Documentation |
L = loss(mdl,X,Y)
L = loss(mdl,X,Y,Name,Value)
L = loss(mdl,X,Y) returns a scalar representing how well mdl classifies the data in X, when Y 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 mdl.
L = loss(mdl,X,Y,Name,Value) returns the loss with additional options specified by one or more Name,Value pair arguments.
L |
Classification error, a scalar. The meaning of the error depends on the values in weights and lossfun. See Classification Error. |
The default classification error is the fraction of data X that mdl misclassifies, where Y represents the true classifications.
The weighted classification error is the sum of weight i times the Boolean value that is 1 when mdl misclassifies the ith row of X, divided by the sum of the weights.
The built-in loss functions are:
'binodeviance' — For binary classification, assume the classes y_{n} are -1 and 1. With weight vector w normalized to have sum 1, and predictions of row n of data X as f(X_{n}), the binomial deviance is
$$\sum {w}_{n}\mathrm{log}\left(1+\mathrm{exp}\left(-2{y}_{n}f\left({X}_{n}\right)\right)\right)}.$$
'exponential' — With the same definitions as for 'binodeviance', the exponential loss is
$$\sum {w}_{n}\mathrm{exp}\left(-{y}_{n}f\left({X}_{n}\right)\right)}.$$
'classiferror' — Predict the label with the largest posterior probability. The loss is then the fraction of misclassified observations.
'hinge' — Classification error measure that has the form
$$L=\frac{{\displaystyle \sum}_{j=1}^{n}{w}_{j}\mathrm{max}\left\{0,1-{y}_{j}\prime f\left({X}_{j}\right)\right\}}{{\displaystyle \sum}_{j=1}^{n}{w}_{j}},$$
where:
w_{j} is weight j.
For binary classification, y_{j} = 1 for the positive class and -1 for the negative class. For problems where the number of classes K > 3, y_{j} is a vector of 0s, but with a 1 in the position corresponding to the true class, e.g., if the second observation is in the third class and K = 4, then y_{2} = [0 0 1 0]′.
$$f({X}_{j})$$ is, for binary classification, the posterior probability or, for K > 3, a vector of posterior probabilities for each class given observation j.
'mincost' — Predict the label with the smallest expected misclassification cost, with expectation taken over the posterior probability, and cost as given by the Cost property of the classifier (a matrix). The loss is then the true misclassification cost averaged over the observations.
To write your own loss function, create a function file in this form:
function loss = lossfun(C,S,W,COST)
N is the number of rows of X.
K is the number of classes in the classifier, represented in the ClassNames property.
C is an N-by-K logical matrix, with one true per row for the true class. The index for each class is its position in the ClassNames property.
S is an N-by-K numeric matrix. S is a matrix of posterior probabilities for classes with one row per observation, similar to the posterior output from predict.
W is a numeric vector with N elements, the observation weights. If you pass W, the elements are normalized to sum to the prior probabilities in the respective classes.
COST is a K-by-K numeric matrix of misclassification costs. For example, you can use COST = ones(K) - eye(K), which means a cost of 0 for correct classification, and 1 for misclassification.
The output loss should be a scalar.
Pass the function handle @lossfun as the value of the LossFun name-value pair.
There are two costs associated with KNN 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 run fitcknn. 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.
There are two costs associated with KNN classification: the true misclassification cost per class, and the expected misclassification cost per observation. The third output of predict is the expected misclassification cost per observation.
Suppose you have Nobs observations that you want to classify with a trained classifier mdl. Suppose you have K classes. You place the observations into a matrix Xnew with one observation per row. The command
[label,score,cost] = predict(mdl,Xnew)
returns, among other outputs, a cost matrix of size Nobs-by-K. Each row of the cost matrix contains the expected (average) cost of classifying the observation into each of the K classes. cost(n,k) is
$$\sum _{i=1}^{K}\widehat{P}\left(i|Xnew(n)\right)C\left(k|i\right)},$$
where
K is the number of classes.
$$\widehat{P}\left(i|Xnew(n)\right)$$ 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.