Grey Model GM(1,1) with Smoothness Optimization of Original Data Sequence

The Grey Model GM(1,1) is a modeling and prediction method based on original data, particularly suitable for system analysis with small sample sizes and incomplete information. Original data often contains noise or irregular fluctuations, which can impact the prediction accuracy of the model. Therefore, before constructing the GM(1,1) model, smoothness optimization of the original data sequence is typically necessary to enhance modeling effectiveness.

The core concept of smoothness optimization involves using mathematical methods to improve the smoothness of raw data, making it better align with the assumptions of the grey model. A common approach is applying Accumulated Generating Operation (AGO) to preprocess the original data - the accumulated sequence generally exhibits stronger exponential patterns, making it more suitable for GM(1,1) modeling. Additionally, methods like moving averages and exponential smoothing can further optimize data smoothness.

From an implementation perspective, AGO transformation can be coded using cumulative sum operations (e.g., numpy.cumsum() in Python), while moving average optimization typically requires window-based convolution calculations. The optimized data can better fit the differential equations of GM(1,1), thereby improving prediction accuracy. This optimization step finds wide applications in financial forecasting, engineering management, and environmental science, particularly in scenarios with significant data fluctuations but stable overall trends.