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REVIEW 3 major objections 5 minor 36 references

Improved GM(1,1) model based on Simpson formula and its applications

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read A discrete Simpson-formula grey model reproduces exponential sequences exactly and beats three standard grey models on real data.

desk verdict A modest but genuine theoretical fix for GM(1,1) with Simpson background value; the 'outperforms' claim is not supported by the tiny empirical differences. read the letter →

arxiv 1908.03493 v2 pith:5GQB6UHT submitted 2019-08-04 stat.AP

Xin Ma , Wenqing Wu , Yuanyuan Zhang This is my paper · ORCID
classification stat.AP MSC 62M1062M2065D30
keywords greyforecastingGM(11)modelSimpsonformulabackgroundvaluediscreteunbiasedpropertyhomogeneousexponentialsequencesmallsample
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper proposes a discrete GM(1,1) grey forecasting model, GMSD(1,1), whose background value is computed with Simpson's rule rather than the usual trapezoid average. The central claim is that this model is unbiased for homogeneous exponential sequences: given data $x^{(0)}(k)=rq^k$, its simulated and predicted values coincide with the true series up to computer round-off. The paper further claims that GMSD(1,1) predicts more accurately than the classical GM(1,1), the discrete DGM(1,1), and the continuous Simpson-based GMSC(1,1) on China's electricity consumption, Lanzhou GDP, and Lanzhou freightage. A sympathetic reader would care because small-sample grey forecasting is widely used in planning, and the improvement is obtained without extra data or complicated optimization.

What carries the argument

The central object is the Simpson background value $z^{(1)}(k)=\frac{x^{(1)}(k-1)+4x^{(1)}(k)+x^{(1)}(k+1)}{3}$, used to discretize the whitening equation $\frac{dx^{(1)}(t)}{dt}+ax^{(1)}(t)=b$ over the interval $[k-1,k+1]$. This produces the second-order difference equation $(a+3)x^{(1)}(k+1)+4ax^{(1)}(k)+(a-3)x^{(1)}(k-1)-6b=0$, which the paper rewrites as a first-order linear difference equation in $x^{(1)}(k+1)-wx^{(1)}(k)$, with $w=\frac{\sqrt{3a^2+9}-2a}{a+3}$. The time response (Eq. 14) is the exact solution of this difference equation, not a sampled exponential, and that exactness is what makes the model unbiased for homogeneous exponential sequences and free of the parameter-dependent error of the continuous GMSC(1,1).

What would settle it

Take the sequence $x^{(0)}(k)=rq^k$ with $q=2.25$ and $r=0.05$ as in Table 2, fit GMSD(1,1), and verify that the restored values equal $rq^k$ to machine precision. Then repeat the comparison on a non-exponential series, such as a linear or quadratic trend or a geometric series with additive noise, and check whether GMSD(1,1) still has the lowest rolling MAPE; if its advantage does not persist, the practical superiority claim is limited to near-exponential data.

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Extended reading notes

Core claim

The paper's central discovery is that replacing the trapezoidal background value of GM(1,1) with the Simpson-rule value, and crucially using the exact solution of the resulting difference equation as the time response, yields a discrete model that is unbiased for homogeneous exponential sequences. In Section 3.4, for $X^{(0)}(k)=rq^k$, the least-squares estimates of $a$ and $b$ reduce to $3(1-q^2)/(1+4q+q^2)$ and $3rq(1+q)/(1+4q+q^2)$, which forces $w=q$; substituting into the restored-value formula gives $\hat{x}^{(0)}(k+1)=x^{(0)}(k+1)$, so simulation and prediction coincide with the true series. The paper also shows that the continuous Simpson-based model GMSC(1,1) does not share this property: its exponential time response satisfies the discretized equation only when $|a|$ is small, and its errors on exact exponential data grow with $|a|$.

