Conditional Run Length Research Articles

Statistical design of phase II exponential chart with estimated parameters under the unconditional and conditional perspectives using exact distribution of median run length

ABSTRACT In this paper, we consider the -quantiles of the conditional run length (CRL) distribution to design and evaluate the phase II exponential chart instead of the widely used metric, the average of CRL. The unconditional and conditional perspectives of the performance evaluation are considered to adjust the control limits of the phase II exponential chart based on the in-control (IC) -quantiles of CRL (denoted by ) distribution. Under the unconditional approach, the mean of the IC is set equal to some pre-specified value, say of the -quantile of run length (QRL) whereas under the conditional perspective, the control limits are adjusted so that that the IC meets or exceeds the nominal value with a high probability. The charts under both perspectives are designed so that they are QRL-unbiased for a nominal value. The IC and out-of-control (OOC) performance studies of the proposed charts are carried out based on the most promising quantile i.e. median run length (MRL). The study shows that the MRL-unbiased exponential chart under the conditional perspective has a better IC performance.

Conditional analysis of Phase II exponential chart for monitoring times to an event

ABSTRACTIt is now well known that the error in estimating parameter(s) adversely affects the performance of control charts. The charts with estimated control limits are usually assessed by some of characteristics of unconditional run length (URL) distribution. However, it is now under criticism because the unconditional analysis based on URL does not give the full impression of the chart’s performance conditioning on a given Phase I sample. In recent years, the focus has been shifted from unconditional performance analysis to conditional analysis which is carried out in terms of conditional run length (CRL) and its associated characteristics and provides deep understanding of conditional performance of the chart given a Phase I sample. In this article, Phase II exponential charts are considered to monitor times to an event and their performance are examined conditional on given Phase I sample by considering several indicators based on conditional average run length (CARL) distribution. In this respect, the exact expressions of the distribution functions of for the estimated exponential control charts are derived and several statistical constants are computed which could be helpful in evaluating and designing the chart.

Phase II Shewhart-type Control Charts for Monitoring Times Between Events and Effects of Parameter Estimation

Monitoring times between events (TBE) is an important aspect of process monitoring in many areas of applications. This is especially true in the context of high-quality processes, where the defect rate is very low, and in this context, control charts to monitor the TBE have been recommended in the literature other than the attribute charts that monitor the proportion of defective items produced. The Shewhart-type t-chart assuming an exponential distribution is one chart available for monitoring the TBE. The t-chart was then generalized to the tr-chart to improve its performance, which is based on the times between the occurrences of r (≥1) events. In these charts, the in-control (IC) parameter of the distribution is assumed known. This is often not the case in practice, and the parameter has to be estimated before process monitoring and control can begin. We propose estimating the parameter from a phase I (reference) sample and study the effects of estimation on the design and performance of the charts. To this end, we focus on the conditional run length distribution so as to incorporate the ‘practitioner-to-practitioner’ variability (inherent in the estimates), which arises from different reference samples, that leads to different control limits (and hence to different IC average run length [ARL] values) and false alarm rates, which are seen to be far different from their nominal values. It is shown that the required phase I sample size needs to be considerably larger than what has been typically recommended in the literature to expect known parameter performance in phase II. We also find the minimum number of phase I observations that guarantee, with a specified high probability, that the conditional IC ARL will be at least equal to a given small percentage of a nominal IC ARL. Along the same line, a lower prediction bound on the conditional IC ARL is also obtained to ensure that for a given phase I sample, the smallest IC ARL can be attained with a certain (high) probability. Summary and recommendations are given. Copyright © 2014 John Wiley & Sons, Ltd.

Conditional Run-Length and Variable-Length Coding of Digital Pictures

In earlier publications, we have presented two coding schemes which take into account the conditional statistics of input signals. In the first scheme, the codewords are assigned in such a way as to provide a signal with long runs of zeros and ones. In the second scheme, each picture element is coded by variable-length codewords according to the values of previously transmitted PEL's. In this paper, by providing further results, we examine these coding schemes in greater detail. The performance of both schemes in terms of entropy and bit rate are compared with an optimum predictive coder. The simulation results indicate that these schemes have a significant advantage over standard predictive encoders. Methods to reduce the storage requirement for the encoder and decoder codebooks are also discussed.

Conditional run-length coding for multilevel graphic image

The concept of conditional run-length coding mainly used for bilevel image is extended for multilevel graphic image. It is shown that the extended coding algorithm is also optimum for the Markov model. Its performance is compared with bit-plane coding and cosine transform coding.

Improved conditional runlength codes based on two composite states

The performances of the conventional CRC (conditional runlength code) and CIRC (conditional incremental runlength code) are improved with some modifications in coding algorithm. These include ZOH (zero-order hold) prediction, modified line-termination, and classification of the incremental runs. Average gains of 15% and 9% are obtained in compression factor for the CRC and CIRC, respectively.

A Conditional Incremental-Runlength Code Based on Two-Dimensional Markov Model

The conditional runlength code is known to be optimum for the Markov source model, but it is complicated in implementation and not efficient for practical line-based transmission. To avoid these disadvantages, a conditional incremental-runlength coding algorithm based on the same model is investigated. The entropy of the incremental runs is shown to be less than that of the runs, but additional information is required for the state identification. Yet, for a practical case with two composite states, the required state information is reduced so much that the new coding is shown to be better than the conditional runlength coding in line-based transmission. The coding efficiency is compared with some other well-known codes for the standard CCITT test documents. It is found to be one of the best codes, especially for the documents of high complexity.