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5.12 Covariance Functions

5.12.1 Exponentiated quadratic covariance function

The exponentiated quadratic kernel defines the covariance between $f(x_i)$ and $f(x_j)$ where $f\colon \mathbb{R}^D \mapsto \mathbb{R}$ as a function of the squared Euclidian distance between $x_i \in \mathbb{R}^D$ and $x_j \in \mathbb{R}^D$: $$\text{cov}(f(x_i), f(x_j)) = k(x_i, x_j) = \alpha^2 \exp \left( - \dfrac{1}{2\rho^2} \sum_{d=1}^D (x_{i,d} - x_{j,d})^2 \right)$$ with $\alpha$ and $\rho$ constrained to be positive.

There are two variants of the exponentiated quadratic covariance function in Stan. One builds a covariance matrix, $K \in \mathbb{R}^{N \times N}$ for $x_1, \dots, x_N$, where $K_{i,j} = k(x_i, x_j)$, which is necessarily symmetric and positive semidefinite by construction. There is a second variant of the exponentiated quadratic covariance function that builds a $K \in \mathbb{R}^{N \times M}$ covariance matrix for $x_1, \dots, x_N$ and $x^\prime_1, \dots, x^\prime_M$, where $x_i \in \mathbb{R}^D$ and $x^\prime_i \in \mathbb{R}^D$ and $K_{i,j} = k(x_i, x^\prime_j)$.

matrix cov_exp_quad(row_vectors x, real alpha, real rho)
The covariance matrix with an exponentiated quadratic kernel of x.

matrix cov_exp_quad(vectors x, real alpha, real rho)
The covariance matrix with an exponentiated quadratic kernel of x.

matrix cov_exp_quad(real[] x, real alpha, real rho)
The covariance matrix with an exponentiated quadratic kernel of x.

matrix cov_exp_quad(row_vectors x1, row_vectors x2, real alpha, real rho)
The covariance matrix with an exponentiated quadratic kernel of x1 and x2.

matrix cov_exp_quad(vectors x1, vectors x2, real alpha, real rho)
The covariance matrix with an exponentiated quadratic kernel of x1 and x2.

matrix cov_exp_quad(real[] x1, real[] x2, real alpha, real rho)
The covariance matrix with an exponentiated quadratic kernel of x1 and x2.