/*
* Invert a toeplitz matrix. Input is the first column of the matrix and also
* is the output.
* IMP - size must be a power of two.
*/
void serial_matrix_inversion(size_t size, const fftw_complex *input, fftw_complex *result) {
if(size < 1) return;
//base case
result[0] = 1 / input[0];
if(size < 2) return;
result[1] = - result[0] * input[1] * result[0];
unsigned int current_size = 2;
while(current_size < size) {
fftw_complex *reversed_input = alloc_reverse_from_vec(current_size * 2 - 1, input);
fftw_complex *temporary = alloc_complex(current_size);
//first convolution
toeplitz_mult(current_size, reversed_input, result, temporary);
fftw_complex *reversed_inverted_pad = alloc_complex(current_size * 2 - 1);
zero_memory(current_size * 2 - 1, reversed_inverted_pad);
memcpy(reversed_inverted_pad + (current_size-1), result,
current_size * sizeof(fftw_complex));
reverse_vec(current_size * 2 - 1, reversed_inverted_pad);
//second convolution
reverse_vec(current_size, temporary);
toeplitz_mult(current_size, reversed_inverted_pad, temporary, reversed_input);
//complement sign
for(int i=0; i < current_size; i++)
reversed_input[i] = - reversed_input[i];
reverse_vec(current_size, reversed_input);
//finally copy in result vec
memcpy(result + current_size, reversed_input, current_size * sizeof(fftw_complex));
fftw_free(temporary);
fftw_free(reversed_input);
fftw_free(reversed_inverted_pad);
current_size *= 2;
}
}
//' @title Discrete Wavelet Transform
//' @description Calculation of the coefficients for the discrete wavelet transformation.
//' @param x A \code{vector} with dimensions \eqn{N\times 1}{N x 1}.
//' @param filter_name A \code{string} indicating the filter.
//' @param nlevels An \code{integer}, \eqn{J}, indicating the level of the decomposition.
//' @param boundary A \code{string} indicating the type of boundary method to use. Either \code{boundary="periodic"} or \code{"reflection"}.
//' @param brickwall A \code{bool} indicating whether the a brick wall procedure should be applied to the coefficients.
//' @return y A \code{field<vec>} that contains the wavelet coefficients for each decomposition level
//' @details
//' Performs a level J decomposition of the time series using the pyramid algorithm
//' @author JJB
//' @keywords internal
//' @examples
//' set.seed(999)
//' x = rnorm(2^8)
//' dwt_cpp(x, filter_name = "haar", nlevels = 4, boundary = "periodic", brickwall = TRUE)
// [[Rcpp::export]]
arma::field<arma::vec> dwt_cpp(arma::vec x, std::string filter_name,
unsigned int nlevels, std::string boundary, bool brickwall) {
if(boundary == "periodic"){
//
}else if(boundary == "reflection"){
unsigned int temp_N = x.n_elem;
arma::vec rev_vec = reverse_vec(x);
x.resize(2*temp_N);
x.rows(temp_N, 2*temp_N-1) = rev_vec;
}else{
Rcpp::stop("The supplied 'boundary' argument is not supported! Choose either periodic or reflection.");
}
unsigned int N = x.n_elem;
unsigned int J = nlevels;
unsigned int tau = pow(2,J);
if(double(N)/double(tau) != floor(double(N)/double(tau))){
Rcpp::stop("The supplied sample size ('x') must be divisible by 2^(nlevels). Either truncate or expand the number of samples.");
}
if(tau > N){
Rcpp::stop("The number of levels [ 2^(nlevels) ] exceeds sample size ('x'). Supply a lower number of levels.");
}
arma::field<arma::vec> filter_info = select_filter(filter_name);
int L = arma::as_scalar(filter_info(0));
arma::vec h = filter_info(1); //check the pulls
arma::vec g = filter_info(2);
arma::field<arma::vec> y(J);
for(unsigned int j = 1; j <= J; j++) {
unsigned int M = N/pow(2,(j-1));
unsigned int M_over_2 = double(M)/2;
arma::vec Wj(M_over_2);
arma::vec Vj(M_over_2);
for(unsigned t = 0; t < M_over_2; t++) {
int u = 2*t + 1;
double Wjt = h(0)*x(u);
double Vjt = g(0)*x(u);
for(int n = 1; n < L; n++){
u -= 1;
if(u < 0){
u = M - 1;
}
Wjt += h(n)*x(u);
Vjt += g(n)*x(u);
}
Wj[t] = Wjt;
Vj[t] = Vjt;
}
y(j-1) = Wj;
x = Vj;
}
// Apply brickwall
if(brickwall){
y = brick_wall(y, filter_info, "dwt");
}
return y;
}
//' @title Maximum Overlap Discrete Wavelet Transform
//' @description
//' Calculation of the coefficients for the discrete wavelet transformation
//' @inheritParams dwt_cpp
//' @return y A \code{field<vec>} that contains the wavelet coefficients for each decomposition level
//' @keywords internal
//' @details
//' Performs a level J decomposition of the time series using the pyramid algorithm.
//' Use this implementation to supply custom parameters instead of modwt(x),
//' which serves as a wrapper function.
//' @author JJB
//' @keywords internal
//' @examples
//' set.seed(999)
//' x = rnorm(100)
//' modwt_cpp(x, filter_name = "haar", nlevels = 4, boundary = "periodic", brickwall = TRUE)
// [[Rcpp::export]]
arma::field<arma::vec> modwt_cpp(arma::vec x, std::string filter_name,
unsigned int nlevels, std::string boundary, bool brickwall){
if(boundary == "periodic"){
//
}else if(boundary == "reflection"){
unsigned int temp_N = x.n_elem;
arma::vec rev_vec = reverse_vec(x);
x.resize(2*temp_N);
x.rows(temp_N, 2*temp_N-1) = rev_vec;
}else{
Rcpp::stop("The supplied 'boundary' argument is not supported! Choose either periodic or reflection.");
}
unsigned int N = x.n_elem;
unsigned int J = nlevels;
unsigned int tau = pow(2,J);
if(tau > N) Rcpp::stop("The number of levels [ 2^(nlevels) ] exceeds sample size ('x'). Supply a lower number of levels.");
arma::field<arma::vec> filter_info = select_filter(filter_name);
int L = arma::as_scalar(filter_info(0));
arma::vec ht = filter_info(1);
arma::vec gt = filter_info(2);
// modwt transform
double transform_factor = sqrt(2);
ht /= transform_factor;
gt /= transform_factor;
arma::field<arma::vec> y(J);
arma::vec Wj(N);
arma::vec Vj(N);
for(unsigned int j = 1; j <= J; j++) {
for(unsigned t = 0; t < N; t++) {
int k = t;
double Wjt = ht(0)*x(k);
double Vjt = gt(0)*x(k);
for(int n = 1; n < L; n++){
k -= pow(2, j-1);
if(k < 0){
k += N;
}
Wjt += ht(n)*x(k);
Vjt += gt(n)*x(k);
}
Wj[t] = Wjt;
Vj[t] = Vjt;
}
y(j-1) = Wj;
x = Vj;
}
// Apply brickwall
if(brickwall){
y = brick_wall(y, filter_info, "modwt");
}
return y;
}