Suppose we have a functional response variable \(y_m(t), \ m=1,\dots,M\), a functional covariate \(x_m(t)\) and also a set of \(p=2\) scalar covariates \(\textbf{u}_m = (u_{m0},u_{m1})^\top\).

A Gaussian process functional regression (GPFR) model used in this example is defined by

\(y_m(t) = \mu_m(t) + \tau_m(x_m(t)) + \varepsilon_m(t)\),

where \(\mu_m(t) = \textbf{u}_m^\top \boldsymbol{\beta}(t)\) is the mean function model across different curves and \(\tau_m(x_m(t))\) is a Gaussian process with zero mean and covariance function \(k_m(\boldsymbol{\theta}|x_m(t))\). That is, \(\tau_m(x_m(t))\) defines the covariance structure of \(y_m(t)\) for the different data points within the same curve.

The error term can be assumed to be \(\varepsilon_m(t) \sim N(0, \sigma_\varepsilon^2)\), where the noise variance \(\sigma_\varepsilon^2\) can be estimated as a hyperparameter of the Gaussian process.

In the example below, the training data consist of \(M=20\) realisations on \([-4,4]\) with \(n=50\) points for each curve. We assume regression coefficient functions \(\beta_0(t)=1\), \(\beta_1(t)=\sin((0.5 t)^3)\), scalar covariates \(u_{m0} \sim N(0,1)\) and \(u_{m1} \sim N(10,5^2)\) and a functional covariate \(x_m(t) = \exp(t) + v\), where \(v \sim N(0, 0.1^2)\). The term \(\tau_m(x_m(t))\) is a zero mean Gaussian process with exponential covariance kernel and \(\sigma_\varepsilon^2 = 1\).

We also simulate an \((M+1)\)th realisation which is used to assess predictions obtained by the model estimated by using the training data of size \(M\). The \(y_{M+1}(t)\) and \(x_{M+1}(t)\) curves are observed on equally spaced \(60\) time points on \([-4,4]\).

```
set.seed(100)
M <- 20
n <- 50
p <- 2 # number of scalar covariates
hp <- list('pow.ex.v'=log(10), 'pow.ex.w'=log(1),'vv'=log(1))
## Training data: M realisations -----------------
tt <- seq(-4,4,len=n)
b <- sin((0.5*tt)^3)
scalar_train <- matrix(NA, M, p)
t_train <- matrix(NA, M, n)
x_train <- matrix(NA, M, n)
response_train <- matrix(NA, M, n)
for(i in 1:M){
u0 <- rnorm(1)
u1 <- rnorm(1,10,5)
x <- exp(tt) + rnorm(n, 0, 0.1)
Sigma <- cov.pow.ex(hyper = hp, input = x, gamma = 1)
diag(Sigma) <- diag(Sigma) + exp(hp$vv)
y <- u0+u1*b + mvrnorm(n=1, mu=rep(0,n), Sigma=Sigma)
scalar_train[i,] <- c(u0,u1)
t_train[i,] <- tt
x_train[i,] <- x
response_train[i,] <- y
}
## Test data (M+1)-th realisation ------------------
n_new <- 60
t_new <- seq(-4,4,len=n_new)
b_new <- sin((0.5*t_new)^3)
u0_new <- rnorm(1)
u1_new <- rnorm(1,10,5)
scalar_new <- cbind(u0_new, u1_new)
x_new <- exp(t_new) + rnorm(n_new, 0, 0.1)
Sigma_new <- cov.pow.ex(hyper = hp, input = x_new, gamma = 1)
diag(Sigma_new) <- diag(Sigma_new) + exp(hp$vv)
response_new <- u0_new + u1_new*b_new + mvrnorm(n=1, mu=rep(0,n_new),
Sigma=Sigma_new)
```

The estimation of mean and covariance functions in the GPFR model is done using the `gpfr`

function:

```
a1 <- gpfr(response = response_train, time = tt, uReg = scalar_train,
fxReg = NULL, gpReg = x_train,
fyList = list(nbasis = 23, lambda = 0.0001),
uCoefList = list(list(lambda = 0.0001, nbasi = 23)),
Cov = 'pow.ex', gamma = 1, fitting = T)
```

Note that the estimated covariance function hyperparameters are similar to the true values:

To visualise all the realisations of the training data:

To visualise three realisations of the training data:

The in-sample fit using mean function from FR model only can be seen:

The GPFR model fit to the training data is visualised by using:

If \(y_{M+1}(t)\) is observed over all the domain of \(t\), the Type I prediction can be seen:

```
b1 <- gpfrPredict(a1, testInputGP = x_new, testTime = t_new,
uReg = scalar_new, fxReg = NULL,
gpReg = list('response' = response_new,
'input' = x_new,
'time' = t_new))
plot(b1, type = 'prediction', colourTrain = 'pink')
lines(t_new, response_new, type = 'b', col = 4, pch = 19, cex = 0.6, lty = 3, lwd = 2)
```

If we assume that \(y_{M+1}(t)\) is only partially observed, we can obtain Type I predictions via:

```
b2 <- gpfrPredict(a1, testInputGP = x_new, testTime = t_new,
uReg = scalar_new, fxReg = NULL,
gpReg = list('response' = response_new[1:20],
'input' = x_new[1:20],
'time' = t_new[1:20]))
plot(b2, type = 'prediction', colourTrain = 'pink')
lines(t_new, response_new, type = 'b', col = 4, pch = 19, cex = 0.6, lty = 3, lwd = 2)
```

Type II prediction, which is made by not including any information about \(y_{M+1}(t)\), is visualised below.