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spde_v3.R
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spde_v3.R
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########################
########################
#### SPDE FUNCTIONS ####
########################
########################
################
# COMMAND LINE #
################
## R CMD SHLIB FFBS_spectral.c -lfftw3
## R CMD SHLIB RML_spectral.c
## R CMD SHLIB propagate_spectral.c
## R CMD SHLIB tkf_spectral.c
## R CMD SHLIB tkf2_spectral.c
################
################
# .C FUNCTIONS #
################
## Define directories ##
main_wd = "/Users/ls616/Google Drive/MPE CDT/PhD/Year 1/Code/Current"
fig_wd = "/Users/ls616/Desktop"
## Set directory ##
setwd(main_wd)
## Load functions from C
dyn.load("FFBS_spectral.so")
dyn.load("RML_spectral.so")
dyn.load("propagate_spectral.so")
dyn.load("tkf_spectral.so")
dyn.load("tkf2_spectral.so")
################
#################
# .R LIBRARIES #
#################
## Additional libraries ##
library(mvtnorm)
library(coda)
library(microbenchmark)
library(lattice)
library(dlm)
library(RColorBrewer)
library(gtools)
#################
#################
# MISCALLENEOUS #
#################
## Remove non functions ##
rm_non_func <- function(){
env <- parent.frame()
rm(list = setdiff(ls(all.names=TRUE, env = env),
lsf.str(all.names=TRUE, env = env)),
envir = env)
}
## Colour Scheme ##
colors <- function(){
return(c("#00008F", "#00009F", "#0000AF", "#0000BF", "#0000CF", "#0000DF",
"#0000EF", "#0000FF", "#0010FF", "#0020FF", "#0030FF", "#0040FF",
"#0050FF", "#0060FF", "#0070FF", "#0080FF", "#008FFF", "#009FFF",
"#00AFFF", "#00BFFF", "#00CFFF", "#00DFFF", "#00EFFF", "#00FFFF",
"#10FFEF", "#20FFDF", "#30FFCF", "#40FFBF", "#50FFAF", "#60FF9F",
"#70FF8F", "#80FF80", "#8FFF70", "#9FFF60", "#AFFF50", "#BFFF40",
"#CFFF30", "#DFFF20", "#EFFF10", "#FFFF00", "#FFEF00", "#FFDF00",
"#FFCF00", "#FFBF00", "#FFAF00", "#FF9F00", "#FF8F00", "#FF8000",
"#FF7000", "#FF6000", "#FF5000", "#FF4000", "#FF3000", "#FF2000",
"#FF1000", "#FF0000", "#EF0000", "#DF0000", "#CF0000", "#BF0000",
"#AF0000", "#9F0000", "#8F0000", "#800000"))
}
## Compute vector L2 norm ##
vec_norm <- function(x) return(sqrt(sum(x^2)))
## Compute linear predictor ##
linear_predictor <- function(x,beta){
return(x%*%beta)
}
## Convert space-time matrix to vector (row = time, column = space) ##
STmatrix_to_vec <- function(mat){
return(as.vector(t(mat)))
}
## Convert vector to space-time matrix (row = time, column = space) ##
vec_to_STmatrix <- function(vec,t=1){
return(matrix(vec,nrow=t,byrow=TRUE))
}
## Map non-gridded observations to a unit n x n grid ##
map_obs_to_grid <- function(n,y_non_grid,coord,
length_x=NULL,length_y=NULL){
## Inputs ##
# n: number of points per axis of the grid onto which the obs. are mapped
# y_non_grid: obs. data in a T x N matrix; coords. of each point in 'coord'
# coord: N x 2 matrix of the coords. of the N obs. data points
# length_x: length of x-axis; must be at least as large as max(coord[,1])
# length_y: length of y-axis; must be at least as large as max(coord[,2])
## Generate points on unit grid ##
cell_x <- rep(1:n, n)/n - 1/(2*n)
cell_y <- as.vector(apply(matrix(1:n), 1, rep, times = n))/n - 1/(2*n)
## Rescale coordinates ##
## If x-axis scale not provided, rescale coordinates using max value
if (is.null(length_x)){
coord[, 1] <- coord[, 1]/max(coord[, 1])
}
## If x-axis scale provided, rescale accordingly
else {
