ARIMA-BP integrated intelligent algorithm for China’s consumer price index forecasting and its applications

Abstract

The autoregressive integrated moving average (ARIMA)–backpropagation (BP) integrated intelligent algorithm for consumer price index (CPI) forecasting was designed based on the ARIMA intelligent forecasting method and the BP intelligent neural network algorithm. The irregular variations in CPI time series data were divided into linear and nonlinear variations. The linear variation was fitted by the ARIMA intelligent forecasting method, and the nonlinear variation was fitted by the BP intelligent neural network. The sum of the fitted linear and nonlinear variations was the CPI forecasted by the ARIMA-BP integrated intelligent algorithm. Results demonstrated that the ARIMA-BP integrated intelligent algorithm could achieve high-precision fitting of the historical CPI data of China. The proposed algorithm showed a forecast error that was smaller than that of the single ARIMA model. Owing to the complexity of the CPI and the combined influence of various factors, achieving accurate CPI forecast is difficult. Such a new integrated intelligent algorithm provides a referent scientific method to forecast the CPI of China in the future. The results can provide government departments reference information of timely price control.

Keywords

ARIMA model BP neural network ARIMA-BP integrated intelligent algorithm intelligent forecast CPI

1 Introduction

Consumer price index (CPI) is closely related to the living standard of people. The rate of change of the CPI is not only often used as a measurement index of inflation but also an important reference basis for government departments to monitor and control the general price level and to make macroeconomic management decisions. Analyzing the development law of CPI and forecasting the CPI level and variation trend accurately in the period ahead are necessary. However, most of the traditional approaches have poor accuracy. Thus, an accurate method should be explored. In this study, an autoregressive integrated moving average (ARIMA)–backpropagation (BP) integrated intelligent algorithm for CPI forecasting was designed based on the ARIMA intelligent forecast method and the BP neural network intelligent algorithm to obtain an accurate CPI forecast.

Section 2 presents the literature review. Section 3 introduces the research methods. Section 4 provides the analysis and discussion of the results. Section 5 concludes the study.

2 Literature review

Controlling price level within a reasonable range is important for a country to adopt appropriate monetary policy and to safeguard a healthy macroeconomic operation [1, 2]. For this reason, scholars have investigated CPI forecasting using various methods.

Several scholars have forecasted the CPI using the traditional regression analysis. Research can be divided into three types. The first type investigates the relationship between CPI and related influencing factors and focuses on forecasting based on the correlations of related factors. For example, Chong Zhang [3] tested the correlation, lead–lag relationship, and co-integration relationship between web search data and CPI. Yongjie Zhang [4] investigated the relationship between information acquisition and stock price index in the Shenzhen Stock Exchange. The second type assesses actual problems based on CPI forecasting, such as CPI convergence reflected by price forecasting [5]. The third type focuses on problems in traditional CPI, as well as its estimation and forecasting processes from the perspective of methods [2, 6, 7].

Considering the shortcomings of traditional regression forecasting, including model complexity, poor accuracy, and requirements of many preconditions and assumptions, scholars have developed a series of models or approaches based on intelligent algorithms and intelligent forecasting methods. Nevertheless, most of these approaches use a single model or method for forecasting during the early research period. For instance, Azhong Ye [8] discussed the CPI data feature of China by the generalized autoregressive conditional heteroscedasticity (GARCH) model. He believed that the GARCH(1,1) model presented a better fitting effect than the traditional regression model. Xin Li [9] and Iqbal Muhammad [10] established ARIMA models for the CPIs of China and Pakistan, respectively. Based on the dynamic model averaging method, Di Filippo [11] forecasted and compared the CPIs of Europe and America.

Due to the low forecast accuracy of a single model or method, several scholars have explored combinations of many methods [12 –24]. In the time series field, a hybrid model or a combination of several models is widely applied in practical forecasting [13]. As an intelligent prediction model, the ARIMA model has many advantages over the traditional prediction model. Therefore, most scholars have established hybrid models based on the ARIMA intelligent prediction model. Mahmood Moosazadeh ea al predicted the incidence of smear positive tuberculosis cases in Iran with seasonal ARIMA [18]. Sowell [19] analyzed the postwar seasonal gross national product of the United States with the fraction ARIMA model. Tseng et al. [20] forecasted the exchange rate of the New Taiwan Dollar to the US Dollar by the fuzzy ARIMA model. Ming Zhou et al. [21] predicted short-term electricity price using the ARIMA model based on wavelet analysis. Kewei Lei et al. [22] conducted a predictive analysis on the inbound tourist amount of China by combining the BP neural network and the ARIMA model. Jisung and Lusk [23] established an ARIMA model with exogenous variables and described and forecasted the CPI of foods in the United States by using the vector autoregression method.

