# The BetaML.Trees Module

BetaML.TreesModule
BetaML.Trees module

Implement the functionality required to build a Decision Tree or a whole Random Forest, predict data and assess its performances.

Both Decision Trees and Random Forests can be used for regression or classification problems, based on the type of the labels (numerical or not). You can override the automatic selection with the parameter forceClassification=true, typically if your labels are integer representing some categories rather than numbers. For classification problems the output of predictSingle is a dictionary with the key being the labels with non-zero probabilitity and the corresponding value its proobability; for regression it is a numerical value.

Please be aware that, differently from most other implementations, the Random Forest algorithm collects and averages the probabilities from the trees, rather than just repording the mode, i.e. no information is lost and the output of the forest classifier is still a PMF.

Missing data on features are supported, both on training and on prediction.

The module provide the following functions. Use ?[type or function] to access their full signature and detailed documentation:

Model definition and training:

• buildTree(xtrain,ytrain): Build a single Decision Tree
• buildForest(xtrain,ytrain): Build a "forest" of Decision Trees

Model predictions and assessment:

• predict(tree or forest, x): Return the prediction given the feature matrix
• oobError(forest,x,y): Return the out-of-bag error estimate
• Utils.accuracy(ŷ,y)): Categorical output accuracy
• Utils.meanRelError(ŷ,y,p): L-p norm based error

Features are expected to be in the standard format (nRecords × nDimensions matrices) and the labels (either categorical or numerical) as a nRecords column vector.

Acknowlegdments: originally based on the Josh Gordon's code

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## Detailed API

BetaML.Trees.DecisionNodeType

DecisionNode(question,trueBranch,falseBranch, depth)

A tree's non-terminal node.

Constructor's arguments and struct members:

• question: The question asked in this node
• trueBranch: A reference to the "true" branch of the trees
• falseBranch: A reference to the "false" branch of the trees
• depth: The nodes's depth in the tree
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BetaML.Trees.ForestType
Forest{Ty}

Type representing a Random Forest.

Individual trees are stored in the array trees. The "type" of the forest is given by the type of the labels on which it has been trained.

Struct members:

• trees: The individual Decision Trees
• isRegression: Whether the forest is to be used for regression jobs or classification
• oobData: For each tree, the rows number if the data that have not being used to train the specific tree
• oobError: The out of bag error (if it has been computed)
• weights: A weight for each tree depending on the tree's score on the oobData (see buildForest)
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BetaML.Trees.LeafType

Leaf(y,depth)

A tree's leaf (terminal) node.

Constructor's arguments:

• y: The labels assorciated to each record (either numerical or categorical)
• depth: The nodes's depth in the tree

Struct members:

• predictions: Either the relative label's count (i.e. a PMF) or the mean
• depth: The nodes's depth in the tree
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BetaML.Api.partitionMethod

partition(question,x)

Dicotomically partitions a dataset x given a question.

For each row in the dataset, check if it matches the question. If so, add it to 'true rows', otherwise, add it to 'false rows'. Rows with missing values on the question column are assigned randomly proportionally to the assignment of the non-missing rows.

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BetaML.Api.predictMethod

predict(forest,x)

Predict the labels of a feature dataset.

For each record of the dataset and each tree of the "forest", recursivelly traverse the tree to find the prediction most opportune for the given record. If the labels the tree has been trained with are numeric, the prediction is also numeric (the mean of the different trees predictions, in turn the mean of the labels of the training records ended in that leaf node). If the labels were categorical, the prediction is a dictionary with the probabilities of each item and in such case the probabilities of the different trees are averaged to compose the forest predictions. This is a bit different than most other implementations where the mode instead is reported.

In the first case (numerical predictions) use meanRelError(ŷ,y) to assess the mean relative error, in the second case you can use accuracy(ŷ,y).

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BetaML.Api.predictMethod

predict(tree,x)

Predict the labels of a feature dataset.

For each record of the dataset, recursivelly traverse the tree to find the prediction most opportune for the given record. If the labels the tree has been trained with are numeric, the prediction is also numeric. If the labels were categorical, the prediction is a dictionary with the probabilities of each item.

In the first case (numerical predictions) use meanRelError(ŷ,y) to assess the mean relative error, in the second case you can use accuracy(ŷ,y).

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BetaML.Trees.buildForestMethod

buildForest(x, y, nTrees; maxDepth, minGain, minRecords, maxFeatures, splittingCriterion, forceClassification)

Builds (define and train) a "forest" of Decision Trees.

Parameters:

See buildTree. The function has all the parameters of bildTree (with the maxFeatures defaulting to √D instead of D) plus the following parameters:

• nTrees: Number of trees in the forest [def: 30]
• β: Parameter that regulate the weights of the scoring of each tree, to be (optionally) used in prediction (see later) [def: 0, i.e. uniform weigths]
• oob: Whether to coompute the out-of-bag error, an estimation of the generalization accuracy [def: false]
• rng: Random Number Generator (see FIXEDSEED) [deafult: Random.GLOBAL_RNG]

Output:

• The function returns a Forest object (see Forest).
• The forest weights default to array of ones if β ≤ 0 and the oob error to +Inf if oob == false.