coord[, 1] <- coord[, 1]/length_x
}
## Similarly for y axis
if (is.null(length_y)){
coord[, 2] <- coord[, 2]/max(coord[, 2])
}
else{
coord[, 2] <- coord[, 2]/length_y
}
## Define 'obs_indices' variable
obs_indices <- rep(0, dim(coord)[1])
## Populate w/ indices of the grid coordinates which are closest to each of
## the observations
for (i in 1:dim(coord)[1]) {
## Compute Euclidean distance
d = abs(cell_x-coord[i, 1])^2 + abs(cell_y-coord[i, 2])^2
## Compute 'obs_indices' as above
obs_indices[i] <- which(d == min(d))
}
## Compute gridded observations values
y = array(NA,c(dim(y_non_grid)[1],n^2))
for(t in 1:dim(y_non_grid)[1]){
for(i in unique(obs_indices)){
y[t,i]=mean(y_non_grid[t,which(i==obs_indices)])
}
}
## Output y
return(y)
}
#################
## ---
############################
# WAVE NUMBERS FOR REAL FT #
############################
## -> Generate all wavenumbers k_j used in real Fourier transform, for a given
## spatial dimension n
## -> Inputs: n is the number of grid points on each (x,y) axis
## -> Output: list containing
## -- 'wave', a 2 by n^2 matrix with wavenumbers
## -- 'cosine_indices', a vector of indices indicating
## column of wavenumbers for cosine terms (first
## four terms not included)
wave_numbers <- function(n){
## Output an error message if n not even ##
if(n%%2!=0){
print("Error: n must be even")
return()
}
## Otherwise generate wavenumbers ##
else{
# Wavenumbers used in the real FT (unscaled by 2pi & unordered)
x_index <- c(rep(0:(n/2),n/2+1),rep(1:(n/2-1),n/2-1))
y_index <- c(as.vector(sapply(0:(n/2),function(i) rep(i,n/2+1))),
as.vector(sapply((-n/2+1):(-1),function(i) rep(i,n/2-1))))
xy_index <- rbind(x_index,y_index)
# Indices of wavenumbers which give a zero sine term
zero_sine_indices <- c(which(x_index==0 & y_index==0),
which(x_index==(n/2) & y_index==0),
which(x_index==0 & y_index==(n/2)),
which(x_index==(n/2) & y_index==(n/2)))
# Indices of cos basis functions (where cos & sin terms both non-zero) are
# 5,7,9,11,...,n^2-1
cosine_indices <- 2*(1:(dim(xy_index)[2]-4))-1+4
## Generate empty array to contain ordered & scaled wavenumbers ##
wave_numbers <- array(0,c(2,n^2))
# First 4 terms are wavenumbers for cosine-only terms
wave_numbers[,1:4] <- xy_index[,zero_sine_indices]
# Remaining n^2-4 terms are wavenumbers of (cosine, sine) terms
wave_numbers[,cosine_indices] <- xy_index[,-zero_sine_indices]
wave_numbers[,cosine_indices+1] <- xy_index[,-zero_sine_indices]
# Scale wavenumbers by 2*pi
wave_numbers <- wave_numbers*2*pi
# Output
return(list(wave=wave_numbers,cosine_indices=cosine_indices))
}
}
############################
###############################
# REAL AND COMPLEX FT INDICES #
###############################
## -> Auxilary function for conversion between complex FFT and the real FT
## -> Inputs: n is the number of grid points on each (x,y) axis
## -> Output: list containing
## -- 'cosine_indices', a vector of the indices of the wave-numbers
## of cosine terms for the real FT (not inc. first 4 cosine only
## terms)
## -- 'exp_indices', a vector of the indices of the wave-numbers in
## the complex FT, whic correspond to those used in the real FT
## -- 'exp_indices_conj', a vector of the indices of the wave-numbers
## in the complex FT not used in the real FT
FFT_index_complex_and_real <- function(n){
# Initialise matrices for complex and real wave numbers
wave_complex <- array(0,c(2,n*n))
wave_real <- array(0,c(2,n*n))