In summary, traditional CPI forecasting approa-ches based on a single model often have low accuracy. Extensive explorations on forecasting based on the combination of several models are conducted in the academic circle; however, only a few of these explorations are about CPI forecasting. Therefore, this study established the ARIMA-BP integrated intelligent algorithm based on the ARIMA intelligent prediction model and the BP neural network intelligent algorithm to forecast the CPI of China in the period ahead.

3 Methodology

3.1 ARIMA intelligent forecasting algorithm

The ARIMA intelligent prediction model is a stochastic process that is composed of autoregression (AR) and moving average (MA) and is called the difference autoregression integrated moving average model. This model is also called the Box–Jenkins model.

The basic idea of the ARIMA intelligent prediction model is that the data series formed by the forecasting object over time is viewed as a random sequence that is described approximately by a certain mathematical model. Once the model is recognized, it can forecast the future value according to the past and current values of the time series.

The ARIMA intelligent prediction model is expressed as follows:

We suppose that Z_t is the d-order integrated sequence, that is, Z_t ∼ I (d). The lag operator L is introduced into the following expression: $ω^{t} = Δ^{d} Z^{t} = (1 - L)^{d} Z_{t}$ (1) where ω_t is a steady sequence, ω_t ∼ I (0). Therefore, the ARMA (p, q) model of ω_t is established as follows:

$\begin{matrix} ω^{t} & = & φ_{1} ω_{t - 1} + φ_{2} ω_{t - 2} + \dots + φ_{t} ω_{t - p} \\ + μ_{t} - θ_{1} μ_{t - 1} - θ_{2} μ_{t - 2} \dots θ_{q} μ_{t - q} \end{matrix}$ (2) where φ₁, φ₂, φ₃ ⋯ φ_p are the AR coefficients, p is the AR order, θ₁, θ₂, θ₃ ⋯ θ_q are the MA coefficients, q is the MA order, and μ_t is the white noise sequence. If one sequence becomes the ARMA (p, q) model after d differential transformations, then it is called the ARIMA (p, d, q) model, where d is the difference order.

The basic forecasting steps of the ARIMA intelligent model are as follows:

Stationarity of the time series is recognized according to the scatter diagram, autocorrelogram, and partial autocorrelogram and augmented Dickey–Fuller unit root test of the time series. Most economic time series are nonstationary sequences.

Smooth processing is implemented to the nonstationary series. Differential treatment is applied to the nonstationary series to transform it into the stationary series.

A corresponding model is established according to the recognition rules of the time series model. Common time series models are AR, MA, and ARMA models.

Parameters are estimated, and statistical test is conducted on parameter estimation. Whether the residual series is white noise is determined.

Predictive analysis is conducted with tested models.

3.2 BP neural network intelligent algorithm

BP neural network is a multilayer feedforward neural network based on the error back propagation algorithm. This neural network was proposed by D.E. Rumelhart et al. in 1969 and has been widely used since then [25 –29]. The structure of the BP neural network is often composed of an input layer, several hidden layers, and an output layer. Each layer contains several nodes, and each node is one neuron. Every neuron only feeds forward to all neurons of the subsequent layer. Nodes among layers adopt full connection, but no intralayer, interlayer, and feedback connections exist. A typical BP neural network is a three-layer network, which has one hidden layer (Fig. 1).

Supposing the neural network has n input neurons, m output neurons, and p neurons in the hidden layer, the output of neurons is expressed as follows: $X_{i}^{j} = σ (\sum_{i = 1}^{n} ω_{ij} x_{i} + b_{j}), j = 1, 2, 3 \dots p$ (3)

The output of neurons in the output layer is derived as follows: $Y_{k} = \sum_{j = 1}^{p} W_{jk} X_{j}, k = 1, 2, 3 \dots m$ (4)

The activation function often uses an S-shaped function, such as tansig function. $tan sig (x) = \frac{2}{1 + e^{- 2 x}} - 1$ (5)

3.3 ARIMA-BP integrated intelligent algorithm

Given that the ARIMA intelligent model and the BP neural network intelligent algorithm have advantages and disadvantages, an integrated intelligent algorithm is designed in this study. The integrated intelligent algorithm combines the ARIMA intelligent model and the BP neural network intelligent algorithm by absorbing their advantages and overcoming their disadvantages. Thus, the proposed model is superior to the ARIMA intelligent model and the BP neural network intelligent algorithm.

