Compute the posterior distribution of a standard GP, using the formalism of Magma. By providing observed data, the prior mean and covariance matrix (by defining a kernel and its associated hyper-parameters), the mean and covariance parameters of the posterior distribution are computed on the grid of inputs that has been specified. This predictive distribution can be evaluated on any arbitrary inputs since a GP is an infinite-dimensional object.

## Usage

```
pred_gp(
data,
grid_inputs = NULL,
mean = NULL,
hp = NULL,
kern = "SE",
get_full_cov = FALSE,
plot = TRUE,
pen_diag = 1e-10
)
```

## Arguments

- data
A tibble or data frame. Required columns: 'Input', 'Output'. Additional columns for covariates can be specified. The 'Input' column should define the variable that is used as reference for the observations (e.g. time for longitudinal data). The 'Output' column specifies the observed values (the response variable). The data frame can also provide as many covariates as desired, with no constraints on the column names. These covariates are additional inputs (explanatory variables) of the models that are also observed at each reference 'Input'.

- grid_inputs
The grid of inputs (reference Input and covariates) values on which the GP should be evaluated. Ideally, this argument should be a tibble or a data frame, providing the same columns as

`data`

, except 'Output'. Nonetheless, in cases where`data`

provides only one 'Input' column, the`grid_inputs`

argument can be NULL (default) or a vector. This vector would be used as reference input for prediction and if NULL, a vector of length 500 is defined, ranging between the min and max Input values of`data`

.- mean
Mean parameter of the GP. This argument can be specified under various formats, such as:

NULL (default). The mean would be set to 0 everywhere.

A number. The mean would be a constant function.

A function. This function is defined as the mean.

A tibble or data frame. Required columns: Input, Output. The Input values should include at least the same values as in the

`data`

argument.

- hp
A named vector, tibble or data frame of hyper-parameters associated with

`kern`

. The columns/elements should be named according to the hyper-parameters that are used in`kern`

. If NULL (default), the function`train_gp`

is called with random initial values for learning maximum-likelihood estimators of the hyper-parameters associated with`kern`

.- kern
A kernel function, defining the covariance structure of the GP. Several popular kernels (see The Kernel Cookbook) are already implemented and can be selected within the following list:

"SE": (default value) the Squared Exponential Kernel (also called Radial Basis Function or Gaussian kernel),

"LIN": the Linear kernel,

"PERIO": the Periodic kernel,

"RQ": the Rational Quadratic kernel. Compound kernels can be created as sums or products of the above kernels. For combining kernels, simply provide a formula as a character string where elements are separated by whitespaces (e.g. "SE + PERIO"). As the elements are treated sequentially from the left to the right, the product operator '*' shall always be used before the '+' operators (e.g. 'SE * LIN + RQ' is valid whereas 'RQ + SE * LIN' is not).

- get_full_cov
A logical value, indicating whether the full posterior covariance matrix should be returned.

- plot
A logical value, indicating whether a plot of the results is automatically displayed.

- pen_diag
A number. A jitter term, added on the diagonal to prevent numerical issues when inverting nearly singular matrices.

## Value

A tibble, representing the GP predictions as two column 'Mean' and
'Var', evaluated on the `grid_inputs`

. The column 'Input' and
additional covariates columns are associated to each predicted values.
If the `get_full_cov`

argument is TRUE, the function returns a list,
in which the tibble described above is defined as 'pred' and the full
posterior covariance matrix is defined as 'cov'.