Load-bearing premise

The exactness result assumes the data are a pure geometric sequence $rq^k$, and the real-data advantage is demonstrated on only three data sets, so the general claim of superior forecasting rests on those examples.

Editorial extensions

If this is right

  • For exact homogeneous exponential data, GMSD(1,1) achieves simulation and prediction errors at the $10^{-13}$ percent level, effectively machine precision, while GMSC(1,1) errors grow with $|a|$.
  • In the electricity consumption example, GMSD(1,1) has the lowest prediction MAPE (3.2669%) among the four models, ahead of GMSC(1,1) (3.5137%), GM(1,1) (5.0428%), and DGM(1,1) (5.1811%).
  • In the Lanzhou GDP application, GMSD(1,1) has the lowest prediction MAPE (7.6118%) and overall MAPE (5.0454%); in the freightage application, the corresponding values are 6.3579% and 4.3325%.
  • Because the time response is the exact solution of the model's own difference equation, the model's accuracy does not degrade for large values of the system parameter $a$, a limitation of the continuous Simpson model.
  • The paper's closing remark proposes that the same discretization idea can be carried over to other grey models such as GM(1,n) and GMC(1,n).

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The unbiasedness theorem is exact only for pure geometric sequences; for real series the reported advantage is empirical, so a natural next test is whether the model keeps its edge on noisy or trended data.
  • Because Simpson's rule uses three consecutive accumulated values, the model effectively incorporates second-order information; this suggests a family of higher-order quadrature grey models, and one could test whether Newton-Cotes formulas of higher order preserve or improve the unbiasedness property.
  • The paper's suggestion to extend the idea to GM(1,n) and GMC(1,n) could be made concrete by checking whether the exact-solution construction removes parameter-dependence in multi-variable grey models as it does here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The paper proposes a discrete grey model, GMSD(1,1), whose background value is constructed with the Simpson numerical integration formula, and whose time response function is the exact solution of the resulting second-order difference equation. The parameters are estimated by least squares, the discrete model is contrasted with the continuous GMSC(1,1) model, and a proof is given that GMSD(1,1) is unbiased for homogeneous exponential sequences. Numerical examples compare GMSD(1,1) with GM(1,1), DGM(1,1), and GMSC(1,1) on a synthetic exponential sequence and on three real datasets (China's electricity consumption, Lanzhou's GDP, and Lanzhou's freightage), with the empirical sections claiming that GMSD(1,1) provides the most accurate predictions.

Significance. The theoretical core of the paper is sound and useful: the discrete time response is internally consistent with the fitted difference equation, the unbiasedness theorem in Section 3.4 is a genuine algebraic result, and Section 3.3 gives a clean diagnostic, the function phi(a), showing why the continuous GMSC(1,1) model is inconsistent when |a| is not small. These contributions are modest but real within the grey-systems literature. The main weakness is that the paper's headline practical claim, that GMSD(1,1) 'outperforms' the other three models, rests on very small point estimates of MAPE on short test sets without any uncertainty quantification. The absence of code and data also makes the numerical comparisons difficult to verify. If the empirical claims are strengthened or suitably weakened, the paper would be a solid contribution to specialized journals in this area.

