## Expand description

This library implements Mixture of Experts method using GP models.

MoE method aims at increasing the accuracy of a function approximation by replacing a single global model by a weighted sum of local gp regression models (experts). It is based on a partition of the problem domain into several subdomains via clustering algorithms followed by a local expert training on each subdomain.

The recombination between the GP models can be either:

`hard`

: one GP model is being responsible to provide the predicted value at the given point. GP selection is done by taking the largest probability of the given point being part of the cluster corresponding to the expert GP. In hard mode, transition between models leads to discontinuity.`smooth`

: all GPs models are taken and their predicted values at a given point are weighted regarding their responsability (probability of the given point being part of the cluster corresponding to the expert GP). In this case the MoE model is continuous. The smoothness is automatically adjusted using a factor, the heaviside factor, which can also be set manually.

## §Implementation

- Clusters are defined by clustering the training data with linfa-clustering gaussian mixture model.
- This library is a port of the SMT MoE method using egobox GP models as experts.
- It leverages on the egobox GP PLS reduction feature to handle high dimensional problems.
- MoE trained model can be save to disk and reloaded. See

## §Features

### §serializable

The `serializable`

feature enables serialization based on serde crate.

### §persistent

The `persistent`

feature enables `save()`

/`load()`

methods for a MoE model
to/from a json file using the serde and serde_json crates.

## §Example

```
use ndarray::{Array2, Array1, Zip, Axis};
use egobox_moe::{GpMixture, Recombination};
use ndarray_rand::{RandomExt, rand::SeedableRng, rand_distr::Uniform};
use rand_xoshiro::Xoshiro256Plus;
use linfa::{traits::Fit, ParamGuard, Dataset};
// One-dimensional test function with 3 modes
fn f3modes(x: &Array2<f64>) -> Array2<f64> {
let mut y = Array2::zeros(x.dim());
Zip::from(&mut y).and(x).for_each(|yi, &xi| {
if xi < 0.4 {
*yi = xi * xi;
} else if (0.4..0.8).contains(&xi) {
*yi = 3. * xi + 1.;
} else {
*yi = f64::sin(10. * xi);
}
});
y
}
// Training data
let mut rng = Xoshiro256Plus::from_entropy();
let xt = Array2::random_using((50, 1), Uniform::new(0., 1.), &mut rng);
let yt = f3modes(&xt);
let ds = Dataset::new(xt, yt);
// Predictions
let observations = Array1::linspace(0., 1., 100).insert_axis(Axis(1));
let predictions = GpMixture::params()
.n_clusters(3)
.recombination(Recombination::Hard)
.fit(&ds)
.expect("MoE model training")
.predict(&observations)
.expect("MoE predictions");
```

## §Reference

Bettebghor, Dimitri, et al. Surrogate modeling approximation using a mixture of experts based on EM joint estimation Structural and multidisciplinary optimization 43.2 (2011): 243-259.

## Structs§

- A structure for clustering
- Flags to specify tested correlation models during experts selection (see
`correlation_spec()`

). - Gaussian mixture is a set of n weigthed multivariate normal distributions of dimension nx This structure is derived from
`linfa::GaussianMixtureModel`

clustering method to handle the resulting multivariate normals and related computations in one go. Moreover an`heaviside factor`

is handled in case of smooth Recombination to control the smoothness between clusters and can be adjusted afterwards - GP surrogate with
`Constant`

regression model and`AbsoluteExponential`

correlation model. - GP surrogate with
`Constant`

regression model and`Matern32`

correlation model. - GP surrogate with
`Constant`

regression model and`Matern52`

correlation model. - GP surrogate with
`Constant`

regression model and`SquaredExponential`

correlation model. - GP surrogate with
`Linear`

regression model and`AbsoluteExponential`

correlation model. - GP surrogate with
`Linear`

regression model and`Matern32`

correlation model. - GP surrogate with
`Linear`

regression model and`Matern52`

correlation model. - GP surrogate with
`Linear`

regression model and`SquaredExponential`

correlation model. - Mixture of gaussian process experts Implementation note: the structure is not generic over ‘F: Float’ to be able to implement use serde easily as deserialization of generic impls is not supported yet See https://github.com/dtolnay/typetag/issues/1
- Mixture of experts parameters
- Mixture of experts checked parameters
- GP surrogate with
`Quadratic`

regression model and`AbsoluteExponential`

correlation model. - GP surrogate with
`Quadratic`

regression model and`Matern32`

correlation model. - GP surrogate with
`Quadratic`

regression model and`Matern52`

correlation model. - GP surrogate with
`Quadratic`

regression model and`SquaredExponential`

correlation model. - Adaptator to implement
`linfa::Predict`

for variance prediction - Flags to specify tested regression models during experts selection (see
`regression_spec()`

). - SGP surrogate with
`AbsoluteExponential`

correlation model. - SGP surrogate with
`Matern32`

correlation model. - SGP surrogate with
`Matern52`

correlation model. - SGP surrogate with
`SquaredExponential`

correlation model.

## Enums§

- SGP inducing points specification
- An error when using MOE algorithm
- Enumeration of recombination modes handled by the mixture
- SGP algorithm method specification
- An enum to represent a n-dim hyper parameter tuning

## Traits§

- A trait to represent clustered structure
- A trait for a GP surrogate.
- A trait for a GP surrogate.
- A trait for a base GP surrogate
- A trait for a GP surrogate with derivatives predictions and sampling
- A trait for Gp surrogate parameters to build surrogate.
- A trait for Mixture of GP surrogates with derivatives using clustering
- A trait for a Sparse GP surrogate.
- A trait for sparse GP surrogate parameters to build surrogate.

## Functions§

- Find the best number of cluster thanks to cross validation

## Type Aliases§

- A result type for Moe algorithm