Merge pull request #272 from stan-dev/feature/gp_function_reference_docs

gp function reference docs (Issue #185)

Author: rok-cesnovar

src/functions-reference/deprecated_functions.Rmd

@@ -9,7 +9,13 @@ if (knitr::is_html_output()) {
 }
 ```
 
-## integrate_ode_rk45, integrate_ode_adams, integrate_ode_bdf ODE integrators {#functions-old-ode-solver}
+```{r results='asis', echo=FALSE}
+if (knitr::is_html_output()) {
+  cat(' * <a href="cov_exp_quad.html">cov_exp_quad</a>\n')
+}
+```
+
+## integrate_ode_rk45, integrate_ode_adams, integrate_ode_bdf ODE Integrators {#functions-old-ode-solver}
 
 These ODE integrator functions have been replaced by those described in:
 
@@ -24,7 +30,7 @@ if (knitr::is_html_output()) {
 A system of ODEs is specified as an ordinary function in Stan within
 the functions block. The ODE system function must have this function
 signature:
-  
+
 ```stan
 array[] real ode(real time, array[] real state, array[] real theta,
                  array[] real x_r, array[] int x_i);
@@ -117,7 +123,7 @@ are as follows.
 
 *   *`ode`*: function literal referring to a function specifying   the
 system of differential equations with signature:
-  
+
 ```
 (real, array[] real, array[] real, data array[] real, data array[] int):array[] real
 ```
@@ -140,7 +146,7 @@ derivatives with respect to time of the state,
 
 For more fine-grained control of the ODE solvers, these parameters can
 also be provided:
-  
+
   *   `data`   *`rel_tol`*: relative tolerance for the ODE  solver, type
 `real`, data only,
 
@@ -159,7 +165,7 @@ with values consisting of solutions at the specified times.
 
 The sizes must match, and in particular, the following groups are of
 the same size:
-  
+
   *   state variables passed into the system function,  derivatives
 returned by the system function, initial state passed  into the
 solver, and rows of the return value of the solver,
@@ -169,3 +175,65 @@ solver,
 
 *   parameters, real data and integer data passed to the solver will
 be passed to the system function
+
+
+## Exponentiated quadratic covariance functions {#cov_exp_quad}
+
+These covariance functions have been replaced by those described in:
+
+```{r results='asis', echo=FALSE}
+if (knitr::is_html_output()) {
+  cat(' * <a href="gaussian-process-covariance-functions.html">Gaussian Process Covariance Functions</a>\n')
+}
+```
+
+With magnitude $\alpha$ and length scale $l$, the exponentiated quadratic kernel is:
+
+$$
+k(x_i, x_j) = \alpha^2 \exp \left(-\dfrac{1}{2\rho^2} \sum_{d=1}^D (x_{i,d} - x_{j,d})^2 \right)
+$$
+
+<!-- matrix; cov_exp_quad; (row_vectors x, real alpha, real rho); -->
+\index{{\tt \bfseries cov\_exp\_quad }!{\tt (row\_vectors x, real alpha, real rho): matrix}|hyperpage}
+
+`matrix` **`cov_exp_quad`**`(row_vectors x, real alpha, real rho)`<br>\newline
+The covariance matrix with an exponentiated quadratic kernel of x.
+`r since("2.16")`
+
+<!-- matrix; cov_exp_quad; (vectors x, real alpha, real rho); -->
+\index{{\tt \bfseries cov\_exp\_quad }!{\tt (vectors x, real alpha, real rho): matrix}|hyperpage}
+
+`matrix` **`cov_exp_quad`**`(vectors x, real alpha, real rho)`<br>\newline
+The covariance matrix with an exponentiated quadratic kernel of x.
+`r since("2.16")`
+
+<!-- matrix; cov_exp_quad; (array[] real x, real alpha, real rho); -->
+\index{{\tt \bfseries cov\_exp\_quad }!{\tt (array[] real x, real alpha, real rho): matrix}|hyperpage}
+
+`matrix` **`cov_exp_quad`**`(array[] real x, real alpha, real rho)`<br>\newline
+The covariance matrix with an exponentiated quadratic kernel of x.
+`r since("2.16")`
+
+<!-- matrix; cov_exp_quad; (row_vectors x1, row_vectors x2, real alpha, real rho); -->
+\index{{\tt \bfseries cov\_exp\_quad }!{\tt (row\_vectors x1, row\_vectors x2, real alpha, real rho): matrix}|hyperpage}
+
+`matrix` **`cov_exp_quad`**`(row_vectors x1, row_vectors x2, real alpha, real rho)`<br>\newline
+The covariance matrix with an exponentiated quadratic kernel of x1 and
+x2.
+`r since("2.18")`
+
+<!-- matrix; cov_exp_quad; (vectors x1, vectors x2, real alpha, real rho); -->
+\index{{\tt \bfseries cov\_exp\_quad }!{\tt (vectors x1, vectors x2, real alpha, real rho): matrix}|hyperpage}
+
+`matrix` **`cov_exp_quad`**`(vectors x1, vectors x2, real alpha, real rho)`<br>\newline
+The covariance matrix with an exponentiated quadratic kernel of x1 and
+x2.
+`r since("2.18")`
+
+<!-- matrix; cov_exp_quad; (array[] real x1, array[] real x2, real alpha, real rho); -->
+\index{{\tt \bfseries cov\_exp\_quad }!{\tt (array[] real x1, array[] real x2, real alpha, real rho): matrix}|hyperpage}
+
+`matrix` **`cov_exp_quad`**`(array[] real x1, array[] real x2, real alpha, real rho)`<br>\newline
+The covariance matrix with an exponentiated quadratic kernel of x1 and
+x2.
+`r since("2.18")`