Notes :

• Each individual decision tree is built using bootstrap over the data, i.e. "sampling N records with replacement" (hence, some records appear multiple times and some records do not appear in the specific tree training). The maxFeature injects further variability and reduces the correlation between the forest trees.
• The predictions of the "forest" (using the function predict()) are then the aggregated predictions of the individual trees (from which the name "bagging": boostrap aggregating).
• This function optionally reports a weight distribution of the performances of eanch individual trees, as measured using the records he has not being trained with. These weights can then be (optionally) used in the predict function. The parameter β ≥ 0 regulate the distribution of these weights: larger is β, the greater the importance (hence the weights) attached to the best-performing trees compared to the low-performing ones. Using these weights can significantly improve the forest performances (especially using small forests), however the correct value of β depends on the problem under exam (and the chosen caratteristics of the random forest estimator) and should be cross-validated to avoid over-fitting.
• Note that this function uses multiple threads if these are available. You can check the number of threads available with Threads.nthreads(). To set the number of threads in Julia either set the environmental variable JULIA_NUM_THREADS (before starting Julia) or start Julia with the command line option --threads (most integrated development editors for Julia already set the number of threads to 4).
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BetaML.Trees.buildTreeMethod

buildTree(x, y, depth; maxDepth, minGain, minRecords, maxFeatures, splittingCriterion, forceClassification)

Builds (define and train) a Decision Tree.

Given a dataset of features x and the corresponding dataset of labels y, recursivelly build a decision tree by finding at each node the best question to split the data untill either all the dataset is separated or a terminal condition is reached. The given tree is then returned.

Parameters:

• x: The dataset's features (N × D)
• y: The dataset's labels (N × 1)
• maxDepth: The maximum depth the tree is allowed to reach. When this is reached the node is forced to become a leaf [def: N, i.e. no limits]
• minGain: The minimum information gain to allow for a node's partition [def: 0]
• minRecords: The minimum number of records a node must holds to consider for a partition of it [def: 2]
• maxFeatures: The maximum number of (random) features to consider at each partitioning [def: D, i.e. look at all features]
• splittingCriterion: Either gini, entropy or variance (see infoGain ) [def: gini for categorical labels (classification task) and variance for numerical labels(regression task)]
• forceClassification: Weather to force a classification task even if the labels are numerical (typically when labels are integers encoding some feature rather than representing a real cardinal measure) [def: false]
• rng: Random Number Generator ((see FIXEDSEED)) [deafult: Random.GLOBAL_RNG]

Notes:

Missing data (in the feature dataset) are supported.

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BetaML.Trees.findBestSplitMethod

findBestSplit(x,y;maxFeatures,splittingCriterion)

Find the best possible split of the database.

Find the best question to ask by iterating over every feature / value and calculating the information gain.

Parameters:

• x: The feature dataset
• y: The labels dataset
• maxFeatures: Maximum number of (random) features to look up for the "best split"
• splittingCriterion: The metric to define the "impurity" of the labels
• rng: Random Number Generator (see FIXEDSEED) [deafult: Random.GLOBAL_RNG]
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BetaML.Trees.infoGainMethod

infoGain(left, right, parentUncertainty; splittingCriterion)

Compute the information gain of a specific partition.

Compare the "information gain" my measuring the difference betwwen the "impurity" of the labels of the parent node with those of the two child nodes, weighted by the respective number of items.

Parameters:

• leftY: Child #1 labels
• rightY: Child #2 labels
• parentUncertainty: "Impurity" of the labels of the parent node
• splittingCriterion: Metric to adopt to determine the "impurity" (see below)

You can use your own function as the metric. We provide the following built-in metrics:

• gini (categorical)
• entropy (categorical)
• variance (numerical)
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BetaML.Trees.matchMethod

match(question, x)

Return a dicotomic answer of a question when applied to a given feature record.

It compares the feature value in the given record to the value stored in the question. Numerical features are compared in terms of disequality (">="), while categorical features are compared in terms of equality ("==").

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BetaML.Trees.oobErrorMethod

oobError(forest,x,y)

Comute the Out-Of-Bag error, an estimation of the validation error.

This function is called at time of train the forest if the parameter oob is true, or can be used later to get the oob error on an already trained forest.

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BetaML.Trees.updateTreesWeights!Method

updateTreesWeights!(forest,x,y;β)

Update the weights of each tree (to use in the prediction of the forest) based on the error of the individual tree computed on the records on which it has not been trained. As training a forest is expensive, this function can be used to "just" upgrade the trees weights using different betas, without retraining the model.

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