Advanced Change Point Detection in Time Series Data

Change Point Detection is an important task in analyzing time series data, where the goal is to find points in the data where the behavior or pattern changes. This task is particularly useful in various fields such as finance, healthcare, and environmental monitoring. Traditionally, change point detection methods have focused on univariate (single variable) data. However, as data becomes more complex and includes multiple variables, detecting changes can be more challenging.

In this work, we present a new approach to detecting change points in time series data that can handle multiple interrelated variables. We focus specifically on cases where there is at most one change point in the data. By using a conceptor matrix, we can learn the underlying behavior of the data within a specified time window. This allows us to compare future observations against this learned behavior to identify potential change points.

#Change Point Detection

Change point detection methods can generally be broken down into three categories: sequential methods, agglomerative methods, and divisive methods. Sequential methods are typically used for real-time applications, where the goal is to detect changes as new data becomes available. Agglomerative methods start with each data point as its own group and merge them based on similarities. Divisive methods, on the other hand, focus on breaking up the data into segments and searching for points that best separate these segments.

Our focus is on divisive methods, which are particularly useful for offline change point detection. These methods look for breaks in the data and attempt to identify segments that are statistically different from one another. Many existing methods assume that the data is independent over time or focus only on changes in mean and variance.

#Proposed Methodology

Our approach utilizes random recurrent neural networks (RNNs) along with conceptor matrices. RNNs are designed to capture the dynamic behavior of time series data by maintaining a state that evolves as new data points are processed. Conceptors act as a way to map these states to a lower-dimensional space, which simplifies the analysis.

By using a conceptor matrix, we can summarize the behavior of the data in a specific training window. This allows us to assess how new data relates to that learned behavior. If a significant deviation from this behavior occurs, it may signal a change point.

#The Role of Conceptors

Conceptors help us recognize patterns in the data. In our case, they serve as a regularized linear mapping of RNN states, allowing us to identify when the current state of the system diverges from the previously learned concepts.

The training process involves initializing the RNN and letting it learn from a specified training period. The conceptor matrix encodes the features learned during this period. Once the conceptor is established, we can process new data through the RNN and compute a distance measure between the current state and the conceptor space. This distance is used to estimate the likelihood of a change point.

#Estimating Change Points

To find the most likely change point, we calculate the angles between the learned conceptor states and the states generated by the RNN. We measure the similarity of these states using Cosine Similarity, which quantifies how close the current state is to the conceptor space. A high similarity value indicates that the RNN is generating states similar to those from the training window, while a low similarity suggests a potential change.

The algorithm utilizes a modified CUSUM statistic, which helps us identify the point in time where the maximum deviation occurs. By systematically analyzing these points, we can determine the most likely change point in the data.

#Bootstrapping for Reliability

One challenge in change point detection is estimating the uncertainty around potential changes. To address this, we employ a Moving Block Bootstrap method. This technique allows us to resample the data in a way that preserves its temporal structure. By generating multiple bootstrapped samples, we can estimate the null distribution of the change point statistic.

This provides a framework for making inferences about change points. Specifically, we can quantify the strength of evidence for a detected change point based on how often it appears across numerous bootstrapped datasets.

#Application to Real Data

We applied our methodology to analyze neural data collected from rats. The data captures local field potentials (LFP), which provide insights into the brain's electrical activity during different states, such as sleep and wakefulness. Our goal was to identify points where the neural activity showed a transition between these states.

By using our conceptor-based change point detection method, we were able to pinpoint moments of transition in the LFP data. The results demonstrated our approach's effectiveness in identifying important changes in the time series, even when traditional methods struggled.

#Simulation Studies

To assess the performance of our method, we conducted extensive simulation studies. We generated time series data with known change points and tested our methodology against existing approaches.

Our findings showed that our conceptor change point (CCP) method outperformed traditional techniques, particularly in cases where the data exhibited temporal dependence or complex patterns. We also evaluated how well our method controlled for false positives when no change point was present.

#Conclusion

Our proposed method for detecting change points in multivariate time series data offers a flexible and robust solution. By leveraging conceptor matrices and RNNs, we can accurately identify change points while accommodating the complexities of nonlinear relationships between variables.

The ability to recognize changes in behavior over time is crucial for many fields, and our approach provides a valuable tool for researchers and practitioners alike. Future work may explore extending this methodology to detect multiple change points and enhance its applicability in real-time settings.