ARIMA Markov Model and Its Application of China’s Total Energy Consumption

Abstract

We propose an auto regressive integrated moving average Markov model (ARIMAMKM) for predicting annual energy consumption in China and enhancing the accuracy of energy consumption forecasts. This novel model extends the traditional auto regressive integrated moving average (ARIMA($p,\mathrm{d},\mathrm{q}$)) model. The stationarity of China’s energy consumption data from 2000 to 2018 is assessed, with an augmented Dickey–Fuller (ADF) test conducted on the $\mathrm{d}$-order difference series. Based on the auto correlation function (ACF) and partial auto correlation function (PACF) plots of the difference time series, the optimal parameters $\mathrm{p}$ and $\mathrm{q}$ are selected using the Akaike information criterion (AIC) and Bayesian information criterion (BIC), thereby determining the specific ARIMA configuration. By simulating real values using the ARIMA model and calculating relative errors, the estimated values are categorized into states. These states are then combined with a Markov transition probability matrix to determine the final predicted values. The ARIMAMKM model is validated using China’s energy consumption data, achieving high prediction accuracy as evidenced by metrics such as mean absolute percentage error (MAPE), root mean square error (RMSE), $\mathrm{S}\mathrm{T}\mathrm{D}$, and ${\mathrm{R}}^2$. Comparative analysis demonstrates that the ARIMAMKM model outperforms five other competitive models: the grey model (GM(1,1)), ARIMA(0,4,2), quadratic function model (QFM), nonlinear auto regressive neural network (NAR), and fractional grey model (FGM(1,1)) in terms of fitting performance. Additionally, the model is applied to Guangdong province’s resident population data to further verify its validity and practicality.

1. Introduction

Energy is fundamental to human survival, social progress, and national economic development. However, excessive energy consumption poses significant challenges to sustainable growth. Consequently, ensuring stable energy supply and curbing accelerated consumption have become urgent priorities for nations and regions worldwide. In recent decades, China’s rapid economic expansion and energy consumption patterns have garnered substantial global attention.

Due to the complexity of the energy consumption system, it is influenced by numerous factors and constraints. In light of international circumstances and domestic developmental requirements, the study of total energy consumption has garnered increasing attention and interest from scholars both domestically and internationally. This research has a long history in the field of statistics, and existing studies on energy consumption demand have employed various methods and models for analysis and forecasting. For instance, Shu et al. [1] utilized the environmental Kuznets curve (EKC) to simulate the relationship between energy consumption and carbon emissions in China, analyzed future trends, and determined the peak years for energy consumption and carbon emissions. Based on the ARIMA model, Li et al. [2] forecasted Beijing’s total energy consumption and proposed corresponding policy recommendations according to the forecast results. Wu et al. [3] combined the nonlinear grey Bernoulli (NGBM(1,1)) model with an interface particle swarm optimization algorithm to optimize parameters and predict short-term renewable energy consumption during China’s “13th Five-Year Plan” period, achieving high prediction accuracy. He [4] compared the prediction effects of two improved GM(1,1) methods and their combination using China’s total energy consumption data from 2007 to 2022 and applied the δRGM(1,1) model to forecast total energy consumption from 2023 to 2027. Shan [5] constructed a multiple regression model based on the stochastic impacts by regression on population, affluence, and technology (STIRPAT) model to analyze China’s energy consumption-related carbon emissions and applied the model to forecast carbon emissions from 2023 to 2035 under different scenarios. Dietz et al. [6,7] introduced the STIRPAT model, which expanded population, per capita wealth, and technology factors to examine the impact of additional factors on carbon emissions.

Ma and Liu [8] discussed a new delay polynomial grey prediction model (TDPGM(1,1)) based on grey system theory, comparing its performance in forecasting economic growth and utilizing commonly used forecasting models to predict China’s gas consumption. Based on statistical data of natural gas consumption in China from 1995 to 2011, Zhang and Zhou [9] employed the Boltzmann model and third-order polynomial curve model to fit historical data, aiming to explore past trends and identify a suitable model for predictions. A combination model based on the Boltzmann model and third-order polynomial was proposed, demonstrating good fitting performance. To maximize the utilization of existing grey system models, a novel strategy for constructing predictive models of China’s electricity supply was developed using ensemble learning [10]. Two numerical validation cases confirmed the proposed method and compared it with other well-known models. Additionally, considering the change in natural gas consumption, a fractional grey prediction model was established to predict the delay effect [11]. Theoretical analysis revealed that it used a more general formula, being unbiased and more flexible than existing models. Wu et al. [12] applied natural gas consumption in the United States, Germany, the United Kingdom, China, and Japan using a new grey Bernoulli equation model, deriving analytic formulas for the time-response function, restoring value, and linear parameter estimates. Chen et al. [13] developed a novel fractional Hausdorff discrete grey model (FHDGM(1,1)) to predict renewable energy consumption from 2021 to 2023 in three regions: Asia-Pacific, Europe, and globally. Zhang et al. [14] constructed a multivariable grey model (EFMGM(1,2)) based on fractional order accumulation and equidimensional recursive optimization to predict the carbon emission trend of fossil energy consumption in China and calculate the carbon emission of fossil energy consumption by fixed-price GDP. Li et al. [15] provided two time series models, namely the improved grey model (IGM(0,n)) and the optimized fractional grey model (OFGM(1,1)), to predict wastewater discharge and energy consumption in China.

