Optimal Weighted Markov Model and Markov Optimal Weighted Combination Model with Their Application in Hunan’s Gross Domestic Product

Abstract

In this paper, we mainly establish an optimal weighted Markov model to predict the GDP of Hunan Province from 2017 to 2023. The new model is composed of a fractional grey model and a quadratic function regression model weighted combination and is obtained through Markov correction. First, the optimal order r of the fractional grey model (FGM) is determined by using the particle swarm optimization (PSO) algorithm, and the FGM model is established. Second, a quadratic regression model is established based on the scatter plot of the data. Then, the optimal weighted Markov model (OWMKM) is obtained by combining the above two sub-models (i.e., the optimal weighted combination model (OWM)) and using Markov correction. Finally, the new model is applied to estimate and predict the GDP of Hunan Province from 2017 to 2023. The forecast results show that the four statistical measures of the optimal weighted Markov model, such as MAPE, RMSE, $R^2$, and STD, are superior to the optimal weighted combination model (OWM), the nonlinear auto regressive model (NAR) and the autoregressive integrated moving average model (ARIMA), which indicates that our new model has strong fitting and higher accuracy. We establish the quadratic regression Markov model (QFRMKM), the fractional grey Markov model (FGMKM), and the optimal combination model of these two sub-models (MKMOWM). The effects of the MKMOWM and OWMKM are compared. This research provides a scientifically reliable reference and has significant importance for understanding the development trends of the economy in Hunan Province, enabling governments and companies to make sound and reliable decisions and plans.

1. Introduction

The gross domestic product (GDP) is not only a crucial indicator that reflects the overall economic scale of a region but also a significant basis for assessing the social development level and the quality of life of its residents. By predicting and analyzing the regional GDP, governments can better understand the local economic and industrial development, formulate relevant policies to guide the adjustment and optimization of the economic structure, achieve sustainable economic growth, and enhance the living standards of the people. As an important economic region in China, the economic development of Hunan Province has a profound impact on the overall economic landscape of the country. By analyzing the GDP of Hunan Province, we can not only grasp the economic development trends in Hunan more accurately but also provide reliable references for government decision-making, in addition to effectively guiding and supporting local economic development.

The research on forecasting GDP traces back to the early 20th century. At that time, economists mainly relied on simple trend analyses and empirical rules to make predictions, such as deducing future GDP growth trends only based on historical data growth rates. With the increasing development of mathematics, statistics, and econometrics, the statistic models and methods have become more complex and precise for forecasting the statistical data.

Generally speaking, the models used to forecast the GDP mainly include time-series analysis methods, machine learning methods, and grey prediction models, and so on. Hyndman and Athanasopoulos [1] provided a comprehensive introduction to the forecasting methods and basic principle and presents enough information about each method for readers to use them sensibly. Examples use R with many data sets taken from the authors’ own consulting experience. Kuhn and Johnson [2] introduced how to perform data preprocessing, model tuning, predictive variables importance measurements, variable selection, and so on. Readers can learn many modeling methods and improve their understanding of many commonly used and modern effective models, such as linear regression, nonlinear regression, and classification models, involving tree methods, support vector machines, etc. Time-series analysis is a method of forecasting based on historical data’s patterns and trends, such as moving average methods, exponential smoothing methods, and ARIMA models. Pierre and Tatsuma [3] utilized the quarterly GDP data in the US to confirm that the existence of breakpoints outweighed the correlation between the trends and the periodic components. Liu and Zhang [4] mainly used the Box–Jenkins model for short-term estimations and forecasting the per capita GDP series. He [5] studied the ARIMA model and conducted an econometric modeling analysis of China’s GDP from 1952 to 2010. Gui and Han [6] computed the sequential growth rate index to monitor and analyze the economic trends, comparing it with the seasonal adjustment model (i.e., TRAMO-SEATS) to identify trend inflection points for relevant policy-making. In [7], the authors conducted linear regression on the macroeconomic data, which include variables, such as the unemployment rate, gold price, and foreign exchange rate. Reducing the data dimension by isolating key features, they enhanced the predictive accuracy of the GDP. Srebro et al. [8] applied the initial Z-score model (a model for manufacturing companies and companies operating in emerging markets) to predict corporate bankruptcy and calculate the Z-score bankruptcy probability.

Machine learning methods primarily utilize the large amounts of data and algorithms for pattern recognition and prediction, including neural networks, decision trees, random forests, and so on. Wang et al. [9] developed the benchmark and extended models; then, by employing BP breakpoint tests and t-tests, they comprehensively studied and forecasted the trends, seasonality, periodicity, types of breakpoints, and intervention effects of the quarterly GDP in China. Subsequently, Geng et al. [10] and Wang et al. [11] advanced the MIDAS model in many aspects, employing it for GDP forecasting. In order to enhance the forecasting accuracy, Li [12] innovatively applied the latest deep learning method, i.e., the Independent Recurrent Neural Network (IndRNN), to predict China’s GDP. Zhang [13] employed exploratory spatial data analysis to evaluate the spatial distribution of per capita GDP across China’s provinces over 25 years. Zhang et al. [14] proposed a Bayesian method, which not only accurately identifies the unknown number of factors but also estimates the latent dynamic factors in dynamic factor models (DFMs) within a real-time forecasting framework. The simulation results indicated that the proposed Bayesian method is effective for the real-time forecasting of the US quarterly GDP. Chu et al. [15] applied a recursive forecasting approach to evaluate performance metrics for different sub-periods. In order to effectively handle multifactor nonlinearity and enhance the convergence speed and accuracy, an economic forecasting model based on the improved RBF neural network was proposed in [16]. Machine learning is useful for macroeconomic forecasting by mostly capturing important nonlinearities that arise in the context of uncertainty and financial frictions [17].

The grey prediction model involves analyzing historical data to deduce the development trends and law of the system, and it can handle systems with incomplete information, uncertainty and instability. For instance, Wang [18] created a variable weight combined prediction model by combining the discrete GM(1,1) model and the new information GM(1,1) model with a composite function-prediction model. This combined model was then applied to forecast the GDP of the Kashgar region. The results indicated that the variable weight GM(1,1) combination prediction model exhibited certain advantages in the time series data prediction. Tian and Liu [19] employed an enhanced Lagrange interpolation method to reconstruct the background value of the GM(1,1) model to predict Xinjiang’s GDP. The results showed that this model eliminates the Runge phenomenon of traditional Lagrange interpolation and has high prediction accuracy and more effectiveness. Gai et al. [20] proposed the grey forecasting method and unveiled the future convergence direction of regional economic development. Zhang and Chen [21] expanded the nonlinear GM(1,1) power model by using both the equal-dimensional grey power model and the principle of new information supplementation. They forecasted China’s GDP during the 13th Five-Year Plan period by prioritizing the latest data. It is known to all that the Nonlinear Grey Bernoulli model (NGBM(1,1)) is successful in control, prediction, and decision-making, especially in forecasting nonlinear small-sample time series. However, enhancing the prediction accuracy of this model poses ongoing challenges. Therefore, Wu et al. [22] developed an optimized nonlinear grey Bernoulli model for forecasting China’s GDP.