The basic idea and steps of this algorithm are as follows. First, the linear variation in the investigated series is fitted based on the ARIMA intelligent model. Second, the forecast error of the ARIMA algorithm or nonlinear variation is simulated using the BP neural network intelligent algorithm. Third, simulation and forecasting are conducted based on the BP neural network, and the forecasting result of the nonlinear part of the investigated series is obtained. Fourth, the linear law fitted by the ARIMA intelligent model and the nonlinear law simulated by the BP neural network intelligent algorithm are combined to generate the forecasting results of the integrated intelligent algorithm.

The flowchart of the ARIMA-BP integrated intelligent algorithm is shown in Fig. 2.

4 Analysis and discussion of the results

First, EViews software was used to fit the linear variation in CPI based on the ARIMA intelligent model. Then, Matlab software was used to simulate the nonlinear variation in CPI based on the BP neural network intelligent algorithm. Finally, the forecasting results of the ARIMA-BP intelligent algorithm are obtained.

4.1 Establishment of the ARIMA intelligent model for CPI

The time series of the monthly CPI data of China from March 2011 to February 2016 was observed through the graphs drawn with EViews, which showed an evident downward trend. The difference method was used to transform the original series into the stationary series. Results of the unit root test after the first difference of the logarithm of the original time series X is shown in Table 1.

T statistics was smaller than the critical value at the 1% significance level, which did not support the original hypothesis and passed the unit root test. The corresponding time series was stationary. Therefore, d = 1 for the ARIMA (p, d, q) model.

The ARIMA(2, 1, 2) model was determined through a comparative analysis by combining the autocorrelogram and partial autocorrelogram of the CPI time series. The Akaike information criterion and Schwarz criterion achieved minimum values of –8.05 and –7.91, respectively. The estimation result of this model was derived as follows, where Y is the logarithmic series of X:

$\begin{matrix} Y_{t} & = & 0.8016 Y_{t - 1} - 0.4956 Y_{t - 2} \\ + ɛ_{t - 1} 1.0533 - ɛ_{t - 2} 0.9948 \end{matrix}$ (6)

Stationary of the residual series was tested, which indicated that the established ARIMA(2, 1, 2) model was reasonable.

4.2 Forecasting result of the ARIMA model

Equation (6) indicated that the forecasting formula of the ARIMA(2, 1, 2) model of X can be expressed as follows:

$\begin{matrix} X_{t} = X_{t - 1} \\ + e^{0.8016 Y_{t - 1} - 0.4956 Y_{t - 2} + ɛ_{t - 1} 1.0533 - ɛ_{t - 2} 0.9948} \end{matrix}$ (7)

The ARIMA(2, 1, 2) model was used to fit the monthly CPI data of China. Statistical data covered 60 months from March 2011 to February 2016. The original data were obtained from the website of the National Bureau of Statistics of China. The forecasting results and errors are listed in Table 2.

4.3 Simulation of the ARIMA forecasting error based on the BP neural network intelligent algorithm

The nonlinear law of the ARIMA error of CPI from July 2014 to February 2016 was simulated by the BP neural network intelligent algorithm.

Input layer: Combining the characteristics of the CPI and China’s convention of five-year planning, the input layer had five neurons, namely, the monthly CPI from February 2014 to June 2014. The monthly CPI of July 2014 was used as the output. A total of 20 groups of data were obtained.

Output layer: The output result had only one index. Therefore, the number of output nodes was 1.

Hidden layer: The number of nodes of the hidden layer was determined by the empirical formula $h = \sqrt{n + m} + a$ , where h is the number of nodes of the hidden layer, n is the number of nodes of the input layer, m is the number of nodes of the output layer, and a is an integer from 0 to 10. According to the experiment, the optimum value of h was 10 when the BP neural network algorithm was stable and could obtain the smallest forecasting error.