Nong [16] analyzed the correlation between crude oil price volatility and clean energy enterprise price volatility in China by establishing a time-varying vector autoregressive model with random volatility, time-varying spillover index, and dynamic condition-related GARCH model. Xiao [17] proposed an integrated energy system multivariate load forecasting method based on variational modal decomposition (VMD) and combined deep neural networks, conducting a case study on the integrated energy system of Arizona State University’s Tempe Campus. Zhao and Guo [18] integrated the multi-task learning (MTL) approach, the bootstrap method, the enhanced salp swarm algorithm (ISSA), and the multi-kernel extreme learning machine (MKELM) to develop an uncertain interval prediction model for electricity, heat, cooling, and gas loads. To enhance the accuracy of multiple load forecasting in a regional integrated energy system, Wang and Zhang [19] introduced a short-term multiple load forecasting model based on the quantum-weighted gated recurrent unit (GRU) neural network and a multi-task learning framework. Leveraging the weight-sharing mechanism in multi-task learning and the principles of least squares support vector machines, a combined forecasting model for electricity, heat, cooling, and gas loads was constructed using multi-task learning and least squares support vector machines [20]. Abid et al. [21] established a new fusion of multi-directional gated cycle units (MD-GRU) and convolutional neural networks (CNN), using global average pooling (GAP) as hybridization for load and energy prediction. The integration of MD-GRU and CNN addressed spatial and temporal aspects and high-dimensional data issues. Alrashidi [22] proposed an ultra-short-term global horizontal irradiance (GHI) prediction model using the CNN algorithm and considering optimal heuristic configuration. Xu et al. [23] designed a grey model with an optimal time-response function, optimizing nonlinear parameters via particle swarm optimization and verifying the model’s reliability using China’s electricity consumption as an example. Wang et al. [24] focused on and proposed an improved grey prediction model based on hybrid prediction stability. They studied a multi-objective ant colony optimization algorithm for dynamically selecting the best input training set and conducted practical evaluations with the annual electricity consumption in various regions of China as the research object. Ding et al. [25] applied the particle swarm optimization algorithm to new initial conditions, combined with a rolling prediction mechanism, obtaining the future trends of China’s total electricity consumption and industrial electricity consumption of 10 billion, respectively.

Liu [26] proposed the ARMI-B-GM(1,1) model, which uses the ARIMA prediction model to fit and predict grey input to realize dynamic changes in gray input, forecasting China’s per capita energy consumption using the ARMI-B-GM(1,1) model. Miao [27] established an ARIMA model to predict future energy consumption, showing that the ARIMA model achieved high prediction accuracy and was suitable for energy consumption forecasting. During the 2012–2020 period, China’s energy consumption was likely to increase, with energy consumption in 2020 expected to reach 4.45 billion tons of standard coal, below the target value. Liu [28] used the X-12-ARIMA model to test whether Spring Festival-related factors affected air quality in Hefei. The autoregressive integrated moving average (ARIMA) time series forecasting approach was employed to predict air pollution in Nanded City, India, focusing on two primary locations [29]. The ARIMA and seasonal autoregressive integrated moving average (SARIMA) methods were utilized for air quality forecasting [30]. Zhang et al. [31] introduced a wavelet-ARMA/ARIMA model for forecasting short-term PM10 concentration series. Compared with traditional ARMA/ARIMA methods, this wavelet-ARMA/ARIMA model effectively reduced forecasting errors, enhanced prediction accuracy, and achieved multi-time-scale predictions. Wavelet reconstruction was applied to the approximate and detail series for PM10 particulate forecasting [32]. Based on experimental results, the wavelet-transform (WT)-based hybrid WT-ARIMA model demonstrated superior performance compared to the conventional ARIMA model. Based on these research findings, many scholars have conducted extensive studies on energy prediction models. However, these models are prone to random errors. To capture the nonlinear trend analysis of China’s annual energy consumption data and improve forecasting accuracy, a time series Markov model (ARIMAMKM) is proposed.

Although time series models excel at handling short-term data in time series, they also have certain limitations, making it challenging to accurately make long-term predictions and mitigate the effects of random errors, especially when capturing relationships in large datasets under nonlinear conditions. To overcome these limitations and enhance prediction accuracy, we propose the ARIMAMKM model to capture nonlinear trends. The prediction accuracy of China’s annual energy consumption data has been improved. The main contributions of this study can be summarized as follows:

(1)

An ARIMA model was established based on time series data. The stationarity of the original time series was evaluated, and the sequence after -order difference was tested using the ADF test. During the model selection process, AIC and BIC criteria were used to determine the values of the model;

(2)

Based on the Markov transition probability matrix and state division, concrete expressions for the estimated and predicted values of the ARIMAMKM model were constructed;

(3)

The validity of the model was verified through numerical examples, and the model was used to predict energy consumption. The proposed model and four comparison models were analyzed, with the validity and robustness of each model expressed using and STD statistics;

(4)

To verify the validity and applicability of our model, the ARIMAMKM model was applied to study Guangdong Province’s permanent population data. This model proved more effective than ARIMA, GM, NAR, and FGM models in the application of Guangdong Province’s permanent population data.

This paper is organized as follows: Section 2 introduces the construction process of the ARIMA and ARIMAMKM models, incorporating Markov model modifications for predictions. Section 3 presents the application of the model, showcasing the results of energy consumption estimation and prediction using the ARIMA model and Markov model and comparing them with those of the other four models. Section 4 applies the ARIMAMKM model to Guangdong Province’s permanent population data to verify the model’s validity and applicability. Finally, Section 5 summarizes the full text and discusses the research prospects of this paper.

