//' Generate a Quantisation Noise (QN) sequence
//'
//' Generate an QN sequence given \eqn{Q^2}
//' @param N An \code{integer} for signal length.
//' @param q2 A \code{double} that contains autocorrection.
//' @return A \code{vec} containing the QN process.
//' @keywords internal
//' @details
//' To generate the quantisation noise, we follow this recipe:
//' First, we generate using a random uniform distribution:
//' \deqn{U_k^*\sim U\left[ {0,1} \right]}{U_k^*~U[0,1]}
//'
//' Then, we multiple the sequence by \eqn{\sqrt{12}}{sqrt(12)} so:
//' \deqn{{U_k} = \sqrt{12} U_k^*}{U_k = sqrt(12)*U_k^*}
//'
//' Next, we find the derivative of \eqn{{U_k}}{U_k}
//' \deqn{{{\dot U}_k} = \frac{{{U_{k + \Delta t}} - {U_k}}}{{\Delta t}}}{U_k^. = (U_(k + (delta)t) - U_k)}
//'
//' In this case, we modify the derivative such that:
//' \eqn{{{\dot U}_k}\Delta t = {U_{k + \Delta t}} - {U_k}}{U_k^. * (delta)t = U_{k + (delta)*t} - U_k}
//'
//' Thus, we end up with:
//' \deqn{{x_k} = \sqrt Q {{\dot U}_k}\Delta t}{x_k = sqrt(Q)*U_k^.*(delta)t}
//' \deqn{{x_k} = \sqrt Q \left( {{U_{k + 1}} - {U_k}} \right)}{x_k = sqrt(Q)* (U_(k+1) - U_(k))}
//'
//' @backref src/gen_process.cpp
//' @backref src/gen_process.h
//' @examples
//' gen_qn(10, 5)
// [[Rcpp::export]]
arma::vec gen_qn(const unsigned int N, double q2 = .1)
{
double sqrt12 = sqrt(12);
arma::vec gu(N+1);
for(unsigned int i=0; i <= N; i++ )
{
gu(i) = sqrt12*R::runif(0.0,1.0);
}
return sqrt(q2)*diff_cpp(gu);
}
//' @title Master Wrapper for the GMWM Estimator
//' @description This function generates WV, GMWM Estimator, and an initial test estimate.
//' @param data A \code{vec} containing the data.
//' @param theta A \code{vec} with dimensions N x 1 that contains user-supplied initial values for parameters
//' @param desc A \code{vector<string>} indicating the models that should be considered.
//' @param objdesc A \code{field<vec>} containing a list of parameters (e.g. AR(1) = c(1,1), ARMA(p,q) = c(p,q,1))
//' @param model_type A \code{string} that represents the model transformation
//' @param starting A \code{bool} that indicates whether the supplied values are guessed (T) or are user-based (F).
//' @param alpha A \code{double} that handles the alpha level of the confidence interval (1-alpha)*100
//' @param compute_v A \code{string} that describes what kind of covariance matrix should be computed.
//' @param K An \code{int} that controls how many times theta is updated.
//' @param H An \code{int} that controls how many bootstrap replications are done.
//' @param G An \code{int} that controls how many guesses at different parameters are made.
//' @param robust A \code{bool} that indicates whether the estimation should be robust or not.
//' @param eff A \code{double} that specifies the amount of efficiency required by the robust estimator.
//' @return A \code{field<mat>} that contains a list of ever-changing estimates...
//' @author JJB
//' @references Wavelet variance based estimation for composite stochastic processes, S. Guerrier and Robust Inference for Time Series Models: a Wavelet-Based Framework, S. Guerrier
//' @keywords internal
//' @export
//' @backref src/gmwm_logic.cpp
//' @backref src/gmwm_logic.h
// [[Rcpp::export]]
arma::field<arma::mat> gmwm_master_cpp(arma::vec& data,
arma::vec theta,
const std::vector<std::string>& desc, const arma::field<arma::vec>& objdesc,
std::string model_type, bool starting,
double alpha,
std::string compute_v, unsigned int K, unsigned int H,
unsigned int G,
bool robust, double eff){
// Obtain counts of the different models we need to work with
std::map<std::string, int> models = count_models(desc);
// HACK METHOD (todo: formalize it)
// Determine if we need to difference
if(models["SARIMA"] > 0){
// Note: s, i, si are 6,7,8 => 5,6,7
for(unsigned int i = 0; i < desc.size(); i ++){
if(objdesc(i).n_elem > 3){
arma::vec sarima_desc = objdesc(i);
// Do we need to difference?
if(sarima_desc(6) > 0 || sarima_desc(7) > 0){
// Perform differencing in specific order...
