Synthesis, characterization, and cytotoxic and antimicrobial activities of ruthenium(II) arene complexes with N,N-bidentate ligands

Package ‘MFPCA’ August 27, 2025 Type Package Title Multivariate Functional Principal Component Analysis for Data Observed on Different Dimensional Domains Version 1.3-11 Date 2025-08-26 Maintainer Clara Happ-Kurz <chk_R@gmx.de> Description Calculate a multivariate functional principal component analysis for data observed on different dimensional domains. The estimation algorithm relies on univariate basis expansions for each element of the multivariate functional data (Happ & Greven, 2018) <doi:10.1080/01621459.2016.1273115>. Multivariate and univariate functional data objects are represented by S4 classes for this type of data implemented in the package 'funData'. For more details on the general concepts of both packages and a case study, see Happ-Kurz (2020) <doi:10.18637/jss.v093.i05>. URL https://github.com/ClaraHapp/MFPCA License GPL-2 Imports abind, foreach, irlba, Matrix(>= 1.5-0), methods, mgcv (>= 1.8-33), plyr, stats Depends R (>= 3.2.0), funData (>= 1.3-4) Suggests covr, fda, testthat (>= 2.0.0) NeedsCompilation yes SystemRequirements libfftw3 (>= 3.3.4) RoxygenNote 7.3.1 Author Clara Happ-Kurz [aut, cre] (ORCID: <https://orcid.org/0000-0003-4737-3835>) Repository CRAN Date/Publication 2025-08-27 20:30:02 UTC 1 2 FCP_TPA Contents FCP_TPA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . MFPCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . multivExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PACE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . plot.MFPCAfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . predict.MFPCAfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . print.MFPCAfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . print.summary.MFPCAfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scoreplot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . scoreplot.MFPCAfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . screeplot.MFPCAfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . summary.MFPCAfit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ttv . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . UMPCA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . univDecomp . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . univExpansion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Index FCP_TPA 2 4 10 11 13 14 15 16 16 17 18 19 19 21 22 23 26 The functional CP-TPA algorithm Description This function implements the functional CP-TPA (FCP-TPA) algorithm, that calculates a smooth PCA for 3D tensor data (i.e. N observations of 2D images with dimension S1 x S2). The results are given in a CANDECOMP/PARAFRAC (CP) model format X= K X dk · uk ◦ vk ◦ wk k=1 where ◦ stands for the outer product, dk is a scalar and uk , vk , wk are eigenvectors for each direction of the tensor. In this representation, the outer product vk ◦wk can be regarded as the k-th eigenimage, while dk · uk represents the vector of individual scores for this eigenimage and each observation. Usage FCP_TPA( X, K, penMat, alphaRange, verbose = FALSE, tol = 1e-04, maxIter = 15, adaptTol = TRUE ) FCP_TPA 3 Arguments X The data tensor of dimensions N x S1 x S2. K The number of eigentensors to be calculated. penMat A list with entries v and w, containing a roughness penalty matrix for each direction of the image. The algorithm does not induce smoothness along observations (see Details). alphaRange A list of length 2 with entries v and w , containing the range of smoothness parameters to test for each direction. verbose Logical. If TRUE, computational details are given on the standard output during calculation of the FCP_TPA. tol A numeric value, giving the tolerance for relative error values in the algorithm. Defaults to 1e-4. It is automatically multiplied by 10 after maxIter steps, if adaptTol = TRUE. maxIter A numeric value, the maximal iteration steps. Can be doubled, if adaptTol = TRUE. adaptTol Logical. If TRUE, the tolerance is adapted (multiplied by 10), if the algorithm has not converged after maxIter steps and another maxIter steps are allowed with the increased tolerance, see Details. Use with caution. Defaults to TRUE. Details The smoothness of the eigenvectors vk , wk is induced by penalty matrices for both image directions, that are weighted by smoothing parameters αvk , αwk . The eigenvectors uk are not smoothed, hence the algorithm does not induce smoothness along observations. Optimal smoothing parameters are found via a nested generalized cross validation. In each iteration of the TPA (tensor power algorithm), the GCV criterion is optimized via optimize on the interval specified via alphaRange$v (or alphaRange$w, respectively). The FCP_TPA algorithm is an iterative algorithm. Convergence is assumed if the relative difference between the actual and the previous values are all below the tolerance level tol. The tolerance level is increased automatically, if the algorithm has not converged after maxIter steps and if adaptTol = TRUE. If the algorithm did not converge after maxIter steps (or 2 * maxIter) steps, the function throws a warning. Value d A vector of length K, containing the numeric weights dk in the CP model. U A matrix of dimensions N x K, containing the eigenvectors uk in the first dimension. V A matrix of dimensions S1 x K, containing the eigenvectors vk in the second dimension. W A matrix of dimensions S2 x K, containing the eigenvectors wk in the third dimension. 