模式识别0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 Density Estimation Basic idea: Probability that a vector x will fall in region R is: P is a smoothed (or averaged) version of the density function p(x) if we have a sample of size n; therefore, the probability that k points fall in R is then: and the expected value for k is: E(k) = nP (3) If Then Therefore, the ratio k/n is a good estimate for the probability P and hence for the density function p. p(x) is continuous and that the region R is so small that p does not vary significantly within it, we can write: where is a point within R and V the volume enclosed by R. Combining equation (1) , (3) and (4) yields: Density Estimation (cont.) Justification of equation (4) We assume that p(x) is continuous and that region R is so small that p does not vary significantly within R. Since p(x) = constant, it is not a part of the sum. Where: ?(R) is: a surface in the Euclidean space R2 a volume in the Euclidean space R3 a hypervolume in the Euclidean space Rn Since p(x) ? p(x’) = constant, therefore in the Euclidean space R3: Condition for convergence The fraction k/(nV) is a space averaged value of p(x). p(x) is obtained only if V approaches zero.  This is the case where no samples are included in R: it is an uninteresting case! In this case, the estimate diverges: it is an uninteresting case! The volume V needs to approach 0 anyway if we want to use this estimation Practically, V cannot be allowed to become small since the number of samples is always limited One will have to accept a certain amount of variance in the ratio k/n Theoretically, if an unlimited number of samples is available, we can circumvent this difficulty To estimate the density of x, we form a sequence of regions R1, R2,…containing x: the first region contains one sample, the second two samples and so on. Let Vn be the volume of Rn, kn the number