Clustering of strata means based on pairwise l1 regularized empirical likelihood

Doctoral Dissertation Clustering of strata means based on pairwise L1 regularized empirical likelihood Department of Statistics Graduate School of Chonnam National University Nong Quynh Van August 2019 Doctoral Dissertation Clustering of strata means based on pairwise L1 regularized empirical likelihood Department of Statistics Graduate School of Chonnam National University Nong Quynh Van Directed by Professor Chi Tim Ng Department of Statistics, Chonnam National University August 2019 Clustering of strata means based on pairwise L1 regularized empirical likelihood Department of Statistics Graduate School of Chonnam National University Nong Quynh Van The dissertation entitled above, by the graduate student Nong Quynh Van, in partial fulfillment of the requirements for the Doctor of Philosophy in Statistics has been deemed by the Professors below Jang Sun Baek, Ph.D Jae Sik Jeong, Ph.D Myung Hwan Na, Ph.D Chi Tim Ng, Ph.D Woo Joo Lee, Ph.D August 2019 i Dedicated to My Parents, My Younger Brother and My Family Acknowledgements First and foremost, I am cordially thankful for my supervisor, Dr Chi Tim Ng who offered the opportunity to study PhD degree for me Without his support, guidance, knowledge and patience, this research would not have been possible I dare to say that he is a good PhD advisor I wish to express my gratitude to the Vietnamese Ministry of Education and Training I would like to thank all students and staff of the Department of Statistics of Chonnam National University for their friendly and good social environment I also would like to thank members of the Mathematical Statistics Research Laboratory, especially Nguyen Van Cuong, Zhang Lili, Zhang Kaimeng, 박진경, 김태경 for their great help and friendship I would like to dedicate this thesis to my parents and my brother who have given me the eternal love and have encouraged me to pursue the long academic journey Without their sacrifices, I would not have finished my studies and my thesis Finally, I would like to take this opportunity to thank my husband for his love and unconditional support The last word goes to my lovely children, Ngo Bao Chau and Ngo Duc Thanh who have brought me the true meaning of love and have given me the strength to get all things done ii Contents List of Tables v List of Figures vi Abstract vii Introduction and Previous Work 1.1 Introduction 1.2 Outline of the dissertation Strata Mean Clustering via Regularized Empirical Likelihood 2.1 Introduction 2.2 L1 Regularized Empirical Likelihood Estimation 2.3 Familywise Error Rate and Bayesian Information Criterion 2.4 Algorithm 2.4.1 One-population m-strata Case 2.4.2 Two-population m-strata Case 11 2.5 Consistency Theory 12 2.5.1 Main Theorems 12 2.5.2 Proofs of Main Theorems 14 2.5.3 Technical Lemmas 19 2.6 Simulation studies 22 2.7 Real Data examples 32 2.7.1 Example 1: Chronic Myelogenous Leukemia Survival Data 32 2.7.2 Example 2: Investigating Structural Change and Monday Effect in the Stock Market 33 iii CONTENTS iv 2.7.3 Example 3: Microarray Data of Breast Cancer Patients 34 2.8 Discussion 39 Deriving hypotheses testing via penalized empirical likelihood 40 3.1 Introduction 40 3.2 One-sample mean test with empirical likelihood 41 3.3 Two samples mean test with empirical likelihood 43 3.4 Simulation studies 45 3.4.1 One-sample Mean Tests based on Empirical Likelihood 45 3.4.2 Two-sample Mean Tests based on Empirical Likelihood 45 3.5 Real data examples 48 3.6 Discussion 49 Discussion and Conclusion 51 References 53 초록 58 Appendix 59 Appendix A Calculation of Derivatives in