Transfer function: For a neural network, if the first layer was an S-shaped function and the second layer was a linear function, then it could be used to simulate any continuous function. Therefore, the transfer function of the hidden layer used the tansig(x) function.

Given that the absolute error of the ARIMA was controlled between [–1, 1], no normalization was needed. In this study, the momentum BP algorithm with variable learning rate was used to train the BP network. The learning step length, target error, learning rate, and displayed iterations were set to 3,000 periods, 0.00001, 0.05, and 10, respectively. The target error was achieved after 893 trainings when the training was ended. The forecasting results are presented in Table 2.

Table 2 shows that the forecasting results of the ARIMA-BP integrated intelligent algorithm are close to the actual values. The maximum error was only 0.008 (June 2015), and the minimum error was close to 0. The forecasting error of the single ARIMA model was high, which ranged from 0.740 (January 2015) to 0.038 (November 2015). Therefore, the proposed ARIMA-BP integrated intelligent algorithm had higher forecasting accuracy and fitting effect than the single ARIMA model.

4.4 CPI forecasting based on the ARIMA-BP integrated intelligent algorithm

Actual CPI data from October 2015 to February 2016 were used as inputs of the trained neural network algorithm to obtain the forecasted CPI of March 2016. Actual CPI data from November 2015 to February 2016 and forecasted CPI data of March 2016 were used as input, thus generating the forecasted CPI of April 2016. In this way, the forecasted CPI from March 2016 to July 2016 could be obtained (Table 2). According to the forecasting results, the CPI of China in the next five months will increase significantly, indicating that the Chinese government has to adopt corresponding economic interventions to maintain a stable price level.

5 Conclusions

We combined the ARIMA intelligent model and the BP neural network intelligent algorithm to explore a new intelligent CPI forecasting algorithm. The ARIMA intelligent model is used to describe the linear variation law of the CPI series, and the BP neural network algorithm is used to describe the nonlinear variation law of the CPI series. The results show that the developed ARIMA-BP integrated intelligent algorithm can achieve effective fitting of CPI fluctuation and has a small forecasting error. The developed algorithm also has a better fitting effect than the single ARIMA intelligent model. The CPI of China is forecasted to increase significantly in the next five months, which deserves the attention of government departments. Appropriate macroeconomic control measures should be adopted to maintain a stable price level.

The proposed algorithm is a good trial of CPI forecasting and sets a precedent for the intelligent forecasting of the CPI of China. Given that CPI is influenced by various factors, absolutely accurate CPI forecasting is difficult to achieve. Intelligent CPI forecasting needs further explorations and continuous efforts.

Footnotes

Acknowledgments

The work was financially supported by the National Planning Office of Philosophy and Social Science of China (No. 12BTJ012), the Technology Foundation for Selected Overseas Scholars in Hebei Province (C201400103), and the Midwest University Comprehensive Strength Promotion Project. The authors declare that there is no conflict of interests.

References

Berardi

, Gautier

and Bihan

H.L.

, More facts about prices: France before and during the great recession, Journal of Money, Credit and Banking47(8) (2015), 1465–1502.

Shapiro

M.D.

and Wilcox

D.W.

, Mismeasurement in the consumer price index: An evaluation, NBER Macroeconomics Annual 1996, Volume 11, MIT Press, 1996, pp. 93–154.

Zhang

, Lv

B.F.

, Peng

and Liu

, Study on the correlation between Internet search data and CPI, Journal of Management Sciences in China15(7) (2012), 50–59+70.

Zhang

Y.J.

, Feng

L.N.

, Jin

, Shen

D.H.

, Xiong

and Zhang

, Internet information arrival and volatility of SME price index, Physica A: Statistical Mechanics and its Applications399 (2014), 70–74.

Cecchetti

S.G.

, Mark

N.C.

and Sonora

R.J.

, Price index convergence among United States cities, International Economic Review43(4) (2002), 1081–1099.

Edvinsson

and Söderberg

, A consumer price index for Sweden 1290-2008, Review of Income and Wealth57(2) (2011), 270–292.

Lin

, Zhang

, Li

X.T.

and Dong

J.C.

, Heterogeneous convergence of regional house prices and the complexity in China, Zbornik Radova Ekonomskog Fakulteta u Rijeci-Proceedings of Rijeka Faculty of Economics33(2) (2015), 325–348.

