Monthly Pork Price Prediction Applying Projection Pursuit Regression: Modeling, Empirical Research, Comparison, and Sustainability Implications
1. Introduction
2. Literature Review
Second, from the perspective of the practical application of the model, the PPAR model established in this paper only uses the pork price data lagging behind 1–3 periods. The established HPPR model, which removed nonsignificant influencing factors and included only six independent variables, greatly simplified the prediction model, making PPAR and HPPR models more practical and obtain higher prediction accuracy. Third, we formulate the basic principles of the regulation and control of pork price according to the best weight size and ranking of the influencing factors obtained, reveal the main factors affecting the fluctuation of pork price and their transmission mechanism, and put forward the principles of strengthening the management of the pork supply chain. The research methods and conclusions in this paper make up for the deficiency of the existing literature and also provide an essential basis for decision making for the relevant government departments to take appropriate measures to stabilize the pork price.
4. Principles of PPR Modeling
This paper mainly establishes the PPAR and the hybrid multivariate prediction pursuit regression (HPPR) models based on the time series data of pork prices and the other independent influencing factors.
4.1. Principle of Establishing the PPAR Model
Two basic assumptions exist for establishing a PPAR prediction model for monthly pork prices based on time series historical data. Firstly, multiple factors affect the monthly pork prices, and the relationship between these factors is very complex, making it difficult to have a mathematical model to represent them. However, the results of these factors are reflected in the changes in monthly pork prices. Secondly, the changes in monthly pork prices have a certain regularity, which autoregressive time series data can represent.
$$R\left(k\right)=\frac{}{{\sum}_{i=k+1}^{n}}$$
where $Ex=\frac{1}{n}$
, $k=\mathrm{1,2},3,......m$, in general, $m<\frac{n}{4},$ n is the number of time series data. With the k increasing, the variance of $R\left(k\right)$ increases, and the estimation accuracy decreases. Therefore, it is usually required to take a smaller value for m. According to the sampling distribution theory, the confidence level is ($1\mathsf{\alpha}$) (generally being 70~80%). When the autoregressive correlation coefficient value meets
$$R\left(k\right)\notin \left[{R}_{L}\left(k\right),{R}_{U}\left(k\right)\right]=\left[\frac{1{\mu}_{\frac{\alpha}{2}}\cdot \sqrt{\left(nk1\right)}}{\left(nk\right)},\frac{1+{\mu}_{\frac{\alpha}{2}}\cdot \sqrt{\left(nk1\right)}}{\left(nk\right)}\right]$$
it can be inferred that delay steps $x\left(ik\right)$ are significantly correlated with $\left\{x\left(i\right)\right\}$, and $x\left(ik\right)$ are used as predictors. The quantile values ${\mu}_{\frac{\alpha}{2}}$ can be found in the standard normal distribution table.
$$$$
where $a\left(j\right)$ is the best projection vector coefficient or weight of the $pdimensional$ autoregressive predictor.
$$Q\left(a,C\right)=$$
where $f\left(i\right)$ is the predicted value of the PPAR model. The formula based on the cubic polynomial ridge function (PRF) is
$$\begin{array}{l}f\left(i\right)=f\left[z\left(i\right)\right]={c}_{0}+{c}_{1}z\left(i\right)+{c}_{2}{\left[z\left(i\right)\right]}^{2}+{c}_{3}{\left[z\left(i\right)\right]}^{3}\\ \hspace{1em}\hspace{1em}={c}_{0}+{c}_{1}{\sum}_{j=1}^{p}a\left(j\right)x(i,j)\end{array}$$
where ${c}_{0}~{c}_{3}$ are the coefficients of the PRF.
In practice, to prevent “overtraining” and “overfitting”, we try the linear ridge function first. The quadratic and cubic polynomial ridge functions are established if the accuracy requirements are unmet.
Step 4: Optimize the objective function (4) to obtain the optimal global solution and obtain the fitting error of the PPAR model based on the first ridge function $\mathrm{e}\left(i\right)=\left[x\left(i\right)f\left(i\right)\right]$. If the appropriate error meets the prediction accuracy requirements, stop building more PRFs and output the model parameters and the performance indicators such as RMSE and MAPE. Otherwise, follow Step 5 to create more dimensional ridge functions.
