README.Rmd

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# PenFFR

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The goal of PenFFR is to design a penalized funtion-on-funtion linear regression with multiples functional and scalar covariates. This package allows you to build two types of functional linear models:
- The concurrent linear model whose equation is given by : $$\mathrm{y}_i(t) = \beta_0(t) + \sum_{\ell=1}^p\beta_\ell(t)\mathrm{x}_i^\ell(t) + \varepsilon_i(t)$$
- The integral linear model given by : $$\mathrm{y}_i(t) = \beta_0(t) + \sum_{\ell=1}^p\int_0^t\beta_\ell(s,t)\mathrm{x}_i^\ell(s)\,ds + \varepsilon_i(t)$$
  
Parameters are estimated using a spline-based expansion with a number of basis functions that you can set.
  
  
To handle heterogeneous data, we also provide a Mixture-of-Experts (MoE) version of the linear concurrent model. The conditional density of $\mathrm{Y}(t)$ according to the function-on-function MoE (FFMoE) model is
$$\begin{eqnarray}
    f(\mathrm{Y}(t)|\mathrm{X}(t),\Psi(t)) &=& \sum_{k=1}^\mathrm{K}\pi_k(\mathrm{X}(t),\alpha_k(t)) \Phi(\mathrm{Y}(t);\mathrm{X}(t)\beta_k(t),\sigma^2_k),\label{ME2}
\end{eqnarray}$$
with
$$\begin{itemize}
    \item $\pi_k(\mathrm{X}(t),\ \alpha_k(t))$ the mixture proportion of group $k$, also called the $k^{\text{th}}$ gated network function, depending on the covariate $\mathrm{X}(t)$ through group specific functional parameter $\alpha_k(t)$. More details will be provided in the next section;
    \item $\Psi_k(t) = (\beta_k(t),\alpha_k(t))$ are the functional parameters;
    \item $\Phi(\mathrm{Y}(t); \mathrm{X}_i(t)\beta_k(t), \sigma^2_k)$ is the Gaussian density probability function of mean $\mathrm{X}(t)\beta_k(t)$ and variance $\sigma^2_k$.
\end{itemize}$$
  
  
## Download
You can download the package
  
* [Version 0.0.1](https://gitlab.tech.orange/rlsoftwarenet/penffr/-/archive/main/penffr-main.zip)
  
Or clone from Gitlab :
```
$ git clone https://gitlab.tech.orange/rlsoftwarenet/penffr.git
```
  
## Cite
If you use this software, please cite the following work :

* Jean Steve Tamo Tchomgui, Julien Jacques, Guillaume Fraysse, Vincent Barriac, Stéphane Chrétien. A mixture of experts regression model for functional response with functional covariates. Statistics and Computing, 2024, 34 (154), [s11222-024-10455-z](https://link.springer.com/article/10.1007/s11222-024-10455-z).
  
* Jean Steve Tamo Tchomgui, Julien Jacques, Vincent Barriac, Guillaume Fraysse, Stéphane Chrétien. A Penalized Spline Estimator for Functional Linear Regression with Functional Response. 2024. [hal-04120709v3](https://hal.science/hal-04120709v3).
  
For any questions, Please contact Jean Steve Tamo Tchomgui at jeanstevetamo at yahoo.fr or Guillaume Fraysse at guillaume dot fraysse at orange.com.
  
## License
Copyright (c) 2021 — 2024 Orange

This code is released under the `GPL-2.0-or-later` license. See the `LICENSE.md` file for more information.
  
  

## Contact
* Homepage: [opensource.orange.com](http://opensource.orange.com/)
* e-mail: guillaume dot fraysse at orange.com
  
  

  
  
## Link documents
- [Contributors list](CONTRIBUTORS.md)
  
  
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