The functional single-index model (FSIM) is given by the expression:
$$Y_i=r(\langle\theta_0,X_i\rangle)+\varepsilon_i, \quad i=1,\dots,n,$$
where \(Y_i\) denotes a scalar response,
\(X_i\) is a functional covariate valued in a separable Hilbert space \(\mathcal{H}\) with an inner product \(\langle \cdot, \cdot\rangle\). The term \(\varepsilon\) denotes the random error, \(\theta_0 \in \mathcal{H}\) is the unknown functional index and \(r(\cdot)\) denotes the unknown smooth link function; \(n\) is the training sample size.
Given \(\theta \in \mathcal{H}\), \(h>0\) and a testing sample {\(X_j,\ j=1,\dots,n_{test}\)}, the predicted responses (see the value y.estimated.test) can be computed using the kernel procedure using
$$
\widehat{r}_{h,\theta}(X_j)=\sum_{i=1}^nw_{n,h,\theta}(X_j,X_i)Y_i,\quad j=1,\dots,n_{test},
$$
with Nadaraya-Watson weights
$$
w_{n,h,\theta}(X_j,X_i)=\frac{K\left(h^{-1}d_{\theta}\left(X_i,X_j\right)\right)}{\sum_{i=1}^nK\left(h^{-1}d_{\theta}\left(X_i,X_j\right)\right)},
$$
where
\(K\) is a kernel function (see the argument kind.of.kernel).
for \(x_1,x_2 \in \mathcal{H}, \) \(d_{\theta}(x_1,x_2)=|\langle\theta,x_1-x_2\rangle|\) is the projection semi-metric.
If the argument y.test is provided to the program (i. e. if(!is.null(y.test))), the function calculates the mean squared error of prediction (see the value MSE.test). This is computed as mean((y.test-y.estimated.test)^2).