MATLAB Implementation Code for Grey GM(1,1) Model

The Grey GM(1,1) model is a widely used grey prediction method suitable for system modeling with limited data and incomplete information. The core concept involves generating new sequences through cumulative summation to weaken the randomness of original data, followed by establishing a first-order differential equation for prediction. When implementing this model in MATLAB, it can be encapsulated as a callable function. The function primarily consists of the following key computational steps: Data Preprocessing: Perform first-order Accumulating Generation Operation (1-AGO) on input data to generate cumulative sequences, enhancing data regularity. Implementation typically uses MATLAB's cumsum() function for efficient cumulative summation. Background Value Construction: Generate background value sequences using adjacent mean values, typically calculated as averages of neighboring data points. This serves as critical intermediate data for differential equation construction. In code, this can be implemented through vectorized operations using mean() and indexing. Parameter Estimation: Solve for the development coefficient (a) and grey input (b) using least squares method, establishing the whitened form of the grey differential equation. The implementation involves matrix operations (pinv() or backslash operator) for efficient parameter calculation. Model Solution: Construct time response functions using the solution of the whitened equation to obtain predicted values for cumulative sequences. The exponential function exp() is typically used for time response implementation. Data Restoration: Convert predicted cumulative values back to original sequence predictions through Inverse Accumulating Generation Operation (IAGO). This is implemented using diff() function for cumulative reduction, producing final prediction results. Accuracy Verification: Evaluate model precision using indicators like posterior variance ratio and small error probability to ensure prediction reliability. Statistical functions like std() and mean() are employed for these calculations. To enhance code versatility, intermediate calculations can be modularized - for instance, separating accumulating generation from parameter estimation. With minor modifications, the code can adapt to other grey system algorithms like GM(1,N) or grey Verhulst models. Function design should standardize inputs and outputs, allowing single or multiple column data input while returning prediction values and model parameters for subsequent analytical calls.