4 MFPCA References G. I. Allen, "Multi-way Functional Principal Components Analysis", IEEE International Workshop on Computational Advances in Multi-Sensor Adaptive Processing, 2013. See Also fcptpaBasis Examples # set.seed(1234) N <- 100 S1 <- 75 S2 <- 75 # define "true" components v <- sin(seq(-pi, pi, length.out = S1)) w <- exp(seq(-0.5, 1, length.out = S2)) # simulate tensor data with dimensions N x S1 x S2 X <- rnorm(N, sd = 0.5) %o% v %o% w # create penalty matrices (penalize first differences for each dimension) Pv <- crossprod(diff(diag(S1))) Pw <- crossprod(diff(diag(S2))) # estimate one eigentensor res <- FCP_TPA(X, K = 1, penMat = list(v = Pv, w = Pw), alphaRange = list(v = c(1e-4, 1e4), w = c(1e-4, 1e4)), verbose = TRUE) # plot the results and compare to true values plot(res$V) points(v/sqrt(sum(v^2)), pch = 20) legend("topleft", legend = c("True", "Estimated"), pch = c(20, 1)) plot(res$W) points(w/sqrt(sum(w^2)), pch = 20) legend("topleft", legend = c("True", "Estimated"), pch = c(20, 1)) MFPCA Multivariate functional principal component analysis for functions on different (dimensional) domains Description This function calculates a multivariate functional principal component analysis (MFPCA) based on i.i.d. observations x1 , . . . , xN of a multivariate functional data-generating process X = (X (1) , . . . X (p) ) MFPCA 5 with elements X (j) ∈ L2 (Tj ) defined on a domain Tj ⊂ IRdj . In particular, the elements can be defined on different (dimensional) domains. The results contain the mean function, the estimated multivariate functional principal components ψ̂1 , . . . , ψ̂M (having the same structure as xi ), the associated eigenvalues ν̂1 ≥ . . . ≥ ν̂M > 0 and the individual scores ρ̂im = < x\ i , ψm >. Moreover, estimated trajectories for each observation based on the truncated Karhunen-Loeve representation x̂i = M X ρ̂im ψ̂m m=1 (1) (p) are given if desired (fit = TRUE). The implementation of the observations xi = (xi , . . . , xi ), i = 1, . . . , N , the mean function and multivariate functional principal components ψ̂1 , . . . , ψ̂M uses the multiFunData class, which is defined in the package funData. Usage MFPCA( mFData, M, uniExpansions, weights = rep(1, length(mFData)), fit = FALSE, approx.eigen = FALSE, bootstrap = FALSE, nBootstrap = NULL, bootstrapAlpha = 0.05, bootstrapStrat = NULL, verbose = options()$verbose ) Arguments mFData A multiFunData object containing the N observations. M The number of multivariate functional principal components to calculate. uniExpansions A list characterizing the (univariate) expansion that is calculated for each element. See Details. weights An optional vector of weights, defaults to 1 for each element. See Details. fit Logical. If TRUE, a truncated multivariate Karhunen-Loeve representation for the data is calculated based on the estimated scores and eigenfunctions. approx.eigen Logical. If TRUE, the eigenanalysis problem for the estimated covariance matrix is solved approximately using the irlba package, which is much faster. If the number M of eigenvalues to calculate is high with respect to the number of observations in mFData or the number of estimated univariate eigenfunctions, the approximation may be inappropriate. In this case, approx.eigen is set to FALSE and the function throws a warning. Defaults to FALSE. bootstrap Logical. If TRUE, pointwise bootstrap confidence bands are calculated for the multivariate functional principal components. Defaults to FALSE. See Details. 6 MFPCA nBootstrap The number of bootstrap iterations to use. Defaults to NULL, which leads to an error, if bootstrap = TRUE. bootstrapAlpha A vector of numerics (or a single number) giving the significance level for bootstrap intervals. Defaults to 0.05. bootstrapStrat A stratification variable for bootstrap. Must be a factor of length nObs(mFData) or NULL (default). If NULL, no stratification is made in the bootstrap resampling, i.e. the curves are sampled with replacement. If bootstrapStrat is not NULL, the curves are resampled with replacement within the groups defined by bootstrapStrat, hence keeping the group proportions fixed. verbose Logical. If TRUE, the function reports extra-information about the progress (incl. timestamps). Defaults to options()$verbose. Details Weighted MFPCA: If the elements vary considerably in domain, range or variation, a weight vector w1 , . . . , wp can be supplied and the MFPCA is based on the weighted scalar product << f, g >>w = p X j=1 Z wj f (j) (t)g (j) (t)dt Tj and the corresponding weighted covariance operator Γw . Bootstrap: If bootstrap = TRUE, pointwise bootstrap confidence bands are generated for the multivariate eigenvalues ν̂1 , . . . , ν̂M as well as for multivariate functional principal components ψ̂1 , . . . , ψ̂M . The parameter nBootstrap gives the number of bootstrap iterations. In each iteration, the observations are resampled on the level of (multivariate) functions and the whole MFPCA is recalculated. In particular, if the univariate basis depends on the data (FPCA approaches), basis functions and scores are both re-estimated. If the basis functions are fixed (e.g. splines), the scores from the original estimate are used to speed up the calculations. The confidence bands for the eigenfunctions are calculated separately for each