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Certifying smoothness of discrete functions and measuring legitimacy of images

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Journal of Complexity 1989年第3期5卷 261-270页

作者： Steele, J.Michael Program in Statistics and Operations Research School of Engineering and Applied Science Princeton University Princeton NJ 08544 United States

Several problems that deal with smoothness of discrete functions and that are motivated by the task of deciding whether an N × N array of distances is a feasible laser radar (LADAR) image are investigated. The main observation is that there is a simple and explicit graph that is tightly related to the smoothness problem. The graph is used to show that a function can be verified to be Lipschitz on a random finite subset of R² by testing the Lipschitz condition at only O(n) pairs of points. © 1989.

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Generalized differentiability for a class of nondifferentiable operators with applications to nonsmooth optimization

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Journal of the Australian Mathematical Society 1989年第1期47卷 114-132页

作者： Reiland, Thomas W. Department of Statistics and Graduate Program Operations Research North Carolina State University Raleigh 27695-8203 Box 8203 United States

A theory of generalized gradients is presented for a class of Lipschitz vector-valued mappings from a Banach space to a locally convex order complete vector lattice. Necessary optimality conditions are obtained for nonconvex programming problems on Banach spaces with vector-valued operator constraints and/or an arbitrary set constraint. Sufficient optimality conditions are also obtained under mild convexity assumptions. 1980 Mathematics subject classification {Amer, Math. Soc.) (1985 Revision): Primary 49 B 27, Secondary 90 C 48. © 1989, Australian Mathematical Society. All rights reserved.

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Fractal and dynamical language methods to construct phylogenetic tree based on protein sequences from complete genomes

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Fractal and dynamical language methods to construct phylogen...

First International Conference on Natural Computation, ICNC 2005

作者： Yu, Zu-Guo Anh, Vo Zhou, Li-Quan Program in Statistics and Operations Research Queensland University of Technology GPO Box 2434 Brisbane QLD 4001 Australia School of Mathematics and Computing Science Xiangtan University Hunan 411105 China

The complete genomes of living organisms have provided much information on their phylogenetic relationships. In the past few years, we proposed three alternative methods to model the noise background in the composition vector of protein sequences from a complete genome. The first method is based on the frequencies of the 20 kinds of amino acids appearing in the genome and the multiplicative model. The second method is based on the iterated function system model in fractal geometry. The last method is based on the relationship between a word and its two sub-words in the theory of symbolic dynamics. Here we introduce these methods. The complete genomes of prokaryotes and eukaryotes are selected to test these algorithms. Our distance-based phylogenetic tree of prokaryotes and eukaryotes agrees with the biologists' "tree of life" based on the 16S-like rRNA genes in a majority of basic branches and most lower taxa. © Springer-Verlag Berlin Heidelberg 2005.

关键词： Genetic engineering

Almost-sure exponential behavior of a stochastic anderson model with continuous space parameter

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Stochastics 1998年第3-4期62卷 251-273页

作者： Ren.A. Carmona[a] Frederi G. Viens[b] [a] Statistics and Operations Research Program Princeton University Princeton [b] Department of Mathematics University of North Texas Denton

We study the almost-sure large time exponential growth of the solution to a linear stochastic parabolic equation with continuous space parameter, and a Gaussian-correlated potential that is white noise in time and homogeneous in space. We use the evolution form of that equation,for which existence and uniqueness are known. We establish a Feynman-Kac formula for the solution. By using a method of discretization of time and space, we prove that for small diffusion parameter κ, there is a deterministic constant c such that almost surely

关键词： Stochastic partial differential equation Lyapunov exponent spatial discretization Gaussian estimates stochastic Feynman-Kac formulastochastic evolution equation AMS Subject Classification Primary: 60H15 Secondary: 60G17

Covariance estimation via fiducial inference

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arXiv 2017年

作者： Shi, W. Jenny Hannig, Jan Lai, Randy C.S. Lee, Thomas C.M. Computational Bioscience Program University of Colorado Department of Statistics & Operations Research University of North Carolina Department of Mathematics & Statistics University of Maine Department of Statistics University of California

As a classical problem, covariance estimation has drawn much attention from the statistical community for decades. Much work has been done under the frequentist and the Bayesian frameworks. Aiming to quantify the uncertainty of the estimators without having to choose a prior, we have developed a fiducial approach to the estimation of covariance matrix. Built upon the Fiducial Berstein-von Mises Theorem (Sonderegger and Hannig 2014), we show that the fiducial distribution of the covariate matrix is consistent under our framework. Consequently, the samples generated from this fiducial distribution are good estimators to the true covariance matrix, which enable us to define a meaningful confidence region for the covariance matrix. Lastly, we also show that the fiducial approach can be a powerful tool for identifying clique structures in covariance matrices.62J10, 62E20, 62F25, 62F12 Copyright © 2017, The Authors. All rights reserved.

