Cepstral Regression

Description

Performs cepstral regression to model frequency domain relationships between a functional response and scalar covariates. Supports ordinary least squares (OLS), reduced-rank regression (RRR), and envelope regression (ENV) methods. Automatically selects the number of cepstral basis functions via AIC.

Usage

CepReg(
  y,
  x,
  method = c("ols", "rrr", "env"),
  number_of_K,
  if_bootstrap = FALSE,
  level = NULL,
  nboot = NULL,
  ind = NULL,
  nrank = NULL
)

Arguments

Value

A list with components:

eff

A list of estimated effect functions (e.g., alpha_effect, beta_effect).

boot

A list of bootstrap results including confidence intervals; NULL if if_bootstrap = FALSE.

fit

A list containing regression coefficients, residuals, smoothed spectral estimates, and other model outputs.

Examples

set.seed(123)
niter <- 5
len <- 10
N <- 3
p <- 2
L <- floor(len/2)-1
frq <- (1:L)/len
mu <- rep(0, p)
rho <- 0
Sigma <- generate_sig(p, rho)

X <- MASS::mvrnorm(N, mu, Sigma)
X[,1] <- runif(N, 0, 1)

spec <- matrix(0,len,N)
for(j in 1:N){
  eta1 <- rnorm(1,0,0.5)
  eta2 <- rnorm(1,0,0.5)
  eta3 <- rnorm(1,0,0.5)
  spec[,j] <- exp(
    2*cos(2*pi*(1:len)/len) +
    X[j,1]*(2*cos(4*pi*(1:len)/len)) +
    eta1 + eta2*cos(2*pi*(1:len)/len) +
    eta3*(cos(4*pi*(1:len)/len))
    )
 }

Z <- data_generater(N,len,sqrt(spec))

res_ols <- CepReg(Z, X, method = "ols", number_of_K = 2,
         if_bootstrap = TRUE, level = 0.95,
         nboot = 2, ind = 1)

eff_ols <- res_ols$eff
boot_ols <- res_ols$boot

plot(frq, eff_ols$alpha_effect, type = 'l', col = "black", xlab = "Frequency", ylab = "",
     ylim = range(c(boot_ols$alpha_ci,
     eff_ols$alpha_effect, 2*cos(2*pi*frq)+0.577)))
title(ylab = expression(alpha(omega)), line = 2, cex.lab = 1.2)
lines(frq, boot_ols$alpha_ci[, 1], col = "black")
lines(frq, boot_ols$alpha_ci[, 2], col = "black")
lines(frq, 2 * cos(2 * pi * frq) + 0.577, col = "red", lty = 1, lwd = 2)
legend("topright",legend = c("True", "Estimated", "CI"),
     col = c("red", "black", "black"), ncol = 1)

Bootstrap Confidence Intervals for Functional Effect Curves

Description

Computes bias-corrected percentile bootstrap confidence intervals for the intercept and covariate effect functions in cepstral-based regression models.

Usage

boot_effect(
  logspect,
  res,
  alpha_effect,
  beta_effect,
  X,
  nbase,
  frq1,
  frq2,
  nrank,
  ind,
  level,
  nboot,
  method = "rrr",
  verb = FALSE
)

Arguments

Value

A list with:

alpha_ci

Matrix with lower and upper CI for intercept effect.

beta_ci

Array or matrix of lower and upper CI for covariate effects.