element as pointwise percentile bootstrap confidence intervals. Analogously, the confidence bands for the eigenvalues are also percentile bootstrap confidence bands. The significance level(s) can be defined by the bootstrapAlpha parameter, which defaults to 5%. As a result, the MFPCA function returns a list CI of the same length as bootstrapAlpha, containing the lower and upper bounds of the confidence bands for the principal components as multiFunData objects of the same structure as mFData. The confidence bands for the eigenvalues are returned in a list CIvalues, containing the upper and lower bounds for each significance level. Univariate Expansions: The multivariate functional principal component analysis relies on a univariate basis expansion for each element X (j) . The univariate basis representation is calculated using the univDecomp function, that passes the univariate functional observations and optional parameters to the specific function. The univariate decompositions are specified via the uniExpansions argument in the MFPCA function. It is a list of the same length as the mFData object, i.e. having one entry for each element of the multivariate functional data. For each element, uniExpansion must specify at least the type of basis functions to use. Additionally, one may add further parameters. The following basis representations are supported: • Given basis functions. Then uniExpansions[[j]] = list(type = "given", functions,scores, ortho), where functions is a funData object on the same domain as mFData, containing MFPCA 7 the given basis functions. The parameters scores and ortho are optional. scores is an N x K matrix containing the scores (or coefficients) of the observed functions for the given basis functions, where N is the number of observed functions and K is the number of basis functions. Note that the scores need to be demeaned to give meaningful results. If scores are not supplied, they are calculated using the given basis functions. The parameter ortho specifies whether the given basis functions are orthonormal orhto = TRUE or not ortho = FALSE. If ortho is not supplied, the functions are treated as non-orthogonal. scores and ortho are not checked for plausibility, use them at your own risk! • Univariate functional principal component analysis. Then uniExpansions[[j]] = list(type = "uFPCA", nbasis, pve, npc, makePD), where nbasis,pve,npc,makePD are parameters passed to the PACE function for calculating the univariate functional principal component analysis. • Basis functions expansions from the package fda. Then uniExpansions[[j]] = list(type = "fda", ...), where ... are passed to funData2fd, which heavily builds on eval.fd. If fda is not available, a warning is thrown. • Spline basis functions (not penalized). Then uniExpansions[[j]] = list(type = "splines1D", bs, m, k), where bs,m,k are passed to the functions univDecomp and univExpansion. For two-dimensional tensor product splines, use type = "splines2D". • Spline basis functions (with smoothness penalty). Then uniExpansions[[j]] = list(type = "splines1Dpen", bs, m,k), where bs,m,k are passed to the functions univDecomp and univExpansion. Analogously to the unpenalized case, use type = "splines2Dpen" for 2D penalized tensor product splines. • Cosine basis functions. Use uniExpansions[[j]] = list(type = "DCT2D",qThresh, parallel) for functions one two-dimensional domains (images) and type = "DCT3D" for 3D images. The calculation is based on the discrete cosine transform (DCT) implemented in the Clibrary fftw3. If this library is not available, the function will throw a warning. qThresh gives the quantile for hard thresholding the basis coefficients based on their absolute value. If parallel = TRUE, the coefficients for different images are calculated in parallel. See univDecomp and univExpansion for details. Value An object of class MFPCAfit containing the following components: values A vector of estimated eigenvalues ν̂1 , . . . , ν̂M . functions A multiFunData object containing the estimated multivariate functional principal components ψ̂1 , . . . , ψ̂M . scores A matrix of dimension N x M containing the estimated scores ρ̂im . vectors A matrix representing the eigenvectors associated with the combined univariate score vectors. This might be helpful for calculating predictions. normFactors The normalizing factors used for calculating the multivariate eigenfunctions and scores. This might be helpful when calculation predictions. meanFunction A multivariate functional data object, corresponding to the mean function. The MFPCA is applied to the de-meaned functions in mFData. fit A multiFunData object containing estimated trajectories for each observation based on the truncated Karhunen-Loeve representation and the estimated scores and eigenfunctions. 