---
title: "Further considerations"
author: "Jorge Cabral"
output:
rmarkdown::html_vignette:
toc: true
toc_depth: 4
link-citations: yes
bibliography: references.bib
csl: american-medical-association-brackets.csl
description: |
Further considerations.
vignette: >
%\VignetteIndexEntry{Further considerations}
%\VignetteEncoding{UTF-8}
%\VignetteEngine{knitr::rmarkdown}
editor_options:
markdown:
wrap: 72
---
```{r, include = FALSE}
knitr::opts_chunk$set(
collapse = TRUE,
comment = "#>"
)
```
## Normalized Entropy
Golan et al. @Golan1996 defined normalized entropy for the signal, $\mathbf{X}\boldsymbol{\beta}$, in the GME framework, as
\begin{align}
\qquad \qquad \qquad \qquad \qquad S(\mathbf{\widehat{p}})=\frac{-\mathbf{\widehat{p}}' \text{ln}\mathbf{\widehat{p}}}{(K+1)\text{ln} M} \qquad \qquad \qquad \qquad \qquad (3)
\end{align}
where $S(\mathbf{\widehat{p}})\in[0,1]$ and $S(\mathbf{\widehat{p}})=1$
indicates perfect uncertainty, and $S(\mathbf{\widehat{p}})=0$ indicates no
uncertainty.
In the GCE framework it can be defined as
\begin{align}
\qquad \qquad \qquad \qquad \qquad S(\mathbf{\widehat{p}})=\frac{-\mathbf{\widehat{p}}' \text{ln}\mathbf{\widehat{p}}}{-\mathbf{\widehat{q}}' \text{ln}\mathbf{\widehat{q}}} \qquad \qquad \qquad \qquad \qquad (4)
\end{align}
but in this case the we can no longer state that $S(\mathbf{\widehat{p}})\in[0,1]$.
`GCEstim` package reports normalized entropies but it uses always the definition
in (3) independently of the framework used.
```{r,echo=FALSE,eval=TRUE}
library(GCEstim)
```
Consider `dataGCE` (see ["Generalized Maximum Entropy framework"](V2_GME_framework.html#Examples) and [Generalized Cross Entropy framework"](V3_GCE_framework.html#Examples)).
The GME estimation can be obtained, for instance, with
```{r,echo=TRUE,eval=TRUE}
res.lmgce.100.GME <-
GCEstim::lmgce(
y ~ .,
data = dataGCE,
cv = TRUE,
cv.nfolds = 5,
support.signal = c(-100, 100),
support.signal.points = 5,
twosteps.n = 0,
seed = 230676
)
```
and the GCE estimation with
```{r,echo=TRUE,eval=TRUE}
res.lmgce.100.GCE <-
GCEstim::lmgce(
y ~ .,
data = dataGCE,
cv = TRUE,
cv.nfolds = 5,
support.signal = c(-100, 100),
support.signal.points =
matrix(
c(
rep(1 / 5, 5),
c(0.1, 0.1, 0.6, 0.1, 0.1),
c(0.1, 0.1, 0.6, 0.1, 0.1),
rep(1 / 5, 5),
rep(1 / 5, 5),
rep(1 / 5, 5)
),
ncol = 5,
byrow = TRUE
),
twosteps.n = 0,
seed = 230676
)
```
The `NormEnt` extracts the normalized entropy from the models by default
(`model=TRUE`).
```{r,echo=TRUE,eval=TRUE}
NormEnt(res.lmgce.100.GME)
NormEnt(res.lmgce.100.GCE)
```
Each estimate has its own normalized entropy associated (`model=FALSE`)
```{r,echo=TRUE,eval=TRUE}
NormEnt(res.lmgce.100.GME, model = FALSE)
```
```{r,echo=TRUE,eval=TRUE}
NormEnt(res.lmgce.100.GCE, model = FALSE)
```
## References
## Acknowledgements
This work was supported by Fundação para a Ciência e Tecnologia (FCT)
through CIDMA and projects
and .