A complete list of titles in this series appears at the end of this volume. 10 Survival of Cooperative, Competitive, and Fuzzy Systems 269 10.1 Introduction: The Concept of Systems and Their Components 269 .
Preface
As the final point of this preface, I feel obligated to be honest about some of the limitations of this book. The material of this book can be used most profitably by practitioners and researchers in the field of reliability and survivability as a source of information and open problems.
Acknowledgements
Introduction and Overview
- RELIABILITY, RISK AND SURVIVAL: STATE-OF-THE-ART
- RISK MANAGEMENT: A MOTIVATION FOR RISK ANALYSIS
- BOOKS ON RELIABILITY, RISK AND SURVIVAL ANALYSIS
- OVERVIEW OF THE BOOK
We have seen that risk management is making decisions under uncertainty using quantified measures of the latter. The more precise the description of the causes, the more credible the uncertainty quantification.
The Quantification of Uncertainty
UNCERTAIN QUANTITIES AND UNCERTAIN EVENTS: THEIR DEFINITION AND CODIFICATION
So, for example, T could denote the number of defects in a particular batch, or the unknown time to failure of an item that survived, or the unknown number of miles to the next failure of a car serviced at, or the remission time of a disease since a medical intervention at, or the amount of radioactive release, or the amount of loss incurred, etc. Because of this synonymy, we can also refer to T≥tas 'the event which is at least'.
PROBABILITY: A SATISFACTORY WAY TO QUANTIFY UNCERTAINTY
- The Rules of Probability
- Justifying the Rules of Probability
To describe the properties of this number called 'probability', it is convenient to introduce an additional notation. We refer to PT=tas 'the probability that the random quantity T takes on the value' or equivalently, 'the probability of the event T=t'.
OVERVIEW OF THE DIFFERENT INTERPRETATIONS OF PROBABILITY
- A Brief History of Probability
- The Different Kinds of Probability
OVERVIEW OF DIFFERENT INTERPRETATIONS OF PROBABILITY 15 big theorem, the law of large numbers (section 3.1.4). Another objection to this notion of probability is that the conditions under which repeatable experiments will be performed are not clear.
EXTENDING THE RULES OF PROBABILITY: LAW OF TOTAL PROBABILITY AND BAYES’ LAW
- Marginalization
If it were different, our own specified probability is also likely to be different. Our final point concerns the issue of consistency and coherence in the specification of personal probabilities.
- Bayes’ Law: The Incorporation of Evidence and the Likelihood
- THE BAYESIAN PARADIGM: A PRESCRIPTION FOR RELIABILITY, RISK AND SURVIVAL ANALYSIS
- PROBABILITY MODELS, PARAMETERS, INFERENCE AND PREDICTION
- The Genesis of Probability Models and Their Parameters
Applying the multiplication rule to the numerator and denominator of the above expression, we get. Then, for any random set, we have, by the law of extension of conversation, .
- Statistical Inference and Probabilistic Prediction
- TESTING HYPOTHESES: POSTERIOR ODDS AND BAYES FACTORS The statistical testing of a hypothesis is usually done to verify a theory or a claim, be it in science,
- Bayes Factors: Weight of Evidence and Change in Odds The absence of evidence is not evidence of absence
- Alternatives to Bayes Factors
- UTILITY AS PROBABILITY AND MAXIMIZATION OF EXPECTED UTILITY
- Utility as a Probability
- Maximization of Expected Utility
- Attitudes to Risk: The Utility of Money
- DECISION TREES AND INFLUENCE DIAGRAMS FOR RISK ANALYSIS
- The Decision Tree
- The Influence Diagram
In the latter case, the Bayes factor cannot be interpreted as a summary of the evidence provided by the data alone. This joint probability is the product of the probabilities associated with all the nodes in the diagram.
