模式识别6.ppt

人有了知识，就会具备各种分析能力， 明辨是非的能力。 所以我们要勤恳读书，广泛阅读， 古人说“书中自有黄金屋。 ”通过阅读科技书籍，我们能丰富知识， 培养逻辑思维能力； 通过阅读文学作品，我们能提高文学鉴赏水平， 培养文学情趣； 通过阅读报刊，我们能增长见识，扩大自己的知识面。 有许多书籍还能培养我们的道德情操， 给我们巨大的精神力量， 鼓舞我们前进。 * DECISION SURFACES-- NETWORK REPRESENTATION A classifier can be visualized as a connected graph with arcs and weights: 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 precomputed. GAUSSIAN CLASSIFIERS -- DISCRIMINANT FUNCTIONS We can use an equivalent linear discriminant function: Decide on the state of nature: GAUSSIAN CLASSIFIERS -- DISCRIMINANT FUNCTIONS Case: ?i = ? Decide on the state of nature: GAUSS