---
title: "eforest(): Random Forests With Energy Trees as Base Learners"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{eforest(): Random Forests With Energy Trees as Base Learners}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>"
)
```


# Introduction

<tt>etree</tt> is the R package where Energy Trees (Giubilei et al., 2022) are 
implemented. The model allows to perform classification and regression with 
covariates that are possibly structured and of different types. 

Energy Trees are used as base learners in ensemble methods that are the 
equivalent of bagging and Random Forests with traditional decision trees. The
two resulting models are bagging of Energy Trees and Random Energy Forests,
respectively.

This vignette explains how these two models can be used in R for classification and 
regression tasks. Both models are implemented using the function 
<tt>eforest()</tt>. The variables' types included in the examples are those 
currently available in the package: numeric, nominal, functional, and in the
form of graphs.

The first thing to do to run the examples is loading the package <tt>etree</tt>.

```{r setup}
library(etree)
```


# Data generation

We need data for our examples. We can use the same covariates for regression and
classification, and then change only the response. Let us consider $100$
observations divided into four equal-sized and disjoint groups
$G_1, G_2, G_3, G_4$. The response variable for regression is defined as:

$$Y^R_i \sim \begin{cases}
      \mathcal{N}(0, 1) & \text{for} \; i \in G_1\\
      \mathcal{N}(1, 1) & \text{for} \; i \in G_2\\
      \mathcal{N}(2, 1) & \text{for} \; i \in G_3\\
      \mathcal{N}(3, 1) & \text{for} \; i \in G_4
    \end{cases}.$$

The corresponding R object must be of class <tt>"numeric"</tt> to be correctly
recognized by <tt>eforest()</tt> as a response variable for regression.

```{r}
# Set seed 
set.seed(123)

# Number of observation
n_obs <- 100

# Response variable for regression
resp_reg <- c(rnorm(n_obs / 4, mean = 0, sd = 1),
              rnorm(n_obs / 4, mean = 2, sd = 1),
              rnorm(n_obs / 4, mean = 4, sd = 1),
              rnorm(n_obs / 4, mean = 6, sd = 1))
```

The response for classification is a nominal variable measured at
four different levels: $1, 2, 3, 4$. For each group, observations are randomly
assigned one of the four levels. The probability distribution is different for 
each group, and it gives the largest probability to the level equal to 
the index of the group, uniformly allocating the rest to the other three levels.
For example, the response variable for the first group is the following:

$$Y^C_i \sim \begin{cases}
      1 & p = 0.85 \\
      2 & p = 0.05 \\
      3 & p = 0.05 \\
      4 & p = 0.05
    \end{cases},
    \quad \text{for} \; i \in G_1.$$
    
The same holds for $G_2, G_3, G_4$, except for switching each time the 
probabilities in such a way that the most probable level matches the index of
the group.

The response variable for classification must be defined as a <tt>factor</tt> to
be correctly identified by <tt>eforest()</tt>.

