goffda (version 0.1.2)

fpc: Computation of functional principal components

Description

Computation of Functional Principal Components (FPC) for equispaced and non equispaced functional data.

Usage

fpc(X_fdata, n_fpc = 3, centered = FALSE, int_rule = "trapezoid",
  equispaced = FALSE, verbose = FALSE)

Value

An "fpc" object containing the following elements:

d

standard deviations of the FPC (i.e., square roots of eigenvalues of the empirical autocovariance estimator).

rotation

orthonormal eigenfunctions (loadings or functional principal components), as an fdata class object.

scores

rotated samples: inner products. between X_fdata and eigenfunctions in
rotation.

l

vector of index of FPC.

equispaced

equispaced flag.

Arguments

X_fdata

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

n_fpc

number of FPC to be computed. If n_fpc > n, n_fpc is set to n. Defaults to 3.

centered

flag to indicate if X_fdata is centered or not. Defaults to FALSE.

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".

equispaced

flag to indicate if X_fdata$data is valued in an equispaced grid or not. Defaults to FALSE.

verbose

whether to show or not information about the fpc procedure. Defaults to FALSE.

Author

Javier Álvarez-Liébana and Gonzalo Álvarez-Pérez.

Details

The FPC are obtained by performing the single value decomposition $$\mathbf{X} \mathbf{W}^{1/2} = \mathbf{U} \mathbf{D} (\mathbf{V}' \mathbf{W}^{1/2})$$ where \(\mathbf{X}\) is the matrix of discretized functional data, \(\mathbf{W}\) is a diagonal matrix of weights (computed by w_integral1D according to int_rule), \(\mathbf{D}\) is the diagonal matrix with singular values (standard deviations of FPC), \(\mathbf{U}\) is a matrix whose columns contain the left singular vectors, and \(\mathbf{V}\) is a matrix whose columns contain the right singular vectors (FPC).

References

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

Examples

## Computing FPC for equispaced data

# Sample data
X_fdata1 <- r_ou(n = 200, t = seq(2, 4, l = 201))

# FPC with trapezoid rule
X_fpc1 <- fpc(X_fdata = X_fdata1, n_fpc = 50, equispaced = TRUE,
              int_rule = "trapezoid")

# FPC with Simpsons's rule
X_fpc2 <- fpc(X_fdata = X_fdata1, n_fpc = 50, equispaced = TRUE,
               int_rule = "Simpson")

# Check if FPC are orthonormal
norms1 <- rep(0, length(X_fpc1$l))
for (i in X_fpc1$l) {

  norms1[i] <- integral1D(fx = X_fpc1$rotation$data[i, ]^2,
                          t = X_fdata1$argvals)

}
norms2 <- rep(0, length(X_fpc2$l))
for (i in X_fpc2$l) {

  norms2[i] <- integral1D(fx = X_fpc2$rotation$data[i, ]^2,
                          t = X_fdata1$argvals)

}

## Computing FPC for non equispaced data

# Sample data
X_fdata2 <- r_ou(n = 200, t = c(seq(0, 0.5, l = 201), seq(0.51, 1, l = 301)))

# FPC with trapezoid rule
X_fpc3 <- fpc(X_fdata = X_fdata2, n_fpc = 5, int_rule = "trapezoid",
              equispaced = FALSE)

# Check if FPC are orthonormal
norms3 <- rep(0, length(X_fpc3$l))
for (i in X_fpc3$l) {

  norms3[i] <- integral1D(fx = X_fpc3$rotation$data[i, ]^2,
                          t = X_fdata2$argvals)

}
# \donttest{
## Efficiency comparisons

# fpc() vs. fda.usc::fdata2pc()
data(phoneme, package = "fda.usc")
mlearn <- phoneme$learn[1:10, ]
res1 <- fda.usc::fdata2pc(mlearn, ncomp = 3)
res2 <- fpc(X_fdata = mlearn, n_fpc = 3)
plot(res1$x[, 1:3], col = 1)
points(res2$scores, col = 2)

microbenchmark::microbenchmark(fda.usc::fdata2pc(mlearn, ncomp = 3),
                               fpc(X_fdata = mlearn, n_fpc = 3), times = 1e3,
                               control = list(warmup = 20))
# }