8 MFPCA CI A list of the same length as bootstrapAlpha, containing the pointwise lower and upper bootstrap confidence bands for each eigenfunction and each significance level in form of multiFunData objects (only if bootstrap = TRUE). CIvalues A list of the same length as bootstrapAlpha, containing the lower and upper bootstrap confidence bands for each eigenvalue and each significance level (only if bootstrap = TRUE). References C. Happ, S. Greven (2018): Multivariate Functional Principal Component Analysis for Data Observed on Different (Dimensional) Domains. Journal of the American Statistical Association, 113(522): 649-659. DOI: doi:10.1080/01621459.2016.1273115 C. Happ-Kurz (2020): Object-Oriented Software for Functional Data. Journal of Statistical Software, 93(5): 1-38. DOI: doi:10.18637/jss.v093.i05 See Also See Happ-Kurz (2020. doi:10.18637/jss.v093.i05) for a general introduction to the funData package and it’s interplay with MFPCA. This file also includes a case study on how to use MFPCA. Useful functions: multiFunData, PACE, univDecomp, univExpansion, summary, plot, scoreplot Examples oldPar <- par(no.readonly = TRUE) set.seed(1) ### simulate data (one-dimensional domains) sim <- simMultiFunData(type = "split", argvals = list(seq(0,1,0.01), seq(-0.5,0.5,0.02)), M = 5, eFunType = "Poly", eValType = "linear", N = 100) # MFPCA based on univariate FPCA uFPCA <- MFPCA(sim$simData, M = 5, uniExpansions = list(list(type = "uFPCA"), list(type = "uFPCA"))) summary(uFPCA) plot(uFPCA) # plot the eigenfunctions as perturbations of the mean scoreplot(uFPCA) # plot the scores # MFPCA based on univariate spline expansions splines <- MFPCA(sim$simData, M = 5, uniExpansions = list(list(type = "splines1D", k = 10), list(type = "splines1D", k = 10)), fit = TRUE) # calculate reconstruction, too summary(splines) plot(splines) # plot the eigenfunctions as perturbations of the mean scoreplot(splines) # plot the scores ### Compare estimates to true eigenfunctions # flip to make results more clear uFPCA$functions <- flipFuns(sim$trueFuns, uFPCA$functions) splines$functions <- flipFuns(sim$trueFuns, splines$functions) MFPCA 9 par(mfrow = c(1,2)) plot(sim$trueFuns[[1]], main = "Eigenfunctions\n1st Element", lwd = 2) plot(uFPCA$functions[[1]], lty = 2, add = TRUE) plot(splines$functions[[1]], lty = 3, add = TRUE) plot(sim$trueFuns[[2]], main = "Eigenfunctions\n2nd Element", lwd = 2) plot(uFPCA$functions[[2]], lty = 2, add = TRUE) plot(splines$functions[[2]], lty = 3, add = TRUE) legend("bottomleft", c("True", "uFPCA", "splines"), lty = 1:3, lwd = c(2,1,1)) # Test reconstruction for the first 10 observations plot(sim$simData[[1]], obs = 1:10, main = "Reconstruction\n1st Element", lwd = 2) plot(splines$fit[[1]], obs = 1:10, lty = 2, col = 1, add = TRUE) plot(sim$simData[[2]], obs = 1:10, main = "Reconstruction\n2nd Element", lwd = 2) plot(splines$fit[[2]], obs = 1:10, lty = 2, col = 1, add = TRUE) legend("bottomleft", c("True", "Reconstruction"), lty = c(1,2), lwd = c(2,1)) # MFPCA with Bootstrap-CI for the first 2 eigenfunctions ### ATTENTION: Takes long splinesBoot <- MFPCA(sim$simData, M = 2, uniExpansions = list(list(type = "splines1D", k = 10), list(type = "splines1D", k = 10)), bootstrap = TRUE, nBootstrap = 100, bootstrapAlpha = c(0.05, 0.1), verbose = TRUE) summary(splinesBoot) plot(splinesBoot$functions[[1]], ylim = c(-2,1.5)) plot(splinesBoot$CI$alpha_0.05$lower[[1]], lty = 2, add = TRUE) plot(splinesBoot$CI$alpha_0.05$upper[[1]], lty = 2, add = TRUE) plot(splinesBoot$CI$alpha_0.1$lower[[1]], lty = 3, add = TRUE) plot(splinesBoot$CI$alpha_0.1$upper[[1]], lty = 3, add = TRUE) abline(h = 0, col = "gray") plot(splinesBoot$functions[[2]], ylim = c(-1,2.5)) plot(splinesBoot$CI$alpha_0.05$lower[[2]], lty = 2, add = TRUE) plot(splinesBoot$CI$alpha_0.05$upper[[2]], lty = 2, add = TRUE) plot(splinesBoot$CI$alpha_0.1$lower[[2]], lty = 3, add = TRUE) plot(splinesBoot$CI$alpha_0.1$upper[[2]], lty = 3, add = TRUE) abline(h = 0, col = "gray") legend("topleft", c("Estimate", "95% CI", "90% CI"), lty = 1:3, lwd = c(2,1,1)) # Plot 95% confidence bands for eigenvalues plot(1:2, splinesBoot$values, pch = 20, ylim = c(0, 1.5), main = "Estimated eigenvalues with 95% CI", xlab = "Eigenvalue no.", ylab = "") arrows(1:2, splinesBoot$CIvalues$alpha_0.05$lower, 1:2, splinesBoot$CIvalues$alpha_0.05$upper, length = 0.05, angle = 90, code = 3) points(1:2, sim$trueVals[1:2], pch = 20, col = 4) legend("topright", c("Estimate", "True value"), pch = 20, col = c(1,4)) ### simulate data (two- and one-dimensional domains) 10 multivExpansion ### ATTENTION: Takes long set.seed(2) sim <- simMultiFunData(type = "weighted", argvals = list(list(seq(0,1,0.01), seq(-1,1,0.02)), list(seq(-0.5,0.5,0.01))), M = list(c(4,5), 20), eFunType = list(c("Fourier", "Fourier"), "Poly"), eValType = "exponential", N = 150) # MFPCA based on univariate spline expansions (for images) and univariate FPCA (for functions) pca <- MFPCA(sim$simData, M = 10, uniExpansions = list(list(type = "splines2D", k = c(10,12)), list(type = "uFPCA"))) summary(pca) plot(pca) # plot the eigenfunctions as perturbations of the mean scoreplot(pca) # plot the scores ### Compare to true eigenfunctions # flip to make results more clear pca$functions <- flipFuns(sim$trueFuns[1:10], pca$functions) par(mfrow = c(5,2), mar = rep(2,4)) for(m in 2:6) # for m = 1, image.plot (used in plot(funData)) produces an error... { plot(sim$trueFuns[[1]], main = paste("True, m = ", m), obs = m) plot(pca$functions[[1]], main = paste("Estimate, m = ", m), obs = m) } par(mfrow = c(1,1)) plot(sim$trueFuns[[2]], main = "Eigenfunctions (2nd element)", lwd = 2, obs= plot(pca$functions[[2]], lty = 2, add = TRUE, obs= 1:5) legend("bottomleft", c("True", "MFPCA"), lty = 1:2, lwd = c(2,1)) 1:5) par(oldPar) multivExpansion Calculate multivariate basis expansion Description Calculate multivariate basis expansion Usage multivExpansion(multiFuns, scores) Arguments multiFuns A multivariate functional data object, containing the multivariate basis functions scores A matrix containing