Research On Statistical Quality Control Based On Sparse Covariance Matrix Detection

In the actual production and processing of products,it is inevitable that there will be some production problems.These production problems may affect the quality of products.Therefore,we must take some measures to eliminate and avoid the impact caused by these production problems,and this process is called statistical quality control.With the rapid development of computer technology,the use of statistical techniques to deal with the quality of products in the production process becomes more and more urgent.Since most of the quality characteristics of a product are interdependent in the production process,the use of univariate control charts for each of these variables may leads to the wrong conclusion.Therefore,compared to one variate control chart,multivariate control chart is much more suitable is in multivariate statistical process control.Initially,the multivariable control chart is mainly used to detect the mean vector.However,we also should note that in the production process,the variance of the variance can also reflect the changes of the product,which can directly reflect the quality of the product.Therefore,in the field of multivariate statistics,the detection of covariance matrix has become even more important.For multivariate or high-dimensional matrices,when a change is detected,the probability of all characteristics changing at the same time is considered to be very small,that is the shifts may only occur in a few elements of the covariance matrix,which brings to sparsity property.In this paper,a new control chart,adaptive LASSO multivariate exponentially weighted moving covariance matrix control chart?ALEWMC?,is proposed to detect the shift of sparse covariance matrix in statistical quality control by using the property of sparsity,and this new control chart is compared with other multivariate control charts.However,as the sparse covariance matrix changes,it is difficult to determine exactly which variable in the covariance matrix is shifting.Therefore,during the analysis of sparse covariance matrix shift,we have to analyze the covariance matrix shift in three categories.The first category is"Variance Shift."Under this category,the shift can only take place at the?1,1?position of the covariance matrix via adding?to?1,1?,while the other elements do not change.The second category is"Correlation Shift."Correlation shift refers to add a correlation?to the?1,2?and?2,1?positions of the covariance matrix,while the other elements do not change.The last category is"simultaneous Variance and Correlation shifts."Variance and Correlation Simultaneous shift refers to add a correlation?to the?1,2?and?2,1?positions of the covariance matrix and add the shift?at?1,1?and?2,2?,while other elements remain unchanged.In this paper,we evaluate the performance of the control chart by the average run length?ARL?.When the process is under control,the average run length?ARL?is recorded as ARL₀.We compute the ARL of different control charts when the process is out of control.When the process is out of control,the smaller ARL is,the more effective this control chart is in detecting this shift.The simulation results show that adaptive LASSO multivariate exponential weighted moving covariance matrix control chart?ALEWMC?have a much faster reaction to shifts than other control charts.