GM Grey Model

The Grey Model (GM) is a mathematical forecasting methodology designed to predict system development trends using incomplete or uncertain data. Widely applied across engineering, economics, environmental science, and social sciences, this model operates on the principle that data sequences follow inherent patterns describable by first-order linear differential equations. Key implementation involves constructing cumulative data sequences (AGO) to weaken randomness, followed by parameter estimation through least squares fitting. The core algorithm solves the whitening equation dx/dt + ax = b, where parameters a (development coefficient) and b (grey action quantity) are derived from accumulated matrix operations. A typical Python implementation would involve: 1. Data preprocessing via cumulative generation 2. Matrix construction for parameter estimation using numpy.linalg.pinv 3. Time-response function computation for prediction 4. Inverse accumulation (IAGO) to restore predicted values The model enables researchers to generate actionable forecasts with minimal data requirements, supporting strategic decision-making for system optimization. Validation techniques often include posterior variance checks and residual analysis to ensure prediction reliability.