Monitoring Profile Trajectories with Dynamic Time Warping Alignment

In conventional profile monitoring problems, profiles for products or process runs are assumed to have the same length. Statistical monitoring cannot be implemented until a complete profile is obtained. However, in certain cases, a single profile may require several days to generate, so it is important to monitor the profile trajectory to detect unexpected changes during the long processing cycle. Motivated by an ingot growth process in semiconductor manufacturing, we propose a method for monitoring growth profile trajectories of unequal lengths. The profiles are first aligned using the dynamic time warping algorithm and then averaged to generate a baseline. Online monitoring of trajectories is performed based on incomplete growth profiles. Both simulations and an actual application are used to demonstrate the use of the proposed method.