Outlier Detection In The Functional Observations With Applications To Profile Monitoring

In some recent SPC applications, a manufacturing process or product is char-acterized by a profile, i.e., responses as a function of a variable which is often time. In order to monitor the manufacturing process, SPC often checks the stability of the profile. However, the presence of outliers has seriously adverse effects on the modeling of profile and accordingly on the properties of control charts. Therefore, outlier detection, aiming at identifying any abnormal profile from a data set, is quite important. In reality, the observations along each profile are not only mea-sured densely but also correlated. Therefore, the detection of abnormal profile can be naturally viewed as identifying outliers in the functional data. In the pre-vious literature, many outliers detection methods were designed for the univariate samples or multivariate samples (the dimension of the data is smaller than the sample size). Comparatively far less work has been done on the detection of out-liers in functional data observations. In this paper, we propose a new functional outlier test based on the functional principal component analysis (PCA). We al-so introduce an efficient stepwise functional outlier detection procedure. After smoothing the raw data, we use the functional PCA to extract a few major and typical features from the functional data. Next, we choose reasonable number of principal components d through the cumulative percentage variance method or the data-based cross validation approach. The test statistics proposed by us only depends on the projections of the difference between each profile and the mean profile on the first d principal components. Under some mild conditions, we have derived some important theoretical properties which include:1) the null distri-bution of the test statistic is asymptotically pivotal with a well-known extreme value distribution; 2) the test is consistent if the number of outliers grow with the sample size. Furthermore, the comparison of different methods in our simulation study indicates that our method is not only fast to compute but also owns the highest power in most cases. Our proposed stepwise functional outlier detection procedure also perform better than the other existing methods. Finally, by il-lustrating the connection between profile monitoring in statistical process control and outlier detections in functional data, we apply the proposed approach to a real-data example from manufacturing processes and show that it performs quite well in applications.