A.Z.

and Li

Z.N.

, GARCH model of inflation in China, Systems Engineering Theory and Practice20(10) (2000), 46–48.

and Huang

D.D.

, Analysis and forecast of CPI based on ARIMA model, Journal of Shenyang University (Social Science)15(3) (2013), 306–310.

10.

Iqbal

and Naveed

, Forecasting inflation: Autoregressive integrated moving average model, European Scientific Journal12(1) (2016), 83–92.

11.

, Filippo, Dynamic model averaging and CPI inflation forecasts: A comparison between the Euro Area and the United States, Journal of Forecasting34(8) (2015), 619–648.

12.

Zhang

and Zhu

C.J.

, Time series analysis of water quality in Hanjiang River, Nature Environment & Pollution Technology13(2) (2014), 413–416.

13.

Singh

and Krishna

V.V.

, Trends in Brain drain, gain and circulation: Indian experience of knowledge workers, Science Technology Society20(3) (2015), 300–321.

14.

Prascevic

and Prascevic

, Application of fuzzy AHP method based on eigenvalues for decision making in construction industry, Technical Gazette23(1) (2016), 57–64.

15.

Madic

, Radovanovic

, Cojbasic

, Nedic

and Gostimirovic

, Fuzzy logic approach for the prediction of dross formation in CO2 laser cutting of Mild Steel, Journal of Engineering Science and Technology Review8(3) (2015), 143–150.

16.

Cheng

J.Y.

, Wan

Z.J.

, Ji

Y.L.

, Li

W.F.

and Wang

Z.M.

, A comprehensive methodology for predicting shield support hazards for a US coal mine, DYNA-Bilbao90(4) (2015), 442–450.

17.

Khashei

, Bijari

and Hejazi

S.R.

, Combining seasonal ARIMA models with computational intelligence techniques for time series forecasting, Soft Computing16(6) (2012), 1091–1105.

18.

Moosazadeh

, Khanjani

, Nasehi

and Bahrampour

, Predicting the incidence of smear positive tuberculosis cases in Iran using time series analysis, Iranian Journal of Public Health44(11) (2015), 1526–1534.

19.

Sowell

, Modeling long-run behavior with the fractional ARIMA model, Journal of Monetary Economics29(2) (1992), 277–302.

20.

Tseng

F.M.

, Tzeng

G.H.

, Yu

H.C.

and Yuan

B.J.C.

, Fuzzy ARIMA model for forecasting the foreign exchange market, Fuzzy Sets and Systems118(1) (2001), 9–19.

21.

Zhou

, Nie

Y.L.

, Li

G.Y.

and Ni

Y.X.

, Short term electricity price prediction method of ARIMA based on wavelet analysis, Power System Technology29(9) (2005), 50–55.

22.

Lei

K.W.

and Chen

, Prediction of inbound tourists in China based on BP neural network and ARIMA combination model, Tourism Tribune22(4) (2007), 20–25.

23.

and Lusk

J.L.

, Predicting food prices using data from consumer surveys and search, Selected paper prepared for presentation at the Southern Agricultural Economics Association (SAEA) Annual Meeting, San Antonio, Texas, 2016.

24.

Yildiz

, Evaluating change in housing for sustainable development: Kosuyolu case in Istanbul, Open House International40(4) (2015), 55–62.

25.

J.Q.

, Wang

, Zhao

D.B.

and Wen

J.T.

, BP neural network prediction-based variable-period sampling approach for networked control systems, Applied Mathematics and Computation185(2) (2007), 976–988.

26.

Ghosh-Dastidar

and Adeli

, Spiking neural networks, International Journal of Neural Systems19(4) (2009), 295–308.

27.

Melin

, Pulido

and Castillo

, Optimization of ensemble neural networks with type-2 fuzzy response integration for predicting the Mackey-Glass time series, Nature and Biologically Inspired Computing (NaBIC), 2013 World Congress, IEEE, 2013, pp. 16–21.

28.

Jovanović

R.Ž.

, Sretenović

A.A.

and Živković

B.D.

, Ensemble of various neural networks for prediction of heating energy consumption, Energy and Buildings94 (2015), 189–199.

29.

Guo

, Prediction of iron ore demand in China based on BP neural network, Land and Resources Information11 (2008), 41–44.