```{r}
# Response variable for classification
resp_cls <- factor(c(sample(x = 1:4, size = n_obs / 4, replace = TRUE, 
                            prob = c(0.85, 0.05, 0.05, 0.05)),
                     sample(x = 1:4, size = n_obs / 4, replace = TRUE, 
                            prob = c(0.05, 0.85, 0.05, 0.05)),
                     sample(x = 1:4, size = n_obs / 4, replace = TRUE, 
                            prob = c(0.05, 0.05, 0.85, 0.05)),
                     sample(x = 1:4, size = n_obs / 4, replace = TRUE, 
                            prob = c(0.05, 0.05, 0.05, 0.85))))
```


For both tasks, covariates are defined as follows:

1. Numeric: 
$$X_{1i} \sim \begin{cases} 
\mathcal{U}[0, 0.55] & \text{for} \; i \in G_1\\
\mathcal{U}[0.45, 1] & \text{for} \; i \in G_2 \cup G_3 \cup G_4
\end{cases};$$

2. Nominal: $$X_{2i} \sim \begin{cases} 
\text{Bern}(0.1) & \text{for} \; i \in G_2\\
\text{Bern}(0.9) & \text{for} \; i \in G_1 \cup G_3 \cup G_4
\end{cases};$$

3. Functions: $$X_{3i} \sim \begin{cases} 
\mathcal{GP}(0, I_{100}) & \text{for} \; \in G_3\\
\mathcal{GP}(1, I_{100}) & \text{for} \; \in G_1 \cup G_2 \cup G_4
\end{cases},$$

where $\mathcal{GP}$ is a Gaussian random process over $100$
evaluation points from $0$ to $1$;

4. Graphs: $$X_{4i} \sim \begin{cases} 
\text{ER}(100, 0.1) & \text{for} \; \in G_4\\
\text{ER}(100, 0.9) & \text{for} \; \in G_1 \cup G_2 \cup G_3
\end{cases},$$

where $\text{ER}(n, p)$ is an Erdős–Rényi random graph with $n$ vertices and $p$
as connection probability.

The structure of the generative process is such that we need at least three
splits with respect to three different covariates to correctly rebuild the four
groups.

The set of covariates must be generated as a list, where each element is a
different variable. Additionally, numeric variables must be numeric vectors, 
nominal variable must be factors, functional variables must be objects of class
<tt>"fdata"</tt>, and variables in the forms of graphs must be lists of objects
having class <tt>"igraph"</tt>.

```{r}
# Numeric
x1 <- c(runif(n_obs / 4, min = 0, max = 0.55),
        runif(n_obs / 4 * 3, min = 0.45, max = 1))

# Nominal
x2 <- factor(c(rbinom(n_obs / 4, 1, 0.1),
               rbinom(n_obs / 4, 1, 0.9),
               rbinom(n_obs / 2, 1, 0.1)))

# Functions
x3 <- c(fda.usc::rproc2fdata(n_obs / 2, seq(0, 1, len = 100), sigma = 1),
        fda.usc::rproc2fdata(n_obs / 4, seq(0, 1, len = 100), sigma = 1,
                             mu = rep(1, 100)),
        fda.usc::rproc2fdata(n_obs / 4, seq(0, 1, len = 100), sigma = 1))

# Graphs
x4 <- c(lapply(1:(n_obs / 4 * 3), function(j) igraph::sample_gnp(100, 0.1)),
        lapply(1:(n_obs / 4), function(j) igraph::sample_gnp(100, 0.9)))

# Covariates list
cov_list <- list(X1 = x1, X2 = x2, X3 = x3, X4 = x4)

```


# Function eforest()

Function <tt>eforest()</tt> allows implementing bagging of Energy Trees (BET)
and Random Energy Forests (REF), depending on the value of the
<tt>random_covs</tt> argument: if it is a positive integer smaller than the
total number of covariates, the function goes for REF; if it is <tt>NULL</tt>
(default), or equal to the number of covariates, BET is used.

As for <tt>etree()</tt>, also <tt>eforest()</tt> is specified in the same way
for regression and classification tasks, automatically distinguishing between
the two based on the type of the response.

Many arguments are inherited from <tt>etree()</tt>. The additional ones are:

* <tt>ntrees</tt>, i.e., the number of Energy Trees to grow in the ensemble;
* <tt>ncores</tt>, i.e., the number of cores to use;
* <tt>perf_metric</tt>, i.e., the performance metric used to compute the
Out-Of_Bag error;
* <tt>random_covs</tt>, i.e., the size of the random subset of covariates to
choose from at each split;
* <tt>verbose</tt>, to visually keep track of the number of trees during the 
fitting process.

The structure of <tt>eforest()</tt> is such that it first generates 
<tt>ntrees</tt> bootstrap samples and then calls <tt>etree()</tt> on each of 
them. Then, it computes the Out-Of-Bag (OOB) score using the performance metric
defined through <tt>perf_metric</tt>.

The object returned by <tt>eforest()</tt> is of class <tt>"eforest"</tt> and
it is composed of three elements: 

* <tt>ensemble</tt>, i.e., a list containing all the fitted trees;
* <tt>oob_score</tt>, i.e., the Out-Of-Bag score;
* <tt>perf_metric</tt>, i.e., the performance metric used for computing the OOB.


# Regression

The first example is regression. The set of covariates is <tt>cov_list</tt> and
the response variable is <tt>resp_reg</tt>. The most immediate way to grow an
ensemble using <tt>eforest()</tt> is by specifying the response and the set of 
covariates only. This corresponds to using the default values for the optional
arguments. Here, we add <tt>ntrees = 12</tt> to reduce the computational burden
and change the performance metric using <tt>perf_metric = 'MSE'</tt> (the default
is <tt>RMSPE</tt> for Root Mean Square Percentage Error).