Exchangeability and Indifference
INTRODUCTION TO EXCHANGEABILITY: DE FINETTI’S THEOREM
- Motivation for the Judgment of Exchangeability
- Relationship between Independence and Exchangeability
- de Finetti’s Representation Theorem for Zero-one Exchangeable Sequences We have seen before that the theory of probability does not concern itself with the interpretation
- Exchangeable Sequences and the Law of Large Numbers
The infinite form of de Finetti's theorem is simply the limiting case, asm→andr/m→, of the above theorem. It is important to note that the infinite form of the theorem fails to hold for finite sequences.
DE FINETTI-STYLE THEOREMS FOR INFINITE SEQUENCES OF NON-BINARY RANDOM QUANTITIES
- Sufficiency and Indifference in Zero-one Exchangeable Sequences
- Invariance Conditions Leading to Mixtures of Other Distributions
The purpose of this section is to explore the above symmetries and produce versions of (2.5) other than those of the mixture of Bernoulli series. This elementary result provides an opportunity to illustrate the practical value of the de Finetti-style theorems we discuss here. Mixtures of Weibull probability distributions with shape and scale arise when all of the above apply except that the N.
Results analogous to the above, but concerning mixtures of the inverse binomial and the binomial, are given by Freedman (1962). All the above results explain the consequences of the judgment of indifference on observables, given a statistic.
ERROR BOUNDS ON DE FINETTI-STYLE RESULTS FOR FINITE SEQUENCES OF RANDOM QUANTITIES
- Bounds for Finitely Extendable Zero-one Random Quantities
- Bounds for Finitely Extendable Non-binary Random Quantities
Bounds analogous to the one above have also been obtained for non-binary random quantities. These limits have their origin in a theorem due to Borel (1914) which relates to the 'total variation distance' (see below) between the joint distribution of a finite number of random variables and the joint distribution of the same number of independent Gaussian variables . Analogous to the above, suppose that the vectorX1 Xnis is evaluated uniformly on the simplex Xi≥0 Sn, given n.
Let Pk be the law of X1 Xn, and let 1 k be independent and identically distributed as. Given Sn, if X1 Xn is judged multinomial in n-tuples of nonnegative integers whose sum is Sn with uniform probabilities 1/n 1/n, then there exists a probability F over 0 such that, for all k≤n /2,.
Stochastic Models of Failure
INTRODUCTION
PRELIMINARIES: UNIVARIATE, MULTIVARIATE AND MULTI-INDEXED DISTRIBUTION FUNCTIONS
While the latter is clear in the expression Rt, the former is implied in; indeed, for clarity, it should really be read as. It is instructive to think approximately the probability that T is between sand+ds, where there is a small increase in T in the vicinity of s. 1 A distribution function is said to be "singular" if it is continuous, not identically zero, and has derivatives that exist, but vanish, almost everywhere.
In the case of one variable, an example of a singular distribution is the "Cantor distribution"; it is a continuous distribution that has no probability density (cf. Singular distributions arise in reliability in the context of multicomponent systems that have dependent lifetimes.
- THE PREDICTIVE FAILURE RATE FUNCTION OF A UNIVARIATE PROBABILITY DISTRIBUTION
- INTERPRETATION AND USES OF THE FAILURE RATE FUNCTION – THE MODEL FAILURE RATE
- The True Failure Rate: Does it Exist?
- Decreasing Failure Rates, Reliability Growth, Burn-in and the Bathtub Curve Rationale for a Decreasing Failure Rate
- The Retrospective (or Reversed) Failure Rate
- MULTIVARIATE ANALOGUES OF THE FAILURE RATE FUNCTION Multivariate analogues of the failure rate function have been considered, among others, by
- The Hazard Gradient
- The Multivariate Failure Rate Function
- The Conditional Failure Rate Functions
- THE HAZARD POTENTIAL OF ITEMS AND INDIVIDUALS
- Hazard Potentials and Dependent Lifelengths
- The Hazard Gradient and Conditional Hazard Potentials
- PROBABILITY MODELS FOR INTERDEPENDENT LIFELENGTHS The notion of independent events was introduced in section 2.2.1, and that of independent
- Preliminaries: Bivariate Distributions
- The Bivariate Exponential Distributions of Gumbel
THE PREDICTED ERROR RATE FUNCTION 65 reliability and survivability (not to be confused with the product integral formula, which will be described shortly). Actuaries use a form of failure rate to determine insurance premiums. The decreasing form of the error rate function does not necessarily mean that the system is actually improving with use.
