斯坦协方差矩阵

Question

我正在试验斯坦和高斯过程。经过一些错误之后，我发现所有事情的根源都是函数cov_exp_quad的奇怪行为。

特别是，我不明白为什么它返回的矩阵是不对称的。

在这里，斯坦脚本：

data {
  int<lower=1> N;
  real x[N];
  real y[N];
}

transformed data {
  matrix[N,N] K = cov_exp_quad(x, y, 1, 1);
  }

generated quantities {
  matrix[N, N] Cov = K;
}

在这里，R代码：

library(rstan)

x <- seq(0, 1, length.out = 3)

X <- expand.grid(x, x)

data_stan <- list(N = dim(X)[1],
                  x = X[, 1],
                  y = X[, 2])

fit <- stan(file = "./stan_script.stan",
            data = data_stan,
            algorithm = "Fixed_param",
            warmup = 0,
            chains = 1,
            iter = 1)

samples <- rstan::extract(fit)

drop(samples$Cov)

这是，而不是输出

> drop(samples$Cov)
           [,1]      [,2]      [,3]      [,4]      [,5]      [,6]      [,7]      [,8]      [,9]
 [1,] 1.0000000 1.0000000 1.0000000 0.8824969 0.8824969 0.8824969 0.6065307 0.6065307 0.6065307
 [2,] 0.8824969 0.8824969 0.8824969 1.0000000 1.0000000 1.0000000 0.8824969 0.8824969 0.8824969
 [3,] 0.6065307 0.6065307 0.6065307 0.8824969 0.8824969 0.8824969 1.0000000 1.0000000 1.0000000
 [4,] 1.0000000 1.0000000 1.0000000 0.8824969 0.8824969 0.8824969 0.6065307 0.6065307 0.6065307
 [5,] 0.8824969 0.8824969 0.8824969 1.0000000 1.0000000 1.0000000 0.8824969 0.8824969 0.8824969
 [6,] 0.6065307 0.6065307 0.6065307 0.8824969 0.8824969 0.8824969 1.0000000 1.0000000 1.0000000
 [7,] 1.0000000 1.0000000 1.0000000 0.8824969 0.8824969 0.8824969 0.6065307 0.6065307 0.6065307
 [8,] 0.8824969 0.8824969 0.8824969 1.0000000 1.0000000 1.0000000 0.8824969 0.8824969 0.8824969
 [9,] 0.6065307 0.6065307 0.6065307 0.8824969 0.8824969 0.8824969 1.0000000 1.0000000 1.0000000

为什么这个不对称？谢谢

Answer 1

它用每个向量的元素之间的距离来构造一个协方差矩阵，而不是使用[x,y]中点之间的欧几里德距离。例如：

#Squared distance between first element of x and all elements of y
(distSq <- (data_stan$x[1] - data_stan$y)^2) 

#Function to create covariance matrix 
#Taken from https://mc-stan.org/docs/2_22/functions-reference/covariance.html
cov_exp_quad <- function(alpha,rho,dist2) (alpha^2)*exp((-1/2*(rho^2))*dist2)

#Compare to first row of covariance matrix from Stan - identical
cbind(samples$Cov[1,1,],cov_exp_quad(1,1,distSq))