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# predict

Predict labels using classification tree

## Syntax

• label = predict(Mdl,X)
• label = predict(Mdl,X,Name,Value)
• [label,score,node,cnum] = predict(___)

## Description

label = predict(Mdl,X) returns a vector of predicted class labels for the predictor data in the table or matrix X, based on the trained, full or compact classification tree Mdl.

label = predict(Mdl,X,Name,Value) uses additional options specified by one or more Name,Value pair arguments. For example, you can specify to prune Mdl to a particular level before predicting labels.

[label,score,node,cnum] = predict(___) uses any of the input argument in the previous syntaxes and additionally returns:

• A matrix of classification scores (score) indicating the likelihood that a label comes from a particular class. For classification trees, scores are posterior probabilities. For each observation in X, the predicted class label corresponds to the minimum expected misclassification cost among all classes.

• A vector of predicted node numbers for the classification (node).

• A vector of predicted class number for the classification (cnum).

## Input Arguments

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Trained classification tree, specified as a ClassificationTree or CompactClassificationTree model object. That is, Mdl is a trained classification model returned by fitctree or compact.

Predictor data to be classified, specified as a numeric matrix or table.

Each row of X corresponds to one observation, and each column corresponds to one variable.

• For a numeric matrix:

• The variables making up the columns of X must have the same order as the predictor variables that trained Mdl.

• If you trained Mdl using a table (for example, Tbl), then X can be a numeric matrix if Tbl contains all numeric predictor variables. To treat numeric predictors in Tbl as categorical during training, identify categorical predictors using the CategoricalPredictors name-value pair argument of fitctree. If Tbl contains heterogeneous predictor variables (for example, numeric and categorical data types) and X is a numeric matrix, then predict throws an error.

• For a table:

• predict does not support multi-column variables and cell arrays other than cell arrays of character vectors.

• If you trained Mdl using a table (for example, Tbl), then all predictor variables in X must have the same variable names and data types as those that trained Mdl (stored in Mdl.PredictorNames). However, the column order of X does not need to correspond to the column order of Tbl. Tbl and X can contain additional variables (response variables, observation weights, etc.), but predict ignores them.

• If you trained Mdl using a numeric matrix, then the predictor names in Mdl.PredictorNames and corresponding predictor variable names in X must be the same. To specify predictor names during training, see the PredictorNames name-value pair argument of fitctree. All predictor variables in X must be numeric vectors. X can contain additional variables (response variables, observation weights, etc.), but predict ignores them.

Data Types: table | double | single

### Name-Value Pair Arguments

Specify optional comma-separated pairs of Name,Value arguments. Name is the argument name and Value is the corresponding value. Name must appear inside single quotes (' '). You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

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Pruning level, specified as the comma-separated pair consisting of 'Subtrees' and a vector of nonnegative integers in ascending order or 'all'.

If you specify a vector, then all elements must be at least 0 and at most max(Mdl.PruneList). 0 indicates the full, unpruned tree and max(Mdl.PruneList) indicates the completely pruned tree (i.e., just the root node).

If you specify 'all', then CompactClassificationTree.predict operates on all subtrees (i.e., the entire pruning sequence). This specification is equivalent to using 0:max(Mdl.PruneList).

CompactClassificationTree.predict prunes Mdl to each level indicated in Subtrees, and then estimates the corresponding output arguments. The size of Subtrees determines the size of some output arguments.

To invoke Subtrees, the properties PruneList and PruneAlpha of Mdl must be nonempty. In other words, grow Mdl by setting 'Prune','on', or by pruning Mdl using prune.

Example: 'Subtrees','all'

## Output Arguments

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Predicted class labels, returned as a vector of the same type as the response data used in training Mdl. Each entry of label corresponds to the class with minimal expected cost for the corresponding row of X.

If Subtrees has T elements, and X has N rows, then labels is an N-by-T matrix. The ith column of labels contains the fitted values produced by the Subtrees(I) subtree.

Posterior probabilities, returned as a numeric matrix of size N-by-K, where N is the number of observations (rows) in X, and K is the number of classes (in Mdl.ClassNames). score(i,j) is the posterior probability that row i of X is of class j.