# Wavenumbers used in the real FT (unscaled by 2pi & unordered)
x_index <- c(rep(0:(n/2),n/2+1),rep(1:(n/2-1),n/2-1))
y_index <- c(as.vector(sapply(0:(n/2),function(i) rep(i,n/2+1))),
as.vector(sapply((-n/2+1):(-1),function(i) rep(i,n/2-1))))
xy_index <- rbind(x_index,y_index)
# Indices of wavenumbers which give a zero sine term
zero_sine_indices <- c(which(x_index==0 & y_index==0),
which(x_index==(n/2) & y_index==0),
which(x_index==0 & y_index==(n/2)),
which(x_index==(n/2) & y_index==(n/2)))
# Indices of cos basis functions (where cos & sin terms both non-zero) are
# 5,7,9,11,...,n^2-1
cosine_indices <- seq(from=5,to=n^2,by=2)
## Real FT ##
# First 4 terms are wavenumbers for cosine-only terms
wave_real[,1:4] <- xy_index[,zero_sine_indices]
# Remaining n^2-4 terms are wavenumbers of (cosine, sine) terms
wave_real[,cosine_indices] <- xy_index[,-zero_sine_indices]
wave_real[,cosine_indices+1] <- xy_index[,-zero_sine_indices]
# Instead of wave-numbers in 0<=x<=n/2 and -n/2+1<=y<=n/2, use (equivalently)
# wave-numbers in 0<=x<=n/2 and 0<=y<=n-1
wave_real[2,wave_real[2,]<0] <- wave_real[2,wave_real[2,]<0]+n
## Complex FT ##
# Wavenumbers used in the complex FT (unscaled by 2pi & unordered), i.e. all
# grid points in 0<=x<=n^2-1, 0<=y<=n^2-1
wave_complex <- rbind(rep(0:(n-1),n),
as.vector(apply(matrix(0:(n-1)),1,rep,times=n)))
# Indices of the wave-numbers in complex FT which correspond to wave-numbers
# in real FT
exp_indices <- match(wave_real[1,]+1i*wave_real[2,],
wave_complex[1,]+1i*wave_complex[2,])
waveCon <- (n-wave_real[,cosine_indices])%%n
# Indices of the wave-numbers in complex FT which don't correspond to wave-
# numbers in real FT
exp_indices_conj <- match(waveCon[1,]+1i*waveCon[2,],
wave_complex[1,]+1i*wave_complex[2,])
## Ouputs ##
return(list(cosine_indices=cosine_indices,exp_indices=exp_indices,
exp_indices_conj=exp_indices_conj))
}
###############################
## ---
################################
# MATERN COV. SPECTRAL DENSITY #
################################
## -> Compute spectrum of the Matern covariance function
## -> Inputs:
## -- 'wave', spatial wavenumbers (from wave_numbers$wave)
## -- 'n', number of grid points on each ax_is (x,y)
## -- 'n_cos', number of cosine only terms (default = 4)
## -- 'rho0', range
## -- 'sigma2', marginal variance
## -- 'nu', smoothness of Matern covariance
## -- 'norm', if TRUE, spectrum is normalised
## -- 'd', the Eucliden distance between two points (default=2)
## -> Output:
## -- 'spectrum', vector of Matern spectrum
matern_spectrum <- function(wave,n,n_cos=4,rho0,sigma2,nu=1,d=2,norm=TRUE){
## Error message if nu<=0 ##
if(nu<=0){
print("Error: nu must be positive")
return()
}
## Error message if sigma^2<=0 ##
if(sigma2<=0){
print("Error: sigma^2 must be positive")
return()
}
## Compute k_j^T k_j ##
w2 <- apply(wave^2,2,sum)
## Compute spectrum (up to proportionality) ##
numerator <- 1
denominator <- ((1/rho0)^2 + w2)^(nu + 1)
spectrum <- numerator/denominator
## Adjustment for cosine only terms ##
spectrum[1:n_cos] <- spectrum[1:n_cos]/2
## Normalise ##
if(norm){
spectrum <- spectrum*(n^2)/sum(spectrum)
}
else{
spectrum <- spectrum/sum(spectrum)
}
## Rescale by sigma^2 ##
spectrum <- sigma2 * spectrum
return(spectrum)
}
################################
###############################
# INNOVATION SPECTRAL DENSITY #
###############################
