Gray Theory (GM) for PM2.5 Prediction in National Graduate Mathematical Modeling Competition

In the National Graduate Mathematical Modeling Competition, PM2.5 prediction problems frequently employ the core model of gray system theory—GM(1,1). This model is particularly suitable for forecasting scenarios with small sample sizes and limited information, which aligns perfectly with the characteristics of atmospheric pollution monitoring data.

The modeling process of gray prediction fundamentally involves transforming discrete PM2.5 monitoring data sequences through accumulation generation to reduce randomness, converting them into new sequences with clear exponential patterns. For complex indicators like PM2.5 that are influenced by multiple factors including meteorological conditions and pollution sources, GM(1,1) excavates intrinsic data patterns through gray differential equations, requiring less data volume and distribution assumptions compared to traditional statistical methods.

Three critical implementation steps require attention: First, performing grade ratio testing on original PM2.5 concentration data to ensure values fall within acceptable coverage ranges; Second, optimizing background value selection when establishing gray differential equations; Finally, conducting residual and posterior variance tests to evaluate model accuracy levels. Common code implementations typically involve creating functions for data preprocessing, parameter estimation using least squares method, and precision validation modules.

The advantage of this modeling approach lies in requiring minimal historical data to build predictive models, making it ideal for environmental monitoring situations with high data acquisition costs. However, special attention must be paid to prediction deviations during PM2.5 sudden changes (like haze outbreaks), where hybrid modeling combining gray models with intelligent algorithms such as neural networks can be considered. Algorithm improvements often include adaptive weight optimization for background values or Markov chain corrections for residuals.