In recent years, there has been a growing research interest in the fractional grey models of the grey system, which is gradually moving away from the study of the traditional grey model based on 1-order accumulation generating operation 1-AGO. For example, Wu et al. [23] proposed the fractional grey model based on the fractional accumulation generating operation (FAGO), while Ma et al. [24] established the FAGO discrete model. Wu et al. [25] constructed the FAGO grey Bernoulli models. Subsequently, Mao et al. [26,27] introduced the fractional derivative based on FAGO. Kang et al. [28] developed a fractional grey model with variable order. Moreover, Xie et al. [29] formulated a generalized fractional grey model by incorporating a generalized fractional derivative, including memory effects.

In terms of combination research, based on the BP neural network and the autoregressive integrated moving average (ARIMA) model, Gao et al. [30] investigated their combined model for forecasting the tourism demand. Furthermore, Wang [31] illustrated the variable weighted model for forecasting the GDP, and they also provided theoretical proofs and practical applications. Lu [32] forecasted the nonlinear GDP residuals, combining the estimated values from both models to derive the final forecast. And by integrating the seasonal autoregressive integrated moving average (SARIMA), GM(1,1), and BP neural networks, Long and Yan [33] applied an ensemble approach for forecasting China’s GDP. Peng and Dang [34] developed the combined ARMA-GM-BP prediction model, which was applied to estimate China’s GDP from 2005 to 2013 and forecasted it for 2014–2015. The results demonstrated that it has superior predictive performance over individual models.

Utilizing the optimal weighted Markov model for analyzing GDP data not only enhances prediction accuracy but also provides robust support for scientific decision-making. Moreover, it propels sustainable economic and social development in the Hunan region, augments people’s well-being, and promotes societal progress. Therefore, this paper will establish this model and apply it to predict and analyze the gross domestic product of Hunan province, which has significant research significance and practical value. The main contributions can be summarized as follows:

(1)

The OWMKM model is developed using the PSO algorithm to determine the optimal order of the FGM model while minimizing mean relative errors.

(2)

According to the Markov transition probability matrix and state division, we formulate precise expressions for the estimated and predicted values of the OWMKM model and MKMOWM model.

(3)

The proposed model’s validity is confirmed through numerical examples and its application in forecasting Hunan’s annual GDP. Our model’s results are compared with those of the ARIMA model and the Nonlinear Auto Regressive Model (NAR), with the robustness of each assessed using the statistics $R^2$ and STD.

(4)

We have validated that the proposed model can more accurately and effectively evaluate the development level of Hunan’s annual GDP compared to the optimal weighted combination model.

The following content of this article is structured as follows. Section 2 outlines the construction of the OWMKM model utilizing quadratic function regression and the FGM model with the PSO algorithm, alongside the steps for adjusting forecasts using the Markov model. Section 3 demonstrates the model’s application, showcasing estimations and predictions of quadratic function regression, the FGM model, OWM, and Markov model modifications based on the GDP. Section 4 shows the estimation and prediction results of the QFRMKM model, FGMKM model, and MKMOWM model for the Hunan GDP. Section 5 provides the outlook and conclusion.

2. The Optimal Weighted Markov Model

In this section, we constructed the OWMKM model by incorporating Markov modifications into the OWM model, composed of the quadratic function regression (QFR) and FGM models. We then compared its performance to ARIMA and conducted four statistical experiments to analyze each model’s results.

2.1. The QFR Model

Based on the Statistical Yearbook of Hunan Province in China, we plotted the GDP scatter plot from 2000 to 2016, observed its growth trend, and established the following quadratic function regression model:

$$x\left(\mathrm{k}\right)={\mathsf{\beta }}_0+{\mathsf{\beta }}_{1}k+{\mathsf{\beta }}_2{k}^2+{\mu }_k , \left(\mathrm{k}=1,\cdots ,\mathrm{n}\right)$$

(1)

where $x\left(\mathrm{k}\right)$ represents GDP,${ \mathsf{\beta }}_0,{\mathsf{\beta }}_1,{ \mathrm{and} \mathsf{\beta }}_2$ are regression parameters, $k$ denotes time, and ${\mu }_k$ is a random error term, typically distributed as ${\mu }_k~N\left(0,{\sigma }^2\right)$.

2.2. The FGM Model

The grey model establishes the grey differential prediction model with a small amount of incomplete information to make a fuzzy long-term description of the development law of things. In the traditional grey model (GM(1,1)), the original data are first accumulated using first-order integration and then fitted into an exponential function. However, a limitation arises when the original data do not adhere to the exponential rule, resulting in poor predictions. Fractional order accumulation addresses this by reflecting the data’s regularity. Adjusting the parameters enhances the prediction accuracy, aligning with the principle of prioritizing new information. The FGM model [23] is a prediction model that extends the traditional grey model and enhances the adaptability of the model to complex data structures. The algorithm comprises three steps.

2.2.1. R-Order Cumulative Generation

Referring to the FGM model [23], $X^{\left(0\right)}=\left\{x^{\left(0\right)}\left(1\right),x^{\left(0\right)}\left(2\right),\cdots ,x^{\left(0\right)}\left(n\right)\right\}$ is provided as a non-negative sequence, using the summation formula:

$$x^{\left(r\right)}={\mathsf{\Sigma }}_{i=1}^k{C}_{k-i+r-1}^{k-i}{x}^{\left(0\right)}\left(i\right)$$

(2)

It is known that the $\mathrm{r}$th-order cumulative sequence is

$$X^{\left(r\right)}=\left\{x^{\left(r\right)}\left(1\right),x^{\left(r\right)}\left(2\right),\cdots ,x^{\left(r\right)}\left(n\right)\right\}$$

where $C_{r-1}^0=1,C_k^{k+1}=0,C_{k-i+r-1}^{k-i}={\displaystyle \frac{\left(k-i+r-1\right)\left(k-i+r-2\right)\cdots \left(r+1\right)r}{\left(k-i\right)!}}$.

2.2.2. Construct a Whitening Differential Equation

The whitening differential equation is

$${\displaystyle \frac{d{x}^{\left(r\right)}\left(t\right)}{\mathrm{dt}}}+a{x}^{\left(\mathrm{r}\right)}\left(t\right)=b,$$

(3)

where $a$ represents the developmental grey number and $b$ signifies the endogenous control grey number.