Rather, the decreasing form of the failure rate function typically describes our improving opinion of the item's survivability. More about the form of the failure percentage function can be found in Aalen and Gjessing (2001).
Thus, copulas provide a graphical way to show the positive dependence between two lifetimes T1 and T2, where the strength of the dependence is a function of the deviation of Cu v from muv. Gumbel (1960) proposed a bivariate distribution system whose marginal distributions are exponential—hence the name "bivariate exponential." Its advantages are that, unlike many other multivariate lifetime distributions, both positive and negative dependence can be represented; the marginal distributions are also uniformly exponential (ie, an exponential distribution with a scale parameter of one).
Under the second version, both positive and negative values of the correlation can be obtained, and the maximum value of the correlation is ±0 25. Thus, like the conditional mean, the conditional variance int2 decreases, ranging from one when =0, to2+ 4t2+t22/1+t24, when=1.
Freund’s Bivariate Exponential Distribution
In contrast to Gumbel's bivariate exponential distributions which do not seem to have a physical motivation, Freund (1961) proposed a bivariate extension of the exponential distribution based on the following construction. The system is considered to function (albeit in a degraded state) even when one of the two units has failed. However, the failed unit increases the stress on the surviving unit, thereby increasing the latter unit's susceptibility to failure.
When the first component fails, the (model) failure percentage of the remaining component increases from ito∗i i=12. For the setup described above, T1 and T2 are not independent, because the failure of one component changes the parameter of the remaining component.
- The Bivariate Exponential of Marshall and Olkin
- A Bivariate Exponential Induced by a Shot-noise Process
- A Bivariate Exponential Induced by a Bivariate Pareto’s Copula
- Other Specialized Bivariate Distributions
- Probabilistic Causality and Causal Failures
- Cascading and Models of Cascading Failures
- FAILURE DISTRIBUTIONS WITH MULTIPLE SCALES
- Model Development
Thus, the survival function of the series system is exponential and the failure rate is constant. This aspect relates to the issue of absolute continuity, which the survival function of the BVE is not. For specified constants 1 2 and 12>0, the survival function of Sarkar's distribution has the form.
However, a numerical regression estimate (with a=1) suggests that ET1 T2=t2 is convex and increases exponentially at t2 (Singpurwalla and Kong, 2004). Let be the failure rate of the item if it were not subjected to any use;.
Parametric Failure Data Analysis
INTRODUCTION AND PERSPECTIVE
When one ignores for the time being the consideration of the 'stopping rule' (section 5.4.2), it is usual to take L nas. These strategies are important in the sense that they can affect a specification of the likelihood (Section 5.4.2). By error data analysis I mean an assessment of the predictive distributions of the form given by (5.4) and (5.7).
While the role of the parameter has been to facilitate the estimation of predictive distributions, it sometimes happens that interest focuses on itself. In this case, the analysis of the error data could also mean the inclusion of an estimate of the posterior distribution (equation (5.5)).
ASSESSING PREDICTIVE DISTRIBUTIONS IN THE ABSENCE OF DATA
- The Exponential as a Chance Distribution
- The Weibull (and Gamma) as a Chance Distribution
- The Bernoulli as a Chance Distribution
- The Poisson as a Chance Distribution
- The Generalized Gamma as a Chance Distribution
- The Inverse Gaussian as a Chance Distribution
LeMT indicates the density of the MT prior distribution; and are the parameters of this density (section 5.3.1). It is important to note that mission time is a parameter above. The Xis here are a generalization of the Bernoulli case of Section 5.2.3, where each Xitoi has only two values 0 and 1.
