fpc_utils {goffda}

Utilities for functional principal components

Description

Computation of coefficients and reconstructions based on Functional Principal Components (FPC). The function fpc_coefs allows to project a functional data sample into a basis of FPC; the reconstruction of the sample from its projections and the FPC is done with fpc_to_fdata. The functions beta_fpc_coefs and fpc_to_beta do analogous operations but for the bivariate kernel \beta and the tensor product of two FPC bases.

Usage

fpc_coefs(X_fdata, X_fpc, ind_X_fpc = 1:3, int_rule = "trapezoid")

beta_fpc_coefs(beta, X_fpc, Y_fpc, ind_X_fpc = 1:3, ind_Y_fpc = 1:3,
  int_rule = "trapezoid")

fpc_to_fdata(coefs, X_fpc, ind_coefs = seq_len(ncol(coefs)))

fpc_to_beta(beta_coefs, X_fpc, Y_fpc,
  ind_coefs_X = seq_len(nrow(beta_coefs)),
  ind_coefs_Y = seq_len(ncol(beta_coefs)))

Arguments

X_fdata

sample of functional data as an fdata object of length n.

X_fpc, Y_fpc

"fpc" objects as resulted from calling fpc.

ind_X_fpc, ind_Y_fpc

vectors giving the FPC indexes for whom the coefficients are computed. Their lengths must be smaller than the number of FPC in X_fpc and Y_fpc, respectively. Default to 1:3.

int_rule

quadrature rule for approximating the definite unidimensional integral: trapezoidal rule (int_rule = "trapezoid") and extended Simpson rule (int_rule = "Simpson") are available. Defaults to "trapezoid".

beta

a matrix containing the bivariate kernel \beta evaluated on a grid. Must be of size c(length(X_fpc$rotation$argvals), length(Y_fpc$rotation$argvals)).

coefs

a vector of coefficients to combine linearly the FPC. Its length must be smaller than the number of FPC in X_fpc.

ind_coefs, ind_coefs_X, ind_coefs_Y

indexes of FPC to associate to the provided coefficients. By default, from the first FPC to the sizes of coefs or beta_coefs.

beta_coefs

a matrix of coefficients to combine linearly the tensor products of FPC. Its size must be smaller than the number of FPC in X_fpc and Y_fpc.

Value

fpc_coefs

a vector of the same length as coefs containing the coefficients of X_fdata in the FPC of X_fpc.

beta_fpc_coefs

a matrix of the same size as beta_coefs containing the coefficients of \beta in the tensor product of the FPC in X_fpc and Y_fpc.

fpc_to_fdata

an fdata object of the same type as X_fpc$rotation.

fpc_to_beta

a matrix with the reconstructed kernel and size
c(length(X_fpc$rotation$argvals), length(Y_fpc$rotation$argvals)).

Author(s)

Eduardo García-Portugués.

References

Jolliffe, I. T. (2002). Principal Component Analysis. Springer-Verlag, New York.

Examples

## Compute FPC coefficients and reconstruct data

# Sample data
X_fdata <- r_ou(n = 200, t = seq(2, 4, l = 201))

# Compute FPC
X_fpc <- fpc(X_fdata = X_fdata, n_fpc = 50)

# FPC coefficients are the same if the data is centered
fpc_coefs(X_fdata = fdata_cen(X_fdata), X_fpc = X_fpc)[1:4, ]
X_fpc$scores[1:4, 1:3]

# Reconstruct the first two curves for an increasing number of FPC
plot(X_fdata[1:2, ], col = 1)
n_fpc <- c(2, 5, 10, 25, 50)
for (j in 1:5) {
  lines(fpc_to_fdata(X_fpc = X_fpc,
                     coefs = X_fpc$scores[, 1:n_fpc[j]])[1:2, ], col = j + 1)
}

## Project and reconstruct beta

# Surface
beta_fun <- function(s, t) sin(6 * pi * s) + cos(6 * pi * t)
s <- seq(0, 1, l = 101)
t <- seq(0, 1, l = 201)
beta_surf <- outer(s, t, FUN = beta_fun)

# Functional data as zero-mean Gaussian process with exponential variogram
X_fdata <- fda.usc::rproc2fdata(n = 100, t = s, sigma = "vexponential",
                                list = list(scale = 2.5))
Y_fdata <- flm_term(X_fdata = X_fdata, beta = beta_surf, t = t) +
  r_ou(n = 100, t = t, sigma = sqrt(0.075) * 2)

# FPC
X_fpc <- fpc(X_fdata = X_fdata, n_fpc = 50)
Y_fpc <- fpc(X_fdata = Y_fdata, n_fpc = 50)

# Coefficients
beta_coefs <- beta_fpc_coefs(beta = beta_surf, X_fpc = X_fpc, Y_fpc = Y_fpc,
                             ind_X_fpc = 1:50, ind_Y_fpc = 1:50)

# Reconstruction
beta_surf1 <- fpc_to_beta(beta_coefs = beta_coefs[1:2, 1:5],
                          X_fpc = X_fpc, Y_fpc = Y_fpc)
beta_surf2 <- fpc_to_beta(beta_coefs = beta_coefs[1:15, 1:10],
                          X_fpc = X_fpc, Y_fpc = Y_fpc)
beta_surf3 <- fpc_to_beta(beta_coefs = beta_coefs[1:50, 1:50],
                          X_fpc = X_fpc, Y_fpc = Y_fpc)

# Show reconstructions
old_par <- par(mfrow = c(2, 2))
col <- viridisLite::viridis(20)
image(s, t, beta_surf, col = col, zlim = c(-2.5, 2.5), main = "Original")
image(s, t, beta_surf1, col = col, zlim = c(-2.5, 2.5), main = "2 x 5")
image(s, t, beta_surf2, col = col, zlim = c(-2.5, 2.5), main = "15 x 10")
image(s, t, beta_surf3, col = col, zlim = c(-2.5, 2.5), main = "50 x 50")
par(old_par)