The exponential solution to this equation is

$$x^{\left(r\right)}\left(t+1\right)=\left(x^{\left(0\right)}\left(1\right)-{\displaystyle \frac{b}{a}}\right)e^{-at}+{\displaystyle \frac{b}{a}}$$

(4)

using the least squares method to compute $\widehat{a}$ and $\widehat{b}$,

$$\left[\begin{array}{c}\widehat{a}\\ \widehat{b}\end{array}\right]={\left(B^{T}B\right)}^{-1}{B}^{T}Y.$$

(5)

where $B=\left[\begin{array}{c}\begin{array}{cc}-0.5\left(x^{\left(r\right)}\left(1\right)+x^{\left(r\right)}\left(2\right)\right)& 1\\ -0.5\left(x^{\left(r\right)}\left(2\right)+x^{\left(r\right)}\left(3\right)\right)& 1\end{array}\\ ⋮\\ \begin{array}{cc}-0.5\left(x^{\left(r\right)}\left(n-1\right)+x^{\left(r\right)}\left(n\right)\right)& 1\end{array}\end{array}\right],$ $Y=\left[\begin{array}{c}{x}^{\left(\mathrm{r}\right)}\left(2\right)-x^{\left(\mathrm{r}\right)}\left(1\right)\\ x^{\left(\mathrm{r}\right)}\left(3\right)-x^{\left(\mathrm{r}\right)}\left(2\right)\\ ⋮\\ x^{\left(\mathrm{r}\right)}\left(\mathrm{n}\right)-x^{\left(\mathrm{r}\right)}\left(\mathrm{n}-1\right)\end{array}\right]$.

2.2.3. Solve the Equation and Obtain the Predicted Value

Through the matrix operation above, the solution of the whitening equation is also referred to as the time corresponding function

$${\widehat{x}}^{\left(\mathrm{r}\right)}\left(k+1\right)=\left(x^{\left(0\right)}\left(1\right)-{\displaystyle \frac{b}{a}}\right)e^{-\widehat{a}k}+{\displaystyle \frac{\widehat{b}}{\widehat{a}}},$$

(6)

where ${\widehat{x}}^{\left(\mathrm{r}\right)}\left(k+1\right)$ is the value of the time $k+1$.

Sequence ${\widehat{X}}^{\left(r\right)}=\left\{{\widehat{x}}^{\left(r\right)}\left(1\right),{\widehat{x}}^{\left(r\right)}\left(2\right),\cdots ,{\widehat{x}}^{\left(r\right)}\left(n\right)\right\},$

its reduction sequence is

$${\alpha }^{\left(r\right)}{\widehat{X}}^{\left(r\right)}=\left\{{\alpha }^{\left(1\right)}{\widehat{x}}^{\left(r\right)\left(1-r\right)}\left(1\right),{\alpha }^{\left(1\right)}{\widehat{x}}^{\left(r\right)\left(1-r\right)}\left(2\right),\cdots ,{\alpha }^{\left(1\right)}{\widehat{x}}^{\left(r\right)\left(1-r\right)}\left(n\right)\right\}$$

(7)

where ${\alpha }^{\left(1\right)}{\widehat{x}}^{\left(r\right)\left(1-r\right)}\left(k\right)={\widehat{x}}^{\left(r\right)\left(1-r\right)}\left(k\right)-{\widehat{x}}^{\left(r\right)\left(1-r\right)}\left(k-1\right)$, the predicted value is {${\widehat{x}}^{\left(0\right)}\left(1\right),{\widehat{x}}^{\left(0\right)}\left(2\right),\dots {\widehat{x}}^{\left(0\right)}\left(n\right)$}.

2.3. Combination Model

From the single model above, the optimal weighting method is to calculate the prediction error of the GDP obtained by the created model at a certain time. Let us suppose that the predicted value of the $j$th estimation model at time $k$ is ${\widehat{x}}_j\left(k\right)$, and $x\left(k\right)$ represents the actual GDP ($k=1,2,\cdots ,n$). The values of the jth model ${\widehat{x}}_j$ are ${\widehat{x}}_j\left(1\right),{\widehat{x}}_j\left(2\right),\cdots ,{\widehat{x}}_j\left(n\right)$. The weights are $U={\left[u_1,u_2,\cdots ,u_m\right]}^T$ with ${\displaystyle \sum }_j^m{u}_j=1.$ The error of the model at this time is $e_{jk}=x\left(k\right)-{\widehat{x}}_j\left(k\right)$ and $e_k={\displaystyle \sum }_j^m{u}_j\left(x\left(k\right)-{\widehat{x}}_j\left(k\right)\right)={\displaystyle \sum }_j^m{u}_j{e}_{jk}$. Then, a residual matrix with $m$ order can be constructed [35],

$$\mathrm{E}={{\displaystyle \sum }}_k^n{e}_{jk}{e}_{tk}\left(j,t=1,2,\cdots ,m\right)$$

(8)

We minimize the GDP forecasting error of the model to enhance the forecasting accuracy. Let $V={\left[1,1,\cdots ,1\right]}^T$ be an $n\times 1$ matrix. The following optimization model is established:

$$\left\{\begin{array}{c}minQ={\displaystyle \sum _k^n}{e}_k^2=U^{T}EU\\ s.t.{\displaystyle \sum _j^m}{u}_j=V^{T}U=1\end{array}\right.$$

(9)

Using the Lagrange multiplier formula, from

$$\left\{\begin{array}{c} {\displaystyle \frac{\partial }{\partial U}}\left(U^{T}EU-2\lambda \left(V^{T}U-1\right)\right)=0\\ {\displaystyle \frac{d}{d\lambda }}\left(U^{T}EU-2\lambda \left(V^{T}U-1\right)\right)=0\end{array}\right.$$

(10)

the weight can be computed as

$$U_0={\left[{\widehat{u}}_1,{\widehat{u}}_2,\cdots ,{\widehat{u}}_m\right]}^T={\displaystyle \frac{E^{-1}V}{V^T{E}^{-1}V}}$$

(11)

The optimal weighted combination model is

$${\widehat{x}}_{OWM}\left(k\right)={\widehat{u}}_1{\widehat{x}}_1\left(k\right)+{\widehat{u}}_2{\widehat{x}}_2\left(k\right)+\cdots +{\widehat{u}}_m{\widehat{x}}_m\left(k\right) (k=1,2,\cdots ,n)$$

(12)

2.4. Markov Model

Markov processes are types of random process. The original model, the Markov chain, was proposed by the Russian mathematician A.A. Markov in 1907. This process possesses the property that its future evolution does not rely on its past evolution, given its present state. For instance, changes in the animal population in a forest exemplify Markov processes. In reality, numerous phenomena follow Markov processes, such as the Brownian motion of particles in liquid, the spread of infectious diseases, and the waiting time for trains at stations. Paper [36] models occasional, discrete shifts in the growth rate of a non-stationary series. Algorithms for inferring these unobserved shifts are presented, and the parameters of an auto-regression are viewed as the outcome of a discrete-state Markov process, a byproduct of which the estimation of parameters is permitted based on maximum likelihood. A related line of research that also relies on MCMC methods concentrates on models in which the volatility dynamics of the latent variables are characterized by a discrete-state first-order Markov chain [37]. Markov processes find wide applications in various fields, including customer assets, intelligent health systems, remote sensing evaluation, and other fields [38,39,40,41,42,43,44,45,46,47].

