Grey System for Predicting Unknown Data

The grey system is an effective method for predicting unknown data, particularly suitable for scenarios with limited data or incomplete information. Its core principle involves constructing mathematical models from small known datasets to forecast future trends.

The most commonly used prediction model in grey systems is GM(1,1), which stands for first-order single-variable grey prediction model. The algorithm follows these key steps:

Data Accumulation Generation (AGO) - The original data sequence undergoes first-order accumulation to enhance regularity patterns, creating a monotonic increasing series suitable for modeling. Code implementation typically uses cumulative sum functions like cumsum() in mathematical programming languages.

Establishing Grey Differential Equation - Using the accumulated data to construct a grey differential equation, calculating the development coefficient (a) and grey input (b) through least squares estimation. This involves matrix operations where parameters are derived from [B^T B]^{-1} B^T Y_n sequences.

Model Solution - Solving the equation to obtain the time-response function, which generates prediction formulas through exponential fitting. The solution typically takes the form x^(1)(k+1) = [x^(0)(1) - b/a]e^{-ak} + b/a.

Data Restoration (IAGO) - Applying inverse accumulation generation to convert predictions back to original scale using differencing operations, implemented through algorithms like diff() or custom cumulative subtraction functions.

Error Testing - Evaluating prediction accuracy through residual analysis, mean relative error calculations, and posterior variance tests. Common validation metrics include mean absolute percentage error (MAPE) and precision grades based on error thresholds.

The method's primary advantage lies in requiring only minimal data (typically 4+ points) to build predictive models without complex statistical assumptions. In practical applications, simply replacing the original data sequence enables rapid forecasting, making it suitable for trend analysis in economics, engineering, environmental studies, and other fields where data scarcity is common.