(s)

Philip Tzvi Reiss <[email protected]>, Noemi Foa, Dror Arbiv and Biplab Paul <[email protected]>

See Also

ccor, ccor_posim

Examples

# see example for ccor_posim

---

Credible interval estimation for curve correlation based on posterior simulation

Description

Inputs raw data representing two curves, and computes a credible interval for the curve correlation between them simulating from the approximate posterior distribution of the joint spline coefficient vector.

Usage

ccor_posim(
  y,
  time,
  curve = NULL,
  method,
  k = 15,
  conf = 0.95,
  ndraw = 999,
  min.overlap = 0
)

Arguments

y, time, curve, k, min.overlap	see ccor
method	"indep" (curves fitted independently) or "mvn" (curves fitted together, with error correlation estimated based on multivariate normal likelihood)
conf	confidence level
ndraw	number of draws from posterior distribution of spline coefficient vector

Value

A list with components

cor	curve correlation
model	the model for the two curves (if method=="mvn"), or a list of the two curves' models (if method=="indep")
bsb	B-spline basis (from package fda) used for the curves
Vc.fda	corrected posterior covariance matrix for the coefficients with respect to the B-spline basis bsb (see the component $Vc in gamObject)
sims	curve correlations for the ndraw draws from the posterior distribution
ci	credible interval for the curve correlation

Author(s)

Philip Tzvi Reiss <[email protected]>, Noemi Foa, Dror Arbiv and Biplab Paul <[email protected]>

See Also

ccor, ccor_boot, mvn

Examples

## Not run:
if (interactive () &&
    requireNamespace("wbwdi", quietly = TRUE) &&
    requireNamespace("ggplot2", quietly = TRUE) &&
    requireNamespace("dplyr", quietly = TRUE)) {

    # Curve correlation of per capita GDP and fertility rate in Paraguay
    wdi_dat <- wbwdi::wdi_get(entities = c("PRY"), start_year=1960, end_year=2023,
                       indicators = c("NY.GDP.PCAP.KD","SP.DYN.TFRT.IN"), format="wide") |>
        dplyr::rename(percapitaGDP = NY.GDP.PCAP.KD, fertility = SP.DYN.TFRT.IN)
    ggplot2::ggplot(wdi_dat, aes(percapitaGDP, fertility, color=year)) + geom_point()

    y <- as.matrix(wdi_dat[ , c("percapitaGDP", "fertility")])

    set.seed(345)

    ci <- list()
    ci[[1]] <- ccor_posim(y=y, time=wdi_dat$year, method="indep")
    ci[[2]] <- ccor_posim(y=y, time=wdi_dat$year, method="mvn")
    ci[[3]] <- ccor_boot(y=y, time=wdi_dat$year, ndraw=399)

    tabl <- matrix(NA, 3, 3)
    for (k in 1:3) tabl[k, ] <- c(ci[[k]]$cor, ci[[k]]$ci)
    dimnames(tabl) <- list(c("Posim_indep", "Posim_MVN", "Bootstrap"), c("Est","Lower95","Upper95"))
    round(tabl, 4)
}
## 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

fdobj

continuous-time multivariate data set of class "fd"

Value

A centered version of the input data.

Author(s)

Philip Tzvi Reiss <[email protected]>

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

fdobj1	continuous-time multivariate data set of class "fd"
fdobj2	an optional second data set
common_trend	logical: centering wrt mean function if TRUE, without centering if FALSE (the default)

Value

If fdobj1 and fdobj2 each consist of a single curve, the (scalar) CT correlation between them. Otherwise a matrix of (cross-) correlations is returned.

Note

When fdobj1==fdobj2, cor.ct(...) is the same as cov2cor(cov.ct(...)).

Author(s)

Biplab Paul <[email protected]> and Philip Tzvi Reiss <[email protected]>

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

fdobj1	continuous-time multivariate data set of class "fd"
fdobj2	an optional second data set
common_trend	logical: centering with respect to the mean function if TRUE, without centering if FALSE (the default)

Value

A matrix of (cross-) covariances

Author(s)

Biplab Paul <[email protected]> and Philip Tzvi Reiss <[email protected]>

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

basis1	basis object from the fda package.
basis2	an optional second basis
rng	time range. By default, the entire range spanned by the basis, or the intersection of the ranges of the two bases.

