Open House, ISM in Tachikawa, 2015.6.19統計数理研究所オープンハウス（立川）、H27.6.19ポスター発

Osamu Komori

小森 理

English

RISM (Repository of the Institute of Statistical Mathematics) / 統計数理研究所 学術研究リポジトリ

2015年6月19日　統計数理研究所　オープンハウスGeneralization of t statistic and AUC by consideringheterogeneity in probability distributionsOsamu Komori Department of Mathematical Analysis and Statistical Inference,　Project Assistant Professor1 Generalized AUCWe discuss a statistical method of a classification problem for two groups. Fora binary class label y 2 f0; 1g and a covariate vector x 2 Rp, we considera statistical situation in which the neither conditional distribution of x giveny = 0 nor given y = 1 are well modelled by a specific distribution.For a sample fx0i : i = 1; : : : ; n0g for y = 0 and a sample fx1j : j =1; : : : ; n1g for y = 1 where n = n0+ n1, we propose a generalized u-statisticdefined byLU (¯) =1n0n1n0Xi=1n1Xj=1U½¯T(x1j ¡ x0i)(¯TS¯)1=2¾; (1)where U is an arbitrary real-valued function: R ! R; S is a normalizingfactor given asS =1nn0Xi=1(x0i ¡ x¯0)(x0i ¡ x¯0)> +1nn1Xj=1(x1j ¡ x¯1)(x1j ¡ x¯1)>: (2)2 Asymptotic consistency and normalityLet us consider the estimator associated with the generalized t-statistic asb¯U = argmax¯2RpLU (¯): (3)Then we consider the following assumption:(A) Ey(gy j wy = a) = 0 for all a 2 R; for y = 0; 1where wy = ¯T0 xy, gy = Qxy, Q = I ¡Σ¯0¯T0 , Σ¤y = QΣyQT, ¹0+ ¹1 = 0,and¯0 =Σ¡1(¹1 ¡ ¹0)f(¹1 ¡ ¹0)TΣ¡1(¹1 ¡ ¹0)g1=2: (4)Theorem 2.1Under Assumption (A), b¯U is asymptotically consistentwith ¯0 for any U .Next we consider the following assumption in addition to (A):(B) vary(gy j wy = a) = Σ¤y for all a 2 R; for y = 0; 1where vary denotes the conditional variance of x given y. Then we assumemixture model for class label y 2 f0; 1g.py(x) =1Xk=1²ykÁ(x; ºyk; Vyk): (5)Theorem 2.2 For y = 0; 1 assumptions (A) and (B) under the infinitemixture model in (5) are equivalent to(A0)Xk2Ky`²k(Q¡Qyk) = 0;Xk2Ky`²ykQykºyk = 0; for8` 2 N; y = 0; 1(B0)Xk2Ky`²yknQykVykQ¡QΣyQo= 0; for 8` 2 N; y = 0; 1where Qyk = Ip ¡ Vyk¯¤¯¤>=(¯¤>Vyk¯¤), Ky` = fk j ¯¤>ºyk =¯¤>ºy`; ¯¤>Vyk¯¤ = ¯¤>Vy`¯¤g.Here we assume the following semiparametric model for probability den-sity functions,py(x) = Ãy(c + ¯>x)(2¼)¡p2jΣyj¡12 exp³¡ x>Σ¡1y x2´; for y = 0; 1; (6)where Ãy is a function from R to R+ and there exists ¸y such thatΣy¯ = ¸y¯; for y = 0; 1: (7)Theorem 2.3The target parameter ¯0 is proportional to ¯ in (6) andboth assumptions (A) and (B) hold for (6).Theorem 2.4Under Assumptions (A) and (B), n1=2( b¯U¡¯0) is asymp-totically distributed as N(0;ΣU ), whereΣU = cUΣ¤0; (8)cU =E0hE1fU 0(w)gi2+ E1hE0fU 0(w)gi2+ 2½EfU 0(w)gEfU 0(w)wg ¡hEfU 0(w)wgi2hEfU 0(w)S(w) + U 0(w)wgi2 ;(9)in which S(w) = @ log f (w)=@w, f (w) is the probability density ofw = w1 ¡ w0, ½ = E(w) and U 0 denotes the first derivative of U .3 Simulation studiesWe consider normal mixtures as follows:x0 » ²0N(0; Ip) + (1¡ ²0)N(º0; Ip)x1 » ²1N(º1;V 1) + ²2N(º2;V 2) + (1¡ ²1 ¡ ²2)N(º3;V 3);where º0 = (¡2;¡0:2; : : : ;¡0:2)>, º1 = (3; 0:3; : : : ; 0:3)>, º2 =(4; 0:4; : : : ; 0:4)>, º3 = (¡1;¡0:1; : : : ;¡0:1)> 2 Rp, V 1 = V 2 = V 3 =Ip,²0 = 0:5, ²1 = ²2 = 0:1. We consider the following U functions.1. optimal-UUopt(w) = Uupper(w) + a1w + a2w2 + ¢ ¢ ¢ + amwm; (10)where the polynomial order m is determined by the cross validation of cU .2. upper-UUupper(w) = log f (w) +12w2 ¡ ½32 + ½2w: (11)3. approx-UUapprox(w) = log f (w) +½2 + ½2w (12)4. auc-UUauc(w) = Φ³w¾´; (13)where ¾ = 0:01.5. linear-U (Fisher)Ulinear(w) = w (14)00.20.40.60.81opt-Uupper-Uapprox-Uauc-Ufishersvmlogisticsquared error0.540.560.580.60.620.640.660.680.70.72opt-Uupper-Uapprox-Uauc-Ufishersvmlogistictest AUCFig1. Squared errors in upper panel and test AUC calculated by independent sample withsize 1000 in lower panel, based on 30 repetitions (p = 20 and n0 = n1 = 50)

Generalization of t statistic and AUC by considering heterogeneity in probability distributions

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Generalization of t statistic and AUC by considering heterogeneity in probability distributions

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RISM (Repository of the Institute of Statistical Mathematics) / 統計数理研究所学術研究リポジトリ

Generalization of t statistic and AUC by considering heterogeneity in probability distributions

Abstract

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RISM (Repository of the Institute of Statistical Mathematics) / 統計数理研究所 学術研究リポジトリ

RISM (Repository of the Institute of Statistical Mathematics) / 統計数理研究所学術研究リポジトリ