major comments (3)
  1. [Section 5; Tables 4-7] The central empirical claim that GMSD(1,1) outperforms GM(1,1), DGM(1,1), and GMSC(1,1) is not supported by the evidence as presented. The reported prediction advantages are extremely small: in Table 5, MAPE_pred is 7.6118% versus 7.6220% for GMSC; in Table 7, it is 6.3579% versus 6.3810%; and in Table 3, it is 3.2669% versus 3.5137% on only three test observations. No confidence intervals, significance tests, or sensitivity analyses are given, and the in-sample MAPE_simu of GMSD(1,1) is the worst among all four models in every application (e.g., Table 3: 1.7387% versus 1.5675% for GM(1,1)). These differences are within the noise of a handful of data points. The authors should either provide proper uncertainty quantification and a test for equality of forecast errors, or substantially weaken the concluding statement to say that GMSD(1,1) performs comparably on these particular datasets rather than claiming general superiority.
  2. [Section 3.4; Eq. (22)] The unbiasedness proof rests on the closed-form least-squares solution stated in Eq. (22), but the derivation is presented with the phrase 'After some calculations' and the algebra is not shown. This is the load-bearing step that identifies the estimated parameters with the values 3(1-q^2)/(1+4q+q^2) and 3rq(1+q)/(1+4q+q^2). The authors should provide the full derivation, including the computation of B^T B and its inverse, and state the conditions under which the inverse exists (for instance, the number of fitting points n and the range of q). Without this, a central result of the paper is not verifiable from the manuscript text.
  3. [Section 3.3; Eq. (18); Tables 3, 4, 6] The numerical implementation of the benchmark models is not transparent. Equation (18) defines the GMSC time response using x(0)(1) as the initial value, and in general this does not reproduce x(0)(2) from the raw data; nevertheless, the GMSC columns in Tables 3, 4, and 6 show exact agreement with the actual value at k=2 (e.g., 28588.0000, 567.0400, and 5973.0000). At the same time, the GMSD response in Eq. (14) uses x(1)(1) and x(1)(2) as initial conditions, so by construction it interpolates the first two original data points. The authors should specify the exact initial-condition convention and the fitted time index set for each of the four models, and state explicitly whether any fitted values are set equal to the observed data by construction. Without this information, the reported advantages could be artifacts of different initialization schemes rather than of the Simpson-based background value.
minor comments (5)
  1. [Section 4.1; Eq. (26)] The text says 'parameters a and b are the provided determined of Eq.(22)'; this should be reworded as 'the provided values determined by Eq. (22)'. Also, Figure 2 depends on r, which is described only as randomly generated in [1,15]; please report a seed or the exact generated values so that the figure is reproducible.
  2. [Introduction, Section 3] The Introduction states that the paper 'study the forecast stability problem' of the discrete GMSD(1,1) model, but no dedicated stability analysis appears in Section 3 or elsewhere. Section 3.3 discusses parameter dependence and inconsistency of GMSC(1,1), which is related but not identical to forecast stability; the text should be aligned with what is actually studied.
  3. [Equations (14)-(15)] The derivation of Eq. (14) is presented for k = 2, 3, ..., n-1, whereas Eq. (15) is stated for k = 1, 2, ..., n-1. Please clarify how the boundary cases k=1 and k=n are defined, since the sums and powers in the formulas involve k-2 and could otherwise be ambiguous.
  4. [Throughout] There are several typos and formatting inconsistencies, for example 'thw development' in Section 1 and the inconsistent spacing of 'GMSC' and 'GMSD' (e.g., 'GM SC(1,1)' versus 'GMSD(1,1)'). A careful proofreading pass is needed.
  5. [Numerical reproducibility] No code or data availability statement is provided. Given that the paper makes quantitative comparisons on real datasets, the raw data and the implementation of the four models should be made available, or at least the data should be provided in an appendix, so that the reported MAPE values can be checked.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: GMSD(1,1) is derived from a Simpson-rule discretization, its time response solves the fitted difference equation, and the homogeneous-exponential unbiasedness theorem is an algebraic self-consistency result rather than a fitted prediction.