```{r}
# Quick fit
ef_fit <- eforest(response = resp_reg,
                  covariates = cov_list,
                  ntrees = 12,
                  perf_metric = 'MSE')
```

By default, <tt>eforest()</tt> uses <tt>minbucket = 1</tt> and 
<tt>alpha = 1</tt> to grow larger and less correlated trees. It employs 
<tt>"cluster"</tt> as <tt>split_type</tt> because it is usually faster. For 
regression, <tt>random_covs</tt> is set to one third (rounded down) of the total
number of covariates.

The returned object, <tt>ef_fit</tt>, stores all the trees in its element
<tt>ensemble</tt>, which is a list. Each tree in the list can be inspected,
plotted, subsetted, and used for predictions just as any other tree produced 
with <tt>etree()</tt> (in fact, also these trees are produced with 
<tt>etree()</tt>!). As an example, we can plot the first tree.

```{r, fig.dim = c(25, 11)}
# Plot of a tree from the ensemble
plot(ef_fit$ensemble[[4]])
```

The Out-Of-Bag score for the whole ensemble is stored in <tt>ef_fit</tt>'s
element <tt>oob_score</tt>. The performance metric used to compute the OOB score
can be retrieved from element <tt>perf_metric</tt>.

```{r}
# Retrieve performance metric
ef_fit$perf_metric

# OOB MSE for this ensemble
ef_fit$oob_score
```

Finally, the ensemble can be used to make predictions using the function
<tt>predict()</tt> on <tt>ef_fit</tt>, and possibly specifying a new set of 
covariates using <tt>newdata</tt>. Any other information, such as the splitting
strategy or the type of the task, is automatically retrieved from the fitted 
object.

Predictions are based either on the fitted values - if <tt>newdata</tt> is not
provided - or on the new set of covariates - if <tt>newdata</tt> is specified.

In both cases, each tree in <tt>ef_fit$ensemble</tt> is used to make predictions
by calling <tt>predict()</tt> on it (with the same specification of 
<tt>newdata</tt>). Then, for regression, individual trees' predictions for 
single observations are combined by arithmetic mean.

Let's start with the case where <tt>newdata</tt> is not provided.

```{r}
# Predictions from the fitted object
pred <- predict(ef_fit)
print(pred)
```

However, this case is rarely of practical utility because the model is usually
evaluated using the OOB score.

Let's see an example where predictions are calculated for a new set of 
covariates. 

```{r}
# New set of covariates
n_obs <- 40
x1n <- c(runif(n_obs / 4, min = 0, max = 0.55),
         runif(n_obs / 4 * 3, min = 0.45, max = 1))
x2n <- factor(c(rbinom(n_obs / 4, 1, 0.1),
                rbinom(n_obs / 4, 1, 0.9),
                rbinom(n_obs / 2, 1, 0.1)))
x3n <- c(fda.usc::rproc2fdata(n_obs / 2, seq(0, 1, len = 100), sigma = 1),
         fda.usc::rproc2fdata(n_obs / 4, seq(0, 1, len = 100), sigma = 1,
                              mu = rep(1, 100)),
         fda.usc::rproc2fdata(n_obs / 4, seq(0, 1, len = 100), sigma = 1))
x4n <- c(lapply(1:(n_obs / 4 * 3), function(j) igraph::sample_gnp(100, 0.1)),
         lapply(1:(n_obs / 4), function(j) igraph::sample_gnp(100, 0.9)))
new_cov_list <- list(X1 = x1n, X2 = x2n, X3 = x3n, X4 = x4n)