Specifically, the previous density is also an IG with parameters and; it is of the form. The prior or is considered a gamma distribution with scale and shape >1, so that the density is equal to the shape.
PRIOR DISTRIBUTIONS IN CHANCE DISTRIBUTIONS
- Using Objective (or Default) Priors
Therefore, what remains to be discussed is the case of the Weibull and the Poisson. In the case of the Weibull, I consider an elicitation of MT, the average lifespan (section 5.2.2). The Bernoulli scenario illustrates a feature of Jeffreys' priors, namely the dependence of the prior on the model for the binomial, and −1/21−−1 for the negative binomial.
Since the limit of observable zero-one sequences has a physical connotation, that is, it is a chance (section 3.1.3), the dependence of the prior on the model for X makes little sense. It is because of the above that many see Jeffreys' priorities as a collection of ad hoc rules.
- Design Strategies for Industrial Life-testing
- Stopping Rules: Non-informative and Informative
- The Total Time on Test
- Exponential Life-testing Procedures
- Life-testing Under the Generalized Gamma and the Inverse Gaussian
- Bernoulli Life-testing Procedures
- Life-testing and Inference Under the BVE
Predictable DISTRIBUTIONS CONTAINING ERROR DATA 149 Bernoulli, knowledge of the stopping process is not essential. In the second case, it will be the product of the right-hand side of (5.30) and the probability thatTr> t. In Section 5.4.5, the nature of these difficulties is illustrated via the case of the Weibull distribution.
Thus, our proposed estimate of the reliability function can be considered as a surrogate for the predictive distribution of T. Here we consider the case of Bernoulli testing when the priorFd is of the form given by (5.12) of Section 5.2.3.
INFORMATION FROM LIFE-TESTS: LEARNING FROM DATA
- Preliminaries: Entropy and Information
- Learning for Inference from Life-test Data: Testing for Confidence
More details regarding the implications of the concave and convex boundaries of Figure 5.4 can be found in Singpurwalla (1996). Suppose he chooses to accept; then 2nature on the random node follows its course of action because the value of the unknown parameter is accidental. Multiple crossings are due to the sample path being close to the accept/reject boundary.
Furthermore, it is assumed that the set-up costs associated with carrying out the life test are low. The predictive distribution of the times to failure of the items on test is based on the preceding alone.
ADVERSARIAL LIFE-TESTING AND ACCEPTANCE SAMPLING Consider the scenario of section 5.5.3 wherein a consumer , who needs a batch of items from a
What is new here is that at the decision node labeled 2, it (doesn't) take the action. In the decision trees in Figures 5.7 and 5.9, it was who performs the actions at all decision nodes. On decision node1, it performs the select action ann (similar to Figure 5.9), and on random node1, nature exposes T*as data.
Note that, unlike the decision trees of Figures 5.7 and 5.9, the terminal utility need not be determined by (or by T) which includes nature's actions. Following the principle of maximization of expected utility, the decision tree of Figure 5.10 will fold backwards, and will provide that sample size that results.
ACCELERATED LIFE-TESTING AND DOSE–RESPONSE EXPERIMENTS
- Formulating Accelerated Life-testing Problems
- The Kalman Filter Model for Prediction and Smoothing
- Inference from Accelerated Tests Using the Kalman Filter
But first I present below another version of the accelerated life testing problem, one that is driven by scenarios in dose-response studies in the biomedical context and in damage assessment experiments in the engineering reliability context. This second version of the accelerated life test problem is also amenable to analysis via a Kalman filter model. Quantitative values of dose (or damage causing stress) will be denoted by X, where X takes values tx0≤x <.
Data from a dose-response experiment (or a damage assessment) involving testing at x1 xm doses will consist of Yx1 Yxm responses. The time-transform functions of Section 5.8.1 are natural candidates for describing the observation equations of the Kalman filter model.
