Algorithm Implementation of Grey Prediction Based on Grey Forecasting Theory

Grey prediction theory is a forecasting method specifically designed for uncertain systems with small sample sizes and limited information, particularly suitable for scenarios with sparse data where establishing precise mathematical models is challenging. Its core principle involves uncovering inherent data patterns through accumulated generating operations and differential equation modeling. The most classic grey prediction model is GM(1,1) (first-order univariate grey model), whose implementation process consists of the following key steps: Data Preprocessing Original non-negative time series data is first transformed into a monotonically increasing sequence through Accumulated Generating Operation (AGO). This process effectively weakens randomness and highlights trend characteristics, addressing issues with significant fluctuations in raw data. In code implementation, this typically involves cumulative sum operations using functions like numpy.cumsum() in Python or cumsum() in MATLAB. Establishing Whitening Equation A first-order ordinary differential equation (whitening equation) is constructed for the accumulated sequence. Parameters including the development coefficient and grey action quantity are solved using the least squares method. These parameters determine the overall shape of the prediction curve. Computational implementation often utilizes matrix operations for parameter estimation, such as numpy.linalg.lstsq() in Python. Model Solution and Restoration The differential equation is solved to obtain predicted values for the accumulated sequence, which are then restored to the original data scale through Inverse Accumulated Generating Operation (IAGO). The key to this step lies in ensuring the restored data satisfies non-negativity constraints. Algorithm implementation requires careful handling of differential equation solutions and inverse accumulation calculations. Model Validation The posterior variance test method is commonly employed, evaluating model accuracy by calculating the mean-square-error ratio and small-error probability. If the test passes, the model can be used for prediction; otherwise, residual correction or other improved models should be considered. Implementation typically involves statistical functions for variance analysis and probability calculations. In practical applications, grey prediction excels at processing data with exponential trends, such as energy consumption and population growth scenarios. Compared to traditional statistical methods, its advantages include no strict requirements for data distribution and lower computational complexity. However, long-term predictions may experience accuracy degradation due to error accumulation, where optimization through Markov chains or neural networks can be beneficial. The algorithm's implementation challenges lie in numerical stability during parameter solving and boundary condition handling. Matrix operation tools in Python or MATLAB can significantly simplify computational processes through efficient linear algebra operations and optimized numerical computations.