2. ARIMAMKM Model

2.1. ARIMA Model

Let the original sequence {${\mathrm{u}}_{\mathrm{t}}$} follow the ARIMA($p,\mathrm{d},\mathrm{q}$) model, then

$$\mathsf{\Phi }\left(\mathrm{B}\right){\left(1-\mathrm{B}\right)}^{\mathrm{d}}{\mathrm{u}}_{\mathrm{t}}=\mathsf{\theta }\left(\mathrm{B}\right){\mathsf{ϵ}}_{\mathrm{t}}$$

(1)

where $\mathrm{B}$ is the lag operator, defined as

$$\mathrm{B}{\mathsf{\mu }}_{\mathrm{t}}={\mathsf{\mu }}_{\mathrm{t}-1}$$

(2)

$$\mathsf{\Phi }\left(\mathrm{B}\right)=1-{\mathsf{\varphi }}_1\mathrm{B}-{\mathsf{\varphi }}_2{\mathrm{B}}^2-\cdots -{\mathsf{\varphi }}_{\mathrm{p}}{\mathrm{B}}^{\mathrm{p}}$$

(3)

is the coefficient polynomial of the autoregressive (AR) part of the ARIMA($p,\mathrm{q}$) model.

$$\mathsf{\theta }\left(\mathrm{B}\right)=1-{\mathsf{\theta }}_1\mathrm{B}-{\mathsf{\theta }}_2{\mathrm{B}}^2-\cdots -{\mathsf{\theta }}_{\mathrm{q}}{\mathrm{B}}^{\mathrm{q}}$$

(4)

is the coefficient polynomial of the moving average (MA) part of the ARIMA($p,\mathrm{d},\mathrm{q}$) model, $\mathrm{d}$ is the order of difference, and ${\mathsf{ϵ}}_{\mathrm{t}}$ is a random error term with zero mean and constant variance.

The main steps to build an ARIMA($p,\mathrm{d},\mathrm{q}$) model are as follows:

Step 1. Stationarity test of the original sequence: This process can be accomplished by drawing the time series graph and using statistical hypothesis testing methods. The test results indicate that {${\mathrm{u}}_{\mathrm{t}}$} is a non-stationary sequence. To ensure the feasibility of subsequent modeling, a series of transformation methods can be applied, such as difference, logarithmic transformation, or regression processing, to convert it into a stationary sequence. This paper primarily employs the difference technique to transform the original sequence into a stationary state.

Step 2. Model order determination: The values of $p$ and $\mathrm{q}$ for the model are determined based on the ACF and PACF plots of the stationary sequence.

Table 1. Order determination criteria for ARMA models.

Model	Autocorrelation Coefficient	Partial Autocorrelation Coefficient
AR($p$)	Dragging tail	$p$-th order dragging tail
MA($\mathrm{q}$)	$\mathrm{q}$-th order chop off the tail	Dragging tail
ARMA($p,\mathrm{q}$)	Dragging tail	Dragging tail

shows the order determination criteria for the ARMA($p,\mathrm{q}$) model.

Step 3. Model selection: When dealing with time series data, multiple suitable models can often be identified for a single time series. Therefore, effective methods are needed to select the optimal model. The focus is primarily on the AIC and BIC criteria for model selection. Under these two criteria, we determine the optimal model by comparing the AIC and BIC values of different models, where the model with the smallest AIC and BIC values is considered the optimal model.

The formula for the AIC criterion is

$$\mathrm{A}\mathrm{I}\mathrm{C}=-2\ln \left(\mathrm{L}\right)+2\mathrm{k}$$

(5)

The formula for the BIC criterion is

$$\mathrm{B}\mathrm{I}\mathrm{C}=-2\ln \left(\mathrm{L}\right)+\mathrm{k}\mathrm{l}\mathrm{n}\left(\mathrm{n}\right)$$

(6)

where $\mathrm{L}$ is the maximum likelihood function value, $\mathrm{k}$ is the number of unknown parameters, and $\mathrm{n}$ is the sample size.

Step 4. Model validation: Next, we perform a white noise test on the residual series. If the test is passed, it is considered that the model has sufficiently extracted the effective information from the stationary series and thus can be further applied to the prediction and analysis of practical problems.

Step 5. Model prediction: We select the optimal model to fit and predict the test set.

2.2. Markov Model and Markov Correction

2.2.1. Partitioning Prediction States

There will always be some relative error between the original total energy consumption and the predicted total energy consumption. These errors can be divided into several intervals after calculating the relative error for each year. The number of intervals is not fixed and generally depends on the amount of raw data and the actual range of maximum and minimum relative errors. The separated state interval can be expressed as

$${\mathrm{E}}_{\mathrm{i}}=\left({\mathrm{Q}}_{\mathrm{i}1},{\mathrm{Q}}_{\mathrm{i}2}\right],(\mathrm{i}=\mathrm{1,2},\cdots ,\mathrm{s})$$

(7)

where ${\mathrm{Q}}_{\mathrm{i}1}$ and ${\mathrm{Q}}_{\mathrm{i}2}$ represent the upper and lower limits of the relative error for the state interval, respectively, and s is the number of state divisions.

2.2.2. Construct the State Transition Probability Matrix

In practical construction, we use

$${\mathrm{P}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{k}\right)={\displaystyle \frac{{\mathrm{m}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{k}\right)}{{\mathrm{M}}_{\mathrm{i}}}}$$

(8)

to approximate the transition probability, where ${\mathrm{M}}_{\mathrm{i}}$ represents the total number of occurrences of state ${\mathrm{E}}_{\mathrm{i}}$ in the transitions, and ${\mathrm{m}}_{\mathrm{i}\mathrm{j}}\left(\mathrm{k}\right)$ denotes the number of transitions from state ${\mathrm{E}}_{\mathrm{i}}$ to state ${\mathrm{E}}_{\mathrm{j}}$ after $\mathrm{k}$ steps.