关键词： Covariance matrix

Bayesian change point detection for functional data

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arXiv 2018年

作者： Li, Xiuqi Ghosal, Subhashis Operations Research Graduate Program North Carolina State University United States Department of Statistics North Carolina State University United States

We propose a Bayesian method to detect change points for functional data. We extract the features of a sequence of functional data by the discrete wavelet transform (DWT), and treat each sequence of feature independently. We believe there is potentially a change in each feature at possibly different time points. The functional data evolves through such changes throughout the sequences of observations. The change point for this sequence of functional data is the cumulative effect of changes in all features. We assign the features with priors which incorporate the characteristic of the wavelet coefficients. Then we compute the posterior distribution of change point for each sequence of feature, and define a matrix where each entry is a measure of similarity between two functional data in this sequence. We compute the ratio of the mean similarity between groups and within groups for all possible partitions, and the change point is where the ratio reaches the minimum. We demonstrate this method using a dataset on climate change. Copyright © 2018, The Authors. All rights reserved.

关键词： Bayesian networks

On the cross-validation bias due to unsupervised preprocessing

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arXiv 2019年

作者： Moscovich, Amit Rosset, Saharon Program in Applied and Computational Mathematics Princeton University PrincetonNJ United States Department of Statistics and Operations Research Tel-Aviv University Tel-Aviv Israel

Cross-validation is the de facto standard for predictive model evaluation and selection. In proper use, it provides an unbiased estimate of a model's predictive performance. However, data sets often undergo various forms of data-dependent preprocessing, such as mean-centering, rescaling, dimensionality reduction, and outlier removal. It is often believed that such preprocessing stages, if done in an unsupervised manner (that does not incorporate the class labels or response values) are generally safe to do prior to cross-validation. In this paper, we study three commonly-practiced preprocessing procedures prior to a regression analysis: (i) variance-based feature selection;(ii) grouping of rare categorical features;and (iii) feature rescaling. We demonstrate that unsupervised preprocessing can, in fact, introduce a substantial bias into cross-validation estimates and potentially hurt model selection. This bias may be either positive or negative and its exact magnitude depends on all the parameters of the problem in an intricate manner. Further research is needed to understand the real-world impact of this bias across different application domains, particularly when dealing with small sample sizes and high-dimensional *** Codes 62-07 Copyright © 2019, The Authors. All rights reserved.

关键词： Regression analysis

Multifractal and correlation analyses of protein sequences from complete genomes

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Physical Review E 2003年第2期68卷 021913-021913页

作者： Zu-Guo Yu Vo Anh Ka-Sing Lau Program in Statistics and Operations Research Queensland University of Technology GPO Box 2434 Brisbane Q4001 Australia Department of Mathematics Xiangtan University Hunan 411105 China Department of Mathematics Chinese University of Hong Kong Shatin Hong Kong China

A measure representation of protein sequences similar to the measure representation of DNA sequences proposed in our previous paper [Yu et al., Phys. Rev. E 64, 031903 (2001)] and another induced measure are introduced. Multifractal analysis is then performed on these two kinds of measures of a large number of protein sequences derived from corresponding complete genomes. From the values of the Dq (generalized dimensions) spectra and related Cq (analogous specific heat) curves, it is concluded that these protein sequences are not completely random sequences. For substrings with length K=5, the Dq spectra of all organisms studied are multifractal-like and sufficiently smooth for the Cq curves to be meaningful. The Cq curves of all bacteria resemble a classical phase transition at a critical point. But the “analogous” phase transitions of higher organisms studied exhibit the shape of double-peaked specific heat function. But for the classification problem, the multifractal property is not sufficient. When the measure representations of protein sequences from complete genomes are considered as time series, a method based on correlation analysis after removing some memory from the time series is proposed to construct a phylogenetic tree. This construction is shown to be reasonably satisfactory.

关键词： Proteins

Bayesian classification of multiclass functional data

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arXiv 2018年

作者： Li, Xiuqi Ghosal, Subhashis Operations Research Graduate Program North Carolina State University RaleighNC27695 United States Department of Statistics North Carolina State University RaleighNC27695 United States

We propose a Bayesian approach to estimating parameters in multiclass functional models. Unordered multinomial probit, ordered multinomial probit and multinomial logistic models are considered. We use finite random series priors based on a suitable basis such as B-splines in these three multinomial models, and classify the functional data using the Bayes rule. We average over models based on the marginal likelihood estimated from Markov Chain Monte Carlo (MCMC) output. Posterior contraction rates for the three multinomial models are computed. We also consider Bayesian linear and quadratic discriminant analyses on the multivariate data obtained by applying a functional principal component technique on the original functional data. A simulation study is conducted to compare these methods on different types of data. We also apply these methods to a phoneme dataset. Copyright © 2018, The Authors. All rights reserved.

关键词： Discriminant analysis