模式识别6剖析.pptVIP

模式识别原理 华中科技大学图像识别与人工智能研究所 图像分析与智能系统研究室 曹治国 Where in case of multi-categories X is an observation for which: (1)if True state of nature = ?i (2)if True state of nature = ?i (3)if True state of nature = ?i (4)if True state of nature = ?i 一个例子 DECISION SURFACES-- LOG PROBABILITIES Some monotonically increasing functions can simplify calculations considerably: What are some of the reasons (3) is particularly useful? Computational complexity (e.g., Gaussian) Numerical accuracy (e.g., probabilities tend to zero) Decomposition (e.g., likelihood and prior are separated and can be weighted differently) Normalization (e.g., likelihoods are channel dependent). DECISION SURFACES-- TWO-CATEGORY CASE A classifier that places a pattern in one of two classes is often referred to as a dichotomizer. We can reshape the decision rule: If we use log of the posterior probabilities: A dichotomizer can be viewed as a machine that computes a single discriminant function and classifies x according to the sign (e.g., support vector machines). DECISION SURFACES-- NORMAL DISTRIBUTIONS Recall the definition of a normal distribution (Gaussian): Mean: Covariance: GAUSSIAN CLASSIFIERS -- DISCRIMINANT FUNCTIONS Recall our discriminant function for minimum error rate classification: For a multivariate normal distribution: GAUSSIAN CLASSIFIERS -- DISCRIMINANT FUNCTIONS Consider the case: ?i = ?2I (statistical independence, equal variance, class-independent variance) GAUSSIAN CLASSIFIERS -- DISCRIMINANT FUNCTIONS The discriminant function can be reduced to: Since these terms are constant w.r.t. the maximization: We can expand this: The term xtx is a constant w.r.t. i, and ?it?i is a constant that can be pr