loss

Classification error

Syntax

L = loss(tree,TBL,ResponseVarName)

L = loss(tree,TBL,Y)

L = loss(tree,X,Y)

L = loss(___,Name,Value)

[L,se,NLeaf,bestlevel] = loss(___)

Description

L = loss(tree,TBL,ResponseVarName) returns a scalar representing how well tree classifies the data in TBL, when TBL.ResponseVarName contains the true classifications.

When computing the loss, loss normalizes the class probabilities in Y to the class probabilities used for training, stored in the Prior property of tree.

L = loss(tree,TBL,Y) returns a scalar representing how well tree classifies the data in TBL, when Y contains the true classifications.

L = loss(tree,X,Y) returns a scalar representing how well tree classifies the data in X, when Y contains the true classifications.

L = loss(___,Name,Value) returns the loss with additional options specified by one or more Name,Value pair arguments, using any of the previous syntaxes. For example, you can specify the loss function or observation weights.

[L,se,NLeaf,bestlevel] = loss(___) also returns the vector of standard errors of the classification errors (se), the vector of numbers of leaf nodes in the trees of the pruning sequence (NLeaf), and the best pruning level as defined in the TreeSize name-value pair (bestlevel).

Input Arguments

tree — Trained classification tree

ClassificationTree model object | CompactClassificationTree model object

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

TBL — Sample data

table

Sample data, specified as a table. Each row of TBL corresponds to one observation, and each column corresponds to one predictor variable. Optionally, TBL can contain additional columns for the response variable and observation weights. TBL must contain all the predictors used to train tree. Multicolumn variables and cell arrays other than cell arrays of character vectors are not allowed.

If TBL contains the response variable used to train tree, then you do not need to specify ResponseVarName or Y.

If you train tree using sample data contained in a table, then the input data for this method must also be in a table.

Data Types: table

X — Data to classify

numeric matrix

Data to classify, specified as a numeric matrix. Each row of X represents one observation, and each column represents one predictor. X must have the same number of columns as the data used to train tree. X must have the same number of rows as the number of elements in Y.

Data Types: single | double

ResponseVarName — Response variable name

name of a variable in TBL

Response variable name, specified as the name of a variable in TBL. If TBL contains the response variable used to train tree, then you do not need to specify ResponseVarName.

If you specify ResponseVarName, then you must do so as a character vector or string scalar. For example, if the response variable is stored as TBL.Response, then specify it as 'Response'. Otherwise, the software treats all columns of TBL, including TBL.ResponseVarName, as predictors.

The response variable must be a categorical, character, or string array, logical or numeric vector, or cell array of character vectors. If the response variable is a character array, then each element must correspond to one row of the array.

Data Types: char | string

Y — Class labels

categorical array | character array | string array | logical vector | numeric vector | cell array of character vectors

Class labels, specified as a categorical, character, or string array, a logical or numeric vector, or a cell array of character vectors. Y must be of the same type as the classification used to train tree, and its number of elements must equal the number of rows of X.

Data Types: categorical | char | string | logical | single | double | cell

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 quotes. You can specify several name and value pair arguments in any order as Name1,Value1,...,NameN,ValueN.

'LossFun' — Loss function

'mincost' (default) | 'binodeviance' | 'classiferror' | 'exponential' | 'hinge' | 'logit' | 'quadratic' | function handle

Loss function, specified as the comma-separated pair consisting of 'LossFun' and a built-in, loss-function name or function handle.

• The following table lists the available loss functions. Specify one using its corresponding character vector or string scalar.

  Value	Description
  'binodeviance'	Binomial deviance
  'classiferror'	Misclassified rate in decimal
  'exponential'	Exponential loss
  'hinge'	Hinge loss
  'logit'	Logistic loss
  'mincost'	Minimal expected misclassification cost (for classification scores that are posterior probabilities)
  'quadratic'	Quadratic loss

  'mincost' is appropriate for classification scores that are posterior probabilities. Classification trees return posterior probabilities as classification scores by default (see predict).

• Specify your own function using function handle notation.

  Suppose that n be the number of observations in X and K be the number of distinct classes (numel(tree.ClassNames)). Your function must have this signature

  lossvalue = lossfun(C,S,W,Cost)

  where:

  • The output argument lossvalue is a scalar.

  • You choose the function name (lossfun).

  • C is an n-by-K logical matrix with rows indicating which class the corresponding observation belongs. The column order corresponds to the class order in tree.ClassNames.

    Construct C by setting C(p,q) = 1 if observation p is in class q, for each row. Set all other elements of row p to 0.

  • S is an n-by-K numeric matrix of classification scores. The column order corresponds to the class order in tree.ClassNames. S is a matrix of classification scores, similar to the output of predict.

  • W is an n-by-1 numeric vector of observation weights. If you pass W, the software normalizes them to sum to 1.

  • Cost is a K-by-K numeric matrix of misclassification costs. For example, Cost = ones(K) - eye(K) specifies a cost of 0 for correct classification, and 1 for misclassification.

