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详细说明： 模式识别第一章作业题，中科院刘成林，Question 1 (Pattern Classification, Chapter 2, Problem 12) Let ωmax(x) be the state of nature for which P(ωmax|x) ≥ P(ωi|x) for all i, i = 1,...,c. (a) Show that P(ωmax|x) ≥ 1/c (b) Show that for the minimum-error-rate decision rule the average probability of error is given by P(error) = 1−RP(ωmax|x)p(x)dx (c) Use these two results to show that P(error) ≤ (c−1)/c (d) Describe a situation for which P(error) = (c−1)/c Question 2 (Pattern Classification, Chapter 2, Problem 13) In many pattern classification problems one has the option either to assign the pattern to one of c classes, or to reject it as being unrecognizable. If the cost for rejects is not too high, rejection may be a desirable action. Let λ(αi|ωi) = 0 i = j i,j = 1,...,c λr i = c + 1 λs otherwise where λr is the loss incurred for choosing the (c + 1)th action, rejection, and λs is the loss incurred for making a substitution error. Show that the minimum risk is obtained if we decide ωi if P(ωi|x) ≥ P(ωi|x) for all j and if P(ωi|x) ≥ 1− λr λs , and reject otherwise. What happens if λr = 0? What happens if λr > λs? Question 3 Now we have N samples, and each sample xi, i = 1,...,N has d-dimensions. Please provide us the proofs and the pseudo-codes of PCA algorithm Question 4 (Pattern Classification, Chapter 2, Problem 10) Consider the following decision rule for a two-category one-dimensional problem: Decide ω1 if x > θ; otherwise decide ω2. (a)Showtheprobabilityoferrorforthisruleisgivenby P(error) = P(ω1)Rθ−∞p(x|ω1)dx+P(ω2)R∞ θ p(x|ω2)dx (b) By differentiating, show that a necessary condition to minimize P(error) is that θ satisfy p(θ|ω1)P(ω1) = p(θ|ω2)P(ω2) (c) Does this equation define θ uniquely? (d) Give an example where a value of θ satisfying the equation actually maximizes the probability of error. Question 5 (Pattern Classification, Chapter 2, Problem 24) Consider the multivariate normal density for which σij = 0 and σii = σ2 i , i.e., Σ = diag(σ2 1,σ2 2,...,σ2 d). (a) Show that the evidence is p(x) = 1 Qd i=1 √2πσi exp[−1 2Pd i=1(xi−µi σi )2]. (b) Plot and describe the contours of constant density. (c) Write an expression for the Mahalanobis distance from x to µ. 1 Question 6 (Pattern Classification, Chapter 2, Problem 32) Let p(x|ωi) ∼ N(µi,σ2I) for a two-category d-dimensional problem with P(ω1) = P(ω2) = 1 2. (a) Show that the minimum probability of error is given by Pe = 1 √2πR∞ a e−µ2 2 dµ, where a = ||µ2 −µ1||/(2σ). (b) Let µ1 = 0 and µ = (µ1,...,µd)t. Use the inequality from [Pattern Classification, Chapter 2, Problem 31] to show that Pe approaches zero as the dimension d approaches infinity. (c) Express the meaning of this result in words. Computer Exercises Several of the computer exercises will rely on the following data: 2.11. COMPUTER EXERCISES 59 Computer exercises Several of the computer exercises will rely on the following data. ω1 ω2 ω3 sample x1 x2 x3 x1 x2 x3 x1 x2 x3 ...展开详情收缩
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