The formula for calculating the transition probability matrix at the $\mathrm{n}$-th step is

$$\mathrm{P}\left(\mathrm{n}\right)={\left[\mathrm{P}\left(1\right)\right]}^{\mathrm{n}}$$

(9)

The one-step transition probability from state ${\mathrm{E}}_{\mathrm{i}}$ to state ${\mathrm{E}}_{\mathrm{j}}$ after one period is ${\mathrm{P}}_{\mathrm{i}\mathrm{j}}$. Writing all such one-step transition probabilities in the form of a matrix, we have

$$\mathrm{P}\left(1\right)=\left[\begin{array}{ccc}{\mathrm{P}}_{11}\left(1\right)& \cdots & {\mathrm{P}}_{1\mathrm{s}}\left(1\right)\\ ⋮& \ddots & ⋮\\ {\mathrm{P}}_{\mathrm{s}1}\left(1\right)& \cdots & {\mathrm{P}}_{\mathrm{s}\mathrm{s}}\left(1\right)\end{array}\right]$$

(10)

2.2.3. Confirmation of Forecast Values

Select the $\mathrm{s}$ most proximate data groups relative to the forecast data point, and assign sequential step indices ($\mathrm{t}=\mathrm{1,2},\cdots ,\mathrm{s}$) in descending order of proximity. A composite matrix is then constructed by aggregating the row vectors from the corresponding $\mathrm{t}$-step state transition matrices. Through column-wise summation of this composite matrix, the maximum-probability state and its confidence interval for the forecast value are determined. The Markov correction value is defined as the midpoint of this interval, denoted as ${\displaystyle \frac{{\mathrm{Q}}_{\mathrm{i}1}+{\mathrm{Q}}_{\mathrm{i}2}}{2}}$. The final forecast value is then obtained as

$${\widehat{\mathrm{u}}}_{\mathrm{M}}\left(\mathrm{k}\right)={\displaystyle \frac{{\widehat{\mathrm{u}}}_{\mathrm{A}\mathrm{R}\mathrm{I}\mathrm{M}\mathrm{A}}\left(\mathrm{k}\right)}{1+\frac{{\mathrm{Q}}_{\mathrm{i}1}+{\mathrm{Q}}_{\mathrm{i}2}}{2}}}$$

(11)

2.3. Model Error Validation and Flow Chart

The model error is evaluated using the mean absolute percentage error (MAPE) and root mean squared error (RMSE). Additionally, the statistical values of STD and ${\mathrm{R}}^2$ are calculated. The formulas for these metrics are defined in Equations (9)–(12) [33,34,35].

The calculation formulas are as follows:

$$\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E}={\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{t}=1}^{\mathrm{n}}\left|{\displaystyle \frac{({\widehat{\mathrm{u}}}_{\mathrm{t}}-{\mathrm{u}}_{\mathrm{t}})}{{\mathrm{u}}_{\mathrm{t}}}}\right|$$

(12)

$$\mathrm{R}\mathrm{M}\mathrm{S}\mathrm{E}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{t}=1}^{\mathrm{n}}{({\widehat{\mathrm{u}}}_{\mathrm{t}}-{\mathrm{u}}_{\mathrm{t}})}^2}$$

(13)

$$\mathrm{S}\mathrm{T}\mathrm{D}=\sqrt{{\displaystyle \frac{1}{\mathrm{n}}}\sum _{\mathrm{t}=1}^{\mathrm{n}}{({\displaystyle \frac{\left|{\widehat{\mathrm{u}}}_{\mathrm{t}}-{\mathrm{u}}_{\mathrm{t}}\right|}{{\mathrm{u}}_{\mathrm{t}}}}-\mathrm{M}\mathrm{A}\mathrm{P}\mathrm{E})}^2}$$

(14)

$${\mathrm{R}}^2=1-{\displaystyle \frac{\sum _{\mathrm{t}=1}^{\mathrm{n}}{({\widehat{\mathrm{u}}}_{\mathrm{t}}-{\mathrm{u}}_{\mathrm{t}})}^2}{\sum _{\mathrm{t}=1}^{\mathrm{n}}{({\widehat{\mathrm{u}}}_{\mathrm{t}}-\overline{\mathrm{u}})}^2}}$$

(15)

To better understand the operating mechanism of ARIMA, we present the flowchart as shown in Figure 1. The flow chart of this paper is as follows:

3. Forecasting China’s Total Energy Consumption

3.1. Selection and Testing of ARIMA($p,d,q$) Model

3.1.1. Stationarity Assessment of ARIMA($p,\mathrm{d},\mathrm{q}$) Model

First, the stationarity of the raw time series is evaluated. As shown in Figure 2 and

Table 2. Augmented Dickey—Fuller test of Raw.

	t-Statistics	Prob
Augmented Dickey—Fuller test statistics	–0.270605	0.9846
Test critical values: 1% level	–4.57159
5% level	–3.690814
10% level	–3.286906

, the total energy consumption data from the China Statistical Yearbook (2000–2018) continuously increases and exhibits a clear trend, indicating that it is a non-stationary series. Therefore, the difference is required to transform it into a stationary series. From Figure 3, it can be seen that after the fourth-order difference, the total energy consumption series becomes stationary.

3.1.2. Stationarity Test of the Difference ARIMA($p,\mathrm{d},\mathrm{q}$) Model

The ADF test is conducted on the series after fourth-order difference to evaluate the stationarity of the difference series. The test results from

Table 3. ADF and white noise test results for the difference series.

	Augmented Dickey–Fuller Test			Ljung-Box Test
Lag order	Dickey—Fuller Test	$p$-value	df	$\mathrm{X}$-squared	$p$-value
2	–4.4732	0.01	1	4.9408	0.02632

show that the $p$-value = 0.01, which is below the significance level of 0.05. Based on this result, we can reject the null hypothesis at the 95% confidence level, confirming that the difference series is stationary. The “Ljung-Box” test is then employed to check the residual sequence for white noise. From the test results in Table 3, it can be observed that the $p$-value for the white noise test is 0.02623, which is below the significance level of 0.05. Therefore, the sequence passes the test at the 95% confidence interval. Thus, it can be concluded that the sequence after the fourth-order difference is a stationary, non-white noise series, meeting the modeling requirements. Based on this conclusion, the ARIMA($p,\mathrm{d},\mathrm{q}$) model can be further established for empirical analysis, with $\mathrm{d}$ = 4.