Value

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

Author(s)

Biplab Paul <[email protected]> and Philip Tzvi Reiss <[email protected]>

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

fdobj	continuous-time multivariate data set of class "fd"
k	number of clusters
common_trend	logical: Should the curves be centered with respect to the mean function? Defaults to FALSE.
init.pts	a set of k time points. The observations at these time points serve as initial values for the k means. Randomly generated if not supplied.
tol	convergence tolerance for the k means
max.iter	maximum number of iterations

Value

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

fdobj	the supplied fdobj
means	means of the k clusters
transitions	transition points between segments
cluster	cluster memberships in the segments defined by the transitions
size	length of each cluster, i.e. sum of lengths of subintervals making up each cluster
totisd	total integrated sum of distances from the overall mean, analogous to totss from kmeans
withinisd	within-cluster integrated sum of distances, i.e. integrated sum of distances from each cluster mean
tot.withinisd	total within-cluster integrated sum of distances, i.e. sum(withinisd)
betweenisd	between-cluster integrated sum of distances, i.e. totisd-tot.withinss

Author(s)

Biplab Paul <[email protected]> and Philip Tzvi Reiss <[email protected]>

See Also

kmeans, plot.kmeans.ct, silhouette.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, xlab=", ylab="Temperature")
axesIntervals(); box()
plot(silhouette.ct(kmtemp3), axes=FALSE, xlab=")
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

fdobj	continuous-time multivariate data set of class "fd"
partition	a priori break points dividing the time interval into segments
part.names	optional character vector of names for the segments

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

means	means of the variables within each segment
scaling	matrix of coefficients defining the discriminants (as in lda)
values	eigenvalues giving the ratios of between to within sums of squares
partition	the supplied partition
fdobj	linear discriminants represented as an "fd" object
nld	number of linear discriminants

Author(s)

Biplab Paul <[email protected]> and Philip Tzvi Reiss <[email protected]>

See Also

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

Examples

## see end of example in ?pca.ct

---

Continuous-time multidimensional scaling

Description

A continuous-time version of classical multidimensional scaling.

Usage

mds.ct(
  d,
  argvals = NULL,
  weights = "equal",
  nbasis = 10,
  norder = 4,
  breaks = NULL,
  k = 2,
  smooth = "sandwich",
  lambda = exp(-10:20),
  gcvplot = TRUE,
  Im.tol = 1e-15,
  recenter = TRUE
)

Arguments

d	matrix of dissimilarities
argvals	time points
weights	quadrature weights for the time points: either "equal" (default) or "trap" (trapezoidal, recommended for unequally spaced times)
nbasis	number of B-spline basis functions
norder	order of B-splines (degree plus 1); default is 4 (cubic splines)
breaks	knots, including the endpoints of the time range
k	number of principal coordinates
smooth	bivariate smoothing method: "sandwich" (default) for the sandwich smoother, "one-step" for a more traditional but slower tensor product smoother
lambda	smoothing parameter
gcvplot	logical: Should a plot of log lambda versus the GCV criterion be produced?
Im.tol	tolerance for imaginary component of eigenvalues: imaginary components with magnitude below this is set to zero, those above it generate an error.
recenter	logical: Should the solution be double-centered?

Value

An object of class "mds.ct", which is a list with components

funcs	an object of class "fd" giving the $k$ principal coordinate functions
evals	eigenvalues
argvals	the given time points
GOF	a $k \times 2$ matrix giving the proportion of dissimilarity explained, according to the two definitions used by the GOF component of the output from cmdscale
sandwich	output of s3, if smooth=="sandwich"
recenter	value of the argument recenter

Author(s)

Philip Tzvi Reiss <[email protected]>

See Also

cmdscale, for the classical version

Examples

if (interactive() && 
    requireNamespace("fda", quietly = TRUE) &&
    requireNamespace("viridisLite", quietly = TRUE) &&
    requireNamespace("vegan", quietly = TRUE)) {
    
    require(ggplot2)
    
    data("handwrit", package = "fda")
    
    fives <- 5 * 0:280  + 1
    hw <- handwrit[ , 1, ]
    sd. <- .015
    
    hh <- cbind(hw[fives,], rnorm(281, sd=sd.))
    classical <- cmdscale(dist(hh), eig=TRUE)
    ctmds <- mds.ct(dist(hh), argvals=handwritTime[fives], nbasis=100, recenter=TRUE,
                    weights="trap", norder=7, lambda=exp(2:40))  
                    # a plot of GCV versus lambda is produced
    
    pro.classical <- vegan::procrustes(hw[fives,], classical, scale=FALSE)
    dat.classical <- as.data.frame(pro.classical$Yrot)
    pro.ctmds <- vegan::procrustes(hw[fives,], fda::eval.fd(ctmds$argvals, ctmds$funcs),
                                   scale=FALSE)
    dat.ctmds <- as.data.frame(matrix(pro.ctmds$translation, length(handwritTime), 2, byrow=TRUE) +
                                   fda::eval.fd(handwritTime, ctmds$funcs) %*% pro.ctmds$rotation)
    names(dat.classical) <- names(dat.ctmds) <- c("x", "y")
    dat.classical$time <- handwritTime[fives]
    dat.ctmds$time <- handwritTime
    