  Specify your function using 'LossFun',@lossfun.

For more details on loss functions, see Classification Loss.

Data Types: char | string | function_handle

'Weights' — Observation weights

ones(size(X,1),1) (default) | name of a variable in TBL | numeric vector of positive values

Observation weights, specified as the comma-separated pair consisting of 'Weights' and a numeric vector of positive values or the name of a variable in TBL.

If you specify Weights as a numeric vector, then the size of Weights must be equal to the number of rows in X or TBL.

If you specify Weights as the name of a variable in TBL, you must do so as a character vector or string scalar. For example, if the weights are stored as TBL.W, then specify it as 'W'. Otherwise, the software treats all columns of TBL, including TBL.W, as predictors.

loss normalizes the weights so that observation weights in each class sum to the prior probability of that class. When you supply Weights, loss computes weighted classification loss.

Data Types: single | double | char | string

Name,Value arguments associated with pruning subtrees:

'Subtrees' — Pruning level

0 (default) | vector of nonnegative integers | 'all'

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(tree.PruneList). 0 indicates the full, unpruned tree and max(tree.PruneList) indicates the completely pruned tree (i.e., just the root node).

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

loss prunes tree 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 tree must be nonempty. In other words, grow tree by setting 'Prune','on', or by pruning tree using prune.

Example: 'Subtrees','all'

Data Types: single | double | char | string

'TreeSize' — Tree size

'se' (default) | 'min'

Tree size, specified as the comma-separated pair consisting of 'TreeSize' and one of the following values:

• 'se' — loss returns the highest pruning level with loss within one standard deviation of the minimum (L+se, where L and se relate to the smallest value in Subtrees).

• 'min' — loss returns the element of Subtrees with smallest loss, usually the smallest element of Subtrees.

Output Arguments

L — Classification loss

vector of scalar values

Classification loss, returned as a vector the length of Subtrees. The meaning of the error depends on the values in Weights and LossFun.

se — Standard error of loss

vector of scalar values

Standard error of loss, returned as a vector the length of Subtrees.

NLeaf — Number of leaf nodes

vector of integer values

Number of leaves (terminal nodes) in the pruned subtrees, returned as a vector the length of Subtrees.

bestlevel — Best pruning level

scalar value

Best pruning level as defined in the TreeSize name-value pair, returned as a scalar whose value depends on TreeSize:

• TreeSize = 'se' — loss returns the highest pruning level with loss within one standard deviation of the minimum (L+se, where L and se relate to the smallest value in Subtrees).

• TreeSize = 'min' — loss returns the element of Subtrees with smallest loss, usually the smallest element of Subtrees.

By default, bestlevel is the pruning level that gives loss within one standard deviation of minimal loss.

Examples

Compute the In-sample Classification Error

Compute the resubstituted classification error for the ionosphere data set.

load ionosphere
tree = fitctree(X,Y);
L = loss(tree,X,Y)

L = 0.0114

Examine the Classification Error for Each Subtree

Unpruned decision trees tend to overfit. One way to balance model complexity and out-of-sample performance is to prune a tree (or restrict its growth) so that in-sample and out-of-sample performance are satisfactory.

Load Fisher's iris data set. Partition the data into training (50%) and validation (50%) sets.

load fisheriris
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));

View the classification tree.

view(Mdl,'Mode','graph');

The classification tree has four pruning levels. Level 0 is the full, unpruned tree (as displayed). Level 3 is just the root node (i.e., no splits).

Examine the training sample classification error for each subtree (or pruning level) excluding the highest level.

m = max(Mdl.PruneList) - 1;
trnLoss = resubLoss(Mdl,'SubTrees',0:m)

trnLoss = 3×1

    0.0267
    0.0533
    0.3067

• The full, unpruned tree misclassifies about 2.7% of the training observations.

• The tree pruned to level 1 misclassifies about 5.3% of the training observations.

• The tree pruned to level 2 (i.e., a stump) misclassifies about 30.6% of the training observations.

Examine the validation sample classification error at each level excluding the highest level.

valLoss = loss(Mdl,meas(idxVal,:),species(idxVal),'SubTrees',0:m)

valLoss = 3×1

    0.0369
    0.0237
    0.3067

• The full, unpruned tree misclassifies about 3.7% of the validation observations.

• The tree pruned to level 1 misclassifies about 2.4% of the validation observations.

• The tree pruned to level 2 (i.e., a stump) misclassifies about 30.7% of the validation observations.

To balance model complexity and out-of-sample performance, consider pruning Mdl to level 1.

pruneMdl = prune(Mdl,'Level',1);
view(pruneMdl,'Mode','graph')

More About

Classification Loss

Classification loss functions measure the predictive inaccuracy of classification models. When you compare the same type of loss among many models, a lower loss indicates a better predictive model.

Consider the following scenario.

• L is the weighted average classification loss.

• n is the sample size.