2.4.1. Status Division

${ \mathrm{F}}_1,{\mathrm{F}}_2,\cdots ,{\mathrm{F}}_{\mathrm{m}}$ represent the data sequence divided into several distinct states by the Markov chain. State transitions occur only at countable moments, such as ${\mathrm{t}}_1,{\mathrm{t}}_2,\cdots ,{\mathrm{t}}_{\mathrm{m}}$.

$$F_i=\left(Q_{i1},Q_{i2}\right], ( i=1,2,\cdots ,j)$$

(13)

The number of states divided is $j$, with lower and upper limits of relative error for the state interval denoted as $Q_{i1}$ and $Q_{i2}.$

2.4.2. State Transition Probability Matrix

The transition probability of the Markov chain from state $E_i$ to state $E_j$ after $k$ steps is denoted as $p_{ij}\left(k\right)$,

$$p_{ij}\left(k\right)={\displaystyle \frac{m_{ij}\left(k\right)}{M_i}}$$

(14)

where $M_i$ represents the total number of occurrences of state $E_i$, and $m_{ij}\left(k\right)$ represent the number of times for the state. $F_i$ is transferred to state $F_j$ via $k$ steps, and $m$ is the number of states divided. The one-step state transition probability matrix is displayed as follows:

$$P\left(1\right)=\left[\begin{array}{ccc}{p}_{11}\left(1\right)& \cdots & p_{1m}\left(1\right)\\ ⋮& ⋮& ⋮\\ p_{m1}\left(1\right)& \cdots & p_{mm}\left(1\right)\end{array}\right]$$

(15)

2.5. Determination of Predicted Values

Select the group data $\mathrm{j}$, closest to the predicted data, and determine the step number $\mathrm{t}$ as $1,2,\cdots ,\mathrm{j}$ in order from near to far. Next, take the row vectors of the $\mathrm{t}$-step state transition matrix corresponding to all data to form a new matrix. Determine the most likely state of the predicted value by summing the column vectors of the new matrixes. After determining the state, establish the state interval. The Markov modification value is the midpoint of this interval, i.e., ${\displaystyle \frac{1}{ 2}}(Q_{i1}+Q_{i2})$. Finally, the Markov predicted value is

$${\widehat{x}}_{OWMKM}\left(k\right)={\displaystyle \frac{{\widehat{x}}_{OWM}\left(k\right)}{1+\frac{1}{2}(Q_{i1}+Q_{i2})}}$$

(16)

Using the Chapman–Kolmogorov equation repeatedly, if the initial vector of the initial state $E_i$ of a variable is $V\left(0\right)$, then the respective k-step transition probability matrix and the state vector are

$$P\left(k\right)=\left(P{\left(1\right)}^k\right)$$

(17)

$$V\left(k\right)=V\left(0\right)·\left(P{\left(1\right)}^k\right)$$

(18)

2.6. Model Test Statistics

In the grey system and weighted model, the mean absolute percentage error (MAPE) and root mean square error (RMSE) are used to evaluate the model errors. With reference [48], we calculate the statistics STD and $R^2$, for which the calculation formulas are as follows:

$$MAPE={\displaystyle \frac{1}{n}}{\displaystyle \sum }_{k=1}^n\left|{\displaystyle \frac{x\left(k\right)-\widehat{x}\left(k\right)}{\mathrm{x}\left(k\right)}}\right|$$

(19)

$$MSE=\sqrt{{\displaystyle \frac{1}{n}}{{\displaystyle \sum }}_{k=1}^n{\left(x\left(k\right)-\widehat{x}\left(k\right)\right)}^2}$$

(20)

$$STD=\sqrt{{\displaystyle \frac{1}{n}}{{\displaystyle \sum }}_{k=1}^n{(\left|{\displaystyle \frac{x\left(k\right)-\widehat{x}\left(k\right)}{x\left(k\right)}}\right|-MAPE)}^2}$$

(21)

$$R^2=1-{\displaystyle \frac{{{\displaystyle \sum }}_{k=1}^n{\left(x\left(k\right)-\widehat{x}\left(k\right)\right)}^2}{{{\displaystyle \sum }}_{k=1}^n{\left(\overline{x}\left(k\right)-\widehat{x}\left(k\right)\right)}^2}}$$

(22)

where $\overline{x}$(k) is the average of training data and $\overline{x}\left(k\right)={\displaystyle \frac{1}{n}}{{\displaystyle \sum }}_{k=1}^{n}x\left(k\right)$.

3. Empirical Research and Result Analysis

The traditional quadratic function model, fractional grey FGM model, optimal weighted combination model, and optimal weighted combination Markov model were developed using Hunan provincial GDP data from 2000 to 2016 (unit: 100 million CNY, data sourced from the Hunan Provincial Bureau of Statistics). Furthermore, forecasts and analyses were conducted on the original data from 2017 to 2023 using the aforementioned four models. The mean of the original GDP data is 22,113, the standard deviation is 15,765, the skewness is 0.3788, the kurtosis is 1.7729, the coefficient of variation is 0.7129, the coefficient of variance is 71.29%, and the range of the sequential growth rate is [2.76%, 24.95%]. Several graphs describing the data are shown in Figure 1.

Figure 1. Trend graph of Hunan GDP data.

3.1. Experiment of QFR Model

Hunan’s GDP data from 2000 to 2016 was inputted into Equation (1), establishing the following quadratic function model:

$$x\left(\mathrm{k}\right)=3375.869-806.783k+67.310k^2$$

(23)

The test statistics for this mode are as follows: $R^2$ = 0.9869, adjusted R-squared, ${\widehat{R}}^2=0.9858$, F-statistic = 904.2 (>3.40 on 2 and 24 degrees of freedom), and $p$-value < 2.2 $\times {10}^{-16}$ ($p$ < 0.05). Estimates ($k=11,12,\cdots ,27$) and predictions ($k=28,29,\cdots ,34$) can be calculated. Refer to Table 1 for specific values.

Table 1. Comparison of QFR, FGM, and OWM results (unit: billion CNY).

