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Rob Hyndman [aut]| Title: | Seasonal Functional Autoregressive Models |
|---|---|
| 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>. |
| Authors: | Hossein Haghbin [aut, cre] ,
Rob Hyndman [aut] |
| Maintainer: | Hossein Haghbin <[email protected]> |
| License: | GPL (>= 2) |
| Version: | 0.0.1 |
| Built: | 2024-11-14 04:29:30 UTC |
| Source: | https://github.com/haghbinh/sfar |
The Rsfar package provides the collection of necessary functions for simulating, estimating and forecasting seasonal functional autoregressive time series of order one.
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.
Atefeh Z., Hossein H., Maryam H., and R.J Hyndman (2021). Seasonal functional autoregressive models. Manuscript submitted for publication. https://robjhyndman.com/publications/sfar/
Create block diagonal matrix
Bdiag(A, k)Bdiag(A, k)
A |
a numeric matrix forming each block. |
k |
an integer value indicating the number of blocks. |
Return a block diagonal matrix from the matrix A.
Bdiag(matrix(1:4,2,2), 3)Bdiag(matrix(1:4,2,2), 3)
Return inverse square root of square positive-definite matrix.
invsqrt(A)invsqrt(A)
A |
a numeric matrix |
inverse square root of square positive-definite matrix.
require(Rsfar) X <- Bdiag(matrix(1:4,2,2), 3) invsqrt(t(X) %*% X)require(Rsfar) X <- Bdiag(matrix(1:4,2,2), 3) invsqrt(t(X) %*% X)
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.
## S3 method for class 'sfar' predict(object, h, ...)## S3 method for class 'sfar' predict(object, h, ...)
object |
an 'sfar' object containing a fitted SFAR(1) model. |
h |
number of steps ahead to predict. |
... |
Other parameters, not currently used. |
An object of class fda.
# 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)$fd # Simulate random SFAR(1) data kr <- function(x, y) { (2 - (2 * x - 1)^2 - (2 * y - 1)^2) / 2 } s <- 5 # the period number X <- rsfar(kr, s, Z) plot(X) # SFAR(1) model parameter estimation: Model1 <- sfar(X, seasonal = s, kn = 1) # Forecasting 3 steps ahead fc <- predict(Model1, h = 3) plot(fc)# Generate Brownian motion noise N <- 300 # the length of the series n <- 200 # the sample rate that each function will be sampled u <- seq(0, 1, length.out = n) # argvalues of the functions d <- 45 # the number of bases basis <- create.fourier.basis(c(0, 1), d) # the basis system sigma <- 0.05 # the std of noise norm Z0 <- matrix(rnorm(N * n, 0, sigma), nrow = n, nc = N) Z0[, 1] <- 0 Z_mat <- apply(Z0, 2, cumsum) # N standard Brownian motion Z <- smooth.basis(u, Z_mat, basis)$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 SFAR(1) process on a Hilbert space of L2[0,1].
rsfar(phi, seasonal, Z)rsfar(phi, seasonal, Z)
phi |
a kernel function corresponding to the seasonal autoregressive operator. |
seasonal |
a positive integer variable specifying the seasonal period. |
Z |
the functional noise object of the class 'fd'. |
A sample of functional time series from a SFAR(1) model of the class 'fd'.
# 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)$value return(norm / A) } # Definition of parabolic integral kernel: norm <- 0.99 kr <- function(x, y) { 2 - (2 * x - 1)^2 - (2 * y - 1)^2 } c0 <- gamma0(norm, kr) phi <- function(x, y) { c0 * kr(x, y) } # Simulating a path from an SFAR(1) process s <- 5 # the period number X <- rsfar(phi, s, Z) plot(X)# Set up Brownian motion noise process N <- 300 # the length of the series n <- 200 # the sample rate that each function will be sampled u <- seq(0, 1, length.out = n) # argvalues of the functions d <- 15 # the number of basis functions basis <- create.fourier.basis(c(0, 1), d) # the basis system sigma <- 0.05 # the stdev of noise norm Z0 <- matrix(rnorm(N * n, 0, sigma), nr = n, nc = N) Z0[, 1] <- 0 Z_mat <- apply(Z0, 2, cumsum) # N standard Brownian motion Z <- smooth.basis(u, Z_mat, basis)$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)
Estimate a seasonal functional autoregressive (SFAR) model of order 1 for a given functional time series.
sfar( X, seasonal, cpv = 0.85, kn = NULL, method = c("MME", "ULSE", "KOE"), a = ncol(Coefs)^(-1/6) )sfar( X, seasonal, cpv = 0.85, kn = NULL, method = c("MME", "ULSE", "KOE"), a = ncol(Coefs)^(-1/6) )
X |
a functional time series. |
seasonal |
a positive integer variable specifying the seasonality parameter. |
cpv |
a numeric with values in [0,1] which determines the cumulative proportion variance explained by the first kn eigencomponents. |
kn |
an integer variable specifying the number of eigencomponents. |
method |
a character string giving the method of estimation. The following values are possible: "MME" for Method of Moments, "ULSE" for Unconditional Least Square Estimation Method, and "KOE" for Kargin-Ontaski Estimation. |
a |
a numeric with value in [0,1]. |
A matrix of size p*p.
# 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)$fd # Simulate random SFAR(1) data kr <- function(x, y) { (2 - (2 * x - 1)^2 - (2 * y - 1)^2) / 2 } s <- 5 # the period number X <- rsfar(kr, s, Z) plot(X) # SFAR(1) model parameter estimation: Model1 <- sfar(X, seasonal = s, kn = 1)# Generate Brownian motion noise N <- 300 # the length of the series n <- 200 # the sample rate that each function will be sampled u <- seq(0, 1, length.out = n) # argvalues of the functions d <- 45 # the number of bases basis <- create.fourier.basis(c(0, 1), d) # the basis system sigma <- 0.05 # the std of noise norm Z0 <- matrix(rnorm(N * n, 0, sigma), nrow = n, nc = N) Z0[, 1] <- 0 Z_mat <- apply(Z0, 2, cumsum) # N standard Brownian motion Z <- smooth.basis(u, Z_mat, basis)$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)