Abstract
In the classical GM(1,1) model, an accumulated generating operation is made on the original non-negative sequence to obtain a monotone increasing 1-AGO sequence, and the forecasting model is established based on the 1-AGO sequence. A great number of scholars have improved the accuracy of grey model prediction through better developed background value and the equation for the time response. In this work, we reconstruct the background value based on a new developed monotonicity-preserving piecewise cubic interpolations spline, and thereby establish a new GM(1,1) model. Numerical examples show that the new GM(1,1) model has better prediction quality of data than the original GM(1,1) model and improves the precision of prediction in practice.
Highlights
Let an original non-negative and uniformly spaced sequence be X (0) =n o x (0) ( 1 ), x (0) ( 2 ), · · ·, x (0) ( n ) . (1)The main idea of the classical grey forecasting GM(1,1) model proposed by Deng [1,2] is to make an accumulated generating operation on the original sequence, so as to reduce the randomization of the original data and obtain an obviously monotone increasing 1-AGO sequence X (1)
We shall give several examples to show that the new GM(1,1) model based on C1 monotonicity-preserving piecewise cubic interpolation spline has better predict accuracy than the classical GM(1,1) model
Numerical examples show that the new GM(1,1) model can improve the forecasting quality, especially in prediction reliability and this model performs better when the original data are presented with convexity in time series
Summary
This method has a shortcoming that when the 1-AGO data sequence varies greatly, the result of prediction will have large error (∆S) with the exponential increasing. In [6], Li and Dai reconstructed x (1) (t) by a high-order Newton interpolation polynomial They estimated the background value z(1) (k + 1) based on the Newton-Cores integral. We shall propose a monotonicity-preserving piecewise cubic interpolation spline to reconstruct the curve x (1) (t) and thereby give a new scheme to estimate the background value z(1) (k + 1).
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