About The Multivariate Weighted Poisson CUSUM Control Chart Study

As an important statistical tool,the CUSUM control chart has the advantages of rapid monitoring of small drifts,real-time tracking and early warning,convenient use and low cost.It is currently widely used in many fields such as industrial production,transportation and disease monitoring.In the industrial production process,the Poisson distribution is often used to characterize the number of defective products.However,in the actual process,multiple count-type data exist widely.Therefore,how to construct and optimize the CUSUM control chart in the case of multiple Poisson has important practical significance.In the control chart model,the concept of weight is often used.The weight indicates the weight of the importance of each variable.In the calculation process,by assigning unequal weights to different observations,it is possible to optimize the control chart and increase the sensitivity of the control chart.Therefore,the paper mainly studies the CUSUM control chart in the case of multivariate Poisson with the weight function.According to the form of the multivariate Poisson CUSUM control chart,a multivariate Poisson CUSUM control chart with log likelihood ratio items with weights is constructed,and the parameters in the weight function wt are discussed.By calculating the average running length in various cases,the influence of the parameters in the weight function wt on the monitoring results of the weighted multivariate Poisson CUSUM control chart during the accumulation process is analyzed.At the same time,the difference between the monitoring effect of the log likelihood ratio weighted multivariate Poisson CUSUM control chart and the traditional multivariate Poisson CUSUM control chart under different weighting functions,different drift amplitudes,and different dimensions of the variables is discussed.The log likelihood ratio weighted multivariate Poisson CUSUM control chart can adjust the size of the parameter in the weight function,a and ?,to make the monitoring ability of the control chart with small drift more sensitive and robust than the traditional multivariate Poisson CUSUM control chart.When the monitoring parameters are constantly approaching the set out-of-control state parameters,the monitoring effect of the log likelihood ratio weighted multivariate Poisson CUSUM control chart begins to deteriorate.The article further compares the monitoring effect of the log likelihood ratio weighted multivariate Poisson CUSUM control chart with the traditional multivariate Poisson CUSUM control chart in the case of single or multiple components drifting.The analysis results are that,when the monitoring parameters are less drifting relative to the controlled state,the monitoring effect of the log likelihood ratio weighted multivariate Poisson CUSUM control chart is still more sensitive and robust than the traditional multivariate Poisson CUSUM control chart.However,when there are more drift components and drift become to be larger,the ARL1 and SDRL1 of the log likelihood ratio weighted multivariate Poisson CUSUM control chart begin to exceed the traditional multivariate Poisson CUSUM control chart.Finally,the article uses two data for empirical analysis,one of which is randomly generated from the trivariate Poisson distribution,and the other is binary count data collected in actual telecommunication communications.Through two empirical analyses,it can be shown that the log likelihood ratio weighted multivariate Poisson CUSUM control chart has a better ability to monitor multivariate count data in practice.