Remaining useful life estimation - A review on the statistical data driven approaches

This book provides a systematic development of tensor methods in statistics, beginning with the study of multivariate moments and cumulants. The effect on moment arrays and on cumulant arrays of making linear or affine transformations of the variables is studied. Because of their importance in statistical theory, invariant functions of the cumulants are studied in some detail. This is followed by an examination of the effect of making a polynomial transformation of the original variables. The fundamental operation of summing over complementary set partitions is introduced at this stage. This operation shapes the notation and pervades much of the remainder of the book. The necessary lattice-theory is discussed and suitable tables of complementary set partitions are provided. Subsequent chapters deal with asymptotic approximations based on Edgeworth expansion and saddlepoint expansion. The saddlepoint expansion is introduced via the Legendre transformation of the cumulant generating function, also known as the conjugate function of the cumulant generating function. A recurring them is that, with suitably chosen notation, multivariate calculations are often simpler and more transparent than the corresponding univariate calculations. The final two chapters deal with likelihood ratio statistics, maximum likelihood estimation and the effect on inferences of conditioning on ancillary or approximately ancillary statistics. The Bartlett adjustment factor is derived in the general case and simplified for certain types of generalized linear models. Finally, Barndorff-Nielsen's formula for the conditional distribution of the maximum liklelihood estimator is derived and discussed. More than 200 Exercises are provided to illustrate the uses of tensor methodology.

Tensor Methods in Statistics

SUMMARY Data analytic procedures are proposed to examine the adequacy of the hypothesized link used in fitting a generalized linear model. Through model expansion and linearization, tests and estimation techniques are provided. These procedures, along with the release of GLIM3, enable the user to examine routinely and objectively the fit of an hypothesized model. Examples are presented to illustrate the testing and fitting procedure.

Goodness of Link Tests for Generalized Linear Models

1 Introduction.- 1.1 Common-cause and special-cause variability.- 1.2 Transient and persistent special causes.- 1.3 The Shewhart and CUSUM charts.- 1.4 Basis for the CUSUM chart for a normal mean.- 1.4.1 Statistical properties of the CUSUM.- 1.5 Out-of-control distribution of the CUSUM.- 1.6 Testing for a shift -the V mask.- 1.7 Estimation following a signal.- 1.8 Using individual readings or rational groups.- 1.9 The decision interval form of the CUSUM.- 1.9.1 Example.- 1.10 Summary.- 1.11 Further reading.- 2 CUSUM design.- 2.1 The choice of k and h.- 2.1.1 Reference value k - "tuning" for a specific shift.- 2.2 Runs, run length, and average run length.- 2.2.1 The choice of h, the decision interval.- 2.2.2 Calculating the k, h, ARL relationship.- 2.2.3 A closer look at the choice of in-control ARL.- 2.2.4 Designing a CUSUM of Xbar.- 2.3 The Shewhart Xbar chart as CUSUM.- 2.4 Summary.- 2.5 Further reading.- 3 More about normal data.- 3.1 In-control ARLs.- 3.2 Out-of-control ARLs.- 3.2.1 Model.- 3.2.2 The ARL following a shift in mean.- 3.3.3 ARL sensitivity to choice of K.- 3.2.4 Out-of-control states and two-sided CUSUMs.- 3.3 FIR CUSUMs: zero start and steady state start.- 3.3.1 Introduction.- 3.3.2 Out-of-control ARL of the FIR CUSUM.- 3.3.3 ARL of two-sided FIR CUSUMS.- 3.3.4 Initial and steady-state ARL.- 3.4 Controlling for the mean within a range.- 3.4.1 Example.- 3.5 The impact of variance shifts.- 3.5.1 Individual data -the approximate normal transform.- 3.5.2 Rational groups-variance CUSUMs.- 3.6 Combined Shewhart and CUSUM charts.- 3.6.1 Example.- 3.7 Effect of model departures.- 3.7.1 Nonnormality.- 3.7.2 Independence.- 3.8 Weighted CUSUMs.- 3.8.1 Example.- 3.9 Summary.- 3.10 Further reading.- 4 Other continuous distributions.- 4.1 The gamma family and normal variances.- 4.1.1 Background.- 4.1.2 Normal variances.- 4.1.3 Design of the CUSUM for scale.- 4.1.4 Example: Sugar bags.- 4.1.5 Shift in the gamma shape parameter ?.- 4.1.6 Example - shift in ss.- 4.2 The inverse Gaussian family.- 4.2.1 Background.- 4.2.2 Shift in mean.- 4.2.3 Shift in scale parameter.- 4.3 Example from General Motors.- 4.3.1 CUSUM chart for location.- 4.3.2 CUSUM chart for ?.- 4.3.3 Remarks.- 4.4 Comments.- 4.5 Further reading.- 5 Discrete data.- 5.1 Types of discrete data.- 5.1.1 Binomial data.- 5.1.2 Count data.- 5.2 The graininess of the ARL function.- 5.3 The Poisson distribution and count data.- 5.3.1 Useful properties of the Poisson distribution.- 5.4 The Poisson and CUSUMs.- 5.4.1 Design for an upward shift.- 5.4.2 Downward shift.- 5.4.3 ARLs.- 5.4.4 Example.- 5.4.5 The effect of departures from Poisson.- 5.4.6 Checking conformity to the Poisson model.- 5.5 Weighted Poisson CUSUMs.- 5.6 The binomial distribution.- 5.6.1 Background.- 5.6.2 Examples.- 5.6.3 The choice of m.- 5.7 Weighted binomial CUSUMs.- 5.7.1 Example.- 5.8 Other discrete distributions.- 5.9 Summary.- 5.10 Further reading.- 6 Theoretical foundations of the CUSUM.- 6.1 General theory.- 6.1.1 Relation of the SPRT to the CUSUM.- 6.1.2 Optimality properties.- 6.2 The general exponential family.- 6.2.1 Derivation of the CUSUM for a normal mean shift.- 6.2.2 The gamma family and normal variance.- 6.2.3 Relation to normal variances.- 6.2.4 The Poisson family.- 6.2.5 The binomial family.- 6.2.6 The negative binomial family.- 6.2.7 The inverse Gaussian family.- 6.2.8 The Weibull distribution.- 6.2.9 Distributions outside the exponential family.- 6.3 The Markov property of CUSUMs.- 6.4 Getting the ARL.- 6.4.1 The renewal equations.- 6.4.2 The Markov chain approach.- 6.4.3 Simulation using variance reduction techniques.- 6.5 Summary.- 6.6 Further reading.- 7 Calibration and short runs.- 7.1 The self-starting approach.- 7.2 The self-starting CUSUM for a normal mean.- 7.2.1 Special features of self-starting charts.- 7.3 Self-starting CUSUMs for gamma data.- 7.3.1 Background.- 7.3.2 The scheme.- 7.3.3 Example.- 7.3.4 Normal data - control of mean and variance.- 7.3.5 Comments.- 7.4 Discrete data.- 7.4.1 The Poisson distribution.- 7.4.2 The binomial distribution.- 7.4.3 Updating the targets.- 7.5 Summary.- 7.6 Further reading.- 8 Multivariate data.- 8.1 Outline of the multivariate normal.- 8.2 Shewhart charting-Hotelling's T2.- 8.3 CUSUM charting - various approaches.- 8.3.1 Collections of unvariate CUSUMs.- 8.4 Regression adjustment.- 8.4.1 Example.- 8.4.2 SPC use of regression-adjusted variables.- 8.4.3 Example - monitoring a carbide plant.- 8.5 Choice of regression adjustment.- 8.6 The use of several regression-adjusted variables.- 8.6.1 Example.- 8.7 The multivariate exponentially weighted moving average.- 8.8 Summary.- 8.9 Further reading.- 9 Special topics.- 9.1 Robust CUSUMs.- 9.2 Recursive residuals in regression.- 9.2.1 Definition and properties.- 9.2.2 Example.- 9.3 Autocorrelated data.- 9.3.1 Example.- 9.4 Summary.- 9.5 Further reading.- 9.5.1 Time series.- 9.5.2 Score methods.- 9.5.3 Robustification.- 9.5.4 Recursive residuals.- 10 Software.- 10.1 Programs and templates.- 10.2 Data files.- References.