section 3.2 59 List of Tables 2.1 Mis-classification Rate for Chisquare distribution 25 2.2 Mis-classification Rate for Gamma distribution with υ = 26 2.3 Compare the influence of variance of Gamma distribution to the performance 27 2.4 The influence of variance of Gamma distribution in small distance between cluster’s means case 27 2.5 Cumulative propotion of number groups in k-th cluster for m=40 28 2.6 Cumulative propotion of number groups in k-th cluster for m=200 29 2.7 Mis-classification Rate for Chi-square distributed Unbalanced Data 30 2.8 Mis-classification Rate for Example 2-Gamma with fixed variance 31 2.9 Detected cluster treatments 36 2.10 Detected cluster absolute returns of years 36 2.11 Detected cluster genes 36 3.1 Powers of one-sample mean penalized empirical likelihood test and empirical likelihood ratio test PLT for Penalized Likelihood Test ELRT for Empirical Likelihood Ratio Test 46 3.2 Powers of two-sample means test via penalized empirical likelihood and empirical likelihood ratio test PLT for Penalized Likelihood Test ELRT for Empirical Likelihood Ratio Test 47 3.3 Statistic values and Critical values of penalized empirical likelihood test and empirical likelihood ratio test PLT for Penalized Likelihood Test ELRT for Empirical Likelihood Ratio Test 49 v List of Figures 2.1 Box plot for comparison average of absolute returns between clusters 36 2.2 Data arranged using Heatmap 37 2.3 Box plot for comparison average of means between clusters 38 vi Abstract To determine the pairwise equalities-in-mean of a vast amount of subsamples with unknown distributions, a clustering approach is developed based on L1 regularized empirical likelihood Under the clustering approach, all possible contradictory conclusions are ruled out automatically On the contrary, the decision rules based on many existing pairwise comparison procedures can generate contradictory results Moreover, under certain mild conditions, the proposed clustering method enjoys the consistency property that with probability going to one, the equalities-in-mean of all pairs of subsamples can be determined correctly An exterior point algorithm is presented for the clustering The applications of the proposed methods are demonstrated using stock market data and microarray data of breast cancer patients vii Chapter Deriving hypotheses testing via penalized empirical likelihood 48 3.5 Real data examples In this example, the traditional empirical likelihood ratio test and penalized em- pirical likelihood test are discussed in two-sample mean test problems Consider the survival data without censoring collected in a randomized trial involving three treatments on Chronic Myelogenous Leukemia, see Hehlmann et al The sample size of the three treatment groups are 120, 195 and 192 respectively The mean survival times of each pair of treatment groups are compared with the level of significance α = 0.05 Let µ1 and µ2 be the mean survival time of treatment and The penalized empirical likelihood test rejects the hypothesis H0 : µ1 = µ2 if |γ(0)| > λ, where γ(0) and λ are defined in section 2.1 The empirical likelihood ratio test rejects H0 if −2 log( ELR) as described in section 2.1 is less than the critical value χ21 (1 − α) The test statistics and their critical values are reported in Table 6.1 Table shows that penalized empirical likelihood test and empirical likelihood ratio traditional test give the same conclusion for three pairs of treatments For both pairs of treatments (1,2) and (1,3), both tests reject the null hypothesis H0 It means that the mean survival time of Treatment is different from those of Treatment and For the