• For binary classification:

  • y_j is the observed class label. The software codes it as –1 or 1, indicating the negative or positive class (or the first or second class in the ClassNames property), respectively.

  • f(X_j) is the positive-class classification score for observation (row) j of the predictor data X.

  • m_j = y_jf(X_j) is the classification score for classifying observation j into the class corresponding to y_j. Positive values of m_j indicate correct classification and do not contribute much to the average loss. Negative values of m_j indicate incorrect classification and contribute significantly to the average loss.

• For algorithms that support multiclass classification (that is, K ≥ 3):

  • y_j^* is a vector of K – 1 zeros, with 1 in the position corresponding to the true, observed class y_j. For example, if the true class of the second observation is the third class and K = 4, then y₂^* = [0 0 1 0]′. The order of the classes corresponds to the order in the ClassNames property of the input model.

  • f(X_j) is the length K vector of class scores for observation j of the predictor data X. The order of the scores corresponds to the order of the classes in the ClassNames property of the input model.

  • m_j = y_j^*′f(X_j). Therefore, m_j is the scalar classification score that the model predicts for the true, observed class.

• The weight for observation j is w_j. The software normalizes the observation weights so that they sum to the corresponding prior class probability. The software also normalizes the prior probabilities so they sum to 1. Therefore,

  $${\displaystyle \sum _{j=1}^n{w}_j}=1.$$

Given this scenario, the following table describes the supported loss functions that you can specify by using the 'LossFun' name-value pair argument.

Loss Function	Value of LossFun	Equation
Binomial deviance	'binodeviance'	$$L={\displaystyle \sum _{j=1}^n{w}_j\log \left\{1+\exp \left[-2m_j\right]\right\}}.$$
Misclassified rate in decimal	'classiferror'	$$L={\displaystyle \sum _{j=1}^n{w}_j}I\left\{{\widehat{y}}_j\ne y_j\right\}.$$ $${\widehat{y}}_j$$ is the class label corresponding to the class with the maximal score. I{·} is the indicator function.
Cross-entropy loss	'crossentropy'	'crossentropy' is appropriate only for neural network models. The weighted cross-entropy loss is $$L=-{\displaystyle \sum _{j=1}^n\frac{{\tilde{w}}_j\log (m_j)}{Kn}},$$ where the weights $${\tilde{w}}_j$$ are normalized to sum to n instead of 1.
Exponential loss	'exponential'	$$L={\displaystyle \sum _{j=1}^n{w}_j\exp \left(-m_j\right)}.$$
Hinge loss	'hinge'	$$L={\displaystyle \sum }_{j=1}^n{w}_j\max \left\{0,1-m_j\right\}.$$
Logit loss	'logit'	$$L={\displaystyle \sum _{j=1}^n{w}_j\log \left(1+\exp \left(-m_j\right)\right)}.$$
Minimal expected misclassification cost	'mincost'	'mincost' is appropriate only if classification scores are posterior probabilities. The software computes the weighted minimal expected classification cost using this procedure for observations j = 1,...,n. Estimate the expected misclassification cost of classifying the observation Xj into the class k: $${\gamma }_{jk}={\left(f{\left(X_j\right)}^{\prime }C\right)}_k.$$ f(Xj) is the column vector of class posterior probabilities for binary and multiclass classification for the observation Xj. C is the cost matrix stored in the Cost property of the model. For observation j, predict the class label corresponding to the minimal expected misclassification cost: $${\widehat{y}}_j=\underset{k=1,\mathrm{...},K}{\text{argmin}}{\gamma }_{jk}.$$ Using C, identify the cost incurred (cj) for making the prediction. The weighted average of the minimal expected misclassification cost loss is $$L={\displaystyle \sum _{j=1}^n{w}_j{c}_j}.$$ If you use the default cost matrix (whose element value is 0 for correct classification and 1 for incorrect classification), then the 'mincost' loss is equivalent to the 'classiferror' loss.
Quadratic loss	'quadratic'	$$L={\displaystyle \sum _{j=1}^n{w}_j{\left(1-m_j\right)}^2}.$$

This figure compares the loss functions (except 'crossentropy' and 'mincost') over the score m for one observation. Some functions are normalized to pass through the point (0,1).

True Misclassification Cost

The true misclassification cost is the cost of classifying an observation into an incorrect class.

You can set the true misclassification cost per class by using the 'Cost' name-value argument when you create the classifier. Cost(i,j) is the cost of classifying an observation into class j when 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 Misclassification Cost

The expected misclassification cost per observation is an averaged cost of classifying the observation into each class.

Suppose you have Nobs observations that you want to classify with a trained classifier, and you have K classes. You place the observations into a matrix X 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

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

where:

• K is the number of classes.

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

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

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 = 10×3

    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.

Extended Capabilities

Tall Arrays

Calculate with arrays that have more rows than fit in memory.

Usage notes and limitations:

• Only one output is supported.

• You can use models trained on either in-memory or tall data with this function.

For more information, see Tall Arrays.