# New response 
new_resp_reg <- c(rnorm(n_obs / 4, mean = 0, sd = 1),
                  rnorm(n_obs / 4, mean = 2, sd = 1),
                  rnorm(n_obs / 4, mean = 4, sd = 1),
                  rnorm(n_obs / 4, mean = 6, sd = 1))

# Predictions
new_pred <- predict(ef_fit,
                    newdata = new_cov_list)
print(new_pred)
```

We can compare the Mean Square Error between the response and the predicted
values with that between the response and its average to see how predictions are
on average closer to the actual values than the response.

```{r}
# MSE between the new response and its average
mean((new_resp_reg - mean(new_resp_reg)) ^ 2)

# MSE between the new response and predictions with the new set of covariates
mean((new_resp_reg - new_pred) ^ 2)
```


# Classification 

The second example is classification. The set of covariates is the same as 
before, i.e., <tt>cov_list</tt>, while the response variable is 
<tt>resp_cls</tt>. Also in this case, the most immediate fit can be obtained by
calling <tt>eforest()</tt> and specifying the response and the set of covariates
only. Here, we also set <tt>ntrees = 12</tt> to reduce the computational burden 
and <tt>split_type = "coeff"</tt> to employ the other splitting strategy.

```{r, fig.dim = c(7, 6)}
# Quick fit
ef_fit <- eforest(response = resp_cls,
                  covariates = cov_list,
                  ntrees = 12,
                  split_type = 'coeff')
```

While everything else works as for regression, for classification tasks the 
default performance metric is Balanced Accuracy, while <tt>random_covs</tt> is 
set to the square root (rounded down) of the number of covariates.

Predictions are obtained using the same commands as for regression. However,
internally, individual trees' predictions for single observations are aggregated
by majority voting rule.

```{r}
# Predictions from the fitted object
pred <- predict(ef_fit)
print(pred)
```

To exemplify the use of <tt>predict()</tt> with <tt>newdata</tt>, we can use the
new set of covariates generated for regression, coupling it with a new response
variable for classification.

```{r}
# New response 
new_resp_cls <- factor(c(sample(x = 1:4, size = n_obs / 4, replace = TRUE, 
                                prob = c(0.85, 0.05, 0.05, 0.05)),
                         sample(x = 1:4, size = n_obs / 4, replace = TRUE, 
                                prob = c(0.05, 0.85, 0.05, 0.05)),
                         sample(x = 1:4, size = n_obs / 4, replace = TRUE, 
                                prob = c(0.05, 0.05, 0.85, 0.05)),
                         sample(x = 1:4, size = n_obs / 4, replace = TRUE, 
                                prob = c(0.05, 0.05, 0.05, 0.85))))

# Predictions
new_pred <- predict(ef_fit,
                    newdata = new_cov_list)
print(new_pred)

# Confusion matrix between the new response and predictions from the fitted tree
table(new_pred, new_resp_cls, dnn = c('Predicted', 'True'))

# Misclassification error for predictions on the new set of covariates
sum(new_pred != new_resp_cls) / length(new_resp_cls)
```

In this case, we obtain a misclassification error of 0.225 on test data, which 
should be however taken with a grain of salt given the very small number of 
trees that have been used in this toy example.


# References

R. Giubilei, T. Padellini, P. Brutti (2022). Energy Trees: Regression and 
Classification With Structured and Mixed-Type Covariates. arXiv preprint.
https://arxiv.org/pdf/2207.04430.pdf.