full rationale

I walked the derivation chain: the model is built by integrating the whitening equation over [k-1, k+1] and applying Simpson's rule to obtain Eq. (9)/(10); Eq. (14) is obtained by algebraic iteration of the resulting second-order difference equation; parameters are estimated by ordinary least squares from Eq. (16); and forecasts are evaluated on held-out samples (Tables 3, 4, 6). No step defines the target as an input or fits a parameter to the quantity being predicted. The unbiasedness theorem in Section 3.4 considers exact homogeneous exponential sequences X(0)(k) = r q^k; substituting the exact 1-AGO forms into B and Y gives the closed-form least-squares solution (22), and substituting (22) into the response formula (15) reduces algebraically to x(0)(k+1) = r q^{k+1}. That is a mathematical consistency property of the estimator, not a circular prediction: the conclusion is derived from the model equations and the assumed data-generating form, not assumed as an input. The paper does cite several earlier papers by the same authors (e.g., refs. 15-19, 26, 33, 34), but these appear as background references for grey-model applications and do not carry the derivation, parameter estimation, or unbiasedness proof. No uniqueness theorem from the authors' prior work is invoked, and no ansatz is smuggled in through a self-citation. The empirical claim that GMSD "outperforms" the other models rests on small MAPE differences and a few test points, and the skeptic's point about lack of statistical testing is a legitimate robustness concern; however, that is a correctness/evaluation issue, not circularity. I therefore find no circular step and set the score to 0.

Assumptions & free parameters 2 free parameters · 4 assumptions · 0 invented entities

The central claim depends only on the two fitted grey parameters (a, b) and the grey modeling framework; no new entities are introduced.

free parameters (2)
  • a = dataset-dependent (least squares, Eq. 17)
    System development coefficient fitted to each dataset; all forecasts depend on it.
  • b = dataset-dependent (least squares, Eq. 17)
    Grey action quantity fitted to each dataset; all forecasts depend on it.
assumptions (4)
  • domain assumption The 1-AGO series x1(k) satisfies the whitening differential equation dx1/dt + a x1 = b.
    Adopted from Deng's grey system theory, Section 2 Eq. (1); it is the modeling assumption that makes the Simpson discretization meaningful.
  • domain assumption The Simpson rule gives an adequate approximation of the integral of x1(t) over [k-1, k+1].
    Used in Eqs. (8)-(9) to replace the integral with (x1(k-1)+4x1(k)+x1(k+1))/3; accuracy depends on x1 being nearly quadratic.
  • domain assumption The unbiasedness proof assumes the original series is exactly the homogeneous exponential sequence X(0)(k)=r q^k.
    Section 3.4 sets the sequence to r q^k; the theorem does not generalize to arbitrary data.
  • standard math Least squares estimation yields the model parameters a and b from the discretized equation.
    Eq. (17) uses standard least squares; no distributional assumptions are stated.

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Cite this review

Pith. "Pith review of Improved GM(1,1) model based on Simpson formula and its applications." pith.science (2026). https://pith.science/paper/5GQB6UHT

@misc{pith2026190803493,
  author       = {Pith},
  title        = {Pith review of: Improved GM(1,1) model based on Simpson formula and its applications},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/5GQB6UHT}},
  note         = {Machine review of arXiv:1908.03493}
}
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 ${\rm GM_{SD}}$(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 ${\rm GM_{SC} }$(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 ${\rm GM_{SD}}$(1,1) model. Finally, this model is used to predict the Gross Domestic Product and the freightage of Lanzhou, and the results illustrate the ${\rm GM_{SD}}$(1,1) model provides accurate prediction.

Figures

Figures reproduced from arXiv: 1908.03493 by the authors.

Figure 1
Figure 1. Function φ(a) for different values of a One checks easily that when |a| approximately to zero, the first term of Eq.(20) is approximately to zero. In this situation, we can say L(t) = R(t). However, when |a| is large (a ≥ 1.5), the errors L(t) − R(t) will be quite large. That implies the function (18) will not coincidence with the difference Eq.(16), and the continuous GMSC(1,1) model may not be accurate. On the oth… view at source ↗
Figure 2
Figure 2. The values of ε for different values of q and r 12 [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Percentage errors among four models for GDP [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Percentage errors among four models for freightag [PITH_FULL_IMAGE:figures/full_fig_p018_4.png]

Discussion (0). Continue with ORCID to comment.

Reference graph

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