[Paper Review] Improved GM(1,1) model based on Simpson formula and its applications

Xin Ma, Wenqing Wu+1 more|arXiv (Cornell University)|Aug 4, 2019

Grey System Theory ApplicationsDecision Sciences28 references55 citations

TL;DR

Introduces a discrete GM(1,1) model GM_SD(1,1) that uses Simpson's formula for the background value, derives its time response and IAGO, proves unbiasedness for homogeneous exponent sequences, and demonstrates superior forecasts over GM(1,1), DGM(1,1), and GM_SC(1,1) in multiple case studies.

ABSTRACT

The classical GM(1,1) model is an efficient tool to {make accurate forecasts} with limited samples. But the accuracy of the GM(1,1) model still needs to be improved. This paper proposes a novel discrete GM(1,1) model, named ${ m GM_{SD}}$(1,1) model, of which the background value is reconstructed using Simpson formula. The expression of the specific time response function is deduced, and the relationship between our model} and the continuous GM(1,1) model with Simpson formula called ${ m GM_{SC} }$(1,1) model is systematically discussed. The proposed model is proved to be unbiased to simulate the homogeneous exponent sequence. Further, some numerical examples are given to validate the accuracy of the new ${ m GM_{SD}}$(1,1) model. Finally, this model is used to predict the Gross Domestic Product and the freightage of Lanzhou, and the results illustrate the ${ m GM_{SD}}$(1,1) model provides accurate prediction.

Motivation & Objective

• Improve the accuracy of GM(1,1) forecasting with limited data by redefining the background value via Simpson's rule.
• Derive the discrete GM_SD(1,1) time response and IAGO restoration formulas.
• Analyze the relationship and differences between GM_SD(1,1) and GM_SC(1,1) continuous model.
• Prove the unbiasedness of GM_SD(1,1) for homogeneous exponent sequences.
• Validate the model through numerical experiments and real-data applications.

Proposed method

• Derive a discrete GM(1,1) model where the background value uses Simpson numerical integration (Eq. 3.1 to 3.3).
• Obtain parameter estimates a and b by least squares on the Simpson-based background (Eq. 30).
• Compute the 1-AGO series and apply first-order IAGO to obtain simulated and predicted original-series values (Eq. 26).
• Compare GM_SD(1,1) with GM_SC(1,1) by analyzing the consistency of the time-response and difference equations (Eq. 40–43).
• Prove unbiasedness for homogeneous exponent sequences via analytical derivation (Eq. 58).
• Evaluate prediction accuracy using APE and MAPE metrics (Eq. 59–60) and conduct numerical and real-data validations (Sections 4–5).

Experimental results

Research questions

• RQ1 Does simpson-based background reconstruction in GM(1,1) improve forecast accuracy for small samples?
• RQ2 How does GM_SD(1,1) compare with GM_SC(1,1) and other grey models in simulation and real-world data?
• RQ3 Is the discrete GM_SD(1,1) unbiased for homogeneous exponent sequences?
• RQ4 What is the impact of using Simpson background values on stability and prediction performance?
• RQ5 Can GM_SD(1,1) outperform GM(1,1) and DGM(1,1) in GDP and freightage forecasting?

Key findings

• GM_SD(1,1) provides accurate simulation and prediction with extremely low numerical error in homogeneous exponent tests.
• GM_SD(1,1) achieves smaller MAPE in GDP forecasting than four competing grey models (mean simu and pred errors).
• GM_SD(1,1) outperforms GM(1,1), DGM(1,1), and GM_SC(1,1) in the Lanzhou GDP and freightage case studies.
• In simulations, GM_SD(1,1) yields near-zero APE for many steps and demonstrates stability across parameter ranges where GM_SC(1,1) may falter.
• The method shows substantial predictive accuracy improvements for real data (electricity consumption example) and offers a framework extendable to GM(1,n) and GMC(1,n).
• MAPE_over and MAPE_pred are consistently favorable for GM_SD(1,1) in reported applications.