GM(1,1) Grey Prediction Function

The GM(1,1) Grey Prediction Function is a forecasting method based on grey system theory, particularly suitable for scenarios with limited data and incomplete information. Its core principle involves accumulating raw data to reduce randomness, uncover underlying patterns, construct a first-order differential equation model, and ultimately perform predictions with validation. In implementation, this typically involves array operations for data transformation and matrix calculations for parameter estimation.

The function generally comprises the following key modules: Data Preprocessing: Perform grade ratio testing on raw data to ensure suitability for grey modeling, applying translation transformation when necessary. This can be implemented using conditional statements and data validation checks. Accumulated Generating Operation (AGO): Generate a more regular new sequence through first-order accumulation (1-AGO), establishing the foundation for differential equations. Code implementation typically uses cumulative sum functions or loop structures. Parameter Estimation: Utilize least squares method to estimate the development coefficient (a) and grey input (b) of the grey differential equation, determining the prediction model. This involves matrix inversion operations and linear algebra computations. Prediction and Inverse Accumulation: Convert predicted values to the original data scale through inverse accumulated generating operation (IAGO), outputting forecast results for future time points. The implementation requires careful handling of index operations and difference calculations.

The validation module is crucial for ensuring model reliability: Residual Test: Calculate absolute/relative errors between predicted and actual values to assess local fitting accuracy. This can be programmed using error calculation functions and statistical measures. Correlation Degree Analysis: Measure geometric similarity between predicted and original curves through grey relational degree, with threshold typically required to be >0.6. Implementation involves normalization and comparison algorithms. Posterior Variance Test: Determine model grade based on residual variance ratio (C) and small error probability (P), where models with C<0.35 are considered first-grade qualified. This requires variance calculations and probability distribution analysis.

The function is suitable for short-term trend forecasting applications such as energy demand and economic indicators, but attention should be paid to the impact of data monotonicity and volatility on prediction accuracy. The implementation should include data quality checks and boundary condition handling.