the scores for each observation in each row. The number of columns must match the number of basis functions. PACE 11 Value A multiFunData object containing the expanded functions for each observation. PACE Univariate functional principal component analysis by smoothed covariance Description This function calculates a univariate functional principal components analysis by smoothed covariance based on code from fpca.sc in package refund. Usage PACE( funDataObject, predData = NULL, nbasis = 10, pve = 0.99, npc = NULL, makePD = FALSE, cov.weight.type = "none" ) Arguments funDataObject An object of class funData or irregFunData containing the functional data observed, for which the functional principal component analysis is calculated. If the data is sampled irregularly (i.e. of class irregFunData), funDataObject is transformed to a funData object first. predData An object of class funData, for which estimated trajectories based on a truncated Karhunen-Loeve representation should be estimated. Defaults to NULL, which implies prediction for the given data. nbasis An integer, representing the number of B-spline basis functions used for estimation of the mean function and bivariate smoothing of the covariance surface. Defaults to 10 (cf. fpca.sc in refund). pve A numeric value between 0 and 1, the proportion of variance explained: used to choose the number of principal components. Defaults to 0.99 (cf. fpca.sc in refund). npc An integer, giving a prespecified value for the number of principal components. Defaults to NULL. If given, this overrides pve (cf. fpca.sc in refund). makePD Logical: should positive definiteness be enforced for the covariance surface estimate? Defaults to FALSE (cf. fpca.sc in refund). 12 PACE cov.weight.type The type of weighting used for the smooth covariance estimate. Defaults to "none", i.e. no weighting. Alternatively, "counts" (corresponds to fpca.sc in refund) weights the pointwise estimates of the covariance function by the number of observation points. Value mu A funData object with one observation, corresponding to the mean function. values A vector containing the estimated eigenvalues. functions A funData object containing the estimated functional principal components. scores An matrix of estimated scores for the observations in funDataObject. Each row corresponds to the scores of one observation. fit A funData object containing the estimated trajectories based on the truncated Karhunen-Loeve representation and the estimated scores and functional principal components for predData (if this is not NULL) or funDataObject (if predData is NULL). npc The number of functional principal components: either the supplied npc, or the minimum number of basis functions needed to explain proportion pve of the variance in the observed curves (cf. fpca.sc in refund). sigma2 The estimated measurement error variance (cf. fpca.sc in refund). estVar The estimated smooth variance function of the data. Warning This function works only for univariate functional data observed on one-dimensional domains. See Also funData, fpcaBasis, univDecomp Examples oldPar <- par(no.readonly = TRUE) # simulate data sim <- simFunData(argvals = seq(-1,1,0.01), M = 5, eFunType = "Poly", eValType = "exponential", N = 100) # calculate univariate FPCA pca <- PACE(sim$simData, npc = 5) # Plot the results par(mfrow = c(1,2)) plot(sim$trueFuns, lwd = 2, main = "Eigenfunctions") # flip estimated functions for correct signs plot(flipFuns(sim$trueFuns,pca$functions), lty = 2, add = TRUE) legend("bottomright", c("True", "Estimate"), lwd = c(2,1), lty = c(1,2)) plot.MFPCAfit 13 plot(sim$simData, lwd = 2, main = "Some Observations", obs = 1:7) plot(pca$fit, lty = 2, obs = 1:7, add = TRUE) # estimates are almost equal to true values legend("bottomright", c("True", "Estimate"), lwd = c(2,1), lty = c(1,2)) par(oldPar) plot.MFPCAfit Plot MFPCA results Description Plots the eigenfunctions as perturbations of the mean (i.e. the mean function plus/minus a constant factor times each eigenfunction separately). If all elements have a one-dimensional domain, the plots can be combined, otherwise the effects of adding and subtracting are shown in two separate rows for each eigenfunction. Usage ## S3 method for class 'MFPCAfit' plot( x, plotPCs = seq_len(nObs(x$functions)), stretchFactor = NULL, combined = FALSE, ... ) Arguments x An object of class MFPCAfit, typically returned by the MFPCA function. plotPCs The principal components to be plotted. Defaults to all components in the MFPCAfit object. stretchFactor The factor by which the principal components are multiplied before adding / subtracting them from the mean function. If NULL (the default), the median absolute value of the scores of each eigenfunction is used. combined Logical: Should the plots be combined? (Works only if all dimensions are onedimensional). Defaults to FALSE. ... Further graphical parameters passed to the plot.funData functions for functional data. Value A plot of the principal components as perturbations of the mean. See Also MFPCA, plot.funData 14 predict.MFPCAfit Examples # Simulate multivariate functional data on one-dimensonal domains # and calculate MFPCA (cf. MFPCA help) set.seed(1) # simulate data (one-dimensional domains) sim <- simMultiFunData(type = "split", argvals = list(seq(0,1,0.01), seq(-0.5,0.5,0.02)), M = 5, eFunType = "Poly", eValType = "linear", N = 100) # MFPCA based on univariate FPCA PCA <- MFPCA(sim$simData, M = 5, uniExpansions = list(list(type = "uFPCA"), list(type = "uFPCA"))) # Plot the results plot(PCA, combined = TRUE) # combine addition and subtraction in one plot predict.MFPCAfit Function prediction based on MFPCA results Description Predict functions based on a truncated multivariate Karhunen-Loeve representation: x̂ = m̂u + M X ρm ψ̂m m=1 with estimated mean function µ̂ and principal components ψm . The scores ρm can be either estimated (reconstruction of observed functions) or user-defined (construction of new functions). Usage ## S3 method for class 'MFPCAfit' predict(object, scores = object$scores, ...) Arguments object An object of class MFPCAfit, typically resulting from a MFPCA function call. scores A matrix containing the score values. The number of columns in scores must equal the number of principal components in object. Each row represents one curve. Defaults to the estimated scores in object, which yields reconstructions of the original data used for the MFPCA calculation. ... Arguments passed to or from other methods. Value A multiFunData object containing the predicted functions. See Also MFPCA print.MFPCAfit 15 Examples #' # Simulate multivariate functional data on one-dimensonal domains # and calculate MFPCA (cf. MFPCA help) set.seed(1) # simulate data (one-dimensional domains) sim <- simMultiFunData(type = "split", argvals = list(seq(0,1,0.01), seq(-0.5,0.5,0.02)), M = 5, eFunType = "Poly", eValType = "linear", N = 100) # MFPCA based on univariate FPCA PCA <- MFPCA(sim$simData, M = 5, uniExpansions = list(list(type = "uFPCA"), list(type = "uFPCA"))) # Reconstruct the original data pred <- predict(PCA) # default reconstructs data used for the MFPCA fit # plot the results: 1st element plot(sim$simData[[1]]) # original data plot(pred[[1]], add = TRUE, lty = 2) # reconstruction # plot the results: 2nd element plot(sim$simData[[2]]) # original data plot(pred[[2]], add = TRUE, lty = 2) # reconstruction print.MFPCAfit Print the results of a Multivariate Functional Principal Component Analysis Description A print function for class MFPCAfit. Usage ## S3 method for class 'MFPCAfit' print(x, ...) Arguments x An object of class MFPCAfit, usually returned by a call to MFPCA. ... Arguments passed to or from other methods. Value No return value, called for side effects 16 scoreplot print.summary.MFPCAfit Print summary of a Multivariate Functional Principal Component Analysis Description A print method for class MFPCAfit.summary Usage ## S3 method for class 'summary.MFPCAfit' print(x, ...) Arguments x An object of class MFPCAfit.summary, usually returned by a call to MFPCAfit.summary. ... Arguments passed to or from other methods. Value No return value, called for side effects scoreplot Scoreplot Generic Description Redirects to plot.default Usage scoreplot(PCAobject, ...) Arguments PCAobject A principal component object ... Arguments passed from or to other methods Value A bivariate plot of scores. scoreplot.MFPCAfit 17 scoreplot.MFPCAfit Plot the Scores of a Multivariate Functional Principal Component Analysis Description This function plots two scores of a multivariate functional principal component analysis for each observation. Usage ## S3 method for class 'MFPCAfit' scoreplot(PCAobject, choices = 1:2, scale = FALSE, ...) Arguments PCAobject An object of class MFPCAfit, typically returned by the MFPCA function. choices The indices of the scores that should by displayed. Defaults to 1:2, i.e. the scores corresponding to the two leading modes of variability in the data. scale Logical. Should the scores be scaled by the estimated eigenvalues to emphasize the proportions of total variance explained by the components. Defaults to FALSE. ... Further parameters passed to the plot.default function. Value A bivariate plot of scores. See Also MFPCA Examples # and calculate MFPCA (cf. MFPCA help) set.seed(1) # simulate data (one-dimensional domains) sim <- simMultiFunData(type = "split", argvals = list(seq(0,1,0.01), seq(-0.5,0.5,0.02)), M = 5, eFunType = "Poly", eValType = "linear", N = 100) # MFPCA based on univariate FPCA PCA <- MFPCA(sim$simData, M = 5, uniExpansions = list(list(type = "uFPCA"), list(type = "uFPCA"))) # Plot the first two scores scoreplot(PCA) # no scaling (default) scoreplot(PCA, scale = TRUE) # scale the scores by the first two eigenvalues 18 screeplot.MFPCAfit screeplot.MFPCAfit Screeplot for Multivariate Functional Principal Component Analysis Description This function plots the proportion of variance explained by the leading eigenvalues in an MFPCA against the number of the principal component. Usage ## S3 method for class 'MFPCAfit' screeplot( x, npcs = min(10, length(x$values)), type = "lines", ylim = NULL, main = deparse(substitute(x)), ... ) Arguments x An object of class MFPCAfit, typically returned by a call to MFPCA. npcs The number of eigenvalued to be plotted. Defaults to all eigenvalues if their number is less or equal to 10, otherwise show only the leading first 10 eigenvalues. type The type of screeplot to be plotted. Can be either "lines" or "barplot". Defaults to "lines". ylim The limits for the y axis. Can be passed either as a vector of length 2 or as NULL (default). In the second case, ylim is set to (0,max(pve)), with pve the proportion of variance explained by the principal components to be plotted. main The title of the plot. Defaults to the variable name of x. ... Other