This function computes the prediction of a new response from the estimation of a SLLiM model, returned by the function sllim.
Indeed, if the inverse conditional density \(p(X | Y)\) and the marginal density \(p(Y)\) are defined according to a SLLiM model (as described in xLLiM-package and sllim), the forward conditional density \(p(Y | X)\) can be deduced.
Under SLLiM model, it is recalled that the inverse conditional \(p(X | Y)\) is a mixture of Student regressions with parameters \((c_k,\Gamma_k,A_k,b_k,\Sigma_k)_{k=1}^K\) and \((\pi_k,\alpha_k)_{k=1}^K\). Interestingly, the forward conditional \(p(Y | X)\) is also a mixture of Student regressions with parameters \((c_k^*,\Gamma_k^*,A_k^*,b_k^*,\Sigma_k^*)_{k=1}^K\) and \((\pi_k,\alpha_k)_{k=1}^K\). These parameters have a closed-form expression depending only on \((c_k,\Gamma_k,A_k,b_k,\Sigma_k)_{k=1}^K\) and \((\pi_k,\alpha_k)_{k=1}^K\).
Finally, the forward density (of interest) has the following expression:
$$p(Y | X=x) = \sum_k \frac{\pi_k S(x; c_k^*,\Gamma_k^*,\alpha_k,1)}{\sum_j \pi_j S(x; c_j^*,\Gamma_j^*,\alpha_j,1)} S(y; A_k^*x + b_k^*,\Sigma_k^*,\alpha_k^y,\gamma_k^y)$$
where \((\alpha_k^y,\gamma_k^y)\) determine the heaviness of the tail of the Generalized Student distribution.
Note that \(\alpha_k^y= \alpha_k + D/2\) and \(\gamma_k^y= 1 + 1/2 \delta(x,c_k^*,\Gamma_k^*)\) where \(\delta\) is the Mahalanobis distance. A prediction of a new vector of responses is computed by:
$$E (Y | X=x) = \sum_k \frac{\pi_k S(x; c_k^*,\Gamma_k^*,\alpha_k,1)}{\sum_j \pi_j S(x; c_j^*,\Gamma_j^*,\alpha_j,1)} (A_k^*x + b_k^*)$$
where \(x\) is a new vector of observed covariates.