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The conformable fractional grey system model

Xin Ma, Wenqing Wu, Bo Zeng, Yong Wang, Xinxing Wu

arXiv:1805.01789v3stat.ME

TL;DR

Existing fractional grey models use computationally complex accumulation and difference definitions, motivating a simpler conformable formulation. The paper defines CFA and CFD, builds CFGM, and evaluates it against FGM. Across benchmark and real-world forecasting studies, CFGM is reported as more effective, while its sensitivity can worsen performance in some cases.

  • Problem

    Existing FOA and FOD definitions are computationally complex, creating difficulties for implementation and deeper theoretical analysis.

  • Method

    The paper defines CFA and CFD from conformable fractional derivatives, builds CFGM, and selects α by a simple brute-force MAPE search.

  • Results

    CFGM outperforms FGM in benchmark validation and natural-gas forecasting across 11 countries, with stronger advantages in longer-term predictions.

  • Takeaways & Limitations

    CFD and CFA provide a simpler alternative for reconstructing fractional grey-model frameworks and may be useful for unstable, non-smooth series.

  • Takeaways & Limitations

    CFGM’s greater sensitivity to new information can also lead to worse performance in some cases.

Abstract

from arXiv · show

The fractional order grey models (FGM) have appealed considerable interest of research in recent years due to its higher effectiveness and flexibility than the conventional grey models and other prediction models. However, the definitions of the fractional order accumulation (FOA) and difference (FOD) is computationally complex, which leads to difficulties for the theoretical analysis and applications. In this paper, the new definition of the FOA are proposed based on the definitions of Conformable Fractional Derivative, which is called the Conformable Fractional Accumulation (CFA), along with its inverse operation, the Conformable Fractional Difference (CFD). Then the new Conformable Fractional Grey Model (CFGM) based on CFA and CFD is introduced with detailed modelling procedures. The feasibility and simplicity and the CFGM are shown in the numerical example. And the at last the comprehensive real-world case studies of natural gas production forecasting in 11 countries are presented, and results show that the CFGM is much more effective than the existing FGM model in the 165 subcases.

1 Introduction

Existing grey models can work with small samples, but mainstream models face limitations for nonlinear series and fractional operations are computationally complex. The paper proposes conformable fractional operators and a CFGM to address these issues and evaluates them against existing FGM approaches.

  • Motivation: Grey models can often produce forecasts from very small samples, unlike many statistical and machine-learning models that require large samples.The passage contrasts grey forecasting with empirical, semiparametric, hybrid, and machine-learning models.
  • Motivation: Most mainstream grey models are linear, although nonlinearities widely occur in real-world fluid-flow, petroleum, economic, and operational systems.The paper argues that linear grey models are insufficient for more general nonlinear applications.
  • Prior fractional grey models: FOA extends grey models as a nonlinear data-preprocessing method and can improve model performance while preserving the basic FGM structure.FOA is described as a general form of the traditional 1-AGO and as usable for other grey models.
  • Research gap: Existing fractional accumulation and differencing definitions are computationally complex, hindering implementation and deeper theoretical analysis.Related fractional models may also yield analytic solutions containing infinite series.
  • Contribution: The paper introduces CFA and CFD from conformable fractional derivatives, builds CFGM from them, and compares CFGM with FGM through numerical, benchmark, and real-world studies.The evaluation includes case studies of natural-gas production forecasting.

2 The conformable fractional accumulation

The paper defines conformable fractional difference and accumulation operators from properties of the conformable derivative. CFA is constructed as the inverse of CFD and is presented as simpler to implement than existing FOA/FOD definitions.

  • Conformable fractional derivative: The conformable derivative’s relationship with the ordinary derivative provides the basis for defining conformable fractional difference.The paper states that this relationship is useful for constructing CFD.
  • Conformable fractional accumulation: CFA is defined as the inverse operator of CFD, extending the ordinary accumulation relationship to fractional orders.The construction starts from the first-order accumulation as the inverse of first-order difference.
  • Conformable fractional difference: CFD is defined for fractional orders α ∈ (n, n + 1], with a uniform formulation that also includes α = 0.At α = 1, the definition yields the corresponding integer-order difference.
  • Higher-order operators: Higher-order CFA and CFD retain uniform definitions for nonnegative α, and CFA reduces to ordinary higher-order accumulation when α = n + 1.The recursive CFA formulation is described as convenient for computer implementation.
  • Comparison with existing operators: Compared with the existing FOA and FOD, CFA and CFD are presented as simpler computational alternatives.The paper identifies the complexity of existing definitions as a motivation for the new operators.

3 The Conformable Fractional Grey Model

The CFGM reconstructs the GM(1,1) framework using CFA and CFD, estimates its parameters from a discrete whitening equation, and selects the fractional order α by minimizing MAPE with brute force.

