loss

Classification error

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

expand all

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

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

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

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

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 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.

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.

ValueDescription
'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)

'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.

For more details on loss functions, see Classification Loss.

Data Types: char | string | function_handle

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:

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

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

expand all

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

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

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

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

expand all

Compute the resubstituted classification error for the ionosphere data set.

tree = fitctree(X,Y);
L = loss(tree,X,Y)
L = 0.0114

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.

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')