Ordinal profile monitoring with random explanatory variables

Profiles characterise the functional relationship between the response variable and one or more explanatory variables and have been playing an important role in many applications. Profile monitoring mainly aims at checking the stability of this relationship. In many situations, we observe that the response variable is categorical with three or more attribute levels, and that there is natural order among the levels. Moreover, the explanatory variables are also random rather than fixed at some predefined values. To fully exploit the ordinal information, it is assumed that there is an unknown latent continuous distribution determining the levels of the ordinal response. Based on this, we propose a novel control chart for jointly monitoring the functional relationship, location shifts in the latent continuous distribution, and the random explanatory variables. Simulation results show that our proposed chart is efficient in detecting abnormalities and is robust to various latent distributions.