		QFR		FGM		OWM
Year	Raw	Predicted Value	Relative Error	Predicted Value	Relative Error	Predicted Value	Relative Error
2000	3551.49	2645.76	−0.2553	3551.49	0	2206.2	−0.3788
2001	3831.90	3387.11	−0.1660	4292.65	0.1202	2947.64	−0.2308
2002	4151.54	4263.08	0.0269	5221.38	0.2577	3798.01	−0.0852
2003	4659.95	5273.66	0.1317	6260.52	0.3435	4794.73	0.0289
2004	5542.62	6418.87	0.1581	7402.21	0.3355	5941.65	0.0720
2005	6369.87	7698.70	0.2086	8650.29	0.3580	7236.89	0.1361
2006	7431.55	9113.14	0.2263	10,012.49	0.3473	8676.68	0.1675
2007	9285.45	10,662.21	0.1483	11,498.52	0.2383	10,256.34	0.1046
2008	11,307.36	12,345.9	0.0918	13,119.55	0.1603	11,970.44	0.0586
2009	12,772.80	14,164.20	0.1089	14,888.03	0.1656	13,812.92	0.0814
2010	15,574.32	16,117.13	0.0349	16,817.67	0.0798	15,777.15	0.0130
2011	18,914.96	18,204.68	−0.0376	18,923.54	0.0005	17,855.81	−0.056
2012	21,207.23	20,426.85	−0.0368	21,222.13	0.0007	20,040.89	−0.0550
2013	23,545.24	22,783.63	−0.0323	23,731.44	0.0079	22,323.65	−0.0519
2014	25,881.28	25,275.04	−0.0234	26,471.20	0.0228	24,694.53	−0.0459
2015	28,538.60	27,901.07	−0.0223	29,462.92	0.0324	27,143.09	−0.0489
2016	30,853.45	30,661.72	−0.0062	32,730.13	0.0608	29,657.9	−0.0387
		RMSE	922.83	RMSE	1499	RMSE	984.02
		MAPE	0.098	MAPE	0.1489	MAPE	0.0973
		STD	7.73%	STD	13.39%	STD	8.81%
		$R^2$	0.9888	$R^2$	0.9721	$R^2$	0.9869
2017	34,590.60	33,556.99	−0.0299	36,298.55	0.0494	32,226.49	−0.0683
2018	36,425.78	36,586.87	0.0044	40,196.25	0.1035	34,835.21	−0.0437
2019	39,752.10	39,751.38	0	44,453.94	0.1183	37,469.19	−0.0574
2020	41,781.49	43,050.51	0.0304	49,105.14	0.1753	40,112.15	−0.0400
2021	46,063.09	46,484.25	0.0091	54,186.5	0.1764	42,746.29	−0.0720
2022	48,670.37	50,052.62	0.0284	59,738.09	0.2274	45,593.14	−0.0400
2023	50,012.85	53,755.60	0.0748	65,803.65	0.3157	48,233.97	−0.0720
		RMSE	1638.9	RMSE	8707.2	RMSE	2383
		MAPE	0.0253	MAPE	0.1666	MAPE	0.0543
		STD	2.34%	STD	8.12%	STD	1.35%
		$R^2$	0.9417	$R^2$	0.5045	$R^2$	0.8222

3.2. Application of FGM Model

From Table 1, it is evident that the original data sequence of Hunan’s GDP from 2000 to 2016 is as follows:

$$X^{\left(0\right)} =\{x^{\left(0\right)}\left(1\right), x^{\left(0\right)}\left(2\right), \cdots , x^{\left(0\right)}\left(17\right)\}=\{3351.49, 3831.90, \cdots , \mathrm{30,853.45}\}$$

According to the PSO algorithm, the optimal order $\mathrm{r}$ is found to be 0.1. The least squares method in regression analysis was then used to estimate the unknown parameters $a$ and $b$,

$$\left[\begin{array}{c}\widehat{a}\\ \widehat{b}\end{array}\right]={\left(B^{T}B\right)}^{-1}{B}^{T}Y=\left(\begin{array}{c}-0.08885\\ 732.76742\end{array}\right)$$

where $B=\left[\begin{array}{cc}-3869.269& 1\\ ⋮& ⋮\\ -36811.746& 1\end{array}\right],Y=\left[\begin{array}{c} 635.559\\ ⋮\\ 3168.927\end{array}\right]$.

The time series function is

$${\widehat{x}}^{\left(0.1\right)}\left(k+1\right)=\left(x^{\left(0\right)}\left(1\right)+8247.24\right)e^{0.08885k}-8247.24$$

The estimated value is

$${\widehat{X}}^{\left(0\right)} =\{{\widehat{x}}^{\left(0\right)}\left(1\right), {\widehat{x}}^{\left(0\right)}\left(2\right), \cdots ,{\widehat{ x}}^{\left(0\right)}\left(17\right)\}=\{3351.49, 4292.65, \cdots , 32730.13\}$$

(24)

The predicted value is

$$\begin{array}{c}{\widehat{X}}^{\left(0\right)} =\{{\widehat{x}}^{\left(0\right)}\left(1\right), {\widehat{x}}^{\left(0\right)}\left(2\right), {\widehat{x}}^{\left(0\right)}\left(3\right), {\widehat{x}}^{\left(0\right)}\left(4\right), {\widehat{x}}^{\left(0\right)}\left(5\right)\} = \{36298.55, 40196.25,\\ 44453.94, 49105.14, 54186.5\}\end{array}$$

(25)

In selecting the fractional order for the FGM model, the decision was based on the MAPE result obtained from running Python code. Ultimately, 0.1 was chosen as the order for cumulative modeling because the 0.1 order cumulative grey model yielded the smallest MAPE, indicating the highest prediction accuracy.

3.3. Application of OWM Model

According to Equations (23)–(25), the fitting residual and their residual matrices for the QFR model and FGM model were calculated,

$$=\left[\begin{array}{cc}{\displaystyle {\displaystyle \sum }_k^n}{e}_{1k}^2& {\displaystyle {\displaystyle \sum }_k^n}{e}_{1k}{e}_{2k}\\ {\displaystyle {\displaystyle \sum }_k^n}{e}_{2k}{e}_{1k}& {\displaystyle {\displaystyle \sum }_k^n}{e}_{2k}^2\end{array}\right]=\left[\begin{array}{cc}14,477,556.60& 20,319,588.12\\ 20,319,588.12& 38,199,409.69\end{array}\right]$$

$$U_0={\displaystyle \frac{E^{-1}V}{V^T{E}^{-1}V}}=\left[\begin{array}{c}{\widehat{u}}_1\\ {\widehat{u}}_2\end{array}\right]=\left[\begin{array}{c}1.4853\\ -0.4853\end{array}\right]$$

The optimal weighted combination model is

$${\widehat{x}}_{OWM}\left(k\right)=1.4853{\widehat{x}}_{QFR}\left(k\right)-0.4853{\widehat{x}}_{FGM}\left(k\right),(k=1,2,\cdots ,n)$$

(26)

Refer to Table 1 for the estimated and predicted values of this model (the data for 2017–2023 are forecasted, and the other tables show the same).

