Nonparametric Change Point Detection Methods for Profile Variability

Due to the importance of seeing profile change in devices such as of medical apparatus, measuring the change point in variability of a different functions is important. In a sequence of functional observations (each of the same length), we wish to determine as quickly as possible when a change in the observations has occurred. Wavelet-based change point methods are proposed that determine when the variability of the noise in a sequence of functional profiles (i.e. the precision profile of medical devices) has occurred; goes out of control from a known, fixed value, or an estimated in-control value. Various methods have been proposed which focus on changes in the form of the function. One method, the NEWMA, based on EWMA, focuses on changes in both. However, the drawback is that the form of the in-control function is known. Others methods, including the χ² for Phase I & Phase II make some assumption about the function. Our interest, however, is in detecting changes in the variance from one function to the next. In particular, we are interested not on differences from one profile to another (variance between), rather differences in variance (variance within). The functional portion of the profiles is allowed to come from a large class of functions and may vary from profile to profile. The estimator is evaluated on a variety of conditions, including allowing the wavelet noise subspace to be substantially contaminated by the profile's functional structure, and is compared to two competing noise monitoring methods. Nikoo and Noorossana (2013) propose a nonparametric wavelet regression method that uses both change point techniques to monitor the variance: a Nonparametric Control Charts, via the mean of m median control charts, and a Parametric Control Charts, via χ²distribution. We propose improvements to their method by incorporating prior data and making use of likelihood ratios. Our methods make use of the orthogonal properties of wavelet projections to accurately and efficiently monitor the level of noise from one profile to the next; detect changes in noise in Phase II setting. We show through simulation results that our proposed methods have better power and are more robust against the confounding effect between variance estimation and function estimation. The proposed methods are shown to be very efficient at detecting when the variability has changed through an extensive simulation study. Extensions are considered that explore the usage of windowing and estimated in-control values for the MAD method; and the effect of the exact distribution under normality rather than the asymptotic distribution. These developments are implemented in the parametric, nonparametric scale, and complete nonparameric settings. The proposed methodologies are tested through simulation and applicable to various biometric and health related topics; and have the potential to improve in computational efficiency and in reducing the number of assumptions required.