graphic parameters passed to plot.default (for type = "lines") or barplot (for type = "barplot"). Value A screeplot, showing the decrease of the principal component score. See Also MFPCA, screeplot summary.MFPCAfit 19 Examples # Simulate multivariate functional data on one-dimensonal domains # and calculate MFPCA (cf. MFPCA help) set.seed(1) # simulate data (one-dimensional domains) sim <- simMultiFunData(type = "split", argvals = list(seq(0,1,0.01), seq(-0.5,0.5,0.02)), M = 5, eFunType = "Poly", eValType = "linear", N = 100) # MFPCA based on univariate FPCA PCA <- MFPCA(sim$simData, M = 5, uniExpansions = list(list(type = "uFPCA"), list(type = "uFPCA"))) # screeplot screeplot(PCA) # default options screeplot(PCA, npcs = 3, type = "barplot", main= "Screeplot") summary.MFPCAfit Summarize a Multivariate Functional Principal Component Analysis Description A summary method for class MFPCAfit Usage ## S3 method for class 'MFPCAfit' summary(object, ...) Arguments object An object of class MFPCAfit, usually returned by a call to MFPCA. ... Arguments passed to or from other methods. Value An object of class summary.MFPCAfit ttv Tensor times vector calculation Description Functionality adapted from the MATLAB tensor toolbox (https://www.tensortoolbox.org/). Usage ttv(A, v, dim) 20 ttv Arguments A An array. v A list of the same length as dim. dim A vector specifying the dimensions for the multiplication. Details Let A be a tensor with dimensions d1 × d2 × . . . × dp and let v be a vector of length di . Then the tensor-vector-product along the i-th dimension is defined as Bj1 ...ji−1 ji+1 ...jd = di X Aj1 ...ji−1 iji+1 ...jd · vi . i=1 It can hence be seen as a generalization of the matrix-vector product. The tensor-vector-product along several dimensions between a tensor A and multiple vectors v_1,...,v_k (k ≤ p) is defined as a series of consecutive tensor-vector-product along the different dimensions. For consistency, the multiplications are calculated from the dimension of the highest order to the lowest. Value An array, the result of the multiplication. References B. W. Bader and T. G. Kolda. Algorithm 862: MATLAB tensor classes for fast algorithm prototyping, ACM Transactions on Mathematical Software 32(4):635-653, December 2006. See Also UMPCA Examples # create a three-mode tensor a1 <- seq(0,1, length.out = 10) a2 <- seq(-1,1, length.out = 20) a3 <- seq(-pi, pi, length.out = 15) A <-a1 %o% a2 %o% a3 dim(A) # multiply along different dimensions dim(ttv(A = A, v = list(rnorm(10)), dim = 1)) dim(ttv(A = A, v = list(rnorm(20)), dim = 2)) dim(ttv(A = A, v = list(rnorm(15)), dim = 3)) # multiply along more than one dimension length(ttv(A = A, v = list(rnorm(10), rnorm(15)), dim = c(1,3))) UMPCA UMPCA 21 UMPCA: Uncorrelated Multilinear Principle Component Analysis Description This function implements the uncorrelated multilinear principal component analysis for tensors of dimension 2, 3 or 4. The code is basically the same as in the MATLAB toolbox UMPCA by Haiping Lu (Link: https://www.mathworks.com/matlabcentral/fileexchange/35432-uncorrelated-multilinear-princip see also references). Usage UMPCA(TX, numP) Arguments TX The input training data in tensorial representation, the last mode is the sample mode. For Nth-order tensor data, TX is of (N+1)th-order with the (N+1)-mode to be the sample mode. E.g., 30x20x10x100 for 100 samples of size 30x20x10. numP The dimension of the projected vector, denoted as P in the paper. It is the number of elementary multilinear projections (EMPs) in tensor-to-vector projection. Value Us The multilinear projection, consisting of numP (P in the paper) elementary multilinear projections (EMPs), each EMP is consisted of N vectors, one in each mode. TXmean The mean of the input training samples TX. odrIdx The ordering index of projected features in decreasing variance. Warning As this algorithm aims more at uncorrelated features than at an optimal reconstruction of the data, hence it might give poor results when used for the univariate decomposition of images in MFPCA. References Haiping Lu, K.N. Plataniotis, and A.N. Venetsanopoulos, "Uncorrelated Multilinear Principal Component Analysis for Unsupervised Multilinear Subspace Learning", IEEE Transactions on Neural Networks, Vol. 20, No. 11, Page: 1820-1836, Nov. 2009. 22 univDecomp Examples set.seed(12345) # define "true" components a <- sin(seq(-pi, pi, length.out = 100)) b <- exp(seq(-0.5, 1, length.out = 150)) # simulate tensor data X <- a %o% b %o% rnorm(80, sd = 0.5) # estimate one component UMPCAres <- UMPCA(X, numP = 1) # plot the results and compare to true values plot(UMPCAres$Us[[1]][,1]) points(a/sqrt(sum(a^2)), pch = 20) # eigenvectors are defined only up to a sign change! legend("topright", legend = c("True", "Estimated"), pch = c(20, 1)) plot(UMPCAres$Us[[2]][,1]) points(b/sqrt(sum(b^2)), pch = 20) legend("topleft", legend = c("True", "Estimated"), pch = c(20, 1)) univDecomp Univariate basis decomposition Description This function calculates a univariate basis decomposition for a (univariate) functional data object. Usage univDecomp(type, funDataObject, ...) Arguments type A character string, specifying the basis for which the decomposition is to be calculated. funDataObject A funData object, representing the (univariate) functional data samples. ... Further parameters, passed to the function for the particular basis to use. Details Functional data Xi (t) can often