## -> Compute spectrum of the innovation term (epsilon)
## -> Inputs:
## -- 'wave', spatial wavenumbers (from wave_numbers$wave)
## -- 'n', number of grid points on each ax_is (x,y)
## -- 'n_cos', number of cosine only terms (default = 4)
## -- 'rho0', range of Matern covariance function for innovation
## -- 'sigma2', marginal variance of Mat. cov. fn. for innovation
## -- 'zeta', damping parameter
## -- 'rho1', range parameter of diffusion term
## -- 'alpha', direction of anisotropy of diffusion term
## -- 'gamma', amount of anisotropy of diffusion term
## -- 'nu', smoothness of Matern covariance for innovations (default = 1)
## -- 'dt', temporal lag between time points (default = 1)
## -- 'norm', if TRUE, spectrum is normalised
## -> Output:
## -- 'Qhat', vector of spectrum of integrated innovation term
innovation_spectrum <- function(wave,n,n_cos=4,rho0,sigma2,
zeta,rho1,alpha,gamma,nu=1,
dt=1,norm=TRUE){
## Error messages if paramaters outside correct range ##
if(nu<0){
print("Error: nu must be positive")
return()
}
if(sigma2<0){
print("Error: sigma^2 must be positive")
return()
}
if(zeta<0){
print("Error: zeta must be positive")
return()
}
if(gamma<0){
print("Error: gamma must be positive")
return()
}
if(alpha<0 & alpha>=(pi/2)){
print("Error: alpha needs must be between 0 and pi/2")
return()
}
## Generate Matern spectrum using 'matern_spectrum' ##
spec <- matern_spectrum(wave=wave,n=n,n_cos=n_cos,rho0=rho0,sigma2=1,
nu=nu,d=2,norm=norm)
## Number of Fourier terms ##
K <- dim(wave)[2]
## Compute Sigma ##
# Special Case if rho1 = 0
if(rho1==0){
Sig <- cbind(c(0,0),c(0,0))
}
# General Case
else{
Sig_Chol <- cbind(c(cos(alpha),-gamma*sin(alpha)),
c(sin(alpha),gamma*cos(alpha)))/rho1
Sig <- solve(t(Sig_Chol)%*%Sig_Chol)
}
## Compute -(k_j^T * Sigma * k_j + zeta) ##
ExpTerm <- -apply(wave*Sig%*%wave,2,sum)-rep(zeta,K)
## Compute Q hat ##
Qhat <- sigma2*spec*(1-exp(2*dt*ExpTerm))/(-2*ExpTerm)
## Output ##
return(Qhat)
}
###############################
## ---
##################
# REAL FT MATRIX #
##################
## -> Compute matrix which applies 2d real FT, for a set of n_obs spatial
## locations
## -> Inputs:
## -- 'wave', matrix of spatial wavenumbers (from wave_numbers$wave)
## -- 'cosine_indices', vector of indices of columns of cos terms
## -- 'n_cos', number of real Fourier terms with only cosine term
## -- 'n', spatial dimensions
## -- 'n_obs', the number of observations
## -- 'coords', a list containing the coordinates of each of
## the 'n_obs' observation locations, scaled by n [e.g. the input
## (1,1) corresponds to actual coordinates (1/n,1/n)]
## -> Output:
## -- 'phi', a n_obs x K matrix which applies the 2D real FT
FFT_real_matrix <- function(wave,cosine_indices,n_cos=4,n,n_obs=NULL,
coords=NULL){
## If n_obs not provided, set n_obs = n^2 ##
if(is.null(n_obs)){
n_obs = n^2
}
## If coords not provided, set coords = uniform n x n grid ##
if(is.null(coords)){
coords = lapply(1:(n^2),
function(i) as.numeric(expand.grid(0:(n-1),0:(n-1))[i,]))
}
## Normalising Constants (Check these?)
A <- sqrt(n^2/2)
B <- sqrt(n^2)
## Number of Fourier basis functions
K = dim(wave)[2]
## Initialise matrices for storage
phi_row <- rep(0, K)
phi <- matrix(0, nrow = n_obs, ncol = K)
## Loop over matrix indices
for(i in 1:n_obs){
# Cosine only terms
phi_row[1:n_cos] <- cos(coords[[i]]%*%(wave[,1:n_cos])/n)/B
# 'Other' cosine terms
phi_row[cosine_indices] <- cos(coords[[i]]%*%(wave[,cosine_indices])/n)/A
# 'Other' sine terms
phi_row[cosine_indices+1] <- sin(coords[[i]]%*%(wave[,cosine_indices+1])/n)/A
# Add to matrix
phi[i,] <- phi_row
}
# Output
return(phi)
