Customized forecasting and change point detection with temporal data

Abstract

Forecasting temporal data is vital in diverse managerial applications. Depending on the application, data can come in different forms, such as real-valued time series, multidimensional time series, functional (infinite dimensional) time series, and other objectoriented data such as shape data, symbolic data, etc. In this dissertation, we propose methods to address customized multi-criteria optimal forecasting and change point detection for real-valued time series processes and parametric modeling of empirically obtained density functions, i.e., functional time series data. The literature widely discusses forecasting methods based on optimizing a single criterion. However, multiple evaluation criteria are vital in most managerial applications of forecasting. In the first essay, we propose an algorithm, titled ‘Adaptive Ensemble Generator’ or AEG, for generating a customized forecast ensemble that considers the user’s preferences across multiple criteria. We demonstrate the usefulness of AEG through realworld forecasting applications of COVID-19 (weekly new cases in the United States) and inventory management for intermittent demand (SKU record from a Walmart – California store). This algorithm’s robustness and other characteristics are examined by applying it to an extensive time-series database. In the second essay, a new family of non-parametric test statistics (namely, I1, I2, and I2f ) collectively termed as the ‘I-statistics’ for change point detection in real-valued time series data is proposed. Using scan-based measures obtained through sliding windows, first and second-order derivative-like discrete sequences are generated from the input time series. The characteristics of these sequences can then be used to detect various types of change points in the original time series. Subsequently, this information can be used for both diagnostics and forecasting purposes. The distributions of the I-statistics under the null hypothesis of no change are estimated empirically. With these test statistics at the core, an algorithm for quick detection of multiple change points of multiple types is proposed, titled ‘Dimorphic Changepoints detector using Istatistics’ or DCI. The performance of the I-statistics is compared with existing tests from the literature by comparing the power of the test(s), false positive rate, and error in the location of detected change point(s). The proposed algorithm’s efficacy is demonstrated in diverse real-world applications such as change points analysis of Bitcoin market price, Brent crude oil spot price, monthly total private construction spending in the US, monthly total no. of passengers at the JFK airport, and more. In the third essay, we work with infinite-dimensional functional time series data in discrete time. We focus our scope on curves of probability density functions evolving in time. We propose a method for parametric modeling and reconstructing the underlying data-generating process of empirically obtained probability density functions evolving in discrete time. This reconstruction can explain the variations in the data and forecast the same. The proposed method, titled ‘Functional Fusion Method’ or FFM, operates by assessing the evolution of parameterized basis functions in both scale and shape. This dual-evolution capturing capability provides improved accuracy and easier interpretability compared to existing methods in the literature. The application of FFM is demonstrated on the annual time series of empirically estimated probability density functions of daily temperatures in New York, Los Angeles, and Washington DC.