Overfitting and Time Series Segmentation: A Locally Adaptive Solution

Time series are unstructured data; they are difficult to monitor, summarize and predict. Weather forecasts, stock market prices, medical data (ECG, EEG) are examples of non-stationary time series we wish to clean, classify and index. Segmentation organizes time series into few intervals having uniform characteristics (flatness, linearity, modality, monotonicity and so on). The popular piecewise linear model can determine where the data goes up or down and at what rate. Unfortunately, when the data does not follow a linear model, the computation of the local slope creates overfitting. We propose an adaptive time series model where the polynomial degree of each interval vary (flat, linear and so on). Given a number of regressors, the cost of each interval is its polynomial degree: flat intervals cost 1 regressor, linear intervals cost 2 regressors, and so on. Our goal is to minimize the Euclidean (l2) error. We present an optimal algorithm running in time O(n) as well as an online (O(n)) top-down heuristic. Over synthetic random walks, historical stock market prices, and electrocardiograms, the adaptive model provides a more accurate segmentation and is a better predictor of missing data points (leave-one-out cross-validation error). In other words, we simultaneously improve the goodnessof-fit and reduce local overfitting.