Type:	Package
Title:	Seasonal Functional Autoregressive Models
Version:	0.0.1
Description:	This is a collection of functions designed for simulating, estimating and forecasting seasonal functional autoregressive time series of order one. These methods are addressed in the manuscript: https://www.monash.edu/business/ebs/research/publications/ebs/wp16-2019.pdf.
License:	GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
URL:	https://github.com/haghbinh/Rsfar
Encoding:	UTF-8
LazyData:	false
RoxygenNote:	7.1.1
Depends:	fda
NeedsCompilation:	no
Packaged:	2021-05-07 20:27:35 UTC; Haghbin
Author:	Hossein Haghbin [aut, cre], Rob Hyndman [aut]
Maintainer:	Hossein Haghbin <haghbinh@gmail.com>
Repository:	CRAN
Date/Publication:	2021-05-10 08:02:27 UTC

Rsfar: A Package for Seasonal Functional Autoregressive Models.

Description

The Rsfar package provides the collection of necessary functions for simulating, estimating and forecasting seasonal functional autoregressive time series of order one.

Details

Functional autoregressive models are popular for functional time series analysis, but the standard formulation fails to address seasonal behavior in functional time series data. To overcome this shortcoming, we introduce seasonal functional autoregressive time series models. For the model of order one, we provide estimation, prediction and simulation methods.

References

Atefeh Z., Hossein H., Maryam H., and R.J Hyndman (2021). Seasonal functional autoregressive models. Manuscript submitted for publication. https://robjhyndman.com/publications/sfar/

See Also

sfar, predict.sfar,

Create block diagonal matrix

Description

Create block diagonal matrix

Usage

Bdiag(A, k)

Arguments

Value

Return a block diagonal matrix from the matrix A.

Examples

Bdiag(matrix(1:4,2,2), 3)

Return inverse square root of square positive-definite matrix.

Description

Return inverse square root of square positive-definite matrix.

Usage

invsqrt(A)

Arguments

Value

inverse square root of square positive-definite matrix.

Examples

require(Rsfar)
X <- Bdiag(matrix(1:4,2,2), 3)
invsqrt(t(X) %*% X)

Prediction of an SFAR model

Description

Compute h-step-ahead prediction for an SFAR(1) model. Only the h-step predicted function is returned, not the predictions for 1,2,...,h.

Usage

## S3 method for class 'sfar'
predict(object, h, ...)

Arguments

Value

An object of class fda.

Examples

# Generate Brownian motion noise
N <- 300 # the length of the series
n <- 200 # the sample rate that each function will be sampled
u <- seq(0, 1, length.out = n) # argvalues of the functions
d <- 45 # the number of bases
basis <- create.fourier.basis(c(0, 1), d) # the basis system
sigma <- 0.05 # the std of noise norm
Z0 <- matrix(rnorm(N * n, 0, sigma), nrow = n, nc = N)
Z0[, 1] <- 0
Z_mat <- apply(Z0, 2, cumsum) # N standard Brownian motion
Z <- smooth.basis(u, Z_mat, basis)$fd

# Simulate random SFAR(1) data
kr <- function(x, y) {
  (2 - (2 * x - 1)^2 - (2 * y - 1)^2) / 2
}
s <- 5 # the period number
X <- rsfar(kr, s, Z)
plot(X)

# SFAR(1) model parameter estimation:
Model1 <- sfar(X, seasonal = s, kn = 1)

# Forecasting 3 steps ahead
fc <- predict(Model1, h = 3)
plot(fc)

Simulation of a Seasonal Functional Autoregressive SFAR(1) process.

Description

Simulation of a SFAR(1) process on a Hilbert space of L2[0,1].

Usage

rsfar(phi, seasonal, Z)

Arguments

Value

A sample of functional time series from a SFAR(1) model of the class 'fd'.

Examples

# Set up Brownian motion noise process
N <- 300 # the length of the series
n <- 200 # the sample rate that each function will be sampled
u <- seq(0, 1, length.out = n) # argvalues of the functions
d <- 15 # the number of basis functions
basis <- create.fourier.basis(c(0, 1), d) # the basis system
sigma <- 0.05 # the stdev of noise norm
Z0 <- matrix(rnorm(N * n, 0, sigma), nr = n, nc = N)
Z0[, 1] <- 0
Z_mat <- apply(Z0, 2, cumsum) # N standard Brownian motion
Z <- smooth.basis(u, Z_mat, basis)$fd

# Compute the standardized constant of a kernel function with respect to a given HS norm.
gamma0 <- function(norm, kr) {
  f <- function(x) {
    g <- function(y) {
      kr(x, y)^2
    }
    return(integrate(g, 0, 1)$value)
  }
  f <- Vectorize(f)
  A <- integrate(f, 0, 1)$value
  return(norm / A)
}
# Definition of parabolic integral kernel:
norm <- 0.99
kr <- function(x, y) {
  2 - (2 * x - 1)^2 - (2 * y - 1)^2
}
c0 <- gamma0(norm, kr)
phi <- function(x, y) {
  c0 * kr(x, y)
}

# Simulating a path from an SFAR(1) process
s <- 5 # the period number
X <- rsfar(phi, s, Z)
plot(X)

Estimation of an SFAR(1) Model

Description

Estimate a seasonal functional autoregressive (SFAR) model of order 1 for a given functional time series.

Usage

sfar(
  X,
  seasonal,
  cpv = 0.85,
  kn = NULL,
  method = c("MME", "ULSE", "KOE"),
  a = ncol(Coefs)^(-1/6)
)

Arguments

Value

A matrix of size p*p.

Examples

# Generate Brownian motion noise
N <- 300 # the length of the series
n <- 200 # the sample rate that each function will be sampled
u <- seq(0, 1, length.out = n) # argvalues of the functions
d <- 45 # the number of bases
basis <- create.fourier.basis(c(0, 1), d) # the basis system
sigma <- 0.05 # the std of noise norm
Z0 <- matrix(rnorm(N * n, 0, sigma), nrow = n, nc = N)
Z0[, 1] <- 0
Z_mat <- apply(Z0, 2, cumsum) # N standard Brownian motion
Z <- smooth.basis(u, Z_mat, basis)$fd

# Simulate random SFAR(1) data
kr <- function(x, y) {
 (2 - (2 * x - 1)^2 - (2 * y - 1)^2) / 2
}
s <- 5 # the period number
X <- rsfar(kr, s, Z)
plot(X)

# SFAR(1) model parameter estimation:
Model1 <- sfar(X, seasonal = s, kn = 1)