Kalman Filtering Tools for Early Detection of Change in Random Processes

There are many areas of science and engineering that involve random processes that are subject to structural changes. Examples of such changes can be caused by a failed machine component, or the onset of an epileptic seizure, to name a few. The Kalman filter (KF) has been a valuable tool for change detection for many years. But only in recent years have advances in computer processing power and software allowed the elements of the KF to be explored in any real depth. Traditionally, the depth in KF research has been focused on settings that were mathematically tractable. In this work we investigate a number of fundamental elements of the KF in settings that are sufficiently realistic to preclude in-depth mathematical analysis. Instead, we rely upon simulations. Two of the most fundamental and important elements of the KF include the covariances of the state and measurement processes. The state process covariance is perhaps the most important single element in the KF setting. Setting it to a value that is too small results in lack of adaptability, but low variability. Setting it too high allows the KF to track more rapid changes, but with greater variability (hence, more false alarms). In this work we explore the performance of the KF in relation to detecting short time changes associated with a slowly time-varying first order autoregressive process [AR(1)]. Such processes are more common that one might think. For example, even if the continuous process is of a high order, if it is sampled fast enough it can be well-modeled by an AR(1) process. And due to major advances in high speed data collection systems, many processes are sampled at much higher rates than is necessary to satisfy sampling laws. Our contribution includes a collection of KF tools that, when used together, can quickly and reliably detect certain types of small changes. To demonstrate the potential of these tools, we apply them to early detection of the onset of an epileptic seizure. The goal is to detect the change early enough, so that action to mitigate it can be applied.