# Continuous-time multidimensional scaling A continuous-time version of classical multidimensional scaling. ```r 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? ## Returns 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` ## Examples ```r 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) } ``` ## See Also `cmdscale`, for the classical version ## Author(s) Philip Tzvi Reiss

mds.ct function