}
##################
###############################
# REAL FT MATRIX SPATIAL GRAD #
###############################
## -> Compute gradient of matrix which applies 2d real FT, for a set of n_obs
## spatial locations
## -> Inputs:
## -- 'wave', matrix of spatial wavenumbers (from wave_numbers$wave)
## -- 'cosine_indices', vector of indices of columns of cos terms
## -- 'n_cos', number of real Fourier terms with only cosine term
## -- 'n', spatial dimensions
## -- 'n_obs', the number of observations
## -- 'coords', a list containing the coordinates of each of
## the 'n_obs' observation locations, scaled by n [e.g. the input
## (1,1) corresponds to actual coordinates (1/n,1/n)]
## -- 'grad_coord_index', the indices of the coordinates with respect
## to which to compute the gradient
## -> Output:
## -- 'phi', a list of length 2*length(grad_coord_index), containing
## the gradients w.r.t x,y of the locations whose index in
## coords is given by grad_coord_index
FFT_real_matrix_grad <- function(wave,cosine_indices,n_cos=4,n,n_obs,coords,
grad_coord_index=1:n_obs){
## Normalising Constants (Check these?)
A <- sqrt(n^2/2)
B <- sqrt(n^2)
## Number of Fourier basis functions
K = dim(wave)[2]
## Initialise matrices
phi_grad_x <- matrix(0,nrow=n_obs,ncol=K)
phi_grad_y <- matrix(0,nrow=n_obs,ncol=K)
## Initalise rows
phi_row_grad_x <- rep(0,K)
phi_row_grad_y <- rep(0,K)
## Loop over matrix indices
for(i in 1:n_obs){
# Grad of cosine only terms
phi_row_grad_x[1:n_cos] <- -wave[1,1:n_cos]*sin(coords[[i]]%*%(wave[,1:n_cos])/n)/B
phi_row_grad_y[1:n_cos] <- -wave[2,1:n_cos]*sin(coords[[i]]%*%(wave[,1:n_cos])/n)/B
# Grad of 'other' cosine terms
phi_row_grad_x[cosine_indices] <- -wave[1,cosine_indices]*sin(coords[[i]]%*%(wave[,cosine_indices])/n)/A
phi_row_grad_y[cosine_indices] <- -wave[2,cosine_indices]*sin(coords[[i]]%*%(wave[,cosine_indices])/n)/A
# Grad of 'other' sine terms
phi_row_grad_x[cosine_indices+1] <- wave[1,cosine_indices+1]*cos(coords[[i]]%*%(wave[,cosine_indices+1])/n)/A
phi_row_grad_y[cosine_indices+1] <- wave[2,cosine_indices+1]*cos(coords[[i]]%*%(wave[,cosine_indices+1])/n)/A
# Add to grad. matrix
phi_grad_x[i,] <- phi_row_grad_x
phi_grad_y[i,] <- phi_row_grad_y
}
## Initialise output ##
output = list()
## Set non grad-coords to zero
if(!is.null(grad_coord_index)){
for (i in grad_coord_index){
phi_grad_x_tmp = phi_grad_x
phi_grad_y_tmp = phi_grad_y
phi_grad_x_tmp[-i,] = rep(0,K)
phi_grad_y_tmp[-i,] = rep(0,K)
output = c(output,list(phi_grad_x_tmp,phi_grad_y_tmp))
}
}
else{
output = list(phi_grad_x,phi_grad_y)
}
# Output
return(output)
}
###############################
## ---
#######################
# WEIGHTING FUNCTIONS #
#######################
## Bivariate Normal ##
bivar_norm = function(x,y,centre_x,centre_y,sigma_x,sigma_y,rho){
z = (x-centre_x)^2/sigma_x^2 + (y-centre_y)^2/sigma_y^2 -
2*rho*(x-centre_x)*(y-centre_y)/(sigma_x*sigma_y)
weights = 1/(2*pi*sigma_x*sigma_y*sqrt(1-rho^2))*exp(-z/(2*(1-rho^2)))
return(weights)
}
bivar_norm_weights = function(n,centre_x,centre_y,sigma_x,sigma_y,rho){
## Generate grid points ##
grid = expand.grid(x = 0:(n-1),y = 0:(n-1))/n
## Compute weights using 'bivar_norm' ##
weights = with(grid,bivar_norm(x = x,y = y, centre_x = centre_x,
centre_y = centre_y, sigma_x = sigma_x,
sigma_y=sigma_y, rho = rho))
## Ouput ##
return(weights)
}
## Sech ##
sech2d = function(x,y,centre_x,centre_y,sigma_x,sigma_y,rho){
z = (x-centre_x)^2/sigma_x^2 + (y-centre_y)^2/sigma_y^2 -
2*rho*(x-centre_x)*(y-centre_y)/(sigma_x*sigma_y)
weights = pracma::sech(sqrt(z))
return(weights)
}
sech2d_weights = function(n,centre_x,centre_y,sigma_x,sigma_y,rho){