3.1.3. Determination of ARIMA($p,\mathrm{d},\mathrm{q}$) Model

By observing the ACF plot (Figure 4) and PACF plot (Figure 5) of the fourth-order difference time series, it is evident that for the difference series, the first, second, and sixth auto-correlation coefficients are significantly outside the confidence interval, while the others fall within the interval. This indicates that the ACF of the series exhibits a truncation property. Similarly, for the PACF, only the first partial auto-correlation coefficient is significant, and the rest fall within the confidence interval, indicating a truncation property in the PACF as well. Therefore, the model can initially be identified as AR(1), MA(1), MA(2), MA(6), ARMA(1,1), ARMA(1,2), or ARMA(1,6).

This study performed residual white noise tests on these seven models. The test results show that all seven models passed the white noise test, qualifying them for further model selection.

During the model selection process, the AIC and BIC criteria were used. The AIC and BIC values of the seven models were calculated and summarized in

Table 4. AIC and BIC values for the seven models.

Model	AR(1)	MA(1)	MA(2)	MA(6)	ARMA(1,1)	ARMA(1,2)	ARMA(1,6)
AIC	333.6152	327.0116	323.9768	327.1382	327.4539	325.7543	328.9624
BIC	335.7394	329.1358	326.809	332.8026	330.2861	329.2945	335.3348

. From Table 4, it is observed that the MA(2) model has the smallest AIC = 323.9768 and BIC = 326.809 among the three models. Since the data are already different by four orders (i.e., $\mathrm{d}$ = 4), the final model is determined to be ARIMA(0,4,2). Next, we perform a significance test on the model.

3.1.4. Test of ARIMA($\mathrm{p},\mathrm{d},\mathrm{q}$)

To validate the model’s effectiveness, a residual white noise test is conducted on the ARIMA(0,4,2) model. The results are presented in

Table 5. Test results of ARIMA(0,4,2) model with white noise.

	Ljung-Box Test
df	$\mathrm{X}$-squared	$p$-value
1	1.4155	0.2342

.

As shown in Table 5, the Ljung-Box test statistic equals $p$ = 0.2342, which exceeds the significance level of 0.05. This indicates that the residual sequence of the ARIMA(0,4,2) model is a white noise sequence, and all effective information from the original sequence has been successfully extracted. Based on this finding, the ARIMA(0,4,2) model is employed for estimating and forecasting China’s energy consumption data.

3.2. Estimation and Forecasting Using the ARIMA(0,4,2) Model

This study selected the total national energy consumption (in million tons of standard coal) as the research object. The dataset originates from the China Statistical Yearbook, covering the period from 2000 to 2023, with annual time series data. By utilizing the Akaike information criterion (AIC), Bayesian information criterion (BIC), and auto correlation function (ACF) and partial auto correlation function (PACF) plots for model order determination, we find that the optimal model is ARIMA(0,4,2). Specifically,

$${\mathrm{u}}_{\mathrm{t}}={{4\mathrm{u}}_{\mathrm{t}-1}-6{\mathrm{u}}_{\mathrm{t}-2}+4{\mathrm{u}}_{\mathrm{t}-3}-{\mathrm{u}}_{\mathrm{t}-4}+{\mathsf{ϵ}}_{\mathrm{t}}+1.0412\mathsf{ϵ}}_{\mathrm{t}-1}-0.0573{\mathsf{ϵ}}_{\mathrm{t}-2},$$

(16)

Using GM(1,1), FGM(1,1) and NAR as comparative models, the estimation and forecasting results are summarized in

Table 6. The comparison results among GM, FGM, NAR, and ARIMA.

		ARIMA		GM		FGM		NAR
Year	Raw	Predicted Value	Relative Error	Predicted Value	Relative Error	Predicted Value	Relative Error	Predicted Value	Relative Error
2000	146,964	146,946.3	–0.01%	146,964	0.00%	146,964	0.00%	146,964	0.00%
2001	155,547	155,633.5	0.06%	209,642	34.78%	193,540	24.43%	155,547	0.00%
2002	169,577	169,418.9	–0.09%	220,954	30.30%	212,861	25.52%	157,982.51	–6.84%
2003	197,083	197,147.1	0.03%	232,876	18.16%	229,490	16.44%	180,992.56	–8.16%
2004	230,281	241,228.8	4.75%	245,442	6.58%	245,178	6.47%	208,834.34	–9.31%
2005	261,369	268,180.5	2.61%	258,685	–1.03%	260,563	–0.31%	242,588.14	–7.19%
2006	286,467	287,843.8	0.48%	272,643	–4.83%	275,966	–3.67%	274,962.32	–4.02%
2007	311,442	303,532.1	–2.54%	287,355	–7.73%	291,577	–6.38%	300,552.97	–3.50%
2008	320,611	333,451.8	4.01%	302,860	–5.54%	307,523	–4.08%	321,490.82	0.27%
2009	336,126	312,804.3	–6.94%	319,201	–5.04%	323,899	–3.64%	349,782.65	4.06%
2010	360,648	356,869.1	–1.05%	336,425	–6.72%	340,780	–5.51%	367,549.22	1.91%
2011	387,043	393,983.9	1.79%	354,577	–8.39%	358,230	–7.44%	385,573.73	–0.38%
2012	402,138	414,705.6	3.13%	373,710	–7.07%	376,304	–6.42%	397,313.35	–1.20%
2013	416,913	405,425.0	–2.76%	393,874	–5.53%	395,055	–5.24%	418,340.43	0.34%
2014	428,334	430,052.1	0.40%	415,127	–3.08%	414,533	–3.22%	434,273.92	1.39%
2015	434,113	435,680.9	0.36%	437,526	0.79%	434,784	0.15%	444,207.63	2.33%
2016	441,492	433,686.3	–1.77%	461,134	4.45%	455,856	3.25%	448,919.26	1.68%
2017	455,827	449,869.9	–1.31%	486,015	6.62%	477,797	4.82%	451,886.75	–0.86%
2018	471,925	476,731.9	1.02%	512,240	8.54%	500,653	6.09%	456,046.63	–3.36%
		RMSE	8635.4	RMSE	27,668.4	RMSE	22,349.85	RMSE	10,709.85
		MAPE	1.85%	MAPE	8.69%	MAPE	7.00%	MAPE	2.98%
		STD	1.82%	STD	9.03%	STD	7.06%	STD	2.85%
		${\mathrm{R}}^2$	99.32%	${\mathrm{R}}^2$	92.26%	${\mathrm{R}}^2$	94.87%	${\mathrm{R}}^2$	99.04%
2019	487,488	489,822.2	0.48%	539,879	10.75%	524,474	7.59%	461,968.45	–5.23%
2020	498,314	509,285.9	2.20%	569,010	14.19%	549,309	10.23%	470,947.09	–5.49%
2021	525,896	530,083.6	0.80%	599,712	14.04%	575,208	9.38%	482,537.05	–8.24%
2022	540,956	551,982.6	2.04%	632,071	16.84%	602,225	11.33%	505,342.54	–6.58%
2023	572,000	574,750.3	0.48%	666,176	16.46%	630,414	10.21%	530,118.52	–7.32%
		RMSE	7382.6	RMSE	77,926.46	RMSE	52,088.79	RMSE	35,503.71
		MAPE	1.20%	MAPE	14.46%	MAPE	9.75%	MAPE	6.57%
		STD	0.76%	STD	2.18%	STD	1.24%	STD	1.12%
		${\mathrm{R}}^2$	94.22%	${\mathrm{R}}^2$	22.52%	${\mathrm{R}}^2$	32.92%	${\mathrm{R}}^2$	30.63%