    # Plot of classical (dots) versus continuous-time (curve) MDS reconstructions
    # of the handwritten "fda" (grey), corrupted by noise
    g1 <- ggplot(hw, aes(x=X, y=Y)) + geom_point(color="darkgrey", size=.6) + coord_fixed() +
        geom_point(data = dat.classical, aes(x, y, color = time), size=1) +
        geom_point(data = dat.ctmds, aes(x, y, color = time), size=.6) +
        scale_color_gradientn(colors=viridisLite::plasma(50)) + 
        labs(x="", y="", color="Time (ms)", 
             title=bquote(sigma == .(sd.) * ", " 
                          ~ .(round(100*ctmds$GOF[2,1], 1)) * "% explained")) 
    print(g1)
    
    g2 <- plot(ctmds)  # uses plot.mds.ct
    print(g2)
    
    g3 <- procrustes.plot.mds(ctmds, dat.classical[ , 1:2])
    print(g3)
    
}

---

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

basis	a basis object of class "basisfd"
rng	time range. By default, the entire interval spanned by the basis. Must be left NULL for Fourier bases.

Value

Vector of means of the basis functions

Author(s)

Biplab Paul <[email protected]> and Philip Tzvi Reiss <[email protected]>

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

fdobj	continuous-time multivariate data set of class "fd"
cor	logical: use correlation matrix if TRUE, covariance if FALSE (the default)
common_trend	logical: Should the curves be centered with respect to the mean function? Defaults to FALSE.

Value

Returns a list including:

var	variances of the principal components.
loadings	the matrix of loadings (i.e., its columns are the eigenvectors of the continuous-time covariance).
scorefd	score functions.

Author(s)

Biplab Paul <[email protected]> and Philip Tzvi Reiss <[email protected]>

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

x	clustering object produced by kmeans.ct
plottype	either "functions" (the default), to display each variable as a smooth function of time, or "distance", to plot distances from the k cluster means versus time.
mark.transitions	logical: Should transitions between clusters be marked with vertical lines? Defaults to TRUE.
col	plot colors
lty	line type
xlab, ylab	x- and y-axis labels
legend	either a logical variable (whether a legend should be included) or a character vector to appear in the legend. Default is TRUE.
ncol.legend	number of columns for legend
cex.legend	character expansion factor for legend
...	other arguments passed to matplot

Value

None; a plot is generated.

Author(s)

Philip Tzvi Reiss <[email protected]> and Biplab Paul <[email protected]>

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

x	linear discriminant analysis object produced by lda.ct
ylab, xlab	y- and x-axis labels
which	which of the linear discrminants to plot
col	color vector
...	other arguments passed to matplot

Value

None; a plot is generated.

Author(s)

Biplab Paul <[email protected]> and Philip Tzvi Reiss <[email protected]>

See Also

lda.ct

Examples

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

---

Plot an mds.ct object

Description

Plots a curve representing of two continuous-time principal coordinates produced by mds.ct.

Usage

## S3 method for class 'mds.ct'
plot(
  x,
  npoints = 500,
  cols = viridis(50),
  title = "",
  size.axes = 11,
  samescale = TRUE,
  coords = 1:2,
  GOF.method = 1,
  digits = 1,
  ...
)

Arguments

x	an object of class "mds.ct"
npoints	number of time points (equally spaced along the range of times) at which to plot the coordinates
cols	color scheme; viridis(50) by default
title	plot title
size.axes	size of axis titles
samescale	logical: Should the coordinates be plotted on the same scale?
coords	which two principal coordinates to plot (default is 1:2)
GOF.method	method to use for computing percent dissimilarity explained (see argument GOF of cmdscale)
digits	number of digits to display for percent dissimilarity explained
...	other arguments, passed to theme

Value

None; just produces a plot.

Examples

# see example for mds.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

x	silhouette object produced by silhouette.ct
mark.transitions	logical: Should transitions between clusters be marked with vertical lines? Defaults to TRUE.
xlab, ylab	x- and y-axis labels
...	other arguments passed to plot

Value

None; a plot is generated.

