Nozer D. Singpurwalla

Understanding the Kalman Filter

Abstract This is an expository article. Here we show how the successfully used Kalman filter, popular with control engineers and other scientists, can be easily understood by statisticians if we use a Bayesian formulation and some well-known results in multivariate statistics. We also give a simple example illustrating the use of the Kalman filter for quality control work.

Software Reliability Modeling

Summary Probability models and statistical methods are a popular technique for evaluating the reliability of computer software. This paper reviews the literature concerning these methods, with an emphasis on the historical perspective. The use of stochastic techniques is justified, and the various probability models that have been proposed, along with any associated statistical estimation and inference procedures, are described. Examples of the models applied to real software failure data are given. A classic software development problem-how long software should be tested before it is released into the marketplace-is analyzed from a decision theoretic standpoint. Finally, the direction of future research is contemplated.

MULTIVARIATE DISTRIBUTIONS FOR THE LIFE LENGTHS OF COMPONENTS OF A SYSTEM SHARING A COMMON ENVIRONMENT

In assessing the reliability of a system of components, it is usual to suppose the components to fail independently of each other. Often this is inappropriate because the common environment acting on all components induces correlation. For example, a harsh environment will encourage early failure of all components. A simple model that incorporates such dependencies is described, and several properties of this model investigated. Calculations are carried out for a parallel system of two components. Inequalities for multicomponent systems are suggested. The results generalize easily.

A Unification of Some Software Reliability Models

In this paper we show how several models used to describe the reliability of computer software can be comprehensively viewed by adopting a Bayesian point of view. We first provide an alternative motivation for a commonly used model, the Jelinski–Moranda model, using notions from shock models. We then show that some alternate models proposed in the literature can be derived by assigning specific prior distributions for the parameters of the above model. We also obtain other structural results such as stochastic inequalities and association, and discuss how these can be interpreted.

Membership Functions and Probability Measures of Fuzzy Sets

The notion of fuzzy sets has proven useful in the context of control theory, pattern recognition, and medical diagnosis. However, it has also spawned the view that classical probability theory is unable to deal with uncertainties in natural language and machine learning, so that alternatives to probability are needed. One such alternative is what is known as “possibility theory.” Such alternatives have come into being because past attempts at making fuzzy set theory and probability theory work in concert have been unsuccessful. The purpose of this article is to develop a line of argument that demonstrates that probability theory has a sufficiently rich structure for incorporating fuzzy sets within its framework. Thus probabilities of fuzzy events can be logically induced. The philosophical underpinnings that make this happen are a subjectivistic interpretation of probability, an introduction of Laplaces famous genie, and the mathematics of encoding expert testimony. The benefit of making probability theory work in concert with fuzzy set theory is an ability to deal with different kinds of uncertainties that may arise within the same problem.

Robustification of Kalman Filter Models

Abstract Kalman filter models based on the assumption of multivariate Gaussian distributions are known to be nonrobust. This means that when a large discrepancy arises between the prior distribution and the observed data, the posterior distribution becomes an unrealistic compromise between the two. In this article we discuss a rationale for how to robustify the Kalman filter. Specifically, we develop a model wherein the posterior distribution will revert to the prior when extreme outlying observations are encountered, and we point out that this can be achieved by assuming a multivariate distribution with Student-t marginals. To achieve fully robust results of the kind desired, it becomes necessary to forsake an exact distribution-theory approach and adopt an approximation method involving “poly-t” distributions. A recursive mechanism for implementing the multivariate-t—based Kalman filter is described, its properties are discussed, and the procedure is illustrated by an example.

An Empirical Stopping Rule for Debugging and Testing Computer Software

Abstract In this article we introduce the problem of computer software reliability and discuss a probabilistic model for describing the failure of software. We suggest a procedure for estimating the parameters of the model and propose a stopping rule for debugging the software. We apply our procedures to some published data on software failures.

The warranty problem: its statistical and game theoretic aspects

Manufacturers of consumer products, such as automobiles, usually offer warranties guaranteeing the product or its parts, for example, for five years or 50,000 miles, whichever comes first. There are at least two issues of interest to applied mathematicians and statisticians that arise from warranty considerations. The first is the specification of an optimum price-warranty combination, and the second is the forecast of a reserve fund to meet warranty claims against the product. The former involves a consideration of the item’s reliability, its rate of usage, the consumer’s attitude toward a specific warranty, and the competitor’s actions, all of which lead towards a game theoretic formulation of the problem. The latter involves the analysis of a special kind of time series in two-dimensions, for which the use of dynamic linear models appears to be natural. This is an expository paper. Due to the timeliness of the topic, the focus here is on problem definition, its scope, and its formulation in a manner th...

Kolmogorov Statistics for Tests of Fit for the Extreme-Value and Weibull Distributions

Abstract Percentage points are given for the Kolmogorov-Smirnov statistics D +, D -, and D and for the Kuiper statistic V for testing fit to the extreme-value distribution with unknown parameters. The statistics may also be used for tests for the Weibull distribution.

Determining an optimal time interval for testing and debugging software

A decision-theoretic procedure for determining an optimal time interval for testing software prior to its release is proposed. The approach is based on the principles of decision-making under uncertainty and involves a maximization of expected utility. Two plausible forms for the utility function, one based on costs and the other involving the realized reliability of the software, are described. Using previous results on probabilistic models for software failure, the ensuing optimization problem (which can be addressed using numerical techniques) is outlined for the case of single-state testing. The sensitivity of the results to the various input parameters is discussed, and some directions for future research are outlined. >