Step 5: Replace $e\left(i\right)$ with $y\left(i\right)$, return to Step 2, repeat Steps 3 and 4, and establish a PPAR model based on the second and third ridge functions until the prediction accuracy requirements are satisfied.
Generally, the higher the order of PRFs or the more the number of PRFs, the more likely it is to have “overtraining” and “overfitting.” Therefore, the verification (test) sample should be set in modeling. The verification sample error decreases gradually and then increases, which indicates that “overtraining” and “overfitting” have occurred; the number of polynomials and the ridge function before “overtraining” and “overfitting” must be taken.
To verify the predictive and generalization capabilities of the PPAR model, we used the monthly data of pork prices in the last 12 months as a validation sample.
4.2. The Principle of Establishing the HPPR Model of Monthly Pork Price Prediction Based on Multivariate Time Series
There are two basic assumptions for establishing an HPPR prediction model for monthly pork prices based on multivariate time series historical data. Firstly, the prices of live pigs, beef, piglets, etc., are the main factors affecting the monthly pork prices, and the effects of other factors can be ignored. Secondly, there is a specific quantitative relationship between the monthly prices of live pigs, piglets, pork, etc., that lags 1–6 periods and the current monthly pork prices.
The PPAR model generally has relatively high fitting accuracy, generalization, and prediction ability. Still, the PPAR model only contains the monthly pork price data so that it can perform multiperiod and inflection point price predictions. Still, it cannot forecast the pork prices that soared rapidly according to the PPAR model. Providing strategic decisions for pig industry development is challenging, and we cannot study the influence mechanism of pork price fluctuation, etc. To achieve these goals, it is generally necessary to establish a nonlinear model between the monthly pork price and its influencing factors. The correlation analysis between the 12 collected factors affecting the monthly pork prices (referred to as the independent variables) and the monthly pork prices show that all the independent variables were significantly correlated with the pork price, and the pork prices lagging 1 to 6 periods were also significantly associated with the current pork price.
Step 1: The monthly piglet price with lagging six periods and the data of the other 11 independent variables lagging one period (from now on referred to as predictors or independent variables $v\left(j\right)$), and the monthly price of pork in the current period $y\left(j\right)$ is not normalized.
$$z\left(i\right)=$$
Steps three to five are the same as those for establishing the PPAR model.
We established two models to predict monthly pork prices: the first is a PPAR model based on the time series data of pork prices, and the second is an HPPR model based on time series data of multiple factors with lagged periods. We compare the performance metrics of two models, BPNN, SVR, LSTM, and other models, and study the applicability, advantages, and disadvantages of the models.
7. Results and Discussion
7.1. Comparison of the PPAR, HPPR, and MLR Models
(2) The HPPR model established has an excellent ability to fit the data, test the prediction and generalization of the samples, and reveal the transmission mechanism and effect of pork price, which can effectively regulate the monthly pork price. Although the HPPR model’s data fitting accuracy and prediction ability are slightly lower than the PPAR model, according to the optimal weight of multiple factors, we can analyze the transmission mechanism of pork price change and judge the pork price fluctuation and changing trend, put forward more targeted measurement, and control pork price fluctuations or soaring, etc. Therefore, establishing the HPPR model is essential for strengthening the pork supply chain management and promoting the healthy development of the pig industry chain. The HPPR model also provides the basis for decision making.
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Comparing Equations (7) and (8), we found that some significant variables differ. The coefficients of the piglet, the mutton, and the compound feed of broiler chickens were less than 0, indicating that these variables adversely affect the pork price, which is difficult to explain in theory. At the same time, the hog–corn ratio has nothing to do with pork prices and is inconsistent with common sense and truth. Therefore, although the fitting and prediction accuracy of the MLR is not low, its results are challenging to explain reasonably, and its practicability is poor.
7.2. Comparison with Xiong et al. [4]
Moreover, we cannot analyze the transmission mechanism of affecting pork prices through the DMA model.