.

As evident from Table 6, the ARIMA(0,4,2) model demonstrates superior forecasting accuracy, with its predictions being closer to the true values and exhibiting smaller relative errors compared to other models. When forecasting test data and estimating training data, the ARIMA(0,4,2) model yields smaller root mean square error (RMSE), standard deviation (STD), and mean absolute percentage error (MAPE) values, as well as a larger ${\mathrm{R}}^2$. However, compared to models such as GM, FGM, and NAR, the ARIMA(0,4,2) model emerges as a better alternative. This model achieves the smallest RMSE, STD, and MAPE when forecasting test data and estimating training data while also attaining the largest ${\mathrm{R}}^2$.

3.3. Markov Model Prediction

3.3.1. Determination of Predicted Values

In general, the state intervals of the ARIMAMKM model are divided based on relative errors. As shown in Table 6, the minimum relative error of the first 19 fitted data points of this model is –6.94%, while the maximum relative error is 4.75%. According to the equal interval rule, the state partitions are divided into four categories: ${\mathrm{E}}_1\left(-6.94\mathrm{\%},-4.01\mathrm{\%}\right], {\mathrm{E}}_2\left(-4.01\mathrm{\%},-1.08\mathrm{\%}\right], {\mathrm{E}}_3\left(-1.08\mathrm{\%},1.85\mathrm{\%}\right],$ and ${\mathrm{E}}_4\left(1.85\mathrm{\%},4.75\mathrm{\%}\right]$.

By combining these state partitions with the probabilities of transitioning from the current state to the next state, the one-step transition probability matrix can be constructed:

$$\mathrm{P}\left(1\right)=\left[\begin{array}{cccc}0& 0& 1& 0\\ 0& 0.25& 0.5& 0.25\\ 0& 0.25& 0.5& 0.25\\ 0.2& 0.2& 0.2& 0.4\end{array}\right]$$

3.3.2. Total Energy Consumption Forecasting

By constructing a new state transition matrix using the most recent four sets of data, the state for 2019 is determined, as presented in

Table 7. The prediction state of 2019.

Year	Initial State	Transferring Steps	${\mathbf{P}}_{\mathbf{i}\mathbf{j}}$	${\mathbf{E}}_1$	${\mathbf{E}}_2$	${\mathbf{E}}_3$	${\mathbf{E}}_4$
2018	3	1	${\mathrm{P}}_{13}$	0	0.25	0.5	0.25
2017	2	2	${\mathrm{P}}_{22}$	0.5	0.2375	0.4250	0.2875
2016	2	3	${\mathrm{P}}_{32}$	0.0575	0.2231	0.4387	0.2806
2015	3	4	${\mathrm{P}}_{43}$	0.0561	0.2216	0.4446	0.2777
Total				0.6136	0.9322	1.8083	1.0958

.

According to the results presented in Table 7, given that the maximum value in the summation corresponds to ${\mathrm{E}}_3$, we infer that the most probable state of total national energy consumption in 2019 is ${\mathrm{E}}_3$. Based on the ARIMA model and the ARIMAMKM model, the predicted values are 489,822.2 and 488,673.81, respectively. Using the same methodology, we derive the forecasting results of the ARIMAMKM model for the total national energy consumption from 2019 to 2023, as detailed in

Table 8. The comparison results among QFM, ARIMA, and ARIMAMKM.