  • Model construction: The CFGM is built by applying CFA and CFD within the classical GM(1,1) framework.When α = 1, CFGM becomes the conventional GM(1,1); using FOA instead yields the existing FGM formulation.
  • Parameter estimation: The continuous whitening equation is discretized with the trapezoid formula before parameter estimation by least squares.The discrete form is used with given samples and α.
  • Prediction and restoration: Estimated parameters generate predicted CFA-series values through the response function, after which CFD restores the predicted original-series values.The computation covers the sample period and additional prediction steps.
  • Selecting α: The model’s procedures assume a given α, which is then optimized by minimizing the model’s MAPE.The optimization is a nonlinear programming problem with nonlinear objective and constraint structure.
  • Selecting α: The paper uses brute force to enumerate α values and repeatedly execute the CFGM computations, prioritizing simplicity over optimization efficiency.The cited procedure evaluates α over [0, 2] in increments of 0.01.

4 Numerical example

The numerical example demonstrates the computational workflow of CFGM, from CFA construction through parameter estimation, prediction, restoration, and selection of α. The example finds α = 0.59 as the optimal order using a brute-force MAPE search.

  • Data and setup: The example uses the first 5 observations to build CFGM and reserves the last 5 for testing.All displayed computations use real-number arithmetic in Matlab R2018a.
  • CFA computation: CFA computation is the first modelling step, illustrated at α = 1.1 through successive accumulation calculations.The computational details are organized in Table 1, including intermediate series and higher-order accumulation terms.
  • CFA behavior: As α increases, each CFA value and its growth speed increase; the series approaches 1-AGO near α = 1 and 2-AGO near α = 2.Figure 1 displays CFA series for α = 0.1, 0.2, ..., 2.
  • CFGM modelling: The CFGM workflow estimates parameters from the CFA series, predicts CFA values with the response function, and restores original-scale values using CFD.The numerical example uses α = 0.59 for constructing the CFA series and matrices B and Y.
  • Selecting α: The brute-force search evaluates fitting MAPE for α from 0 to 2 in increments of 0.01 and identifies α = 0.59 as optimal.MAPE values often jump when α is near an integer, and the authors report the search as available for selecting the order.

5 Validation of CFGM with benchmark data sets

The benchmark validation compares CFGM with FGM and AR across 21 small, varied data sets using 1-step, 2-step, and 3-step forecasts. CFGM achieves the strongest accuracy and stability, with slower error growth as the prediction horizon increases.

  • Benchmark data: The validation uses 21 data sets from the Time Series Data Library, covering different sizes and backgrounds.The data-set information is listed in Table 2.
  • Evaluation criteria: Performance is evaluated with MSE, MAE, and MAPE together with their standard deviations.These criteria are defined for assessing modelling accuracy and stability.
  • Validation scheme: The study compares CFGM, FGM, and AR using classical 1-step, 2-step, and 3-step prediction tests.The prediction procedure builds longer-horizon forecasts from previous predicted points.
  • Validation results: Across the benchmark cases, CFGM has smaller criteria than FGM and AR, indicating the highest accuracy and stability.The result is reported for all three prediction horizons, with bold values marking the best results in Table 4.
  • Prediction-horizon effects: 0.4%: CFGM’s MAPE increases by only 0.4% for one additional prediction step, whereas FGM’s MAPE increases more than two times.The authors also report much smaller increases in CFGM’s MAE and MSE than in FGM’s.

6 Applications and analysis

The paper applies CFGM to annual natural-gas production forecasting in 11 countries, using the case study to compare its real-world performance with the existing FGM model.

  • Applications and analysis: Annual natural-gas production forecasting is conducted for 11 countries to assess CFGM against FGM in real-world applications.The case study focuses on countries identified in the paper as mid-sized gas producers.

6.1 Background and data collection

The case study examines annual natural-gas production in 11 countries using data from 2008 to 2016. It targets mid- and small-sized economic entities whose production and consumption may be affected by changing market, industrial, and policy conditions.

  • Data collection: The data cover annual natural-gas production from 2008 to 2016 for 11 mid- or small-sized economic entities.The production data are listed in Table 6 and sourced from the BP Statistical Review of World Energy.
  • Background: The study addresses a research focus that has mainly emphasized larger gas producers such as the USA, China, and Russia.The selected countries broaden the application beyond the largest producers discussed in recent research.
  • Background: Changes in energy markets, industrial development, and domestic policies may significantly affect natural-gas consumption in the selected countries.These factors motivate examining the selected countries as a distinct application setting.