pairs (2,3), both tests suggest that Treatment and Treatment share the same mean survival time Chapter Deriving hypotheses testing via penalized empirical likelihood 49 Table 3.3: Statistic values and Critical values of penalized empirical likelihood test and empirical likelihood ratio test PLT for Penalized Likelihood Test ELRT for Empirical Likelihood Ratio Test Paired Treatment Method (T1 , T2 ) (T1 , T3 ) (T2 , T3 ) 3.6 Statistic Value Critical Value PLT 10.0084 8.0427 ELRT 5.4960 3.8414 PLT 10.3759 8.1178 ELRT 5.6849 3.8414 PLT 0.0174 10.2080 ELRT 1.11-05 3.8414 Discussion Recently, the penalized empirical likelihood method has been widely used in sta- tistical inference In this chapter, we show that by choosing tuning parameter appropriately, the penalized empirical likelihood lead to testing procedures whose probability of committing Type I error can be controlled at a predetermined level The resulting penalized empirical likelihood tests perform comparably to the traditional empirical likelihood ratio tests in terms of the power of the tests It is an interesting future research direction to extend the idea of penalized empirical likelihood method to the multiple test problems In some situations, one would be interested in two or more null hypotheses involving two or more scalar-valued functions, say H01 : g1 (µ) = , H02 : g2 (µ) = , , H0p : g p (µ) = In a simultaneous test, one either accept all nulls or reject all nulls That is to consider the null H0 : all H01 , H02 , , H0p hold against H1 : not H0 In a multiple test, accepting some of the nulls is allowed Let G (µ) = ( g1 (µ), g2 (µ), , g p (µ))T The penalized empirical likelihood func- Chapter Deriving hypotheses testing via penalized empirical likelihood 50 tion for the simultaneous test can be defined as p ni ∑ ∑ log{1 + τ T f (Xij ; µ)} − λ G (µ) i =1 j =1 Here, · is the Euclidean norm A similar penalty is considered in Yuan and Lin (2006) for grouped variable selections in the context of regression analysis To allow higher degree of flexibility, p ni ∑ ∑ log{1 + τ T f (Xij ; µ)} − λ[GT (µ)ΩG(µ)]1/2 i =1 j =1 can also be used Here, Ω is some positive-definite matrix and is allowed to be depending on the nuisance parameter θ The penalized likelihood function for the multiple test can be defined as p ni ∑ ∑ log{1 + τ i =1 j =1 p T f (Xij ; µ)} − λ ∑ | g j (µ)| j =1 The critical value λ can be selected so as to control certain error rates at a fixed level of 0.05 or 0.01 The family-wise error rate and the false discovery rate as defined in Benjamini and Hochberg (1995) are commonly used General discussion on the multiple comparison methods can be found in Hochberg and Tamhane (1987) and Miller (1981) The full mathematical treatment of the error control under penalized likelihood framework worths another paper and is not given here This can be an interesting future research direction Chapter Discussion and Conclusion In this dissertation, we reformulated and implemented a new pairwise L1 regularized empirical likelihood method to cluster strata means Our proposed method used a penalty on pairwise differences between the strata means to achieve the sparsity of pair-differences estimation and the merging of the cluster strata means To avoid having to specify a parametric family for data, non-parametric empirical likelihood approach of Owen (1998) is adopted We also have derived selection consistency of the proposed penalized empirical likelihood method (see Theorem and Theorem 2) To illustrate the method, we simulated data from Gamma distribution, Chi-square distribution The influence of variance to the performance is considered