7.3. Comparison of PPR with SVR, BPNN, etc.
Through the above comparison, we can conclude the following. Firstly, the PPAR and HPPR models not only have simple structures but also have explicit structures with precise mathematical meanings, and their prediction accuracy is higher than other machine learning models such as SVR and BPNN. Secondly, the PPAR and HPPR models are semiparametric models. When establishing the models, only the coefficients (weights) of multiple factors or autoregressive terms and the coefficients of the ridge function need to be optimized, which is not easy to cause “overtraining”. Thirdly, based on the established models, the importance of multiple influencing factors or autoregressive terms can be directly judged, making it easier to analyze the transmission mechanism of pork price, build a pork price control mechanism, and strengthen pork supply chain management. This promotes the sustainable development of pig farming, as well as upstream and downstream industries such as cattle, sheep, and chickens, agricultural product production, feed processing, and sales, and lastly, promotes sustainable agricultural and regional development.
7.4. To Predict the Pork Price Using the Latest Data Available
8. Conclusions, Policy Recommendations, Limitations, and Future Research
8.1. Conclusions
(1) The sustainable development of the pig industry is an important component of animal husbandry, feed processing industry, and agriculture, which significantly impacts achieving sustainable economic, social, and environmental development. The reliable and accurate prediction and risk warning of pork price fluctuations are the foundation and guarantee for achieving the sustainable development of the pig industry, playing a leading role. Establishing PPAR and HPPR models and accurately and reliably predicting the pork price changing trend help the Chinese government to establish a longterm mechanism to promote the sustainable development of the pig industry, improve and strengthen the system for pork (pig) price prediction and warning mechanisms, collect the information about feed prices such as corn and finishing pig feed as well as piglet prices in a timely manner, strengthen monitoring of African swine fever and other diseases, strengthen the management of the pig industry chain, ensure controllable price fluctuations and stable production, and achieve the sustainable development of the pig industry (animal husbandry) and its related industries, laying a solid foundation for sustainable agricultural development.
(2) There is important theoretical significance and practical value in establishing the PPAR and HPPR models to forecast the monthly pork price and expand the method. We collect the time series data of the monthly pork prices from January 2000 to September 2020 as well as the other 12 influencing factors (variables), such as the piglet and corn prices. For the monthly pork price, the studied results of the PPAR model with one linear or quadratic PRF show that the pork price lagged by 1–3 periods has a significant influence, and the lagged period of one has the most and positive impact, while the lagged period of two plays is of secondary significance and has a reverse and harmonic impact. The PPAR model possesses high fitting accuracy and good generalization ability. According to the time series data of the piglet price with a lagged period of six, the other variables, and the pork price with lagged period one, we established an HPPR model with one linear PRF. We found that seven variables, including the hog price, beef price, pork price, finishing pig feed price, piglet price, hog–corn ratio, and corn price, are important influencing factors. Among them, the hog price had the most significant impact, playing a decisive and positive role, followed by the beef and pork prices with a lagged period of one. The influence impacts of other variables are almost the same. Therefore, we established the PPAR and HPPR models to expand a method for monthly pork price prediction.
(3) The generalization ability and applicability of the established PPAR and HPPR models are better than SVR, BPNN, DMA, and other methods. Compared with SVR, BPNN, and DMA models, the PPAR and HPPR models are semiparametric and “white box” models. We established the PPAR and HPPR models with a few parameters, which are more straightforward, more explicit in mathematical meaning, and more convenient for applications than the other models. According to the best weights of the established PPR models, we can directly judge the importance of the lagged periods of the pork price, the importance of each variable, and its ranking, put forward the practical measurement of adjusting the pork price, and study the transmission mechanism and effectiveness of the pork price. According to market surveys or collected data, if the hog price has risen significantly in a month, we should increase the pork and beef supply to stabilize the pork price. Otherwise, the pork price will dramatically increase in the next month. Similarly, if the corn and beef prices in a month have increased significantly, it indicates that the pork prices in the next month will also rise significantly. If the monitoring finds that the piglet price increases significantly, the monthly pork price will rise considerably in the sixth month.
(4) According to the PPAR model, we can forecast the monthly pork price in multiperiods with higher accuracy, and the government departments can conveniently judge the changing trend of the pork price. With the HPPR model, we can forecast the monthly pork price with a lagged period of one and study the transmission mechanism and effectiveness of the pork price. The related government departments take adequate measures to strengthen pork supply chain management and take steps to control the pork price. The studied results of the PPAR model show that only the periods lagged by 1–3 of the monthly pork price have an important impact on the current pork price; it is not necessary to introduce more lagged periods into models, and it is beneficial to simplify the model, improving its practicability. The prediction accuracy of the PPAR model is even higher than the HPPR model. Still, its shortcomings are not suitable for studying the pork price transmission mechanism and the measures and suggestions to control the pork price. According to the results of the HPPR model, we can analyze the transmission mechanism and effectiveness of the monthly pork price, and the government authorities can strengthen the management of the pork supply chain and promote the healthy development of the pig industry chain. We established the HPPR model to delete seven factors with lower influence, although this does not mean that these seven factors are unrelated to the monthly pork price. Their influence impact has been reflected by factors such as hog–corn ratio, corn price, etc. The transmission mechanism of the monthly pork price is very complex and needs to be studied further.
