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Copy pathMCMC pred.stan
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MCMC pred.stan
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functions{
// BM kernel
real kern_bm(
real x_i,
real y_i){
real result=0.5 *(abs(x_i)+abs(y_i)-abs(x_i-y_i));
return result;
}
// gram matrix
matrix sq_bm_K_gram(int N,vector x){
matrix[N,N] K_gram;
matrix[N,N] A;
matrix[N,N] result;
matrix[N,N] C = diag_matrix(rep_vector(1.0, N)) - (1.0 / N) * rep_matrix(1.0, N, N);
for (i in 1:N) {
for (j in 1:N) {
K_gram[i, j] = kern_bm(x[i], x[j]);
}
}
A = C * K_gram * C;
result = A*A;
return result;
}
// kernel vector
matrix sq_bm_k_vec(int N, vector x, vector x_new){
matrix[N,N] K_gram;
matrix[N,N] C = diag_matrix(rep_vector(1.0, N)) - (1.0 / N) * rep_matrix(1, N, N);
matrix[N,N] K_cent;
int M = num_elements(x_new);
matrix[N,M] k_vec;
matrix[N,M] centered;
matrix[N,M] result;
for (i in 1:N) {
for (j in 1:N) {
K_gram[i, j] = kern_bm(x[i], x[j]);
}
}
K_cent = C' * K_gram * C;
for (i in 1:N) {
for (j in 1:M) {
k_vec[i, j] = kern_bm(x[i], x_new[j]);
}
}
{
vector[N] first_element = K_gram * rep_vector(1.0,N);
row_vector[M] second_element = rep_row_vector(1.0,N)*k_vec;
real third_element = rep_row_vector(1.0,N) * K_gram * rep_vector(1.0,N);
centered = k_vec - (1.0/N * first_element * rep_row_vector(1.0,M))- (1.0/N * rep_vector(1.0,N)*second_element)+ (1.0/(square(N)) * third_element * rep_matrix(1, N, M));
result = K_cent * centered;
}
return result;
}
matrix sq_bm_k_star(int N, vector x, vector x_new){
matrix[N,N] K_gram;
matrix[N,N] C = diag_matrix(rep_vector(1.0, N)) - (1.0 / N) * rep_matrix(1, N, N);
matrix[N,N] K_cent;
int M = num_elements(x_new);
matrix[N,M] k_vec;
matrix[N,M] centered;
matrix[M,M] result;
for (i in 1:N) {
for (j in 1:N) {
K_gram[i, j] = kern_bm(x[i], x[j]);
}
}
K_cent = C' * K_gram * C;
for (i in 1:N) {
for (j in 1:M) {
k_vec[i, j] = kern_bm(x[i], x_new[j]);
}
}
{
vector[N] first_element = K_gram * rep_vector(1.0,N);
row_vector[M] second_element = rep_row_vector(1.0,N)*k_vec;
real third_element = rep_row_vector(1.0,N) * K_gram * rep_vector(1.0,N);
centered = k_vec - (1.0/N * first_element * rep_row_vector(1.0,M))- (1.0/N * rep_vector(1.0,N)*second_element)+ (1.0/(square(N)) * third_element * rep_matrix(1, N, M));
result = centered' * centered;
}
return result;
}
}
data {
int<lower=1> N;
vector[N] x1;
vector[N] x2;
vector[N] y;
int<lower=1> M;
matrix[M,1] x_new1;
matrix[M,1] x_new2;
}
transformed data {
vector[N] mu = rep_vector(0, N);
matrix[N,N] K_gram1 = sq_bm_K_gram(N,x1);
matrix[N,N] K_gram2 = sq_bm_K_gram(N,x2);
matrix[N,M] k_vec1 = sq_bm_k_vec(N, x1, x_new1[,1]);
matrix[N,M] k_vec2 = sq_bm_k_vec(N, x2, x_new2[,1]);
matrix[M,M] k_star1 = sq_bm_k_star(N,x1, x_new1[,1]);
matrix[M,M] k_star2 = sq_bm_k_star(N,x2, x_new2[,1]);
}
parameters {
real<lower=0> sigma;
real<lower=0> lambda_1;
real<lower=0> lambda_2;
real<lower=0> alpha0;
}
model{
matrix[N,N] K = square(alpha0)*(rep_matrix(1,N,N)+lambda_1*K_gram1+lambda_2*K_gram2);
//matrix[N,N] capital_sigma = K + square(sigma) * diag_matrix(rep_vector(1, N));
// diagonal elements
matrix[N,N] L;
for (n in 1:N)
K[n, n] = K[n, n] + square(sigma);
L=cholesky_decompose(K);
//prior
// sigma ~ normal(0, 1);
// lambda_1 ~ normal(0, 1);
// lambda_2 ~ normal(0, 1);
// alpha0 ~ normal(0, 1);
target += exponential_lpdf(alpha0| 2);
target += exponential_lpdf(lambda_1| 2);
target += exponential_lpdf(lambda_2| 2);
target += exponential_lpdf(sigma| 2);
//likelihood
//target += -0.5 * y'*inverse(K)*y -sum(log(diagonal(L)))-N/2.0*log(2.0*pi());//y'*mdivide_left_tri(capital_sigma,y)
//target += multi_normal_lpdf(y | mu, capital_sigma); // they should be the same
//y ~ multi_normal_cholesky(mu, L);
target += multi_normal_cholesky_lpdf(y | mu, L);
}
generated quantities{
real mloglik;
vector[M] mu_predicted;
matrix[M,M] var_predicted;
vector[N] f_one;
vector[N] f_two;
{
matrix[N,N] K_eval;
matrix[N,N] L_eval;
matrix[N,M] k_vec_eval;
matrix[M,M] k_star_eval;
vector[N] alpha_eval;
matrix[N,M] v;
K_eval = square(alpha0)*(rep_matrix(1,N,N)+lambda_1*K_gram1+lambda_2*K_gram2);
for (n in 1:N)
K_eval[n, n] = K_eval[n, n] + square(sigma);
//matrix[N,N] capital_sigma_eval = (K_eval + square(sigma) * diag_matrix(rep_vector(1, N)));
L_eval=cholesky_decompose(K_eval);
k_vec_eval = square(alpha0)*(rep_matrix(1,N,M)+lambda_1*k_vec1+lambda_2*k_vec2);
k_star_eval = square(alpha0)*(rep_matrix(1,M,M)+lambda_1*k_star1+lambda_2*k_star2);
//Rasmussen
alpha_eval=L_eval'\(L_eval\y);
v = L_eval\k_vec_eval;
mu_predicted = k_vec_eval'*alpha_eval;
var_predicted = k_star_eval - v'*v;
mloglik = -0.5 * y'*alpha_eval - sum(log(diagonal(L_eval)))- N/2.0*log(2.0*pi());
f_one =square(alpha0)*lambda_1*K_gram1*alpha_eval;
f_two =square(alpha0)*lambda_2*K_gram2*alpha_eval;
}
}