In all cases, simulation results showed very good finite sample performance of the selection consistency and classification We also applied the new method for Breast Cancer data, Chronic Myelogenous Leukemia Survival data and Stock Market data Overall,the highlight of this approach is avoiding intransitive conclusions and all strata are classified correctly with large probability Both theory and numerical examples confirm the merits of the new pairwise comparisons approach One interesting thing that hypothesis testing problem is equivalent to penalized likelihood estimation problem Based on this point of view, we believe that clustering can be considered as multiple hypothesis testing problem In Chapter of this dissertation, we also reformulated one hypothesis testing using L1 regularized empirical likelihood The tuning parameter in the L1 penalty plays the same role as the level of significance in the traditional hypothesis testing problem Additionally, the link between the proposed method and the well-known empirical likelihood ratio test Owen (2001) is furnished Simulation studies and real data examples further confirm 51 Chapter Discussion and Conclusion 52 the usefulness of this method One interesting extension of the approach is to explore L1 regularized empirical likelihood to cluster strata means in high-dimensional problem The other interesting research direction is to explore the selection consistency of BIC for pair differences estimation References Andreou E., Ghysels E (2002), Detecting multiple breaks in financial market volatility dynamics, Journal of Applied Econometrics, 17, 579–600 Agresti, A., Bini, M., Bertaccini, B., and Ryu, E (2008), Simultaneous conficence interval for comparing binomial parameters, Biometrics, 64, 1270–1275 Cao, R and Van Keilegom, I (2006), Empirical likelihood tests for two-sample problems via nonparametric density estimation, The Canadian Journal of Statistics, 34, 61— 77 Coutts, J A and Hayes, P A (1999), The Weekend Effect, the Stock Exchange Account and the Financial Times 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B., and Cui, X (2010), General solutions to consistency problems in multiple hypothesis testing, Biometrical Journal, 52, 735–746 Zhu, X and Qu, A (2018) Cluster analysis of longitudinal profiles with subgroups Electronic Journal of Statistics, 12, 171–193 Zou, H (2006), The Adaptive Lasso and its oracle properties, Journal of American Statistical Association, 101, 1418–1429 초록 알려지지 않은 분포를 가진 엄청난 양의 서브 샘플의 쌍의 등식을 결정하기 위해, L1 정규화 된 경험적 우도를 기반으로 클러스터링 접근법이 개발됩니다 클러스터링 접근법에 서 가능한 모든 모순되는 결론은 자동으로 배제됩니다 반대로, 많은 기존 쌍 비교 절차를 기반으로 한 결정 규칙은 모순 된 결과를 생성 할 수 있습니다 더욱이, 특정 온화한 조건 하에서 제안 된 클러스터링 방법은 일관성 특성을 가지므로, 확률이 1이되면 모든 서브 샘플 쌍들의 평균이 동일하게 정확하게 결정될 수있다 외부 포인트 알고리즘은 클러스터링을 위 해 제공됩니다 제안 된 방법의 응용은 유방암 환자의 주식 시장 데이터 및 마이크로 어레이 데이터를 사용하여 시연된다 58 Appendix A Calculation of Derivatives in section 3.2 In this section, the formulas necessary to the computations in Step of the algorithm in section 3.2 are provided Step 3: Fix all θij update τ1 , , τm at the same time Use Newton method to solve unconstrained minimization problem Q∗∗∗ (τ, θ ) = m β ∑ fi (τi ) + ∑(ai − a j − θij )2 i =1 i< j Gradient of Q∗∗∗ : τi Q ∗∗∗ = m ∂ f i (τi ) + β ∑( − a j − θij ) ∂τi i< j Let f i = k1i + g1i + k2i + g2i , where n (1) k1i = ∑ log (1) (1) n(1) + ηi ( xij − µi ) , j =1 n (2) k2i = ∑ log (1) n (2 ) + ηi ( µ i (2) − xij + ) , j =1 g1i = g2i = β h ; h1i = 1i β h ; h2i = 2i (1) (1) ni n(1) ( xij − µi ) j =1 n(1) + ηi ( xij − µi ) ∑ (1) (1) n (2) ( µ i j =1 n (2 ) + ηi ( µ i − xij + ) (1) 59 , (2) ni ∑ (1) (2) − xij + ) Chapter A Calculation of Derivatives