3.4. Hunan’s GDP Analysis of OWMKM

3.4.1. Construct Transition Matrix

According to the relative error of the optimal weighted combination Markov model (OWMM), the state intervals are determined. Table 1 displays the minimum relative error of the first 17 fitted data points of this model as −0.3788 and the maximum as 0.1675. Therefore, five state intervals are created based on the equal spacing rule, respectively.

$$F_1\left(-0.4,-0.2865\right], F_2\left(-0.2865,-0.173\right], F_3\left(-0.173,-0.0595\right], F_4\left(-0.0595, 0.054\right]$$

$$F_5\left(0.054, 0.1675\right]$$

The probability of transitioning from the current state to the next state and the corresponding state intervals can be obtained as follows: the one-step state transition probability matrices are

$$\mathrm{P}(1)=\left[\begin{array}{ccccc}0& 1& 0& 0& 0\\ 0& 0& 1& 0& 0\\ 0& 0& 0& 1& 0\\ 0& 0& 0& 0.857& 0.143\\ 0& 0& 0& 0.167& 0.833\end{array}\right]$$

3.4.2. Forecast Hunan’s GDP

A new state transition matrix is constructed using the latest data sets. The status of 2017 is outlined below, with the results presented in Table 2.

Table 2. Forecasting the status of 2017.

Year	Initial State	Transfer Steps	${\mathit{p}}_{\mathit{i}\mathit{j}}$	${\mathit{F}}_1$	${\mathit{F}}_2$	${\mathit{F}}_3$	${\mathit{F}}_4$	${\mathit{F}}_5$
2016	4	1	$p_{14}$	0	0	0	0.8570	0.1430
2015	4	2	$p_{24}$	0	0	0	0.7583	0.2417
2014	4	3	$p_{34}$	0	0	0	0.6903	0.3097
2013	4	4	$p_{44}$	0	0	0	0.6433	0.3567
2012	4	5	$p_{54}$	0	0	0	0.6109	0.3891
Total				0	0	0	3.5598	1.4402

Note: this table displays the transition probability magnitude for each step and the final transition state for the five years leading up to 2017.

According to Table 2, the most probable GDP state for Hunan in 2017 is $F_4$, as it holds the highest value. The predicted value for the optimal weighted combination model in 2017 is 32,226.49 billion CNY, while for the OWMKM model, as per Equation (16), it is 32,315.35 billion CNY. Using the same method, predicted values for the Markov model from 2018 to 2021 are obtained and detailed in Table 3. Figure 2 presents the estimated and predicted values for each sub-model, the optimal weighted model, and the Markov model.

Table 3. The comparison results among OWM, ARIMA, and OWMKM (unit: billion CNY).

		OWM			ARIMA		OWMKM
Year	Raw	Predicte Value	Relative Error	Station Value	Predicte Value	Relative Error	Predicted Value	Relative Error
2000	3551.49	2206.20	−0.3788	1	3549.90	−0.0004	3359.26	−0.0541
2001	3831.90	2947.64	−0.2308	2	3836.03	0.0011	3826.86	−0.0013
2002	4151.54	3798.01	−0.0852	3	4117.36	−0.0082	4297.6	0.0352
2003	4659.95	4794.73	0.0289	4	4495.76	−0.0352	4807.95	0.0318
2004	5542.62	5941.65	0.0720	5	5191.54	−0.0633	5349.22	−0.0349
2005	6369.87	7236.89	0.1361	5	6390.58	0.0033	6515.31	0.0228
2006	7431.55	8676.68	0.1675	5	7104.84	−0.0440	7811.55	0.0511
2007	9285.45	10,256.34	0.1046	5	8618.94	−0.0718	9233.70	−0.0056
2008	11,307.36	11,970.44	0.0586	5	11,074.59	−0.0206	10,776.89	−0.0469
2009	12,772.8	13,812.92	0.0814	5	13,123.72	0.0275	12,435.66	−0.0264
2010	15,574.32	15,777.15	0.0130	4	14,237.59	−0.0858	15,820.65	0.0158
2011	18,914.96	17,855.81	−0.0560	4	18,727.54	−0.0099	17,905.04	−0.0534
2012	21,207.23	20,040.89	−0.0550	4	21,630.52	0.0200	20,096.15	−0.0524
2013	23,545.24	22,323.65	−0.0519	4	23,597.46	0.0022	22,385.2	−0.0493
2014	25,881.28	24,694.53	−0.0459	4	25,993.23	0.0043	24,762.62	−0.0432
2015	28,538.6	27,143.09	−0.0489	4	28,186.08	−0.0124	27,217.93	−0.0463
2016	30,853.45	29,657.90	−0.0387	4	31,300.04	0.0145	29,739.68	−0.0361
		RMSE	984.02		RMSE	435.23	RMSE	710.33
		MAPE	0.0973		MAPE	0.0250	MAPE	0.0357
		STD	8.81%		STD	2.57%	STD	1.67%
		$R^2$	0.9869		$R^2$	0.9978	$R^2$	0.9932
2017	34,590.60	32,226.49	−0.0683	3	33,210.16	−0.0485	32,315.35	−0.0658
2018	36,425.78	34,835.21	−0.0437	4	35,242.42	−0.0325	34,931.27	−0.0410
2019	39,752.10	37,469.19	−0.0574	4	37,467.45	−0.0575	37,572.51	−0.0548
2020	41,781.49	40,112.15	−0.0400	4	39,732.76	−0.0490	40,222.76	−0.0373
2021	46,063.09	42,746.29	−0.072	3	41,982.50	−0.0886	42,846.16	−0.0698
2022	48,670.37	45,593.14	−0.0400	3	44,238.26	−0.0911	45,478.32	−0.0656
2023	50,012.85	48,233.97	−0.072	4	46,491.69	−0.0704	48,040.79	−0.0394
		RMSE	23,830		RMSE	2962.10	RMSE	2360.90
		MAPE	0.0543		MAPE	0.0613	MAPE	0.0543
		STD	1.35%		STD	2.13%	STD	1.30%
		$R^2$	0.8222		$R^2$	0.6769	$R^2$	0.8298

Figure 2. The predicted results of different models for the GDP of Hunan in China.