Year	Raw	QFM		ARIMA			ARIMAMKM
		Predicted Value	Relative Error	Predicted Value	Relative Error	Station Value	Predicted Value	Relative Error
2000	146,964	128,263.95	–12.72%	146,946.3	–0.01%	3	146,601.78	–0.25%
2001	155,547	156,619.24	0.69%	155,633.5	0.06%	3	155,268.62	–0.18%
2002	169,577	183,868.37	8.43%	169,418.9	–0.09%	3	169,021.69	–0.33%
2003	197,083	210,011.34	6.56%	197,147.1	0.03%	3	196,684.89	–0.20%
2004	230,281	235,048.15	2.07%	241,228.8	4.75%	4	233,522.55	1.41%
2005	261,369	258,978.79	–0.91%	268,180.5	2.61%	4	259,613.26	–0.67%
2006	286,467	281,803.28	–1.63%	287,843.8	0.48%	3	297,168.95	3.74%
2007	311,442	303,521.61	–2.54%	303,532.1	–2.54%	2	311,458.72	0.01%
2008	320,611	324,133.77	1.10%	333,451.8	4.01%	4	322,799.41	0.68%
2009	336,126	343,639.78	2.24%	312,804.3	–6.94%	1	330,922.29	–1.55%
2010	360,648	362,039.62	0.39%	356,869.1	–1.05%	3	356,032.42	–1.28%
2011	387,043	379,333.30	–1.99%	393,983.9	1.79%	4	381,397.77	–1.46%
2012	402,138	395,520.83	–1.65%	414,705.6	3.13%	4	401,457.50	–0.17%
2013	416,913	410,602.19	–1.51%	405,425.0	–2.76%	2	416,012.51	–0.22%
2014	428,334	424,577.39	–0.88%	430,052.1	0.40%	3	429,043.84	0.17%
2015	434,113	437,446.43	0.77%	435,680.9	0.36%	3	434,659.45	0.13%
2016	441,492	449,209.32	1.75%	433,686.3	–1.77%	2	445,011.85	0.80%
2017	455,827	459,866.04	0.89%	449,869.9	–1.31%	2	461,618.08	1.27%
2018	471,925	469,416.60	–0.53%	476,731.9	1.02%	3	475,614.20	0.78%
		RMSE	7801.5	RMSE	8635.4		RMSE	3807.5
		MAPE	2.59%	MAPE	1.85%		MAPE	0.80%
		STD ${\mathrm{R}}^2$	3.11%	STD	1.82%		STD	0.86%
			99.45%	${\mathrm{R}}^2$	99.32%		${\mathrm{R}}^2$	99.87%
2019	487,488	477,860.99	–1.97%	489,822.2	0.48%	3	488,673.81	0.24%
2020	498,314	485,199.23	–2.63%	509,285.9	2.20%	4	508,091.88	1.96%
2021	525,896	491,431.31	–6.55%	530,083.6	0.80%	3	528,840.82	0.56%
2022	540,956	496,557.23	–8.21%	551,982.6	2.04%	4	550,688.48	1.80%
2023	572,000	500,576.99	–12.49%	574,750.3	0.48%	3	573,402.80	0.25%
		RMSE	41,292.6	RMSE	7382.6		RMSE	6362.0
		MAPE	6.37%	MAPE	1.20%		MAPE	0.96%
		STD	3.85%	STD	0.76%		STD	0.76%
		${\mathrm{R}}^2$	35.01%	${\mathrm{R}}^2$	94.22%		${\mathrm{R}}^2$	95.63%

. As can be observed from Table 8, the RMSE, MAPE, STD, and ${\mathrm{R}}^2$ of the training set data fitted by the ARIMA(0,4,2) model are 8635.4, 1.85%, 1.82%, and 99.32%, respectively. Conversely, the RMSE, MAPE, STD, and ${\mathrm{R}}^2$ of the training set data fitted by the ARIMAMKM(0,4,2) model are 37.08, 0.80%, 0.86%, and 99.87%, respectively. This demonstrates that the fitting performance of the ARIMAMKM(0,4,2) model surpasses that of the ARIMA(0,4,2) model. Furthermore, the RMSE, MAPE, STD, and R-squared ${\mathrm{R}}^2$ of the test set data fitted by the ARIMA(0,4,2) model are 7382.6, 1.20%, 0.76%, and 94.22%, respectively. In contrast, the RMSE, MAPE, STD, and ${\mathrm{R}}^2$ of the test set data fitted by the ARIMAMKM(0,4,2) model are 6362.0, 0.96%, 0.76%, and 95.63%, respectively. This confirms that the prediction accuracy of the ARIMAMKM(0,4,2) model is superior to that of the ARIMA(0,4,2) model. Both the estimation and prediction accuracy of the ARIMAMKM(0,4,2) model outperform those of the QFM model. In the QFM model, the least square method is used to determine the regression coefficients to minimize the sum of the squares of the errors between the observed values and the predicted values of the model. Model evaluation uses evaluation indicators such as RMSE and ${\mathrm{R}}^2$.

As depicted in Figure 6, the curve generated by the ARIMAMKM(0,4,2) model aligns more closely with the true value (Raw) compared to the ARIMA(0,4,2) model. The ARIMAMKM(0,4,2) model predicts China’s energy consumption for the years 2024–2027 to be 593,456.87, 616,089.29, 638,723.69, and 661,125.55 (unit: 10,000 t), respectively.

4. Research on the Prediction of Permanent Resident Population in Guangdong

To validate the effectiveness and adaptability of our proposed model, following the same procedure outlined in Section 3, we establish ARIMA(1,2,1), GM, FGM, NAR, and ARIMAMKM models to predict the permanent resident population in Guangdong Province. The permanent resident population data are from the Guangdong Statistical Yearbook. The results are summarized in

Table 9. The comparison results among GM, FGM(1,1), NAR, ARIMA, and ARIMAMKM in permanent resident population of Guangdong.