If Subtrees has T elements, and X has N rows, then score is an N-by-K-by-T array, and node and cnum are N-by-T matrices.

Node numbers for the predicted classes, returned as a numeric vector. Each entry corresponds to the predicted node in Mdl for the corresponding row of X.

Class numbers corresponding to the predicted labels, returned as a numeric vector. Each entry of cnum corresponds to a predicted class number for the corresponding row of X.

## Definitions

### Predicted Class Label

predict classifies so as to minimize the expected classification cost:

$\stackrel{^}{y}=\underset{y=1,...,K}{\mathrm{arg}\mathrm{min}}\sum _{k=1}^{K}\stackrel{^}{P}\left(k|x\right)C\left(y|k\right),$

where

• $\stackrel{^}{y}$ is the predicted classification.

• K is the number of classes.

• $\stackrel{^}{P}\left(k|x\right)$ is the posterior probability of class k for observation x.

• $C\left(y|k\right)$ is the cost of classifying an observation as y when its true class is k.

### Score (tree)

For trees, the score of a classification of a leaf node is the posterior probability of the classification at that node. The posterior probability of the classification at a node is the number of training sequences that lead to that node with the classification, divided by the number of training sequences that lead to that node.

For example, consider classifying a predictor X as true when X < 0.15 or X > 0.95, and X is false otherwise.

Generate 100 random points and classify them:

rng(0,'twister') % for reproducibility
X = rand(100,1);
Y = (abs(X - .55) > .4);
tree = fitctree(X,Y);
view(tree,'Mode','Graph')

Prune the tree:

tree1 = prune(tree,'Level',1);
view(tree1,'Mode','Graph')

The pruned tree correctly classifies observations that are less than 0.15 as true. It also correctly classifies observations from .15 to .94 as false. However, it incorrectly classifies observations that are greater than .94 as false. Therefore, the score for observations that are greater than .15 should be about .05/.85=.06 for true, and about .8/.85=.94 for false.

Compute the prediction scores for the first 10 rows of X:

[~,score] = predict(tree1,X(1:10));
[score X(1:10,:)]
ans =

0.9059    0.0941    0.8147
0.9059    0.0941    0.9058
0    1.0000    0.1270
0.9059    0.0941    0.9134
0.9059    0.0941    0.6324
0    1.0000    0.0975
0.9059    0.0941    0.2785
0.9059    0.0941    0.5469
0.9059    0.0941    0.9575
0.9059    0.0941    0.9649

Indeed, every value of X (the right-most column) that is less than 0.15 has associated scores (the left and center columns) of 0 and 1, while the other values of X have associated scores of 0.91 and 0.09. The difference (score 0.09 instead of the expected .06) is due to a statistical fluctuation: there are 8 observations in X in the range (.95,1) instead of the expected 5 observations.

### True Misclassification Cost

There are two costs associated with classification: the true misclassification cost per class, and the expected misclassification cost per observation.

You can set the true misclassification cost per class in the Cost name-value pair when you create the classifier using the fitctree method. Cost(i,j) is the cost of classifying an observation into class j if its true class is i. By default, Cost(i,j)=1 if i~=j, and Cost(i,j)=0 if i=j. In other words, the cost is 0 for correct classification, and 1 for incorrect classification.

### Expected Cost

There are two costs associated with classification: the true misclassification cost per class, and the expected misclassification cost per observation.

Suppose you have Nobs observations that you want to classify with a trained classifier. Suppose you have K classes. You place the observations into a matrix Xnew with one observation per row.

The expected cost matrix CE has size Nobs-by-K. Each row of CE contains the expected (average) cost of classifying the observation into each of the K classes. CE(n,k) is

$\sum _{i=1}^{K}\stackrel{^}{P}\left(i|Xnew\left(n\right)\right)C\left(k|i\right),$

where

• K is the number of classes.

• $\stackrel{^}{P}\left(i|Xnew\left(n\right)\right)$ is the posterior probability of class i for observation Xnew(n).

• $C\left(k|i\right)$ is the true misclassification cost of classifying an observation as k when its true class is i.

### Predictive Measure of Association

The predictive measure of association is a value that indicates the similarity between decision rules that split observations. Among all possible decision splits that are compared to the optimal split (found by growing the tree), the best surrogate decision split yields the maximum predictive measure of association. The second-best surrogate split has the second-largest predictive measure of association.