Author(s)

Philip Tzvi Reiss <[email protected]>

See Also

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

---

Procrustes transformation for continuous-time multidimensional scaling

Description

Matches classical principal coordinates to continuous-time principal coordinates produced by mds.ct, via Procrustes transformation.

Usage

procrustes.mds.ct(obj, points, coord = NULL)

Arguments

obj	an object of class "mds.ct"
points	matrix of classical principal coordinates, e.g. as produced by cmdscale
coord	which coordinates to transform.

Value

The output of procrustes.

Note

The function uses procrustes, from the vegan package, to transform the classical principal coordinates given by points to the continuous-time principal coordinates defined by obj, evaluated at the time points given therein. By default, all of the coordinates are used. If obj and points do not include the same number of coordinates, the smaller number is used, and a warning is issued.

Author(s)

Philip Tzvi Reiss <[email protected]>

See Also

procrustes.plot.mds

Examples

# see call to plot.procrustes.ct in mds.ct example

---

Plot of classical and continuous-time principal coordinates

Description

This function plots two continuous-time principal coordinates along with the corresponding classical principal coordinates, where the latter is aligned with the former by Procrustes transformation

Usage

procrustes.plot.mds(
  obj,
  points,
  npoints = 500,
  cols = viridis(50),
  procoord = 1:2,
  plotcoord = 1:2,
  linewidth = 1.3,
  ltitle = "time",
  title = "",
  samescale = FALSE,
  cex.legend = 1,
  GOF.method = 1,
  digits = 1,
  xlim = NULL,
  ylim = NULL,
  size.axis.title = 11,
  size.axis.text = 11,
  size.ltitle = 11,
  size.ltext = 11,
  coord_fixed = TRUE,
  xlab = NULL,
  ylab = NULL,
  colourbar = TRUE,
  ...
)

Arguments

obj	an object of class "mds.ct"
points	matrix of classical principal coordinates, e.g. as produced by cmdscale
npoints	number of time points (equally spaced along the range of times) at which to plot the coordinates
cols	color scheme; viridis(50) by default
procoord	coordinates to which Procrustes transformation should be applied
plotcoord	which coordinates to plot
linewidth	line width for the principal coordinate curve
ltitle	legend title
title	title of the graph
samescale	logical: Should the x- and y-axes have the same range?
cex.legend	scaling factor for legend key
GOF.method	method to use for computing percent dissimilarity explained (see argument GOF of cmdscale)
digits	number of digits to display for percent dissimilarity explained
xlim, ylim	x- and y-limits. Ignored if samescale==TRUE.
size.axis.title	size for axis titles
size.axis.text	size for axis text
size.ltitle	size for color legend title
size.ltext	size for color legend text
coord_fixed	logical: Should aspect ratio be fixed to 1?
xlab, ylab	x- and y-labels. If these are NULL, the principal coordinate numbers and the percent dissimilarity explained are used as the axis labels.
colourbar	logical: Should a color bar (legend) be included?
...	arguments passed to labs

Value

None; a plot is generated

Author(s)

Philip Tzvi Reiss <[email protected]>

See Also

procrustes.mds.ct

Examples

# see example for mds.ct

---

Symmetric sandwich smoother

Description

This function is used by mds.ct to fit a symmetric version of the sandwich smoother of Xiao et al. (2013).

Usage

s3(Y, B, P, lambda = exp(-10:20))

Arguments

Y	a square symmetric matrix
B	a B-spline basis evaluation matrix
P	a penalty matrix
lambda	a vector of smoothing parameter values

Value

A list with components

gcv	generalized cross-validation (GCV) criterion for each value of lambda
bestlam	value of lambda minimizing the GCV
coefmat	matrix of tensor product B-spline coefficients
Yhat	the smoothed version of Y

Author(s)

Philip Tzvi Reiss <[email protected]>

References

Xiao, L., Li, Y. & Ruppert, D. (2013). Fast bivariate P-splines: the sandwich smoother. Journal of the Royal Statistical Society Series B, 75(3), 577–599.

See Also

mds.ct

Examples

# see example for mds.ct (which calls s3)

---

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

kmobj	continuous-time k-means clustering from kmeans.ct
ngrid	number of equally spaced grid points at which to compute the silhouette index

Value

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

grid	grid of ngrid points spanning the time range
value	silhouette index at each point along the grid
transitions	transition points between segments
cluster	cluster memberships in the segments defined by the transitions
mean	mean silhouette index

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 <[email protected]>

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

fdobj

continuous-time multivariate data set of class "fd"

Value

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

Author(s)

Philip Tzvi Reiss <[email protected]> and Biplab Paul <[email protected]>

See Also

center.ct

---