There is, I think, something ethereal about i —the square root of minus one. I remember first hearing about it at school. It seemed an odd beast at that time—an intruder hovering on the edge of reality.

Usually familiarity dulls this sense of the bizarre, but in the case of i it was the reverse: over the years the sense of its surreal nature intensified. It seemed that it was impossible to write mathematics that described the real world in …

I and i

Probability Distributions.- Linear Models for Regression.- Linear Models for Classification.- Neural Networks.- Kernel Methods.- Sparse Kernel Machines.- Graphical Models.- Mixture Models and EM.- Approximate Inference.- Sampling Methods.- Continuous Latent Variables.- Sequential Data.- Combining Models.

Pattern Recognition and Machine Learning

A review on machinery diagnostics and prognostics implementing condition-based maintenance

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

A survey of maintenance policies of deteriorating systems

In this paper, a multivariate Bayesian control chart is designed for a condition-based maintenance application. The system deterioration process is modeled as a 3-state hidden Markov process, with good, warning and failure states. Bayesian control chart is then applied to monitor the process by plotting the posterior probability that the system is in the warning state. It has been shown in the literature that Bayesian control chart is an optimal tool for statistical process control unlike traditional control charts. In this paper, a new fault detection scheme is developed based on the average run length criterion. Comparison results with former control chart methods are provided to illustrate the effectiveness of this approach.

ARL criterion in Bayesian process control using hidden Markov model

We study the optimal control of a production process subject to a deterministicdrift and to random shocks. The process mean is observable at discrete points of time after producing a batch and, at each such point, a decision is made whether to reset the process mean to some initial value or to continue with the production. The objective is to find the initial setting of the process mean and the resetting time that minimizes the expected average cost per unit time. It is shown that the optimal control policy is of a control limit type. An algorithm for finding the optimal control parameters is presented.

Optimal control of a deteriorating production process

This article presents a model of a deteriorating concrete structure subject to external load and inspection. The structure naturally deteriorates in time due to the environmental effects and the changes in the construction materials. The deterioration process is modeled as a gamma process. The stochastic process of load arrivals is generated by a Poisson process and the variability of the random loads is assumed to be governed by a generalized Pareto distribution. Upon inspection, the parameters of the gamma distribution are updated using the Bayesian method. Thus, after each inspection, the resistance of the concrete structure is modified and the corresponding lifetime distribution of the structure can be estimated. An example of concrete girders is provided to illustrate the whole methodology.

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Bayesian update of the residual life of a concrete structure using inspection data

For processes subject to deterioration, process control, generally referred to as EPC (engineering process control) or APC (automatic process control), should be applied in conjunction with the process monitoring, SPC (statistical process control), to identify the occurrence of assignable causes and try to eliminate them. However, a certain degree of autocorrelation is present in almost all real processes which has a major impact on the performance of classical control charts. To improve this performance, we propose a new statistic for the monitoring of the process mean to compensate for the lost portion of the process deviation due to the autocorrelation. The classical Shewhart chart is applied to this statistic. Using the Markov chain approach, we obtained explicit formulas for the run length distribution, average run length and the standard deviation of the run length.

Optimal Control and Monitoring of Deteriorating Production Processes

In this paper, we propose a method to model the relationship between failure time and degradation for a simple step stress test where underlying degradation path is linear and different causes of failure are possible. It is assumed that the intensity function depends only on the degradation value. No assumptions are made about the distribution of the failure times. A simple step-stress test is used to shorten failure time of products and a tampered failure rate (TFR) model is proposed to describe the effect of the changing stress on the intensities. We assume that some of the products that fail during the test have a cause of failure that is only known to belong to a certain subset of all possible failures. This case is known as masking. In the presence of masking, the maximum likelihood estimates (MLEs) of the model parameters are obtained through an expectation-maximization (EM) algorithm by treating the causes of failure as missing values. The effect of incomplete information on the estimation of parameters is studied through a Monte-Carlo simulation. Finally, a real example is analyzed to illustrate the application of the proposed methods. Keywords—Expectation-maximization (EM) algorithm, cause of failure, intensity, linear degradation path, masked data, reliability function.

Viliam Makis

Papers

ARL criterion in Bayesian process control using hidden Markov model

Optimal control of a deteriorating production process

Bayesian update of the residual life of a concrete structure using inspection data

Optimal Control and Monitoring of Deteriorating Production Processes

A Partially Accelerated Life Test Planning with Competing Risks and Linear Degradation Path under Tampered Failure Rate Model