Suppose xj and xk are predictor variables j and k, respectively, and jk. At node t, the predictive measure of association between the optimal split xj < u and a surrogate split xk < v is

${\lambda }_{jk}=\frac{\text{min}\left({P}_{L},{P}_{R}\right)-\left(1-{P}_{{L}_{j}{L}_{k}}-{P}_{{R}_{j}{R}_{k}}\right)}{\text{min}\left({P}_{L},{P}_{R}\right)}.$

• PL is the proportion of observations in node t, such that xj < u. The subscript L stands for the left child of node t.

• PR is the proportion of observations in node t, such that xju. The subscript R stands for the right child of node t.

• ${P}_{{L}_{j}{L}_{k}}$ is the proportion of observations at node t, such that xj < u and xk < v.

• ${P}_{{R}_{j}{R}_{k}}$ is the proportion of observations at node t, such that xju and xkv.

• Observations with missing values for xj or xk do not contribute to the proportion calculations.

λjk is a value in (–∞,1]. If λjk > 0, then xk < v is a worthwhile surrogate split for xj < u.

## Examples

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Examine predictions for a few rows in a data set left out of training.

Partition the data into training (50%) and validation (50%) sets.

n = size(meas,1);
rng(1) % For reproducibility
idxTrn = false(n,1);
idxTrn(randsample(n,round(0.5*n))) = true; % Training set logical indices
idxVal = idxTrn == false;                  % Validation set logical indices

Grow a classification tree using the training set.

Mdl = fitctree(meas(idxTrn,:),species(idxTrn));

Predict labels for the validation data. Count the number of misclassified observations.

label = predict(Mdl,meas(idxVal,:));
label(randsample(numel(label),5)) % Display several predicted labels
numMisclass = sum(~strcmp(label,species(idxVal)))
ans =

5×1 cell array

'setosa'
'setosa'
'setosa'
'virginica'
'versicolor'

numMisclass =

3

The software misclassifies three out-of-sample observations.

Partition the data into training (50%) and validation (50%) sets.

n = size(meas,1);
rng(1) % For reproducibility
idxTrn = false(n,1);
idxTrn(randsample(n,round(0.5*n))) = true; % Training set logical indices
idxVal = idxTrn == false;                  % Validation set logical indices

Grow a classification tree using the training set, and then view it.

Mdl = fitctree(meas(idxTrn,:),species(idxTrn));
view(Mdl,'Mode','graph')

The resulting tree has four levels.

Estimate posterior probabilities for the test set using subtrees pruned to levels 1 and 3.

[~,Posterior] = predict(Mdl,meas(idxVal,:),'SubTrees',[1 3]);
Mdl.ClassNames
Posterior(randsample(size(Posterior,1),5),:,:),...
% Display several posterior probabilities
ans =

3×1 cell array

'setosa'
'versicolor'
'virginica'

ans(:,:,1) =

1.0000         0         0
1.0000         0         0
1.0000         0         0
0         0    1.0000
0    0.8571    0.1429

ans(:,:,2) =

0.3733    0.3200    0.3067
0.3733    0.3200    0.3067
0.3733    0.3200    0.3067
0.3733    0.3200    0.3067
0.3733    0.3200    0.3067

The elements of Posterior are class posterior probabilities:

• Rows correspond to observations in the validation set.

• Columns correspond to the classes as listed in Mdl.ClassNames.

• Pages correspond to the subtrees.

The subtree pruned to level 1 is more sure of its predictions than the subtree pruned to level 3 (i.e., the root node).

## Algorithms

predict generates predictions by following the branches of Mdl until it reaches a leaf node or a missing value. If predict reaches a leaf node, it returns the classification of that node.

If predict reaches a node with a missing value for a predictor, its behavior depends on the setting of the Surrogate name-value pair when fitctree constructs Mdl.

• Surrogate = 'off' (default) — predict returns the label with the largest number of training samples that reach the node.

• Surrogate = 'on'predict uses the best surrogate split at the node. If all surrogate split variables with positive predictive measure of association are missing, predict returns the label with the largest number of training samples that reach the node. For a definition, see Predictive Measure of Association.