## Generate grid points ##
grid = expand.grid(x = 0:(n-1),y = 0:(n-1))/n
## Compute weights using 'bivar_norm' ##
weights = with(grid,sech2d(x = x,y = y, centre_x = centre_x,
centre_y = centre_y, sigma_x = sigma_x,
sigma_y=sigma_y))
## Ouput ##
return(weights)
}
#######################
####################
# WEIGHTING MATRIX #
####################
weights_mat_func = function(weights){
weights_mat = diag(weights)
return(weights_mat)
}
####################
#########################
# SPEC WEIGHTING MATRIX #
#########################
spec_weights_mat_func = function(n, K, weights = NULL,
weights_mat = NULL){
## SPDE_FT Object ##
SPDE_FT = spde_initialise(n,t,K)
wave = SPDE_FT$wave
cosine_indices = SPDE_FT$cosine_indices
n_cos = SPDE_FT$n_cos
n_cos_sin = length(cosine_indices)
K = SPDE_FT$K
## Weighting Matrix ##
if(is.null(weights_mat)){
weights_mat = weights_mat_func(weights)
}
## Phi ##
Phi = FFT_real_matrix(wave,cosine_indices,n_cos,n)
## Spectral Weighting Matrix ##
spec_weights_mat = t(Phi)%*%weights_mat%*%Phi
## Output ##
return(spec_weights_mat)
}
#########################
## ---
##############################
# SPECTRAL PROPAGATOR MATRIX #
##############################
## -> Compute spectral propagator matrix G of vector autoregressive model
## -> Inputs:
## -- 'wave', matrix of spatial wavenumbers (from wave_numbers$wave)
## -- 'cosine_indices', vector of indices of columns of cos terms
## -- 'zeta', damping parameter
## -- 'rho1', range parameter of diffusion term
## -- 'gamma', amount of anisotropy in diffusion
## -- 'alpha', direction of anisotropy in diffusion
## -- 'mu_x', x component of drift
## -- 'mu_y', y component of drift
## -- 'dt', temporal lag between time points (default = 1)
## -- 'n_cos', number of real Fourier terms with only cosine term
## -> Output:
## -- 'G', the propagator matrix
propagator_matrix <- function(wave,cosine_indices,zeta,rho1,gamma,alpha,
mu_x,mu_y,dt=1,n_cos=4){
## Number of terms in Fourier expansion
K <- dim(wave)[2]
## Special case when rho = 0
if(rho1==0){
Sig <- cbind(c(0,0),c(0,0))
}
## Compute Sigma (see pg 4 of r_spate documentation)
else{
Sig_Chol <- cbind(c(cos(alpha),-gamma*sin(alpha)),
c(sin(alpha),gamma*cos(alpha)))/rho1
Sig <- solve(t(Sig_Chol)%*%Sig_Chol)
}
## Compute -Delta*(k_j^T * Sigma * k_j + zeta)
H_diag <- -dt*apply(wave*Sig%*%wave,2,sum)-dt*rep(zeta,K)
## Compute -Delta * mu * k_j
H_off_diag <- dt*c(mu_x,mu_y)%*%wave
## Compute G
# Initialise
G <- matrix(0,ncol=K,nrow=K)
# Diagonal elements
diag(G) <- c(exp(H_diag[1:n_cos]),
exp(H_diag[(n_cos+1):K])*cos(H_off_diag[(n_cos+1):K]))
# Off diagonal elements
diag(G[cosine_indices,cosine_indices+1]) <-
-exp(H_diag[cosine_indices])*sin(H_off_diag[cosine_indices])
diag(G[cosine_indices+1,cosine_indices]) <-
exp(H_diag[cosine_indices])*sin(H_off_diag[cosine_indices])
## Alternative: straightforward but MUCH slower
# Compute Delta*H matrix
# H = matrix(0,ncol=K,nrow=K)
# diag(H) = H_diag
# for (i in cosine_indices){
# H[i,i+1]=-H_off_diag[i]
# H[i+1,i]=H_off_diag[i]
#}
# Compute G = exp(Delta*H)
# library(expm)
# G <- expm(H)
## Output
return(G)
}
##############################
####################################
# SPECTRAL PROPAGATOR MATRIX (VEC) #
####################################