3.5. Comparative Analysis of the Results of Each Model

From Table 1 and Table 3, it is evident that the QFR model outperforms the FGM model in estimating the original GDP data, with smaller RMSE, MAPE, and STD values and a higher $R^2$ value. The OWM model shares similar MAPE and $R^2$ values with the QFR model but exhibits larger RMSE and STD values. However, the OWMKM model surpasses all others, demonstrating superior performance across all metrics. In summary, the OWMKM model yields the best estimation, followed closely by the QFR and OWM models, while the ARIMA model and FGM model lag behind in effectiveness.

In terms of the estimation, it can be found that the RMSE, MAPE, and STD values of the OWM model are all smaller than the corresponding values of the FGM model. The $R^2$ value of the OWM model is greater than that of the FGM model. The MAPE value and RMSE value of the QFR model are smaller than the corresponding values of the OWM model, and the $R^2$ value of the QFR model is larger than the corresponding values of the OWM model. The STD values of OWMKM model, QFR model, and OWM model are equal. The RMSE, MAPE, STD, and $R^2$ values of the OWMKM model are better than those of the OWM and FGM model. This shows that among the four models, the QFR model has a better prediction effect, the OWMKM model has a better prediction effect than that of the OWM model and ARIMA model, and the FGM model has a weaker effect.

In terms of prediction, the OWM model outperforms the FGM model with smaller RMSE, MAPE, and STD values and a higher $R^2$ value. The QFR model exhibits smaller MAPE and RMSE values compared to the OWM model, with a higher $R^2$ value. The STD values of the OWMKM, QFR, and OWM models are equal. However, the OWMKM model surpasses all others with better RMSE, MAPE, STD, and $R^2$ values. In summary, the QFR model demonstrates the best prediction effect, followed by the OWMKM model, while the OWM and FGM models show weaker effects.

In summary, the OWMKM model outperforms the OWM model in both estimation and prediction. The results depicted in Figure 2 indicate that the curve of the OWMKM model closely aligns with the true values compared to the OWM model.

4. Research on the Prediction of Hunan’s GDP Based on MKMOWM

In the same way as in Part 3, the QFRMKM model, FGMKM model, and MKMOWM model are established to predict Hunan’s GDP of China. The Markov optimal weighted combination model is

$${\widehat{x}}_{MKMOWM}\left(k\right)=1.1264{\widehat{x}}_{QFRMKM}\left(k\right)-0.1264{\widehat{x}}_{FGMKM}\left(k\right),(k=1,2,\cdots ,n)$$

(27)

The results are shown in Figure 3 and Table 4 and Table 5.

Figure 3. The predicted results of different Markov models for the GDP of Hunan in China.

Table 4. The comparison results among QFR, NAR, and QFRMKM (unit: billion CNY).

		QFR			NAR		QFRMKM
Year	Raw	Predicted Value	Relative Error	Station Value	Predicted Value	Relative Error	Predicted Value	Relative Error
2000	3551.49	2645.76	−0.2553	1	3551.49	0000	3322.31	−0.0645
2001	3831.90	3387.11	−0.1660	1	3831.90	0000	4253.23	0.1100
2002	4151.54	4263.08	0.0269	3	4151.54	0000	4325.80	0.0420
2003	4659.95	5273.66	0.1317	5	4660.31	0.0001	4476.25	−0.0394
2004	5542.62	6418.87	0.1581	5	5543.82	0.0002	5448.31	−0.0170
2005	6369.87	7698.70	0.2086	5	6590.70	0.0347	6534.62	0.0259
2006	7431.55	9113.14	0.2263	5	7432.25	0.0001	7735.19	0.0409
2007	9285.45	10,662.21	0.1483	5	9285.73	0.0000	9050.03	−0.0254
2008	11,307.36	12,345.90	0.0918	4	11,307.29	0.0000	11,412.15	0.0093
2009	12,772.80	14,164.20	0.1089	4	13,350.84	0.0453	13,092.93	0.0251
2010	15,574.32	16,117.13	0.0349	4	15,574.57	0.0000	14,898.16	−0.0434
2011	18,914.96	18,204.68	−0.0376	3	18,915.83	0.0000	18,472.53	−0.0234
2012	21,207.23	20,426.85	−0.0368	3	21,207.04	−0.0000	20,727.39	−0.0226
2013	23,545.24	22,783.63	−0.0323	3	23,545.58	0.0000	23,118.85	−0.0181
2014	25,881.28	25,275.04	−0.0234	3	26,301.74	0.0162	25,646.92	−0.0091
2015	28,538.60	27,901.07	−0.0223	3	28,539.97	0.0000	28,311.58	−0.0080
2016	30,853.45	30,661.72	−0.0062	3	30,942.76	0.0029	31,112.85	0.0084
		RMSE	922.83		RMSE	182.73	RMSE	328.09
		MAPE	0.0980		MAPE	0.0058	MAPE	0.0313
		STD	7.73%		STD	1.31%	STD	2.47%
		$R^2$	0.9888		$R^2$	0.9995	$R^2$	0.9987
2017	34,590.6	33,556.99	−0.0299	3	32,720.35	−0.0541	34,050.72	−0.0156
2018	36,425.78	36,586.87	0.0044	3	34,384.71	−0.0560	37,125.18	0.0192
2019	39,752.10	39,751.38	0.0000	3	35,905.24	−0.0968	40,336.05	0.0147
2020	41,781.49	43,050.51	0.0304	3	36,937.82	−0.1159	43,683.92	0.0455
2021	46,063.09	46,484.25	0.0091	3	37,797.83	−0.1794	47,168.18	0.0240
2022	48,670.37	50,052.62	0.0284	3	39,096.34	−0.1967	50,789.06	0.0435
2023	50,012.85	53,755.60	0.0748	4	39,572.78	−0.2087	49,689.96	−0.0065
		RMSE	1638.90		RMSE	6707.08	RMSE	1228.00
		MAPE	0.0253		MAPE	0.1296	MAPE	0.0241
		STD	2.34%		STD	6.05%	STD	1.38%
		$R^2$	0.9417		$R^2$	0.1417	$R^2$	0.9576

Table 5. The comparison results among FGM, FGMKM, and MKMOWM (unit: billion CNY).