Examples

set.seed(123)
N <- 10
len <- 12
nbase <- 2
nrank <- 1
nboot <- 10
level <- 0.95
p <- 2
ind <- 1:p

Y <- matrix(rnorm(N * nbase), nrow = N, ncol = nbase)

X <- matrix(rnorm(N * p), nrow = N, ncol = p)


frq <- seq(1, nbase) / len

rrr_out <- rrr_get(Y, X, frq, nbase, nrank)

eff <- effect_get(rrr_out$alph, rrr_out$bet, frq, nbase, ind)
alpha_eff <- eff$alpha_effect
beta_eff <- eff$beta_effect

logspect <- matrix(rnorm(N * length(frq)), nrow = N, ncol = length(frq))

boot_ci <- boot_effect(
  logspect = logspect,
  res = rrr_out$res,
  alpha_effect = alpha_eff,
  beta_effect = beta_eff,
  X = X,
  nbase = nbase,
  frq1 = frq,
  frq2 = frq,
  nrank = nrank,
  ind = ind,
  level = level,
  nboot = nboot,
  method = "rrr",
  verb = TRUE
)

plot(frq, beta_eff[, 1], type = "l", col = "blue", lwd = 2,
     ylab = "Effect", xlab = "Frequency",
     main = paste("Effect Function and", level*100, "% CI for Covariate", ind[1]))
lines(frq, boot_ci[[ind[1]]][, 1], col = "red", lty = 2)
lines(frq, boot_ci[[ind[1]]][, 2], col = "red", lty = 2)
legend("topright", legend = c("Effect", "Bootstrap CI"), col = c("blue", "red"),
       lty = c(1, 2), lwd = c(2, 1))

Estimate Cepstral Coefficients from Periodogram

Description

Estimates replicate-specific cepstral coefficients and smoothed log-spectra using a Fourier cosine basis and Whittle-type approximation.

Usage

cep_get(perd, k0, frq)

Arguments

Value

A list with:

f

An N × k0 matrix of estimated cepstral coefficients.

ff

An N × K matrix of smoothed log-spectra.

Examples

set.seed(123)
Y <- matrix(rnorm(20 * 5), nrow = 20, ncol = 5)
len <- nrow(Y)
L <- floor(len/2)-1
frq <- (1:L)/len
perd <- perd_get(Y)
result <- cep_get(perd = perd, k0 = 3, frq = frq)

Generate Time Series

Description

Simulates real-valued time series using the Cramér spectral representation and inverse FFT.

Usage

data_generater(N, nobs, spec)

Arguments

Value

Matrix of size nobs × N of generated time series

Examples

set.seed(123)
N    <- 3
nobs <- 20
freqs <- (1:nobs) / nobs

spec <- matrix(NA, nrow = nobs, ncol = N)
for (i in 1:N) {
  spec[, i] <- exp(2 * cos(2 * pi * freqs) + rnorm(1, sd = 0.1))
}

data_generater(N = N, nobs = nobs, spec = spec)

Compute Functional Effects of Intercept and Covariates

Description

Projects cepstral coefficient intercept and covariate effects onto the frequency domain using the cepstral basis functions.

Usage

effect_get(alpha, beta, frq, nbase, ind)

Arguments

Value

A list containing:

alpha_effect

Functional intercept across frequency.

beta_effect

Matrix of functional covariate effects.

Examples

frq <- seq(0, 1, length.out = 16)[2:8]
alpha <- rnorm(3)
beta <- matrix(rnorm(2 * 3), 2, 3)
result <- effect_get(alpha, beta, frq, nbase = 3, ind = c(1, 2))

Envelope Estimator for Log-Spectral Regression

Description

Fits an envelope regression model to predict cepstral coefficients from covariates.

Usage

env_get(X, f, frq, nbase)

Arguments

Value

A list containing:

alph

Intercept vector.

bet

Envelope regression coefficient matrix.

spechat

Estimated smoothed log-spectra.

res

Residuals from envelope model.

Examples

library(Renvlp)

set.seed(123)
frq <- seq(0, 1, length.out = 16)[2:8]
n <- 20
p <- 3
nbase <- 5
X <- matrix(rnorm(n * p), n, p)
f <- matrix(rnorm(n * nbase), n, nbase)

u_max <- min(ncol(X), ncol(f))
cv_errors <- numeric(u_max)
for (j in 1:u_max) {
  cv_errors[j] <- cv.xenv(X, f, j, m = 5, nperm = 50)
}
optimal_u <- which.min(cv_errors)

env_result <- env_get(X, f, frq, nbase = nbase)

Generate Exponential Correlation Covariance Matrix

Description

Creates an n × n covariance matrix with entries \rho^{|i - j|}.