(5) We establish a PPAR model using the latest pork price data from January 2000 to November 2023 to forecast the trend of pork prices changing in the following months. The results show that the pork price will rise in the future. The departments of price management and business administration should closely monitor the changes in pork prices and take timely measurements to adjust pork, hog, beef, etc., supply to ensure stable prices and increased efficiency in the pig farming industry.
8.2. Policy Recommendations
 (1)

To improve the monitoring of the monthly pork price, piglet price, other information, and the timeliness of monthly pork price prediction.
The pork price is the center of the whole price system of the pig industry chain. There is a lagged effect in the price transmission of pig breeding, and the transmission effectiveness of slaughtering and sales links also has information asymmetry, as well as sudden situations such as swine fever, which is highly likely to lead to drastic price fluctuations. Therefore, if the monthly pork price is to be controlled within a reasonable range, the relevant government departments must further improve the daily monitoring of the monthly pork price, piglet price, and other information and timely feedback on the drastic changes in relevant prices, to improve the timeliness and reliability of the monthly pork price forecast.
Many factors influence the pork price. According to the results of this paper, the pork price monitoring system mainly involves primary data collection, management, processing, etc. It should focus on monitoring the baby cost (piglets price), feeding cost (corn, pig ratio, pig, and chicken feed prices, etc.), alternative production prices (such as beef, mutton, live chicken, etc.), and hog price index etc. We must apply the timely data to establish the PPAR and HPPR models to ensure the timeliness of the monthly pork price prediction. Based on timely pork price predictions, the market subject can make good decisions and take corresponding measures to keep the pork price fluctuation within a reasonable range, ensuring the orderly operation of the market mechanism.
 (2)

To standardize the release of the pork price information and to realize realtime information sharing.
Information asymmetry is a fundamental reason for the risk of the pork market. The regulatory information department should promptly release the pork price forecast results and the price information of related products, simplify the information query process, and realize information sharing. In this way, the market administrators, producers, and operators in the pig industry chain can, in a timely and accurate manner, grasp the market development trend and reliably guide the market administrators, producers, and operators to adjust the production and operation decisions according to the forecasting information, and actively adapt to the changes in the market situation.
 (3)

To improve the risk early warning system of the monthly pork price and the government’s coordinating ability.
Relevant government departments should establish an emergency control mechanism for pork prices to ensure market supply and price stability. Sudden outbreaks such as African swine fever are unpredictable and quickly lead to drastic changes in pork prices in the short term. Therefore, in addition to monitoring the price information, the relevant government departments must also closely monitor the epidemic situation of pigs, coordinate the release and storage of frozen pork meat from the central reserve in a timely fashion, and ensure the essential balance between the supply and demand of pork, to reduce the adverse impact of the pig epidemics.
8.3. Limitations and Future Research
Theoretically, the relationship between pork supply and demand should be one of the essential factors in determining the monthly price change of pork. Data composition techniques, such as VMD, EEMD, etc., have been widely applied in modeling time series data, and there are still some differences in their effectiveness. So, there are two limitations in this paper. First, without complete data on pork’s supply and demand, similar to the other literature, we do not consider the monthly supply and demand of pork in our modeling. Furthermore, infectious and sow reproductive diseases have always threatened the sustainable development of the pig farming industry; there is a shortage of related information, so we do not consider these factors. Second, we establish PPAR and HPPR models using the original data, do not decompose the pork price time series data into independent components, and do not compare whether the data decomposition will improve the generalization ability, applicability, and reliability. In future research, we should collect and consider the pork supply, demand, and disease factors to establish HPPR models. Secondly, we will decompose the pork price into independent components using VMD and EEMD, etc., and study whether data decomposition techniques will improve the model performance or not.
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