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CUSUM control charts based on likelihood ratio for preliminary analysis

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Science China Mathematics 2007年第1期50卷 47-62页

作者： Yi DAI Zhao-jun WANG Chang-liang ZOU lpmc and Department of Statistics School of Mathematical SciencesNankai UniversityTianjin 300071China

To detect and estimate a shift in either the mean and the deviation or both for the preliminary analysis, the statistical process control (SPC) tool, the control chart based on the likelihood ratio test (LRT), is the most popular method. Sullivan and woodall pointed out the test statistic lrt(n1, n2) is approximately distributed as x2(2) as the sample size n,n1 and n2 are very large, and the value of n1 = 2,3,..., n - 2 and that of n2 = n - n1. So it is inevitable that n1 or n2 is not large. In this paper the limit distribution of lrt(n1, n2) for fixed n1 or n2 is figured out, and the exactly analytic formulae for evaluating the expectation and the variance of the limit distribution are also obtained. In addition, the properties of the standardized likelihood ratio statistic slr(n1, n) are discussed in this paper. Although slr(n1, n) contains the most important information, slr(i, n)(i≠n1) also contains lots of information. The cumulative sum (CUSUM) control chart can obtain more information in this condition. So we propose two CUSUM control charts based on the likelihood ratio statistics for the preliminary analysis on the individual observations. One focuses on detecting the shifts in location in the historical data and the other is more general in detecting a shift in either the location and the scale or both. Moreover, the simulated results show that the proposed two control charts are, respectively, superior to their competitors not only in the detection of the sustained shifts but also in the detection of some other out-of-control situations considered in this paper.

关键词： preliminary analysis false alarm probability CUSUM chart likelihood ratio test 62P30

Lower Bound for Quantum Integration Error on Anisotropic Sobolev Classes

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Acta Mathematica Sinica,English Series 2010年第4期26卷 669-678页

作者： Pei Xin YE School of Mathematical Sciences and lpmc Nankai University Tianjin 300071 P. R. China

We study the approximation of the integration of multivariate functions in the quantum model of computation. Using a new reduction approach we obtain a lower bound of the n-th minimal query error on anisotropic Sobolev class R（Wpr（[0, 1]d））（r R＋d）. Then combining this result with our previous one we determine the optimal bound of n-th minimal query error for anisotropic Hblder- Nikolskii class R（H∞r（[0,1]d）） and Sobolev class R（W∞r（[0,1]d））. The results show that for these two types of classes the quantum algorithms give significant speed up over classical deterministic and randomized algorithms.

关键词： quantum integration anisotropic Sobolev classes Holder-Nikolskii classes n-th minimal query error

Learning Rates for Least Square Regressions with Coefficient Regularization

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Acta Mathematica Sinica,English Series 2012年第11期28卷 2205-2212页

作者： Bao Huai SHENG Pei Xin YE Jian Li WANG Department of Mathematics Shaoxing College of Arts and Sciences School of Mathematical Sciences and lpmc Nankai University

We analyze the learning rates for the least square regression with data dependent hy- pothesis spaces and coefficient regularization algorithms based on general kernels. Under a very mild regularity condition on the regression function, we obtain a bound for the approximation error by esti- mating the corresponding K：-functional. Combining this estimate with the previous result of the sample error, we derive a dimensional free learning rate by the proper choice of the regularization parameter.

关键词： Least square regressions coefficient regularization general kernel M-functional learningrates

Chaotic Dynamics of Monotone Twist Maps

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Acta Mathematica Sinica,English Series 2022年第1期38卷 179-204页

作者： Guo Wei YU Chern Institute of Mathematics and lpmc Nankai UniversityTianjin 300071P.R.China

For a monotone twist map,under certain non-degenerate condition,we showed the existence of infinitely many homoclinic and heteroclinic orbits between two periodic neighboring minimal orbits with the same rotation number,which indicates chaotic *** results also apply to geodesics of smooth Riemannian metrics on the two-dimension torus.

关键词： Monotone twist maps homoclinic and heteroclinic orbits topological entropy

The isometric extension of “into” mappings on unit spheres of AL-spaces

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Science China Mathematics 2008年第10期51卷 1904-1918页

作者： DING GuangGui School of Mathematics Science, Chern Institute of Mathematics and lpmc, Nankai University, Tianjin 300071, China School of Mathematics Science Chern Institute of Mathematics and LPMC Nankai University Tianjin China

In this paper, we show that if V0 is an isometric mapping from the unit sphere of an AL-space onto the unit sphere of a Banach space E, then V0 can be extended to a linear isometry defined on the whole space.