Usage

generate_sig(n, rho)

Arguments

Value

An n × n positive definite covariance matrix.

Examples

S <- generate_sig(5, 0.5)

Ordinary Least Squares Estimator for Log-Spectral Regression

Description

Performs OLS regression to estimate the association between covariates and cepstral coefficients.

Usage

ols_get(X, f, frq, nbase)

Arguments

Value

A list containing:

alph

Intercept vector.

bet

OLS coefficient matrix.

spechat

Estimated smoothed log-spectra.

res

Matrix of residuals.

Examples

frq <- seq(0, 1, length.out = 16)[2:8]
n <- 10
p <- 3
nbase <- 5
X <- matrix(rnorm(n * p), n, p)
f <- matrix(rnorm(n * nbase), n, nbase)

ols_result <- ols_get(X, f, frq, nbase)

Compute the Periodogram of Multivariate Time Series

Description

This function computes the periodogram for each time series in the input matrix.

Usage

perd_get(Y)

Arguments

Value

A numeric matrix of dimension N x L, where each row is the periodogram of a time series.

Examples

set.seed(123)
Y <- matrix(rnorm(20), ncol = 4)
perd <- perd_get(Y)

Generate a Fourier Cosine Basis Matrix for Log-Spectral Modeling

Description

Constructs a matrix of Fourier cosine basis functions evaluated at a given frequency grid. Used in cepstral smoothing of log-spectra.

Usage

psi_get(k0, frq)

Arguments

Value

A k0 x length(frq) matrix of basis function.

Examples

set.seed(123)
frq<-seq(0,1, length.out=5)
psi<-psi_get(k0=3, frq)

Reduced-Rank Regression on Cepstral Coefficients

Description

Fits a reduced-rank regression (RRR) between covariates and cepstral coefficients using a specified maximum rank, and reconstructs log-spectra.

Usage

rrr_get(X, f, frq, nbase, nrank)

Arguments

Value

A list containing:

alph

Estimated intercept vector.

bet

Estimated coefficient matrix.

spechat

Estimated log-spectra.

res

Matrix of residuals.

Examples

set.seed(123)
frq <- seq(0, 1, length.out = 16)[2:8]
n <- 5
p <- 2
nbase <- 2

X <- matrix(rnorm(n * p), n, p)

psi <- psi_get(nbase, frq)

true_beta <- matrix(rnorm(p * nbase), p, nbase)
alph <- rnorm(nbase)
f <- X %*% true_beta + matrix(alph, n, nbase, byrow = TRUE) +
     matrix(rnorm(n * nbase), n, nbase)

rrr <- rrr_get(X, f, frq, nbase = nbase, nrank = 1)

Fisher Scoring Algorithm For Estimating Cepstral Coefficients

Description

Estimates replicate-specific cepstral coefficients and corresponding smoothed log-spectra using a Whittle likelihood approximation.

Usage

spec_regress(perd, psi, Wmat, k0)

Arguments

Value

A list with:

f

An N × k0 matrix of estimated cepstral coefficients.

ff

An N × K matrix of smoothed log-spectra.

Examples

set.seed(123)
N <- 5
len <- 20
L <- floor(len/2) - 1
frq <- (1:L) / len

Y <- matrix(rnorm(len * N), nrow = len, ncol = N)

perd <- perd_get(Y)

k0 <- 3
psi <- psi_get(k0, frq)

Wmatin <- matrix(0, k0, k0)
for (j in 1:ncol(psi)) {
  Wmatin <- Wmatin + psi[, j] %*% t(psi[, j])
}
Wmat <- solve(Wmatin)

out <- spec_regress(perd, psi, Wmat, k0)