Workflow summary

This is a very concise summary of the general workflow.

  • Without functional model, from a dataset:

    gllim = xllim.GLLiM(K, D, L, gamma_type="full", sigma_type="diag", n_hidden_variables=n_hidden_variables)
gllim.initialize(x_gen, y_gen, *initialisation args*)
gllim.train(x_gen, y_gen, *training args*)
predictions = gllim.inverseDensities(y_obs, y_obs_noise)

  • With functional model:

    model = xllim.TestModel()
x_gen, y_gen = model.genData(N, *args*)
gllim = xllim.GLLiM(K, D, L, gamma_type="full", sigma_type="diag", n_hidden_variables=n_hidden_variables)
gllim.initialize(x_gen, y_gen, *initialisation args*)
gllim.train(x_gen, y_gen, *training args*)
predictions = gllim.inverseDensities(y_obs, y_obs_noise)
is_results = model.importanceSampling(predictions.fullGMM, y_obs, y_obs_noise, N_0, *kwargs*)

Useful dimension parameters

Symbol

Parameter

Description

K

Number of Gaussians in GLLiM model

It corresponds to the number of affine transformations in the GLLiM model.

D

Output (Y) dimension

The dimension of the model output corresponds to the Y vector dimension such that the forward model is represented by Y=F(X).

L

Input (X) dimension

The dimension of the model output corresponds to the X vector dimension such that the forward model is represented by Y=F(X). It also represents the number of features composed of observed and latent variables such that L = L_t + L_w.

L_t

Observed input dimension

It corresponds to the number of observed features.

L_w

Latent input dimension

It corresponds to the number of unobserved/hidden features.

  • Lw = 0. This is the fully supervised case, and is equivalent to the mixture of local linear experts (MLE) model.

  • Lw = D. Σ takes the form of a general covariance matrix and we obtain the JGMM model. This is the most general GLLiM model, which requires the estimation of K full covariance matrices of size (D + L) × (D + L). This model becomes over-parameterized and intractable in high dimensions.

  • 0 < Lw < D. This corresponds to the hybrid GLLiM model, and yields a wide variety of novel regression models in between MLE and JGMM.

N

Dataset size

It is the number of pair (X,Y) in the dataset used to train the GLLiM model.

N_obs

Number of observation

It is related to the number of observations, usually the model output Y. It refers to the number of observed vector we want to apply the trained GLLiM model on.

Shape of all ndarrays

Name

Shape

Note

Pi

(K,)

GLLiMParameters.Pi

A

(K, D, L)

GLLiMParameters.A

B

(K, D)

GLLiMParameters.B

C

(K, L)

GLLiMParameters.C

Gamma

(K, L, L)

According to the covariance matrix type the shape can be adapted in order to not have useless zeros. (‘full’,)

  • In the case of Full covariance matrix (gamma_type = ‘full’), Gamma is of shape (K, L, L).

  • In the case of Diagonal covariance matrix (gamma_type = ‘diag’), Gamma is of shape (K, L) with Gamma[k] representing the variances vector of the k^{th} gaussian.

  • In the case of Isotropic covariance matrix (gamma_type = ‘iso’), Gamma is of shape (K) with Gamma[k] representing the unique variance of the k^{th} gaussian.

Sigma

(K, D, D)

Idem to Gamma.

X

(L, N)

It applies to X_gen, X_obs and X_pred with the corresponding N_gen, N_obs dimension. X_pred can be estimated by PredictionResult.mergedGMM.mean or ImportanceSamplingResult.predictions

Y

(D, N)

Idem to X.