sllim_inverse_map {xLLiM}

Inverse Mapping from sllim parameters

Description

This function computes the prediction of a new response from the estimation of the SLLiM model, returned by the function sllim.

Usage

sllim_inverse_map(y,theta,verb=0)

Arguments

y

An D x N matrix of input observations with variables in rows and subjects on columns

theta

An object returned by the sllim function

verb

Verbosity: print out the progression of the algorithm. If verb=0, there is no print, if verb=1, the progression is printed out. Default is 0.

Details

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.

Value

Returns a list with the following elements:

x_exp

An L x N matrix of predicted responses by posterior mean. If L_w latent factors are added to the model, the first Lt rows (1:Lt) are predictions of responses and rows (L_t+1):L (recall that L=L_t+L_w) are estimations of latent factors.

alpha

Weights of the posterior Gaussian mixture model

Author(s)

Emeline Perthame ([e-mail]), Florence Forbes ([e-mail]), Antoine Deleforge ([e-mail])

References

[1] A. Deleforge, F. Forbes, and R. Horaud. High-dimensional regression with Gaussian mixtures and partially-latent response variables. Statistics and Computing, 25(5):893–911, 2015.

[2] E. Perthame, F. Forbes, and A. Deleforge. Inverse regression approach to robust nonlinear high-to-low dimensional mapping. Journal of Multivariate Analysis, 163(C):1–14, 2018. https://doi.org/10.1016/j.jmva.2017.09.009

See Also

xLLiM-package,sllim

Examples

data(data.xllim)

## Setting 5 components in the model
K = 5

## the model can be initialized by running an EM algorithm for Gaussian Mixtures (EMGM)
r = emgm(data.xllim, init=K); 
## and then the sllim model is estimated
responses = data.xllim[1:2,] # 2 responses in rows and 100 observations in columns
covariates = data.xllim[3:52,] # 50 covariates in rows and 100 observations in columns
mod = sllim(responses,covariates,in_K=K,in_r=r);

# Prediction on a test dataset
data(data.xllim.test)
pred = sllim_inverse_map(data.xllim.test,mod)
## Predicted responses
print(pred$x_exp)