View source: R/sllim_inverse_map.R
sllim_inverse_map | R Documentation |
This function computes the prediction of a new response from the estimation of the SLLiM model, returned by the function sllim
.
sllim_inverse_map(y,theta,verb=0)
y |
An |
theta |
An object returned by the |
verb |
Verbosity: print out the progression of the algorithm. If |
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.
Returns a list with the following elements:
x_exp |
An |
alpha |
Weights of the posterior Gaussian mixture model |
Emeline Perthame (emeline.perthame@inria.fr), Florence Forbes (florence.forbes@inria.fr), Antoine Deleforge (antoine.deleforge@inria.fr)
[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
xLLiM-package
,sllim
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)
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