How to Build and Validate a GM(1,1) Grey Prediction Model Step‑by‑Step
This article explains the GM(1,1) grey prediction model, covering data preprocessing, model construction, parameter estimation, error testing methods, and how to generate forecasts, providing a practical guide for applying the technique to time‑series data.
1 GM(1,1) Prediction Model
When a data series shows a monotonic exponential increase after accumulation, it suggests a differential equation with an exponential solution. Therefore a first‑order grey differential equation model is proposed, denoted GM(1,1), where the first “1” indicates a first‑order differential equation and the second “1” indicates a single‑variable grey model.
Given the reference data series, the first‑order accumulated generating operation (1‑AGO) is performed to obtain the accumulated series, and the mean‑generated series is then computed for further modeling.
2 GM(1,1) Model Prediction Steps
2.1 Data Validation and Processing
To ensure the feasibility of the modeling method, the known data series must be examined. Compute the level ratio of the reference sequence. If all level ratios fall within the admissible range, the series can be used for GM(1,1) prediction; otherwise, apply a suitable translation (adding a constant) to bring the series into the admissible range.
2.2 Model Construction
Establish the first‑order, single‑variable differential equation model. To identify the parameters, set up the equation on the interval and apply the least‑squares method, yielding estimates for the parameters. The discrete form of the model is derived, and solving it provides the predicted values.
2.3 Error Testing
Two verification methods are available:
Relative error test: compute the relative error; if it is less than 0.1, the model meets general requirements, and if less than 0.05, it meets higher requirements.
Level‑ratio deviation test: calculate the level ratios of the reference series, then use the development coefficient to obtain the level‑ratio deviation; the same thresholds apply for assessing the model.
2.4 Forecasting
Using the GM(1,1) model, obtain the predicted value at the desired point and present the forecast according to the specific problem.
References
Si Shoukui, Sun Xijing. Python Mathematics Experiment and Modeling.
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