// First non-seasonal and then seasonal.
if(sarima_desc(6) > 0){
// Lag is always 1, number of differences is (i)
data = diff_cpp(data, 1, sarima_desc(6));
}
if(sarima_desc(7) > 0){
// Lag is always seasonality (s) and differences is (si).
data = diff_cpp(data, sarima_desc(5), sarima_desc(7));
}
// Kill loop. We only handle the tsmodel object with the first difference.
break;
}
}
}
}
// ------ Variable Declarations
// Length of the Time Series
unsigned int N = data.n_elem;
// Number of Scales (J)
unsigned int nlevels = floor(log2(N));
// Number of parameters
unsigned int np = theta.n_elem;
// Take the mean of the first difference
double expect_diff = mean_diff(data);
// Guessed values of Theta (user supplied or generated)
arma::vec guessed_theta = theta;
// MODWT decomp
arma::field<arma::vec> modwt_decomp = modwt_cpp(data, "haar", nlevels, "periodic", true);
// Obtain WV and confidence intervals
arma::mat wvar = wvar_cpp(modwt_decomp, robust, eff, alpha, "eta3");
// Extract
arma::vec wv_empir = wvar.col(0);
arma::vec ci_lo = wvar.col(1);
arma::vec ci_hi = wvar.col(2);
//-------------------------
// Obtain Covariance Matrix
//-------------------------
arma::mat V;
// compute_cov_cpp is the hard core function. It can only be improved by using parallelization.
if(compute_v == "diag" || compute_v == "full"){
arma::field<arma::mat> Vout = compute_cov_cpp(modwt_decomp, nlevels, compute_v, robust, eff);
if(robust){
V = Vout(1);
}else{
V = Vout(0);
}
}else{
V = fast_cov_cpp(wvar.col(2), wvar.col(1));
}
// Obtain the Omega matrix
arma::mat omega = arma::inv(diagmat(V));
// Store the original V matrix (in case of bootstrapping) for use in the update function
arma::mat orgV = V;
// Calculate the values of the Scales
arma::vec scales = scales_cpp(nlevels);
// Min-Max / N
double ranged = dr_slope(data);
// Guess starting values for the theta parameters
if(starting){
// Always run guessing algorithm
theta = guess_initial(desc, objdesc, model_type, np, expect_diff, N, wvar, scales, ranged, G);
// If under ARMA case and only ARMA is in the model,
// then see how well these values are.
if(desc.size() == 1 && (desc[0] == "SARIMA" || desc[0] == "ARMA11") && N <= 1000){
// Use R's ARIMA function to estimate parameter space
arma::vec theta2 = Rcpp_ARIMA(data, objdesc(0)); // Only 1 objdesc in the available.
// Obtain the obj function under omega with these initial guesses
// DO >>NOT<< USE Yannick's to optimize starting values!!!!
double mle_css_obj = getObjFun(theta2, desc, objdesc, model_type, omega, wv_empir, scales);
// Obtain the objective function under Yannick's starting algorithm
double init_guess_obj = getObjFunStarting(theta, desc, objdesc, model_type, wv_empir, scales);
// What performs better?
if(mle_css_obj < init_guess_obj){
// Disable starting value optimization if using MLE.
theta = theta2;
starting = false;
}
}
guessed_theta = theta;
}
// Obtain the GMWM estimator's estimates.
theta = gmwm_engine(theta, desc, objdesc, model_type,
wv_empir, omega, scales, starting);
// Optim may return a very small value. In this case, instead of saying its zero (yielding a transform issue), make it EPSILON.
theta = code_zero(theta);
// Enable bootstrapping
if(compute_v == "bootstrap"){
for(unsigned int k = 0; k < K; k++){
// Create the full V matrix
V = cov_bootstrapper(theta, desc, objdesc, N, robust, eff, H, false);
// Update the omega matrix
omega = arma::inv(diagmat(V));
// The theta update in this case MUST not use Yannick's starting algorithm. Hence, the false value.
theta = gmwm_engine(theta, desc, objdesc, model_type, wv_empir, omega, scales, false);
// Optim may return a very small value. In this case, instead of saying its zero (yielding a transform issue), make it EPSILON.
theta = code_zero(theta);
}
}
if(desc[0] == "SARIMA" && desc.size() == 1){
arma::vec temp = objdesc(0);
unsigned int p = temp(0);
if(p != 0 && invert_check(arma::join_cols(arma::ones<arma::vec>(1), -theta.rows(0, p - 1))) == false){
Rcpp::Rcout << "WARNING: This ARMA model contains AR coefficients that are NON-STATIONARY!" << std::endl;
}
}
// Order AR1s / GM so largest phi is first!
if(models["AR1"] > 1 || models["GM"] > 1){
theta = order_AR1s(theta, desc, objdesc);
}
// Obtain the objective value function
arma::vec obj_value(1);
obj_value(0) = getObjFun(theta, desc, objdesc, model_type, omega, wv_empir, scales);
arma::vec dr_s(1);
dr_s(0) = ranged;
// Decomposition of the WV.
arma::mat decomp_theo = decomp_theoretical_wv(theta, desc, objdesc, scales);
arma::vec theo = decomp_to_theo_wv(decomp_theo);
// Export information back
arma::field<arma::mat> out(13);
out(0) = theta;
out(1) = guessed_theta;
out(2) = wv_empir;
out(3) = ci_lo;
out(4) = ci_hi;
out(5) = V;
out(6) = orgV;
out(7) = expect_diff;
out(8) = theo;
out(9) = decomp_theo;
out(10) = obj_value;
out(11) = omega;
out(12) = dr_s;
return out;
}