in section 3.2 Then, v1 = 60 f i = (v1 , v2 , v3 ) The derivatives of f i are as follows, ∂ fi (1) ∂h1i = βh1i ∂µi (1) ∂µi n (1) −∑ j =1 ηi (1) (1) n(1) + ηi ( xij − µi ) v2 = ∂h ∂h ∂ fi = h1i + βh1i 1i + h2i + βh2i 2i , ∂ηi ∂ηi ∂ηi v3 = ∂ fi ∂k ∂g = 2i + 2i = ∂ai ∂ai ∂ai n (2) ∂h2i + βh2i (1) ∂µi n (2) ηi ( (2) + η ( µ ) − x (2) j =1 n i i ij ∑ +∑ (1) n (2 ) + ηi ( µ i (2) − xij + ) ∂h2i , ∂ai + βh2i + ) j =1 ηi and ∂h1i ∂ηi = −n ∂h1i (1) (1) ∂h2i ∂ai j =1 n(1) + ηi ( xij − µi ) ∑ (n(1) + ηi ( xij − µi ))2 (1) (1) = = −∑n (1) , n (2) ( n (2) ) j =1 (n(2) + ηi (µi − xij + ))2 ∑ (1) (1) ni , (1) j =1 ∂µi ∂h2i ∂ηi (1) ( n (1) ) ∂h2i = (1) ( xij − µi ) n (1) = −∑ ∂µi (1) n (1) µi (2) (1) , (2) − xij + n (2 ) + ηi ( µ i j =1 (2) (2) − xij + ) Hessian matrix of Q∗∗∗ = ∂2 f i ∂ηi2 = (1) ∂ ( µ i )2 ∂2 f i (τi ) + β ( m − 1) , ∂τi2 Q∗∗∗ τi ,τj = − β , for i = j , (2) ∂2 f i ∂a2i ∂2 f i Q∗∗∗ τi ,τi = n ∂2 k2i ∂2 g2i + = − ∑ ∂a2i ∂a2i j =1 ∂h1i +β ∂ηi n (1) = −∑ j =1 ∂h1i ∂ηi j =1 (1) n (2 ) + ηi ( µ i + h1i ∂2 h1i ∂ηi2 + (1) (1) n(1) + ηi ( xij − µi ) +β (1) (2) − xij + ) + h2i + h1i (1) +β ∂h2i ∂ηi ∂µi ηi n (2 ) + ηi ( µ i (2) ∂h1i ∂h2i (1) ∂µi ∂h2i ∂ai +β − xij + ) ∂h2i +β ∂ηi ηi n (2) −∑ ηi ∂2 h2i ∂ηi2 + h2 , ∂2 h1i (1) ∂ ( µ i )2 + h2i ∂2 h2i (1) ∂ ( µ i )2 , ∂2 h2i ∂a2i , , Chapter A Calculation of Derivatives in section 3.2 ∂2 f i ∂ai ∂ηi ∂2 f i (1) ∂ai ∂µi ∂2 f i (1) ∂h2i ∂h2i ∂h2i ∂2 h2 +β · + h2 ∂ai ∂ηi ∂ai ∂ai ∂ηi = n (2) (1) n (2 ) + ηi ( µ i j =1 ∂h1i = (1) ∂ηi ∂µi , ηi = −∑ ∂h1i +β · (1) ∂µi ∂µi 61 ∂h2i +β (2) − xij + ) ∂h2i ∂2 h2i + h2 (1) ∂ai ∂ai ∂µ · (1) ∂µi ∂h1i ∂2 h1i + h1i (1) ∂ηi ∂ηi ∂µ i ∂h2i + +β (1) ∂µi i ∂h2i (1) · ∂µi ∂h2i ∂2 h2i + h2i (1) ∂ηi ∂ηi ∂µ i where ∂2 h1i ∂a2i ∂2 h1i ∂ηi2 ∂2 h1i ∂ ( µ i )2 ∂2 h1i (1) = 0, = 2n (1) (1) n (1) ∑ (1) , (1) n(1) + ηi ( xij − µi ) j =1 = −2( n (1) xij − µi (1) ) n (1) ∑ j =1 ηi (1) (1) n(1) + ηi ( xij − µi ) , = 0, ∂ai ∂µi ∂2 h1i ∂ai ∂ηi ∂2 h1i (1) ∂ηi ∂µi ∂2 h2i ∂a2i ∂2 h2i ∂ηi2 ∂2 h2i ∂ai ∂ηi ∂2 h2i (1) ∂ai ∂µi = 0, = ( xij − µi ) j =1 n(1) + ηi ( xij − µi ) ∂2 h2i (1) ∂ ( µ i )2 = 2n (2) (1) ∑ ∂2 h2i (1) (2) ) (2) (1) = −2( n (2) ) ∑ ∑ j =1 , (1) , (2) (µi − xij + ) (1) n (2 ) + ηi ( µ i (2) − xij + ) ηi n (2) , (2) n (2) (2) − xij + ) − xij + ) j =1 n (2) ηi (1) n (2 ) + ηi ( µ i − xij + n (2 ) + ηi ( µ i ∂ηi ∂µi = −2( n ∑ ) µi , n (2) (2) j =1 n (2) (1) (1) = −2( n j =1 = (1) (1) n (1) = ( n (1) ) ∑ (1) + ηi ( µ i (2) , − xij + ) , Publications Quynh Van, Nong and Chi Tim, Ng (2017), Pairwise multiple comparison with empirical likelihood, Contributed publication and presentation in: The Korean Statistical Society Autumn Conference 2017 Proceedings Nong, Q.V., Ng, C.T., Lee, W et al (2018), Hypothesis testing via a penalized-likelihood approach, Journal of the Korean Statistical Society, https://doi.org/10.1016/j.jkss.2018.11.005 Quynh Van, Nong and Chi Tim, Ng (2018), Clustering of subsample means based on pairwise L1 regularized empirical likelihood, Annals of the Institute of Statistical Mathematics, has been submitted for review 62 ... Dissertation Clustering of strata means based on pairwise L1 regularized empirical likelihood Department of Statistics Graduate School of Chonnam National University Nong Quynh Van Directed by Professor... Department of Statistics, Chonnam National University August 2019 Clustering of strata means based on pairwise L1 regularized empirical likelihood Department of Statistics Graduate School of Chonnam... developed based on L1 regularized empirical likelihood Under the clustering approach, all possible contradictory conclusions are ruled out automatically On the contrary, the decision rules based on