关键词： isometric extension (atomic) AL-space

Large Deviation Principle for the Fourth-order Stochastic Heat Equations with Fractional Noises

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Acta Mathematica Sinica,English Series 2010年第1期26卷 89-106页

作者： Yi Ming JIANG Ke Hua SHI Yong Jin WANG School of Mathematical Sciences and lpmc Nankai University Tianjin 300071 P. R. China

In this paper, we shall study a fourth-order stochastic heat equation driven by a fractional noise, which is fractional in time and white in space. We will discuss the existence and uniqueness of the solution to the equation. Furthermore, the regularity of the solution will be obtained. On the other hand, the large deviation principle for the equation with a small perturbation will be established through developing a classical method.

关键词： fourth-order stochastic heat equation fractional noise existence and uniqueness regularity large deviation principle

BOUND ON THE MAXIMUM NUMBER OF CLEAR TWO-FACTOR INTERACTIONS FOR 2^(n-(n-k)) DESIGNS

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Acta Mathematica Scientia 2008年第4期28卷 949-954页

作者：赵胜利张润楚 School of Mathematical Sciences Qufu Normal University lpmc and Department of Statistics School of Mathematical SciencesNankai University

Clear effects criterion is an important criterion for selecting fractional factorial designs[1].Tang et al.[2]derived upper and lower bounds on the maximum number of clear two-factor interactions（2fi＇s）in 2^n-（n-k）designs of resolution Ⅲ and Ⅳ by constructing 2^n-（n-k）*** the method in[2]does not perform well sometimes when the resolution is Ⅲ.This article modifies the construction method for 2^n-（n-k） designs of resolution Ⅲ in[2].The modified method is a great improvement on that used in[2].

关键词： Random fractional design clear resolution

On Construction of Optimal Two-Level Designs with Multi Block Variables

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Journal of Systems Science & Complexity 2018年第3期31卷 773-786页

作者： ZHAO Yuna ZHAO Shengli LIU Minqian lpmc and Institute of Statistics Nankai University Tianjin 300071 China School of Statistics Qufu Normal University Qufu 273165 China

When running an experiment, inhomogeneity of the experimental units may result in poor estimations of treatment effects. Thus, it is desirable to select a good blocked design before running the experiment. Mostly, a single block variable was used in the literature to treat the inhomogeneity for simplicity. However, in practice, the inhomogeneity often comes from multi block variables. Recently, a new criterion called B2-GMC was proposed for two-level regular designs with multi block variables. This paper proposes a systematic theory on constructing some B^2-GMC designs for the first time. Experimenters can easily obtain the B^2-GMC designs according to the construction method. Pros of B^2-GMC designs are highlighted in Section 4, and the designs with small run sizes are tabulated in Appendix B for practical use.

关键词： Blocked design general minimum lower order confounding multi block variables Yatesorder.

Linear recurring sequences and subfield subcodes of cyclic codes

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Science China Mathematics 2013年第7期56卷 1413-1420页

作者： GAO ZhiHan FU FangWei Department of Combinatorics and Optimization University of Waterloo Chern Institute of Mathematics and lpmc Nankai University

Linear recurring sequences over finite fields play an important role in coding theory and cryptography. It is known that subfield subcodes of linear codes yield some good codes. In this paper, we study linear recurring sequences and subfield subcodes. Let Mqm(f(x)) denote the set of all linear recurring sequences over Fqm with characteristic polynomial f(x) over Fqm . Denote the restriction of Mqm(f(x)) to sequences over Fq and the set after applying trace function to each sequence in Mqm(f(x)) by Mqm(f(x)) | Fq and Tr( Mqm(f(x))), respectively. It is shown that these two sets are both complete sets of linear recurring sequences over Fq with some characteristic polynomials over Fq. In this paper, we firstly determine the characteristic polynomials for these two sets. Then, using these results, we determine the generator polynomials of subfield subcodes and trace codes of cyclic codes over Fqm .

关键词： linear recurring sequences characteristic polynomial cyclic codes subfield subcodes trace codes