Prediction Using Discriminant Analysis Models - MATLAB & Simulink (original) (raw)
predict uses three quantities to classify observations: posterior probability, prior probability, and cost.
predict classifies so as to minimize the expected classification cost:
where
- y^ is the predicted classification.
- K is the number of classes.
- P^(k|x) is the posterior probability of class k for observation x.
- C(y|k) 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.
Posterior Probability
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 Σ_k at a 1-by-d point x is
where |Σk| is the determinant of Σ_k_, and Σk−1 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).
Prior Probability
The prior probability is one of three choices:
'uniform'— The prior probability of classkis 1 over the total number of classes.'empirical'— The prior probability of classkis the number of training samples of classkdivided by the total number of training samples.- A numeric vector — The prior probability of class
kis thejth element of thePriorvector. See fitcdiscr.
After creating a classifier obj, you can set the prior using dot notation:
where 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.
Cost
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, 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.
You can set any cost matrix you like when creating a classifier. Pass the cost matrix in the Cost name-value pair in fitcdiscr.
After you create a classifier obj, you can set a custom cost using dot notation:
B is a square matrix of size K-by-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 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
where
- K is the number of classes.
- P^(i|X(n)) is the posterior probability of class i for observation Xnew(n).
- C(k|i) is the cost of classifying an observation as k when its true class is i.