Assessing Convergence and Mixing of MCMC Implementations via Stratification

Some posterior distributions lead to Markov chain Monte Carlo (MCMC) chains that are naturally viewed as collections of subchains. Examples include mixture models, regime-switching models, and hidden Markov models. We obtain MCMC-based estimators of posterior expectations by combining different subgroup (subchain) estimators using stratification and poststratification methods. Variance estimates of the limiting distributions of such estimators are developed. Based on these variance estimates, we propose a test statistic to aid in the assessment of convergence and mixing of chains. We compare our diagnostic with other commonly used methods. The approach is illustrated in two examples: a latent variable model for arsenic concentration in public water systems in Arizona and a Bayesian hierarchical model for Pacific sea surface temperatures. Supplementary materials, which include MATLAB codes for the proposed method, are available online.     

The European options hedge perfectly in a Poisson-Gaussian stock market model

It is shown that n + 1 European call options written on a stock S with different strike prices (or the stock and n calls) are non-redundant assets in a model for the stock driven by a Brownian motion and n independent Poisson processes. That extends the result obtained for n = 1 by Pham and implies that the proposed model can price and perfectly hedge any integrable derivative on S.     

The distance of a subdivision surface to its control polyhedron

For standard subdivision algorithms and generic input data, near an extraordinary point the distance from the limit surface to the control polyhedron after m subdivision steps is shown to decay dominated by the mth power of the subsubdominant (third largest) eigenvalue. Conversely, for Loop subdivision we exhibit generic input data so that the Hausdorff distance at the mth step is greater than or equal to the mth power of the subsubdominant eigenvalue.In practice, it is important to closely predict the number of subdivision steps necessary so that the control polyhedron approximates the surface to within a fixed distance. Based on the above analysis, two such predictions are evaluated. The first is a popular heuristic that analyzes the distance only for control points and not for the facets of the control polyhedron. For a set of test polyhedra this prediction is remarkably close to the true distance. However, a concrete example shows that the prediction is not safe but can prescribe too few steps. The second approach is to first locally, per vertex neighborhood, subdivide the input net and then apply tabulated bounds on the eigenfunctions of the subdivision algorithm. This yields always safe predictions that are within one step for a set of typical test surfaces.     

The growth, oscillation and fixed points of solutions of complex linear differential equations in the unit disc

We consider the complex differential equations of the form

Ak(z)f(k)+Ak−1(z)f(k−1)+?+A₁(z)f^′+A₀(z)f=F(z),     

Nonnegative iterations with asymptotically constant coefficients

Let A_k,k=0,1,2,…, be a sequence of real nonsingular n×n matrices which converge to a nonsingular matrix A. Suppose that A has exactly one positive eigenvalue λ and there exists a unique nonnegative vector u with properties Au=λu and u=1. Under further additional conditions on the spectrum of A, it is shown that if x₀≠0 and the iterates

are nonnegative, then converges to u and converges to λ as k→∞.     

A convex optimization approach for minimizing the ratio of indefinite quadratic functions over an ellipsoid

We consider the nonconvex problem (RQ) of minimizing the ratio of two nonconvex quadratic functions over a possibly degenerate ellipsoid. This formulation is motivated by the so-called regularized total least squares problem (RTLS), which is a special case of the problem’s class we study. We prove that under a certain mild assumption on the problem’s data, problem (RQ) admits an exact semidefinite programming relaxation. We then study a simple iterative procedure which is proven to converge superlinearly to a global solution of (RQ) and show that the dependency of the number of iterations on the optimality tolerance grows as . This research is partially supported by the Israel Science Foundation, ISF grant #489-06.     

How does a stochastic optimization/approximation algorithm adapt to a randomly evolving optimum/root with jump Markov sample paths

Stochastic optimization/approximation algorithms are widely used to recursively estimate the optimum of a suitable function or its root under noisy observations when this optimum or root is a constant or evolves randomly according to slowly time-varying continuous sample paths. In comparison, this paper analyzes the asymptotic properties of stochastic optimization/approximation algorithms for recursively estimating the optimum or root when it evolves rapidly with nonsmooth (jump-changing) sample paths. The resulting problem falls into the category of regime-switching stochastic approximation algorithms with two-time scales. Motivated by emerging applications in wireless communications, and system identification, we analyze asymptotic behavior of such algorithms. Our analysis assumes that the noisy observations contain a (nonsmooth) jump process modeled by a discrete-time Markov chain whose transition frequency varies much faster than the adaptation rate of the stochastic optimization algorithm. Using stochastic averaging, we prove convergence of the algorithm. Rate of convergence of the algorithm is obtained via bounds on the estimation errors and diffusion approximations. Remarks on improving the convergence rates through iterate averaging, and limit mean dynamics represented by differential inclusions are also presented. The research of G. Yin was supported in part by the National Science Foundation under DMS-0603287, in part by the National Security Agency under MSPF-068-029, and in part by the National Natural Science Foundation of China under #60574069. The research of C. Ion was supported in part by the Wayne State University Rumble Fellowship. The research of V. Krishnamurthy was supported in part by NSERC (Canada).     

A fourth order iterative method for solving nonlinear equations

The present paper illustrates an iterative numerical method to solve nonlinear equations of the form f(x) = 0, especially those containing the partial and non partial involvement of transcendental terms. Comparative analysis shows that the present method is faster than Newton-Raphson method, hybrid iteration method, new hybrid iteration method and others. Cost is also found to be minimum than these methods. The beauty in our method can be seen because of the optimization in important effecting factors, i.e. lesser number of iteration steps, lesser number of functional evaluations and lesser value of absolute error in final as well as in individual step as compared to the other methods. This work also demonstrates the higher order convergence of the present method as compared to others without going to the computation of second derivative.     

A Steffensen-like method and its higher-order variants

For solving nonlinear equations, we suggest a second-order parametric Steffensen-like method, which is derivative free and only uses two evaluations of the function in one step. We also suggest a variant of the Steffensen-like method which is still derivative free and uses four evaluations of the function to achieve cubic convergence. Moreover, a fast Steffensen-like method with super quadratic convergence and a fast variant of the Steffensen-like method with super cubic convergence are proposed by using a parameter estimation. The error equations and asymptotic convergence constants are obtained for the discussed methods. The numerical results and the basins of attraction support the proposed methods.     

A descent algorithm without line search for unconstrained optimization

In this paper, a new descent algorithm for solving unconstrained optimization problem is presented. Its search direction is descent and line search procedure can be avoided except for the first iteration. It is globally convergent under mild conditions. The search direction of the new algorithm is generalized and convergence of corresponding algorithm is also proved. Numerical results show that the algorithm is efficient for given test problems.