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MCMC.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] x3;
vector[N] x4;
vector[N] x5;
vector[N] x6;
vector[N] y;
int<lower=1> M;
matrix[M,1] x_new1;
matrix[M,1] x_new2;
matrix[M,1] x_new3;
matrix[M,1] x_new4;
matrix[M,1] x_new5;
matrix[M,1] x_new6;
}
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,N] K_gram3 = sq_bm_K_gram(N,x3);
matrix[N,N] K_gram4 = sq_bm_K_gram(N,x4);
matrix[N,N] K_gram5 = sq_bm_K_gram(N,x5);
matrix[N,N] K_gram6 = sq_bm_K_gram(N,x6);
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[N,M] k_vec3 = sq_bm_k_vec(N, x3, x_new3[,1]);
matrix[N,M] k_vec4 = sq_bm_k_vec(N, x4, x_new4[,1]);
matrix[N,M] k_vec5 = sq_bm_k_vec(N, x5, x_new5[,1]);
matrix[N,M] k_vec6 = sq_bm_k_vec(N, x6, x_new6[,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]);
matrix[M,M] k_star3 = sq_bm_k_star(N,x3, x_new3[,1]);
matrix[M,M] k_star4 = sq_bm_k_star(N,x4, x_new4[,1]);
matrix[M,M] k_star5 = sq_bm_k_star(N,x5, x_new5[,1]);
matrix[M,M] k_star6 = sq_bm_k_star(N,x6, x_new6[,1]);
}
parameters {
real<lower=0> sigma;
real<lower=0> lambda_1;
real<lower=0> lambda_2;
//real<lower=0> lambda_3;
//real<lower=0> lambda_4;
//real<lower=0> lambda_5;
//real<lower=0> lambda_6;
real<lower=0> alpha0;
}
model{
matrix[N,N] K = square(alpha0)*(rep_matrix(1,N,N)+lambda_1*K_gram1+lambda_2*K_gram2);//+lambda_3*K_gram3+lambda_4*K_gram4+lambda_5*K_gram5);
// 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 += std_normal_lpdf(alpha0);
target += std_normal_lpdf(lambda_1);
target += std_normal_lpdf(lambda_2);
//target += std_normal_lpdf(lambda_3);
//target += std_normal_lpdf(lambda_4);
//target += std_normal_lpdf(lambda_5);
//target += std_normal_lpdf(lambda_6);
target += std_normal_lpdf(sigma);
//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_zero;
// vector[M] f_one;
// vector[M] f_two;
// vector[M] f_three;
// vector[N] f_four;
// vector[N] f_five;
// vector[N] f;
{
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);//+lambda_3*K_gram3+lambda_4*K_gram4+lambda_5*K_gram5);
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);//+lambda_3*k_vec3+lambda_4*k_vec4+lambda_5*k_vec5);
k_star_eval = square(alpha0)*(rep_matrix(1,M,M)+lambda_1*k_star1+lambda_2*k_star2);//+lambda_3*k_star3+lambda_4*k_star4+lambda_5*k_star5);
//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_zero = square(alpha0)*sum(alpha_eval)*rep_vector(1,N);
// f_one =square(alpha0)*lambda_1*K_gram1*alpha_eval;//K_gram1*alpha_eval
// f_two =square(alpha0)*lambda_2*K_gram2*alpha_eval;
// f_three =square(alpha0)*lambda_3*K_gram3*alpha_eval;
// f_four =square(alpha0)*lambda_4*K_gram4*alpha_eval;
// f_five =square(alpha0)*lambda_5*K_gram5*alpha_eval;
// f = square(alpha0)*(rep_matrix(1,N,N)+lambda_1*K_gram1+lambda_2*K_gram2+lambda_3*K_gram3+lambda_4*K_gram4+lambda_5*K_gram5)*alpha_eval;
}
}