6.2 Overall performance in comparison to the existing fractional grey model

Time-series cross validation compares CFGM and FGM across varied training windows and initial points, covering 165 model fits across 11 cases. CFGM shows better overall fitting and prediction accuracy, greater stability, and a narrower parameter-search requirement.

  • Cross-validation design: Time-series cross validation builds models across different initial points and training sample sizes, with 15 subcases per case and 165 fits across 11 cases.The 15 subcases arise from varying initial points and modelling lengths.
  • Fitting performance: CFGM has smaller MSE, MAE, and MAPE than FGM in 9, 7, and 7 fitting cases, respectively.Most CFGM standard deviations are also smaller, indicating better fitting stability.
  • Prediction performance: CFGM has smaller MSE, MAE, and MAPE than FGM in all prediction cases.Nearly all CFGM standard deviations are smaller, with exceptions for case 1's AE and APE.
  • Prediction performance: The maximum prediction MAPE is 32.2738% for CFGM versus 59.8024% for FGM.The reported upper-bound comparison accompanies the authors' assessment of CFGM's robustness.
  • Parameter selection: FGM requires searching α over a wider range than CFGM, implying that CFGM's optimal parameter is easier to select in applications.The comparison uses [−2, 2] for FGM and [0, 2] for CFGM in the cross-validation procedure.

6.3 About the parameter α

The optimal α values for CFGM are concentrated almost entirely below 1, whereas FGM uses a wider distribution extending substantially above 1. This concentration makes CFGM's parameter optimization easier and informs the design of more precise search algorithms.

  • CFGM parameter range: For CFGM, 84% of optimal α values lie in (0, 1), 15% equal 0, and 99% lie in [0, 1).Only 1.22% of CFGM optimal α values fall in (1, 2].
  • FGM parameter range: For FGM, 72% of optimal α values lie in (0, 1), while 25% lie in (1, 2).This distribution indicates a wider range is needed for FGM than for CFGM.
  • Implications for optimization: CFGM applications almost only need α searched in [0, 1), making α easier to optimize and supporting more precise algorithms or alternative optimizers.The reported range restriction narrows the available search space for parameter selection.

6.4 Some typical cases and analysis

Typical cases show that CFGM responds more strongly than FGM to newly added information, improving on some non-smooth series but becoming vulnerable when additional points distort the fitted trend. The analysis attributes this behavior to differences in how restored-value errors depend on accumulated-series errors.

  • 6.4.1 The non-smooth series with one inflexion point: In case 8, adding one non-inflexion point improved both models, while adding two made CFGM much better but left FGM poor.The case concerns Turkmenistan natural-gas production and has one main inflexion point.
  • 6.4.1 The non-smooth series with one inflexion point: The authors identify the inflexion point as important for CFGM, while additional non-inflexion points can weaken that dependence.They also report that CFGM is more sensitive than FGM to newly added non-inflexion points.
  • 6.4.2 The non-smooth series with two inflexion points: In case 9, CFGM was slightly more accurate than FGM and correctly captured the overall trend after one non-inflexion point was added.The series had two inflexion points, and the case used Brunei natural-gas production.
  • 6.4.3 The non-smooth series with four inflexion points: In case 7, CFGM performance worsened as additional inflexion or non-inflexion points were added, whereas FGM continued following the original series’ overall increasing trend.The series had four inflexion points and approached an oscillatory pattern.
  • 6.4.4 Further discussions on the typical cases: The CFGM’s restored-value error depends on only a few nearby accumulated-series errors, whereas FGM’s error depends on all errors in the FOA series.For α in (0,1], the CFGM error is often affected by the former two points, while FGM can propagate one large error forward.
  • 6.4.4 Further discussions on the typical cases: This local error dependence makes CFGM more responsive to new points and can help it outperform FGM on non-smooth natural-gas series, but the sensitivity can also worsen performance.The paper’s typical-case analysis links CFGM’s stronger performance to unstable time series and its sensitivity to new information.

7 Conclusions

The paper introduces CFA and CFD to construct CFGM, emphasizing simpler implementation and fractional-order tuning. Benchmark and real-world evaluations report stronger performance than FGM, while broader extensions remain future work.

  • CFGM is built from novel conformable fractional accumulation and difference definitions, making the model easier to implement and tune.The optimal fractional order α can be obtained with a simple Brute Force method.
  • CFGM outperforms FGM in benchmark tests and natural-gas-consumption forecasting across 11 countries, especially for longer-term predictions.The evaluation compares 1-step, 2-step, and 3-step predictions and uses time-series cross-validation.
  • CFD and CFA provide a methodology for reconstructing existing fractional grey-model frameworks and developing new conformable grey models.The authors also identify possible applications to difference equations, statistical models, dynamical-system simulation, and control as future work.
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