Multivariate CUSUM Daubechies Discrete Wavelet Transformation Coefficients Charts for Quality Control

The multivariate CUSUM chart was used to control and monitor several measurable qualitative characteristics of production processes through the accumulated sum of Mahalanobis distance. In this article, new charts correspond to the CUSUM chart based on wavelet analysis. The discrete wavelet transform of the Daubechies wavelet of order 40 was used to obtain approximation coefficients (proportional to the sum of observations) at each level and detail coefficients (proportional to the differences of observations) at each level. The approximation coefficients were used to calculate the cumulative sum statistic of Mahalanobis distance to construct the MCUSUM Daubechies wavelet approximate coefficients chart and to find the optimal reference value for target average run length, and the detail coefficients were used to calculate the cumulative sum statistic of Mahalanobis distance values and to construct the MCUSUM Daubechies wavelet detail coefficients chart to find the optimal reference value for target average run length. Therefore, two charts were obtained, the first of which is used to control the averages of the qualitative properties of the produced material, and the second is used to control the standard deviation of the qualitative properties. The proposed charts were more efficient than conventional charts while maintaining average run length, depending on the simulation study and real data for the Erbil Steel company.