Convergence of Probability Measures. By P. Billingsley. Chichester, Sussex, Wiley, 1968. xii, 253 p. 9 1/4“. 117s.

Convergence of Probability Measures

Two convergence aspects of the EM algorithm are studied: (i) does the EM algorithm find a local maximum or a stationary value of the (incomplete-data) likelihood function? (ii) does the sequence of parameter estimates generated by EM converge? Several convergence results are obtained under conditions that are applicable to many practical situations Two useful special cases are: (a) if the unobserved complete-data specification can be described by a curved exponential family with compact parameter space, all the limit points of any EM sequence are stationary points of the likelihood function; (b) if the likelihood function is unimodal and a certain differentiability condition is satisfied, then any EM sequence converges to the unique maximum likelihood estimate A list of key properties of the algorithm is included

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On the convergence properties of the em algorithm

The problem of estimating the parameters which determine a mixture density has been the subject of a large, diverse body of literature spanning nearly ninety years. During the last two decades, the...

Mixture densities, maximum likelihood, and the EM algorithm

We present an integral equation which generalizes a variety of known rendering algorithms. In the course of discussing a monte carlo solution we also present a new form of variance reduction, called Hierarchical sampling and give a number of elaborations shows that it may be an efficient new technique for a wide variety of monte carlo procedures. The resulting rendering algorithm extends the range of optical phenomena which can be effectively simulated.

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The rendering equation

Surrogate-based Analysis and Optimization

The central intention of this survey is to review dynamic programming models for water resource problems and to examine computational techniques which have been used to obtain solutions to these problems. Problem areas surveyed here include aqueduct design, irrigation system control, project development, water quality maintenance, and reservoir operations analysis. Computational considerations impose severe limitation on the scale of dynamic programming problems which can be solved. Inventive numerical techniques for implementing dynamic programming have been applied to water resource problems. Discrete dynamic programming, differential dynamic programming, state incremental dynamic programming, and Howard's policy iteration method are among the techniques reviewed. Attempts have been made to delineate the successful applications, and speculative ideas are offered toward attacking problems which have not been solved satisfactorily.

Dynamic programming applications in water resources

This paper describes a modification of differential dynamic programming (DDP) which makes that technique applicable to certain constrained sequential decision problems such as multireservoir control problems discussed in the hydrology literature. The authors contend that the method proffered here is superior to available alternatives. This belief is supported by analysis (wherein it transpires that constrained DDP does not suffer the ‘curse of dimensionality’ and requires no discretization) and computational experimentation (wherein DDP is found to quickly locate solutions of 4-reservoir problems introduced by other investigations as well as the solution of a 10-reservoir problem thought to be beyond the capability of alternative methods).

Constrained Differential Dynamic Programming and Its Application to Multireservoir Control

A new statistically based approach to the problem of estimating spatially varying aquifer transmissivities on the basis of steady state water level data is presented. The method involves solving a family of generalized nonlinear regression problems and then selecting one particular solution from this family by means of a comparative analysis of residuals. A linearized error analysis of the solution is included. This analysis allows one to estimate the covariance of the transmissivity estimates as well as the square error of the estimates of hydraulic head. In addition to the explicitly statistical orientation of the method, it has an additional feature of permitting the user to incorporate a priori information about the transmissivities. This information may be based on actual field data such as pumping tests, or on statistical data accumulated from similar aquifers elsewhere in the world. A highly efficient explicit numerical scheme for solving the inverse problem in an approximate manner when errors in water level data are sufficiently small is also described. When these errors are large, the explicit scheme may still be useful for obtaining a rapid initial idea about the approximate location of the optimum solution. Paper 1 presents the theory and illustrates it by a theoretical example. The purpose of this example is to demonstrate the effectiveness of our method in dealing with noisy data obtained from a known model. Application of the method to real data will be described in paper 2.

A statistical approach to the inverse problem of aquifer hydrology: 1. Theory

An Introduction to Numerical Computations

This investigation is concerned with recovering a regression function $g(x_i)$ on the basis of noisy observations taken at uniformly spaced design points $x_i$. It is presumed that the corresponding observations are corrupted by additive dependent noise, and that the noise is, in fact, induced by a general linear process in which the summand law can be discrete, as well as continuously distributed. Discreteness induces a complication because such noise is not known to be strong mixing, the postulate by which regression estimates are often shown to be asymptotically normal. In fact, as cited, there are processes of this character which have been proven not to be strong mixing. The main analytic result of this study is that, in general circumstances which include the non-strong mixing example, the smoothers we propose are asymptotically normal. Some motivation is offered, and a simple illustrative example calculation concludes this investigation. The innovative elements of this work, mainly, consist of compassing models with discrete noise, important in practical applications, and in dispensing with mixing assumptions. The ensuing mathematical difficulties are overcome by sharpening standard arguments.

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Sidney Yakowitz

Papers

Dynamic programming applications in water resources

Constrained Differential Dynamic Programming and Its Application to Multireservoir Control

A statistical approach to the inverse problem of aquifer hydrology: 1. Theory

An Introduction to Numerical Computations

Fixed-design regression for linear time series