be approximated by a linear combination of basis functions bk (t) Xi (t) = K X k=1 θik bk (t), i = 1, . . . , N. univExpansion 23 The basis functions may be prespecified (such as spline basis functions or Fourier bases) or can be estimated from the data (e.g. by functional principal component analysis) and are the same for all observations X1 (t), . . . , Xn (t). The coefficients (or scores) θik reflect the weight of each basis function bk (t) for the observed function Xi (t) and can be used to characterize the individual observations. Value scores A matrix of scores (coefficients) for each observation based on the prespecified basis functions. B A matrix containing the scalar products of the basis functions. Can be NULL if the basis functions are orthonormal. ortho Logical. If TRUE, the basis functions are all orthonormal. functions A functional data object, representing the basis functions. Can be NULL if the basis functions are not estimated from the data, but have a predefined form. See Details. Warning The options type = "DCT2D" and type = "DCT3D" have not been tested with ATLAS/MKL/OpenBLAS. See Also MFPCA, univExpansion, fpcaBasis, splineBasis1D, splineBasis1Dpen, splineBasis2D, splineBasis2Dpen, umpcaBasis, fcptpaBasis, fdaBasis, dctBasis2D, dctBasis3D Examples # generate some data dat <- simFunData(argvals = seq(0,1,0.01), M = 5, eFunType = "Poly", eValType = "linear", N = 100)$simData # decompose the data in univariate functional principal components... decFPCA <- univDecomp(type = "uFPCA", funDataObject = dat, npc = 5) str(decFPCA) # or in splines (penalized) decSplines <- univDecomp(type = "splines1Dpen", funDataObject = dat) # use mgcv's default params str(decSplines) univExpansion Calculate a univariate basis expansion Description This function calculates a univariate basis expansion based on given scores (coefficients) and basis functions. 24 univExpansion Usage univExpansion( type, scores, argvals = ifelse(!is.null(functions), functions@argvals, NULL), functions, params = NULL ) Arguments type A character string, specifying the basis for which the decomposition is to be calculated. scores A matrix of scores (coefficients) for each observation based on the given basis functions. argvals A list, representing the domain of the basis functions. If functions is not NULL, the usual default is functions@argvals. See funData and the underlying expansion functions for details. functions A functional data object, representing the basis functions. Can be NULL if the basis functions are not estimated from observed data, but have a predefined form. See Details. params A list containing the parameters for the particular basis to use. Details This function calculates functional data Xi (t), i = 1 . . . N that is represented as a linear combination of basis functions bk (t) Xi (t) = K X θik bk (t), i = 1, . . . , N. k=1 The basis functions may be prespecified (such as spline basis functions or Fourier bases) or can be estimated from observed data (e.g. by functional principal component analysis). If type = "default" (i.e. a linear combination of arbitrary basis functions is to be calculated), both scores and basis functions must be supplied. Value An object of class funData with N observations on argvals, corresponding to the linear combination of the basis functions. Warning The options type = "spline2Dpen", type = "DCT2D" and type = "DCT3D" have not been tested with ATLAS/MKL/OpenBLAS. univExpansion 25 See Also MFPCA, splineFunction1D, splineFunction2D, splineFunction2Dpen, dctFunction2D, dctFunction3D, expandBasisFunction Examples oldPar <- par(no.readonly = TRUE) par(mfrow = c(1,1)) set.seed(1234) ### Spline basis ### # simulate coefficients (scores) for N = 10 observations and K = 8 basis functions N <- 10 K <- 8 scores <- t(replicate(n = N, rnorm(K, sd = (K:1)/K))) dim(scores) # expand spline basis on [0,1] funs <- univExpansion(type = "splines1D", scores = scores, argvals = list(seq(0,1,0.01)), functions = NULL, # spline functions are known, need not be given params = list(bs = "ps", m = 2, k = K)) # params for mgcv plot(funs, main = "Spline reconstruction") ### PCA basis ### # simulate coefficients (scores) for N = 10 observations and K = 8 basis functions N <- 10 K <- 8 scores <- t(replicate(n = N, rnorm(K, sd = (K:1)/K))) dim(scores) # Fourier basis functions as eigenfunctions eFuns <- eFun(argvals = seq(0,1,0.01), M = K, type = "Fourier") # expand eigenfunction basis funs <- univExpansion(type = "uFPCA", scores = scores, argvals = NULL, # use argvals of eFuns (default) functions = eFuns) plot(funs, main = "PCA reconstruction") par(oldPar) Index splineBasis2Dpen, 23 splineFunction1D, 25 splineFunction2D, 25 splineFunction2Dpen, 25 summary, 8 summary.MFPCAfit, 19 barplot, 18 dctBasis2D, 23 dctBasis3D, 23 dctFunction2D, 25 dctFunction3D, 25 eval.fd, 7 expandBasisFunction, 25 ttv, 19 UMPCA, 20, 21 umpcaBasis, 23 univDecomp, 6–8, 12, 22 univExpansion, 7, 8, 23, 23 FCP_TPA, 2 fcptpaBasis, 4, 23 fdaBasis, 23 fpcaBasis, 12, 23 funData, 11, 12, 24 funData2fd, 7 irregFunData, 11 MFPCA, 4, 13–15, 17–19, 23, 25 multiFunData, 5, 7, 8 multivExpansion, 10 optimize, 3 PACE, 7, 8, 11 plot, 8 plot.default, 16–18 plot.funData, 13 plot.MFPCAfit, 13 predict.MFPCAfit, 14 print.MFPCAfit, 15 print.summary.MFPCAfit, 16 scoreplot, 8, 16 scoreplot.MFPCAfit, 17 screeplot, 18 screeplot.MFPCAfit, 18 splineBasis1D, 23 splineBasis1Dpen, 23 splineBasis2D, 23 26