## -> Compute spectral propagator matrix G of vector autoregressive model, in
## vector form
## -> Inputs:
## -- 'wave', matrix of spatial wavenumbers (from wave_numbers$wave)
## -- 'cosine_indices', vector of indices of columns of cos terms
## -- 'zeta', damping parameter
## -- 'rho1', range parameter of diffusion term
## -- 'gamma', amount of anisotropy in diffusion
## -- 'alpha', direction of anisotropy in diffusion
## -- 'mu_x', x component of drift
## -- 'mu_y', y component of drift
## -- 'dt', temporal lag between time points (default = 1)
## -- 'n_cos', number of real Fourier terms with only cosine term
## -> Output: list containing
## -- 'G_cos', first n_cos diagonal entries of G, corresponding to
## cosine only terms
## -- 'G_1', remaining diagonal entries of G, corresponding to other
## cosine terms
## -- 'G_2', off (upper) diagonal entries of G, corresponding to
## G_1 entries
propagator_vec <- function(wave,cosine_indices,zeta,rho1,gamma,alpha,mu_x,mu_y,
dt=1,n_cos=4){
## Number of terms in Fourier expansion
K <- dim(wave)[2]
## Special case when rho = 0
if(rho1==0){
Sig <- cbind(c(0,0),c(0,0))
}
## Compute Sigma (see pg 4 of r_spate documentation)
else{
Sig_Chol <- cbind(c(cos(alpha),-gamma*sin(alpha)),
c(sin(alpha),gamma*cos(alpha)))/rho1
Sig <- solve(t(Sig_Chol)%*%Sig_Chol)
}
## Compute -Delta*(k_j^T * Sigma * k_j + zeta)
H_diag <- -dt*apply(wave*Sig%*%wave,2,sum)-dt*rep(zeta,K)
## Compute -Delta * mu * k_j
H_off_diag <- dt*c(mu_x,mu_y)%*%wave
## First n_cos diagonal entries, corresponding to first n_cos
## cosine terms
G_cos <- exp(H_diag[1:n_cos])
## Other diagonal entries, corresponding to other cosine terms
G_1 <- exp(H_diag[cosine_indices]) * cos(H_off_diag[cosine_indices])
## Off (upper) diagonal entries, corresponding to diagonal entries above
G_2 <- -exp(H_diag[cosine_indices]) * sin(H_off_diag[cosine_indices])
## Alternative: more straightforward, but SLIGHTLY slower
## Compute propagator matrix
# G_matrix = propagator_matrix(wave,cosine_indices,zeta,rho1,gamma,alpha,mu_x,mu_y,dt=1,n_cos=4)
## First n_cos diagonal entries, corresponding to first n_cos cosine terms
# G_cos = diag(G_matrix)[1:n_cos]
## Other diagonal entries, corresponding to other cosine terms
# G_1 = diag(G_matrix)[cosine_indices]
## Off (upper) diagonal entries, corresponding to diagonal entries above
# G_2 = diag(G_matrix[cosine_indices,cosine_indices+1])
## Output list of vectors
G <- list(G_cos=G_cos,G_1=G_1,G_2=G_2)
return(G)
}
####################################
## ---
######################
# SPDE OBJECT FOR FT #
######################
## -> Generate 'SPDE_FT' objects used for two-dim Fourier transform
## -> Inputs:
## -- 'n', number of points on each ax_is (x,y)
## -- 't', number of time points
## -- 'K', number of Fourier basis functions (default = n^2)
## -> Output: list containing
## -- 'n', number of points on each ax_is (x,y)
## -- 't', number of time points
## -- 'wave', 2 x n^2 matrix of wave numbers
## -- 'cosine_indices', vector of indices of cosine terms
## -- 'n_cos', number of cosine-only terms
## -- 'FFT_indices', indices used for conversion between complex FFT
## and real FT
spde_initialise <- function(n,t,K=n^2){
## Generate wave_numbers object (contains all wave numbers, and
## cosine indices)
wave_list <- wave_numbers(n)
## Add element to list: number of cosine only terms equals 4
wave_list$n_cos <- 4
## Indices of wavenumbers of basis functions: 1,...,n^2
basis_indices <- 1:(n^2)
## If number of Fourier basis functions is less than n^2 (the full basis):
if(K<(n^2)){
## Sort wave-numbers in ascending order of distance from the origin
## Compute 'cut' as the [K/n^2]^th sample quantile of the
## set of distances of all wave-numbers from the origin
cut <- quantile(apply(wave_list$wave,2,vec_norm)
[order(apply(wave_list$wave,2,vec_norm))],
probs=K/(n^2))
## Indices of wave-numbers of new set of basis functions = wave-numbers
## whose distance to origin is less than 'cut'
basis_indices <- which(apply(wave_list$wave,2,vec_norm)<=cut)
