## Motivation

There are dozens of machine learning algorithms out there. It is impossible to learn all their mechanics, however, many algorithms sprout from the most established algorithms, e.g. ordinary least squares, gradient boosting, support vector machines, tree-based algorithms, and neural networks. At STATWORX we discuss algorithms daily to evaluate their usefulness for a specific project. In any case, understanding these core algorithms is key to most machine learning algorithms in the literature.

While I like reading machine learning research papers, the math is sometimes hard to follow. That is why I like implementing the algorithms in R by myself. Of course, this means digging through the maths and the algorithms as well. However, you can challenge your understanding of the algorithm directly.

In my two subsequent blog post, I will introduce two machine learning algorithms in 150 lines of R Code. This blog post will be about regression trees, which are the foundation of most tree-based algorithms. You can find the other blog post about coding gradient boosted machines from scratch on our blog. The algorithms will cover all core mechanics while being very generic. You can find all code on my GitHub.

## Gathering all puzzle pieces

Surely, there are tons of great articles out there which explain regression trees theoretically accompanied with a hands-on example. This is not the objective of this blog post. If you are interested in a hands-on tutorial with all the necessary theories I strongly recommend this tutorial. The objective of this blog post is to establish the theory of the algorithm by writing simple R code. You do not need any prior knowledge of the algorithm to follow. The only thing you need to know is our objective: We want to estimate our real-valued target (`y`

) with a set of real-valued features (`X`

).

Most probably you are already familiar with decision trees, which is a machine learning algorithm to solve classification tasks. As the name itself states regression trees solve regressions, i.e. estimation with continuously scaled targets. These kinds of trees are the key part of every tree-based method, since the way you grow a tree is more of the same really. The differing parts among the implementations are mostly about the splitting rule. In this tutorial, we will program a very simple, but generic implementation of a regression tree.

Fortunately, we do not have to cover much maths in this tutorial, because the algorithm itself is rather a technical than a mathematical challenge. With that said, the technical path I have chosen might not be the most efficient way, but I tried to trade off efficiency with simplicity.

Anyway, as most of you might know decision or regression trees are rule-based approaches. Meaning, we are trying to split the data into partitions conditional to our feature space. The data partitioning is done with the help of a splitting criterion. There is no common ground on how to do those splits, there are rather multiple different splitting criteria with different pros and cons. We will focus on a rather simple criterion in this tutorial. Bear with me, here comes some math.

Ok, so what does this state? This is the sum of squared errors determined in two different subsets ( and ). As the name suggests, that should be something we want to minimize. In fact, it is the squared distance between the mean and the target within this data subset. In every node of our regression tree we calculate the SSE for every potential split we could do in our data for every feature we have to figure out the best split we can achieve.

Let us have a look at the R Code:

```
# This is the splitting criterion we minimize (SSE [Sum Of Squared Errors]):
# SSE = sum{i in S_1} (y_i - bar(y)_1)^2 + sum{i in S_2} (y_i - bar(y)_2)^2
sse_var <- function(x, y) {
splits <- sort(unique(x))
sse <- c()
for (i in seq_along(splits)) {
sp <- splits[i]
sse[i] <- sum((y[x < sp] - mean(y[x < sp]))^2) +
sum((y[x >= sp] - mean(y[x >= sp]))^2)
}
split_at <- splits[which.min(sse)]
return(c(sse = min(sse), split = split_at))
}
```

The function takes two inputs our numeric feature `x`

and our target real-valued `y`

. We then go ahead and calculate the SSE for every unique value of our `x`

. This means we calculate the SSE for every possible data subset we could obtain conditional on the feature. Often we want to cover more than one feature in our problem, which means that we have to run this function for every feature. As a result, the best splitting rule has the lowest SSE among all possible splits of all features. Once we have determined the best splitting rule we can split our data into these two subsets according to our criterion, which is nothing else than feature `x <= split_at`

and `x > split_at`

. We call these two subsets children and they again can be split into subsets again.

Let us lose some more words on the SSE though, because it reveals our estimator. In this implementation, our estimator in the leaf is simply the average value of our target within this data subset. This is the simplest version of a regression tree. However, with some additional work, you can apply more sophisticated models, e.g. an ordinary least squares fit.