Cumulative Sum Charts and Charting for Quality Improvement

The probability-distributed principle in basin-scale hydrology considers the frequency of occurrence of hydrological variables (model inputs, parameters or elements) of certain magnitudes over the basin without regard to the location of a particular occurrence within the basin. The random assemblage of different parts is considered more important than the relation of the parts, one to another. Rainfall-runoff models based on probability-distributed infiltration capacity and storage capacity concepts, and which generate runoff according to Hortonian and saturation overland flow mechanisms respectively, are distinguished. Two types of probability-distributed storage capacity model are identified, one based on an assumption that storage elements at points in the basin respond independently of their neighbours, and the other where storage elements interact so as to equalize the depth of stored water over the basin. Allowing redistribution of water leads to simplification of the model equations. The p...

The probability-distributed principle and runoff production at point and basin scales

https://rss.onlinelibrary.wiley.com/doi/pdf/10.1111/j.2517-6161.1978.tb01039.x

The Inverse Gaussian Distribution and its Statistical Application—A Review

Early occurrence of certain events such as failure or repairs is a common phenomenon in the lifetime of industrial products. Often, the log normal distribution has been found as a useful model to be applicable whenever the early occurrences dominate a lifetime distribution. In this paper we suggest the use of the inverse Gaussian distribution for a model of such lifetime behavior and discuss different reliability features of the distribution. It is shown that its failure rate is nonmonotonic, initially increasing and then decreasing. Advantages in the use of the inverse Gaussian over the log normal are given. Certain numerical results are presented for illustration.

The Inverse Gaussian Distribution as a Lifetime Model

A common problem in many areas is to achieve a combination of control variables which will simultaneously maximize all of the responses which are functions of the control variables. The fact that in general we can not simultaneously maximize all of the responses has led to the formulation of compromise solutions to this vector maximum problem. A point in the control variable space is better than a second point if each response at the first point is greater than or equal to the corresponding response at the second point. If a point is such that there is no better point, it is said to be admissible, or efficient. A set of points is complete if given any point not in the set there is a point in the set which is better. In this paper this formulation is used to consider the allocation of sample size to several strata when there are several characteristics of interest.

http://www.jstor.org/stable/pdfplus/2283148.pdf

Optimum Allocation of Sampling Units to Strata When There are R Responses of Interest

Minimum variance unbiased estimates are given of the cumulative distribution function for the normal, binomial, Poisson, and negative exponential distributions. Although diverse areas of application might lead to interest in estimating the cumulative distribution function, most of the examples in this paper suppose that one is interested in the fraction of product meeting fixed specification limits.

Estimating the Fraction of Acceptable Product

This paper presents a method of constructing conservative joint confidence regions for the parameters of a linear model and the volumes of these regions are compared with the volume of an exact confidence region. Confidence regions on the entire surface are obtained from the confidence region on the parameters. The procedures are illustrated for the case of a simple linear model.

John Leroy Folks

Papers

The Inverse Gaussian Distribution and its Statistical Application—A Review

The Inverse Gaussian Distribution as a Lifetime Model

Optimum Allocation of Sampling Units to Strata When There are R Responses of Interest

Estimating the Fraction of Acceptable Product

Straight Line Confidence Regions for Linear Models