## Print warning message if number of basis functions to be used (i.e.
## number of wave-numbers whose distance to the origin is less than 'cut')
## isn't equal to the number specified by the user
if(length(basis_indices)!=K){
print(paste("Warning: ",length(basis_indices)," Fourier functions",
" (instead of ",K,") are used since when",
" using only ",K," there is anisotropy in",
" the reduced dimensional basis.",sep=""))
}
## Recompute 'wave_list' using only the required wave-numbers
wave_list$wave <- wave_list$wave[,basis_indices]
wave_list$n_cos <- 4-sum(is.na(match(1:4,basis_indices)))
wave_list$cosine_indices <-
match(basis_indices[match(wave_list$cosine_indices,basis_indices)
[!is.na(match(wave_list$cosine_indices,
basis_indices))]],basis_indices)
K <- length(basis_indices)
}
## Compute FFT indices
FFT_indices <- FFT_index_complex_and_real(n)
## Generate 'SPDE_FT' object
SPDE_FT <- list(n=n,t=t,wave=wave_list$wave,
cosine_indices=wave_list$cosine_indices,
n_cos=wave_list$n_cos,basis_indices=basis_indices,
FFT_indices=FFT_indices, K = length(basis_indices))
class(SPDE_FT) <- "SPDE_FT"
## Output
return(SPDE_FT)
}
######################
## ---
########################
# SPECTRAL PROPAGATION #
########################
## -> Propagate `alpha_t’ vector in time.
## -> Inputs:
## -- 'alpha_t', vector of spectral coefficients.
## -- 'SPDE_FT', SPDE_FT object from `spde_initialise'
## (require this or 'n')
## -- 'n', number of points on each axis (x,y) (require
## this or 'SPDE_FT')
## -- 'G_vec', spectral propagator matrix in vector form
## (require this or 'param')
## -- 'param', parameters from the SPDE (require this or
## 'G_vec')
## -> Output:
## -- 'alpha_t_plus_1', vector of propagated spectral
## coefficients (G*alpha_t)
## NB:
## -- 'spectral_propagate_matrix' provides same functionality,
## but with G in matrix form. This fn. is considerably slower.
## -- 'spectral_propagate' is a wrapper function for 'propagate_function'
propagate_function = function(xtp1,xt,G_cos,G_1,G_2,n_cos_sin,n_cos){
for(i in 1:n_cos){
xtp1[i] = G_cos[i]*xt[i]
}
for(i in 1:n_cos_sin){
xtp1[n_cos + 2*i - 1] = G_1[i] * xt[n_cos + 2*i - 1] + G_2[i] * xt[n_cos + 2*i]
xtp1[n_cos + 2*i] = G_1[i] * xt[n_cos + 2*i] - G_2[i] * xt[n_cos + 2*i - 1]
}
return(xtp1)
}
spectral_propagate <- function(alpha_t,SPDE_FT=NULL,n=NULL,G_vec=NULL,
param=NULL){
## If SPDE_FT not given: create SPDE_FT object
if(is.null(SPDE_FT)){
SPDE_FT <- spde_initialise(n=n,t=1)
}
## If G_vec not given:
if(is.null(G_vec)){
## If also param. not given: report error message
if(is.null(param)){
print("Either 'G_vec' or 'param' is required.")
return()
}
## Else create G (vector form) using input param.
else{
G_vec <- propagator_vec(wave=SPDE_FT$wave,
cosine_indices=SPDE_FT$cosine_indices,
zeta=param[3],rho1=param[4],gamma=param[5],
alpha=param[6],mu_x=param[7],mu_y=param[8],
dt=1,n_cos=SPDE_FT$n_cos)
}
}
## Compute alpha_t_plus_1 = G*alpha_t
#alpha_t_plus_1 <- propagate_function(xtp1=(rep(0,SPDE_FT$n*SPDE_FT$n)),
# xt = alpha_t,
# G_cos = G_vec$G_cos,
# G_1 = G_vec$G_1,
# G_2 = G_vec$G_2,
# n_cos_sin = length(SPDE_FT$cosine_indices),
# n_cos = SPDE_FT$n_cos)