## The Algorithm

Enough with the talking, let’s get to the juice. In the following, you will see the algorithm in all of its beauty. Afterward, we will break down the algorithm into easy-to-digest code chunks.

```
#' reg_tree
#' Fits a simple regression tree with SSE splitting criterion. The estimator function
#' is the mean.
#'
#' @param formula an object of class formula
#' @param data a data.frame or matrix
#' @param minsize a numeric value indicating the minimum size of observations
#' in a leaf
#'
#' @return itemize{
#' item tree - the tree object containing all splitting rules and observations
#' item fit - our fitted values, i.e. X %*% theta
#' item formula - the underlying formula
#' item data - the underlying data
#' }
#' @export
#'
#' @examples # Complete runthrough see: www.github.com/andrebleier/cheapml
reg_tree <- function(formula, data, minsize) {
# coerce to data.frame
data <- as.data.frame(data)
# handle formula
formula <- terms.formula(formula)
# get the design matrix
X <- model.matrix(formula, data)
# extract target
y <- data[, as.character(formula)[2]]
# initialize while loop
do_splits <- TRUE
# create output data.frame with splitting rules and observations
tree_info <- data.frame(NODE = 1, NOBS = nrow(data), FILTER = NA,
TERMINAL = "SPLIT",
stringsAsFactors = FALSE)
# keep splitting until there are only leafs left
while(do_splits) {
# which parents have to be splitted
to_calculate <- which(tree_infoNODE)
# paste filter rules
tmp_filter <- c(paste(names(tmp_splitter), ">=",
splitting[2,tmp_splitter]),
paste(names(tmp_splitter), "<",
splitting[2,tmp_splitter]))
# Error handling! check if the splitting rule has already been invoked
split_here <- !sapply(tmp_filter,
FUN = function(x,y) any(grepl(x, x = y)),
y = tree_infoTERMINAL != "SPLIT")
} # end for
} # end while
# calculate fitted values
leafs <- tree_info[tree_infoTERMINAL == "SPLIT")
for (j in to_calculate) {
# handle root node
if (!is.na(tree_info[j, "FILTER"])) {
# subset data according to the filter
this_data <- subset(data, eval(parse(text = tree_info[j, "FILTER"])))
# get the design matrix
X <- model.matrix(formula, this_data)
} else {
this_data <- data
}
# estimate splitting criteria
splitting <- apply(X, MARGIN = 2, FUN = sse_var, y = y)
# get the min SSE
tmp_splitter <- which.min(splitting[1,])
# define maxnode
mn <- max(tree_infoFILTER)
# append the splitting rules
if (!is.na(tree_info[j, "FILTER"])) {
tmp_filter <- paste(tree_info[j, "FILTER"],
tmp_filter, sep = " & ")
}
# get the number of observations in current node
tmp_nobs <- sapply(tmp_filter,
FUN = function(i, x) {
nrow(subset(x = x, subset = eval(parse(text = i))))
},
x = this_data)
# insufficient minsize for split
if (any(tmp_nobs <= minsize)) {
split_here <- rep(FALSE, 2)
}
# create children data frame
children <- data.frame(NODE = c(mn+1, mn+2),
NOBS = tmp_nobs,
FILTER = tmp_filter,
TERMINAL = rep("SPLIT", 2),
row.names = NULL)[split_here,]
# overwrite state of current node
tree_info[j, "TERMINAL"] <- ifelse(all(!split_here), "LEAF", "PARENT")
# bind everything
tree_info <- rbind(tree_info, children)
# check if there are any open splits left
do_splits <- !all(tree_infoTERMINAL == "LEAF", ]
fitted <- c()
for (i in seq_len(nrow(leafs))) {
# extract index
ind <- as.numeric(rownames(subset(data, eval(parse(
text = leafs[i, "FILTER"])))))
# estimator is the mean y value of the leaf
fitted[ind] <- mean(y[ind])
}
```

At the end of our calculation, we have a filter rule for every leaf in our tree. With the help of these filters, we can easily calculate the fitted values by simply applying the filter on our data and calculating our fit, i.e. the mean of our target in this leaf. I am sure by now you can think of a way to implement more sophisticated estimators, which I would leave up to you.

Well, that’s a regression tree with minimum size restriction. I have created a little run-through with data from my simulation package on my GitHub, which you can check out and try everything on your own. Make sure to check out my other blog post about coding gradient boosted machines from scratch.