		FGM			FGMKM		MKMOWM
Year	Raw	Predicted Value	Relative Error	Station Value	Predicted Value	Relative Error	Predicted Value	Relative Error
2000	3551.50	3551.49	0	1	3428.74	−0.0346	3308.86	−0.0683
2001	3831.90	4292.65	0.1202	2	3875.42	0.0114	4300.99	0.1224
2002	4151.50	5221.38	0.2577	4	4428.65	0.0667	4312.80	0.0388
2003	4660.00	6260.52	0.3435	5	4853.19	0.0415	4428.60	−0.0496
2004	5542.60	7402.21	0.3355	5	5738.23	0.0353	5411.66	−0.0236
2005	6369.90	8650.29	0.3580	5	6705.75	0.0527	6512.99	0.0225
2006	7431.60	10,012.49	0.3473	5	7761.74	0.0444	7731.83	0.0404
2007	9285.50	11,498.52	0.2383	4	9752.77	0.0503	8961.20	−0.0349
2008	11,307	13,119.55	0.1603	3	11,476.16	0.0149	11,404.06	0.0086
2009	12,773	14,888.03	0.1656	3	13,023.11	0.0196	13,101.76	0.0258
2010	15,574	16,817.67	0.0798	2	15,186.62	0.0249	14,861.70	−0.0458
2011	18,915	18,923.54	0.0005	1	18,269.49	0.0341	18,498.19	−0.0220
2012	21,207	21,222.13	0.0007	1	20,488.63	0.0339	20,757.57	−0.0212
2013	23,545	23,731.44	0.0079	1	22,911.21	0.0269	23,145.10	−0.0170
2014	25,881	26,471.20	0.0228	1	25,556.28	0.0126	25,658.38	−0.0086
2015	28,539	29,462.92	0.0324	1	28,444.60	0.0033	28,294.77	−0.0085
2016	30,853	32,730.13	0.0608	1	31,598.88	0.0242	31,051.41	0.0064
		RMSE	1499.00		RMSE	409.89	RMSE	334.40
		MAPE	0.1489		MAPE	0.0313	MAPE	0.0332
		STD	13.39%		STD	1.62%	STD	2.76%
		$R^2$	0.9721		$R^2$	0.9979	$R^2$	0.9986
2017	34,591	36,298.55	0.0494	1	35,043.97	0.0131	32,292.51	−0.0192
2018	36,426	40,196.25	0.1035	2	38,806.96	0.0654	36,912.60	0.0134
2019	39,752	44,453.94	0.1183	2	42,917.49	0.0796	40,009.75	0.0065
2020	41,781	49,105.14	0.1753	3	47,407.93	0.1347	43,213.19	0.0343
2021	46,063	54,186.50	0.1764	3	52,313.67	0.1357	46,517.77	0.0099
2022	48,670	59,738.09	0.2274	4	57,673.38	0.1850	49,918.86	0.0257
2023	50,013	65,803.65	0.3157	5	63,529.30	0.2703	47,974.62	−0.0414
		RMSE	8707.20		RMSE	7074.80	RMSE	1124.75
		MAPE	0.1666		MAPE	0.1262	MAPE	0.0214
		STD	8.12%		STD	7.84%	STD	1.20%
		$R^2$	0.5045		$R^2$	0.5948	$R^2$	0.9582

From Table 4 and Table 5, it is evident that the QFRMKM model outperforms the QFR model in estimating the original GDP data, with smaller RMSE, MAPE, and STD values and a higher $R^2$ value, the same as for FGMKM and FGM. However, the MKMOWM model surpasses all others, demonstrating superior performance across all metrics. In summary, the OWMKM model yields the best estimation, followed closely by the QFRMKM and FGMKM models, while the ARIMA model lags behind in effectiveness. The results depicted in Figure 3 indicate that the curve of the MKMOWM model closely aligns with the true values compared to the FGMKM model and QFRMKM model.

In terms of the estimation and prediction, it can be found that the RMSE, MAPE, and STD values of the MKMOWM model are all smaller than the corresponding values of the OWMKM model. In terms of prediction, the results of the OWMKM model are better than those of the NAR model. The $R^2$ value of the MKMOWM model is greater than that of the OWMKM model.

To see the sensitivity and stability of the models, the statistics Augmented Dickey–Fuller (ADF), p-value, and Lag order were used. The specific results are shown in the following Table 6.

Table 6. The sensitivity and stability of the models.

Statistic	Raw	OWM	OWMKM	MKMOWM	QFRMKM	FGMKM
ADF	−2.6396	−3.6657	−3.8905	−3.0440	−3.000	−2.1295
p-value	0.3287	0.04531	0.0295	0.1747	0.1914	0.5230
Lag order	2	2	2	2	2	2

As can be seen from the table above, the sensitivity and stability of the OWMKM model and the MKMOWM model established by us are superior to those of their sub-models.

In order to validate the robustness of the obtained results, we performed pairwise cross-validation between the models: OWMKM and MKMOWM, OWMKM and QFRMKM, OWMKM and FGMKM, OWMKM and OWM, MKMOWM and QFRMKM, MKMOWM and FGMKM, MKMOWM and OWM, QFRMKM and FGMKM, QFRMKM and OWM, and FGMKM and OWM. The specific heatmaps are shown in the figure below.

As can be seen from Figure 4, the model established by us is robust.

Figure 4. Heatmap of calculated results between different models.

The forecasted GDP of Hunan by OWMKM from 2024 to 2026 is 50,586.27, 52,986.55, and 55,260.74 (unit: billion CNY). The projected values of the GDP in Hunan by MKMOWM from 2024 to 2026 are as follows: 51,141.54, 54,395.67, and 57,708.27 (unit: billion CNY).

5. Conclusions

The optimal weighted combination model is one of the models that has been widely studied in the field of statistical data prediction. This paper focuses on utilizing the GDP data of Hunan Province from 2000 to 2021 as historical data, to establish and investigate the optimal weighted Markov model for predicting the subsequent five years’ (i.e., 2017–2023) development trend of the GDP in Hunan Province. The predictive results indicate that the new model established in this paper exhibits higher forecasting accuracy and better stability compared to traditional grey prediction models, such as the optimal combination model (OWM), NAR model, and ARIMA model. Additionally, the new model, to some extent, mitigates the randomness of the original data, reducing the disturbance bounds of the grey prediction model solutions. In summary, the MKMOWM model outperforms the OWMKM model in both estimation and prediction. The model developed in this paper not only enables more precise forecasting of the regional gross domestic product (GDP) but also can be effectively applied to predict statistical data, such as the regional permanent population and real estate prices. In this paper, there is no research on the influencing factors and variables of the GDP. The use of longitudinal data to model the GDP was not considered. We will use a neural network model to study the GDP and its influencing factors. We will apply the combined model to analyze the direct and indirect effects of corporate bankruptcy on the GDP, the impact of corporate bankruptcy. First of all, the increase in non-performing loans and the bankruptcy of enterprises will lead to their loans becoming non-performing loans, increasing the bad debt rate of banks. Second, the impact on household consumption and corporate insolvency not only affects the financial sector but also indirectly affects household consumption by reducing employment opportunities and wage income. Third, the loss of wages and corporate bankruptcy may lead to a large number of employees unemployed or reduced working hours, which directly affects the income level and consumption capacity of households, further depressing the GDP. Finally, recession expectations, such expectations, can affect consumer spending behavior and business investment decisions, thus restraining economic growth, with less investment in businesses, further depressing GDP growth.