Importance Sampling for p-value Computations in Multivariate Tests

Abstract

An importance sampling procedure is developed to approximate the distribution of an arbitrary function of the eigenvalues for a matrix beta random matrix or a Wishart random matrix. The procedure is easily implemented and provides confidence intervals for the p-values of many of the commonly used test statistics in multivariate analysis. An adaptive procedure allows for the control of either absolute error or relative error in this p-value estimation through the choice of importance sample size.     

Control of time-periodic systems via symbolic computation with application to chaos control

A general method for the control of linear time-periodic systems employing symbolic computation of Floquet transition matrix is considered in this work. It is shown that this method is applicable to chaos control. Nonlinear chaotic systems can be driven to a desired periodic motion by designing a combination of a feedforward controller and a feedback controller. The design of the feedback controller is achieved through the symbolic computation of fundamental solution matrix of linear time-periodic systems in terms of unknown control gains. Then, the Floquet transition matrix (state transition matrix evaluated at the end of the principal period) can determine the stability of the system owing to classical techniques such as pole placement, Routh–Hurwitz criteria, etc. Thus it is possible to place the Floquet multipliers in the desired locations to determine the control gains. This method can be applied to systems without small parameters. The Duffing’s oscillator, the Rössler system and the nonautonomous parametrically forced Lorenz equations are chosen as illustrative examples to demonstrate the application.     

An Efficient Algorithm for Rare-event Probability Estimation,Combinatorial Optimization,and Counting

Although importance sampling is an established and effective sampling and estimation technique, it becomes unstable and unreliable for high-dimensional problems. The main reason is that the likelihood ratio in the importance sampling estimator degenerates when the dimension of the problem becomes large. Various remedies to this problem have been suggested, including heuristics such as resampling. Even so, the consensus is that for large-dimensional problems, likelihood ratios (and hence importance sampling) should be avoided. In this paper we introduce a new adaptive simulation approach that does away with likelihood ratios, while retaining the multi-level approach of the cross-entropy method. Like the latter, the method can be used for rare-event probability estimation, optimization, and counting. Moreover, the method allows one to sample exactly from the target distribution rather than asymptotically as in Markov chain Monte Carlo. Numerical examples demonstrate the effectiveness of the method for a variety of applications.      

Importance Sampling Simulations of Markovian Reliability Systems Using Cross-Entropy

This paper reports simulation experiments, applying the cross entropy method such as the importance sampling algorithm for efficient estimation of rare event probabilities in Markovian reliability systems. The method is compared to various failure biasing schemes that have been proved to give estimators with bounded relative errors. The results from the experiments indicate a considerable improvement of the performance of the importance sampling estimators, where performance is measured by the relative error of the estimate, by the relative error of the estimator, and by the gain of the importance sampling simulation to the normal simulation.     

Computational Aspects of Bayesian Spectral Density Estimation

Gaussian time-series models are often specified through their spectral density. Such models present several computational challenges, in particular because of the nonsparse nature of the covariance matrix. We derive a fast approximation of the likelihood for such models. We propose to sample from the approximate posterior (i.e., the prior times the approximate likelihood), and then to recover the exact posterior through importance sampling. We show that the variance of the importance sampling weights vanishes as the sample size goes to infinity. We explain why the approximate posterior may typically be multimodal, and we derive a Sequential Monte Carlo sampler based on an annealing sequence to sample from that target distribution. Performance of the overall approach is evaluated on simulated and real datasets. In addition, for one real-world dataset, we provide some numerical evidence that a Bayesian approach to semiparametric estimation of spectral density may provide more reasonable results than its frequentist counterparts. The article comes with supplementary materials, available online, that contain an Appendix with a proof of our main Theorem, a Python package that implements the proposed procedure, and the Ethernet dataset.     

A useful technique for piecewise deterministic Markov decision processes

This note presents a technique that is useful for the study of piecewise deterministic Markov decision processes (PDMDPs) with general policies and unbounded transition intensities. This technique produces an auxiliary PDMDP from the original one. The auxiliary PDMDP possesses certain desired properties, which may not be possessed by the original PDMDP. We apply this technique to risk-sensitive PDMDPs with total cost criteria, and comment on its connection with the uniformization technique.     

Deviation matrix and asymptotic variance for GI/M/1-type Markov chains

We investigate deviation matrix for discrete-time GI/M/1-type Markov chains in terms of the matrix-analytic method, and revisit the link between deviation matrix and the asymptotic variance. Parallel results are obtained for continuous-time GI/M/1-type Markov chains based on the technique of uniformization. We conclude with A. B. Clarke＇s tandem queue as an illustrative example, and compute the asymptotic variance for the queue length for this model.     

Adaptive Monte Carlo methods for matrix equations with applications

This paper discusses empirical studies with both the adaptive correlated sequential sampling method and the adaptive importance sampling method which can be used in solving matrix and integral equations. Both methods achieve geometric convergence (provided the number of random walks per stage is large enough) in the sense: e_ν≤cλ^ν, where e_ν is the error at stage ν, λ∈(0,1) is a constant, c>0 is also a constant. Thus, both methods converge much faster than the conventional Monte Carlo method. Our extensive numerical test results show that the adaptive importance sampling method converges faster than the adaptive correlated sequential sampling method, even with many fewer random walks per stage for the same problem. The methods can be applied to problems involving large scale matrix equations with non-sparse coefficient matrices. We also provide an application of the adaptive importance sampling method to the numerical solution of integral equations, where the integral equations are converted into matrix equations (with order up to 8192×8192) after discretization. By using Niederreiter’s sequence, instead of a pseudo-random sequence when generating the nodal point set used in discretizing the phase space Γ, we find that the average absolute errors or relative errors at nodal points can be reduced by a factor of more than one hundred.     

The cross-entropy method with patching for rare-event simulation of large Markov chains

There are various importance sampling schemes to estimate rare event probabilities in Markovian systems such as Markovian reliability models and Jackson networks. In this work, we present a general state-dependent importance sampling method which partitions the state space and applies the cross-entropy method to each partition. We investigate two versions of our algorithm and apply them to several examples of reliability and queueing models. In all these examples we compare our method with other importance sampling schemes. The performance of the importance sampling schemes is measured by the relative error of the estimator and by the efficiency of the algorithm. The results from experiments show considerable improvements both in running time of the algorithm and the variance of the estimator.     

Sparse regression Chebyshev polynomial interval method for nonlinear dynamic systems under uncertainty

This paper proposes a new higher-efficiency interval method for the response bound estimation of nonlinear dynamic systems, whose uncertain parameters are bounded. This proposed method uses sparse regression and Chebyshev polynomials to help the interval analysis applied on the estimation. It is also a non-intrusive method which needs much fewer evaluations of original nonlinear dynamic systems than the other Chebyshev polynomials based interval methods. By using the proposed method, the response bound estimation of nonlinear dynamic systems can be performed more easily, even if the numerical simulation in nonlinear dynamic systems is costly or the number of uncertain parameters is higher than usual. In our approach, the sparse regression method “elastic net” is adopted to improve the sampling efficiency, but with sufficient accuracy. It alleviates the sample size required in coefficient calculation of the Chebyshev inclusion function in the sampling based methods. Moreover, some mature technologies are adopted to further reduce the sample size and to guarantee the accuracy of the estimation. So that the number of sampling, which solves the certain ordinary differential equations (ODEs), can be reduced significantly in the Chebyshev interval method. Three numerical examples are presented to illustrate the efficiency of proposed interval method. In particular, the last two examples are high dimension uncertain problems, which can further exhibit the ability to reduce the computational cost.