		ARIMA			NAR		GM		FGM		ARIMAMKM
Year	Raw	Predicted Value	Relative Error	Station Value	Predicted Value	Relative Error	Predicted Value	Relative Error	Predicted Value	Relative Error	Predicted Value	Relative Error
2001	8733.18	8724.45	–0.10%	3	8733.18	0.00%	8733.18	0.00%	8733.18	0.00%	8718.78	–0.16%
2002	8842.08	8812.97	–0.33%	3	8842.08	0.00%	8719.83	–1.38%	7793.29	–11.86%	8807.24	–0.39%
2003	8962.69	8947.45	–0.17%	3	8992.16	0.33%	8916.77	–0.51%	8367.22	–6.64%	8941.63	–0.23%
2004	9110.66	9077.85	–0.36%	3	9102.76	–0.09%	9118.16	0.08%	8899.98	–2.31%	9071.95	–0.42%
2005	9194	9248.87	0.60%	4	9242.68	0.53%	9324.11	1.42%	9355.25	1.75%	9219.36	0.28%
2006	9442.07	9289.95	–1.61%	1	9459.41	0.18%	9534.7	0.98%	9744.78	3.21%	9415.61	–0.28%
2007	9659.52	9645.07	–0.15%	3	9532.94	–1.31%	9750.05	0.94%	10,081.33	4.37%	9638.8	–0.21%
2008	9893.48	9902.23	0.09%	4	9869.17	–0.25%	9970.26	0.78%	10,374.85	4.87%	9899.06	0.06%
2009	10,130.19	10,092.1	–0.38%	3	10,137.34	0.07%	10,195.45	0.64%	10,632.86	4.96%	10,085.54	–0.44%
2010	10,440.94	10,379.71	–0.59%	2	10,490.13	0.47%	10,425.72	–0.15%	10,861.06	4.02%	10,461.83	0.20%
2011	10,756	10,714.49	–0.39%	3	10,850.3	0.88%	10,661.2	–0.88%	11,063.88	2.86%	10,707.53	–0.45%
2012	11,041	11,080.37	0.36%	4	11,136.08	0.86%	10,901.99	–1.26%	11,244.8	1.85%	11,045.02	0.04%
2013	11,270	11,305.26	0.31%	4	11,242.36	–0.25%	11,148.22	–1.08%	11,406.61	1.21%	11,269.19	–0.01%
2014	11,489	11,507.86	0.16%	4	11,533.14	0.38%	11,400.01	–0.77%	11,551.61	0.54%	11,471.15	–0.16%
2015	11,678	11,689.21	0.10%	4	11,706.54	0.24%	11,657.49	–0.18%	11,681.69	0.03%	11,651.92	–0.22%
2016	11,908	11,873.7	–0.29%	3	11,895.01	–0.11%	11,920.79	0.11%	11,798.44	–0.92%	11,865.98	–0.35%
2017	12,141	12,114.25	–0.22%	3	12,136.02	–0.04%	12,190.03	0.40%	11,903.19	–1.96%	12,106.38	–0.29%
2018	12,348	12,378.04	0.24%	4	12,459.88	0.91%	12,465.35	0.95%	11,997.1	–2.84%	12,338.55	–0.08%
		RMSE	48.1		RMSE	52.2	RMSE	84.66	RMSE	394.71	RMSE	27.9
		MAPE	0.0036		MAPE	0.0038	MAPE	0.0069	MAPE	0.0312	MAPE	0.0024
		STD	0.0034		STD	0.0037	STD	0.0045	STD	0.0277	STD	0.0014
		${\mathrm{R}}^2$	0.9984		${\mathrm{R}}^2$	0.9978	${\mathrm{R}}^2$	0.9949	${\mathrm{R}}^2$	0.9057	${\mathrm{R}}^2$	0.9994
2019	12,489	12,542.53	0.43%	4	12,619.88	1.05%	12,746.89	2.06%	12,081.16	–3.27%	12,534.38	0.36%
2020	12,624	12,736.49	0.89%	4	12,667.62	0.35%	13,034.79	3.25%	12,156.23	–3.71%	12,728.21	0.83%
2021	12,684	12,919.34	1.86%	4	12,680.12	–0.03%	13,329.19	5.09%	12,223.08	–3.63%	12,910.94	1.79%
2022	12,656.8	13,101.12	3.51%	4	12,685.88	0.23%	13,630.24	7.69%	12,282.37	–2.96%	13,092.6	3.44%
2023	12706	13,272.96	4.46%	4	12,687.49	–0.15%	13,938.09	9.70%	12,334.7	–2.92%	13,264.33	4.39%
		RMSE	343.44		RMSE	63.62	RMSE	789.57	RMSE	418.48	RMSE	336.47
		MAPE	0.0223		MAPE	0.0036	MAPE	0.0556	MAPE	0.033	MAPE	0.0216
		STD	0.0154		STD	0.0036	STD	0.0281	STD	0.0033	STD	0.0154
		${\mathrm{R}}^2$	0.1949		${\mathrm{R}}^2$	0.0802	${\mathrm{R}}^2$	0.0734	${\mathrm{R}}^2$	0.0351	${\mathrm{R}}^2$	0.2012

and depicted in Figure 7. The ARIMAMKM model forecasts the resident population of Guangdong for the years 2024–2027 to be 13,434.61, 13,595.98, 13,755.60, and 13,907.23 (unit: 10,000), respectively.

5. Research Results and Discussion

This paper proposes a novel ARIMAMKM model for predicting national annual total energy consumption. The Markov transition probability matrix is employed to determine the appropriate state, and the information criteria such as AIC and BIC are utilized to identify the optimal model parameters. Based on the predicted results derived from Chinese Statistical Yearbook data spanning 2000 to 2018, four statistical metrics—MAPE, RMSE, STD, and ${\mathrm{R}}^2$—are computed and compared across different models. According to the values of these statistics, it is evident that the estimation accuracy of the ARIMAMKM model surpasses that of the QFM, GM, NAR, and FGM models. Moreover, the fitting performance of the ARIMAMKM model outperforms the traditional ARIMA model. Subsequently, the proposed model is applied to forecast the national annual total energy consumption from 2019 to 2023. The findings indicate that the ARIMAMKM model demonstrates significant advantages in predicting China’s annual total energy consumption. Additionally, validation using Guangdong’s permanent resident population data yields favorable outcomes. By incorporating Markov chain adjustments, the ARIMAMKM model effectively captures nonlinear trends and random errors within the data. Nevertheless, the ARIMAMKM model exhibits certain limitations. First, its predictive accuracy may be compromised in the presence of extreme or sudden events. Second, while the ARIMAMKM model performs admirably with the energy data presented herein, further optimization may be required for extremely complex multivariate datasets. Future research endeavors can enhance the prediction accuracy of the model by integrating other optimization algorithms, such as particle swarm optimization and genetic algorithms. Furthermore, potential applications of the model could extend to additional domains, including climate forecasting and financial market prediction.