Simple Implementation Method of Gray Prediction

Gray prediction is an effective method for time series forecasting, particularly suitable for scenarios with limited data volume and clear trends. Its core principle involves processing raw data through cumulative generation to reduce randomness, followed by establishing a differential equation model for prediction.

The fundamental steps for implementing gray prediction are as follows:

First, perform cumulative generation on the original one-dimensional data. This step helps weaken the randomness of raw data and enhance its regularity. The cumulative sequence exhibits a more pronounced exponential growth trend, facilitating subsequent modeling. In code implementation, this can be achieved using a simple loop or vectorized operations to create an accumulated sequence.

Then, establish a GM(1,1) model, which is a first-order univariate gray model. The key to this step involves constructing a data matrix and solving model parameters using the least squares method. The obtained parameters include the development coefficient and gray action quantity, which determine the model's predictive capability. Algorithmically, this requires building a coefficient matrix from the cumulative sequence and solving the normal equations.

Next, establish the whitening equation for the gray differential equation. By solving this whitening equation, prediction values for the cumulative sequence can be obtained. This step involves solving differential equations and requires substituting previously obtained parameters. The implementation typically uses exponential function solutions with the calculated coefficients.

Finally, perform cumulative reduction. Since the previous prediction results apply to the cumulative sequence, reduction operations must be applied to restore the original sequence's prediction values. This step yields the final required prediction results. Code-wise, this involves subtracting adjacent elements in the predicted cumulative sequence.

For cases requiring prediction of N values, the model extrapolates N time-point values based on the trend patterns of existing data. The prediction accuracy largely depends on the quality and stationarity of the original data. The implementation requires extending the time parameter in the whitening equation solution.

The advantage of gray prediction lies in its low data requirements, suitability for small-sample forecasting, and relatively small computational load. However, it's important to note that prediction effectiveness may decrease when data exhibits significant fluctuations or clear periodicity. In practical applications, it can be combined with other forecasting methods or data preprocessing techniques to improve prediction accuracy.