Prediction Using Discriminant Analysis Models
predict classifies so as to minimize the expected classification cost:
is the predicted classification.
K is the number of classes.
is the posterior probability of class k for observation x.
is the cost of classifying an observation as y when its true class is k.
The space of
X values divides into regions where a classification
Y is a particular value. The regions are separated by straight lines for linear discriminant analysis, and by conic sections (ellipses, hyperbolas, or parabolas) for quadratic discriminant analysis. For a visualization of these regions, see Create and Visualize Discriminant Analysis Classifier.
The posterior probability that a point x belongs to class k is the product of the prior probability and the multivariate normal density. The density function of the multivariate normal with 1-by-d mean μk and d-by-d covariance Σk at a 1-by-d point x is
where is the determinant of Σk, and is the inverse matrix.
Let P(k) represent the prior probability of class k. Then the posterior probability that an observation x is of class k is
where P(x) is a normalization constant, namely, the sum over k of P(x|k)P(k).
The prior probability is one of three choices:
'uniform'— The prior probability of class
kis 1 over the total number of classes.
'empirical'— The prior probability of class
kis the number of training samples of class
kdivided by the total number of training samples.
A numeric vector — The prior probability of class
jth element of the
After creating a classifier
obj, you can set the prior using dot notation:
obj.Prior = v;
v is a vector of positive elements representing the frequency with which each element occurs. You do not need to retrain the classifier when you set a new prior.
There are two costs associated with discriminant analysis classification: the true misclassification cost per class, and the expected misclassification cost per observation.
True Misclassification Cost per Class
Cost(i,j) is the cost of classifying an observation into class
j if its true class is
i. By default,
i=j. In other words, the cost is
0 for correct classification, and
1 for incorrect classification.
You can set any cost matrix you like when creating a classifier. Pass the cost matrix in the
Cost name-value pair in
After you create a classifier
obj, you can set a custom cost using dot notation:
obj.Cost = B;
B is a square matrix of size
K when there are
K classes. You do not need to retrain the classifier when you set a new cost.
Expected Misclassification Cost per Observation
Suppose you have
Nobs observations that you want to classify with a trained discriminant analysis classifier
obj. Suppose you have
K classes. You place the observations into a matrix
Xnew with one observation per row. The command
[label,score,cost] = predict(obj,Xnew)
returns, among other outputs, a cost matrix of size
K. Each row of the cost matrix contains the expected (average) cost of classifying the observation into each of the