Time Series Analysis with Exponential Smoothing and Grey GM(1,1) Models

Time series refers to a sequence of numerical values arranged in chronological order according to statistical indicators, commonly used to study temporal variation patterns. Exponential smoothing is a fundamental time series forecasting method where the core algorithm calculates weighted averages of historical data to make future predictions, typically implemented through recursive formulas with smoothing coefficients (alpha values) that assign decreasing weights to older observations. The Grey GM(1,1) model represents a specialized form of exponential smoothing particularly suitable for small-sample, nonlinear, non-stationary, and uncertain data forecasting scenarios. This model operates through accumulating generation operations (AGO) to construct differential equations, requiring minimal data points while handling systems with partial unknown information. Beyond these individual methods, optimal weighted combination models and empirical weighted combination models can be applied to refine and optimize forecasting outcomes. These combination approaches involve algorithmic weight optimization techniques (like quadratic programming for optimal weighting) or domain-knowledge-driven empirical weight assignments to improve prediction accuracy through multi-model integration.