# Monthly Pork Price Prediction Applying Projection Pursuit Regression: Modeling, Empirical Research, Comparison, and Sustainability Implications

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials: Data Resource

#### 3.1. Collecting the Monthly Pork Price

#### 3.2. Preliminary Determination of the Main Variables Affecting the Fluctuation of Pork Prices and Collecting the Data

## 4. Principles of PPR Modeling

#### 4.1. Principle of Establishing the PPAR Model

#### 4.2. The Principle of Establishing the H-PPR Model of Monthly Pork Price Prediction Based on Multivariate Time Series

## 5. An Empirical Study on Establishing a PPAR Model for Monthly Pork Price Prediction

#### 5.1. Determination of the Reasonable Number of Time Series Lagged Periods

#### 5.2. Establishment of the Optimal PPAR Model

## 6. Establishment of the H-PPR Model of Monthly Pork Price Prediction with Mixed Multivariate Time Series Data

#### 6.1. Selection of the Critical Variables

#### 6.2. Establishment of an H-PPR Model Based on the Monthly Pork Price and Multivariate Price Time Series

## 7. Results and Discussion

#### 7.1. Comparison of the PPAR, H-PPR, and MLR Models

#### 7.2. Comparison with Xiong et al. [4]

#### 7.3. Comparison of PPR with SVR, BPNN, etc.

#### 7.4. To Predict the Pork Price Using the Latest Data Available

## 8. Conclusions, Policy Recommendations, Limitations, and Future Research

#### 8.1. Conclusions

#### 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.

- (2)
- To standardize the release of the pork price information and to realize real-time information sharing.

- (3)
- To improve the risk early warning system of the monthly pork price and the government’s coordinating ability.

#### 8.3. Limitations and Future Research

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Schematic diagram of pork price change over time. (The ordinate is pork price, unit CNY; the horizontal coordinate is time, starting from January 2000, the same below).

**Figure 2.**Schematic diagram of the AE and RE changes of the PPAR model with one linear PRF for the monthly pork price.

**Figure 3.**The schematic diagram of the changing relationship between the pork price and the price of the hog, corn, the piglet with a lag period of one, and the piglet price with a lag period of six (the piglet’s price with lag periods one and six are on the right ordinate).

Refs. | Model Used | Samples | Univariate/Multivariate Time Series | Simple/ Challenging to Implement | Price |
---|---|---|---|---|---|

[2] | ARIMA, SARIMA | 2010–2018, 100 obs. | Univariate | Simple ^{&} | Monthly, Sugar, Russian |

[4] | DMA, TVP, AR | January 2000–March 2019, 321 obs. | Multivariate, 11 factors | Challenging | Monthly, Pork, China |

[5] | PCA-GM-BPNN | January 2010–December 2018, 108(96/12) * | Multivariate, 12 factors | Challenging | Monthly, Pork, China |

[6] | RBFNN, GA-SVR, EMD-GA-SVR, EMD-GA-SVR, EEMD-GA-SVR, CEEMD-GA-SVR, CEEMD-PEFFT-GCD-SVR | January 2006–June 2018, 150(120/30) | Univariate | Challenging | Monthly, Pork, China |

[11] | GRNN, BPNN | 1 March 2011–25 March 2014, 732(502/110/110) | Univariate | Challenging | Daily, pork, China |

[13] | GM, TS, BPNN, TS-GM, TS-BPNN, BPNN-BPNN, GM-GM, BPNN-GM, GM-BPNN | January 2000–June 2008, 102(90/12) | Multivariate, 4 factors | Challenging | Monthly, hog, China |

[17] | MLR | January 1897–December 1916 | Multivariate, 4 factors | Simple | Monthly, hog, USA |

[18] | Empirical formula, Demand-curve method | January 1903–December 1914 | Multivariate | Simple | Monthly, hog, USA |

[19] | ARIMA, BPNN, Econometric model | January 1974–December 1996 | Multivariate, Univariate | Challenging | Quarterly, monthly, hog, USA |

[23] | ARIMA, SARIMAX, RF, SVR, Ridge, LGBM, XGBoost, RNN, LSTM, CatBOOST | January 2016–February 2022, 322(80:20) ** | Univariate | Challenging | Weekly, pork, Spain |

[25] | LDA, Deep learning (LSTM), RF, BPNN, CNN, Gradient, Boosting, Ridge | January 2010–December 2019, 1175(987/188) | Univariate | Challenging | Daily, pork, South Korea |

[26] | CAR, CART, SA-CART, SSA-CART, WSO-CART | January 2011–December 2015, 257(80:10:10) | Multivariate | Challenging | Weekly, pork, China |

[27] | LSTM, STL-ATTLSTM, BERTLSTM, GCNLSTM, HGLSTM | January 2013–December 2020 | Multivariate | Challenging | Weekly, hog, China |

[28] | ARIMA, EMD-ARIMA, VMD-ARIMA, SVR, EMD-SVR, VMD-SVR, RNN, EMD-RNN, VMD-RNN, LSTM, EMD-LSTM, VMD-LSTM | January 1974–December 2017, 11,085(80:10:10) | Univariate | Challenging | Daily, corn and soybean, USA |

[29] | ARIMA, ETS, SVM, LSTM, BPNN, Other 12 combined models | January 2013–December 2018 | Univariate | Challenging | Monthly, onion and potato, India |

[30] | SARIMA, TDNN, ELM, STL-ELM | January 2010–December 2020, January 2005–December 2020 | Univariate | Challenging | Monthly, potato, India |

[31] | ARMA, GM, BPNN, GRNN, RBFNN, FOA-GRNN, FOA-RBFNN | January 2010–April 2020, 112(80/30) | Univariate | Challenging | Monthly, vegetables, China |

[32] | Lasso, SARIMA, STL-SVR-ARMA, SVR, RF, LSTM, VMD-LR-ARMA, STL-SVR-SNN-ARMA | January 2006–December 2018, 678(658/20) | Univariate | Challenging | Weekly, pork and hog, China |

[33] | RF, XGB, LGBM, BPNN, RNN, LSTM, EEMD-BPNN, EEMD-RNN, EMD-MiLSTM, EEMD-MiLSTM | 240(90:10) | Univariate | Challenging | Monthly, hog, China |

[34] | SVR-cyclical component, BPNN, Wavelet-SVR | January 2011–March 2017 | Univariate | Challenging | Monthly, pork and piglet, China |

[35] | SVR, SVR-CIs, SVR-CIs-CIp, SVR-CIs+p, SVR-WD, SVR-EMD, SVR-EEMD, SVR-SSA | January 2011–December 2017, 84(80:20) | Univariate | Challenging | Monthly, pork, China |

^{&}Simple means that it is simple or easy to implement the model. Challenging implies that it is challenging or complex to implement the model and train it with over-training or overfitting easily, which possesses good flexibility and nonlinear approximation ability.

**Table 2.**Comparison of the optimal weights, polynomial coefficients, and objective function values of different PPAR models for monthly pork prices.

Model | $\mathbf{The}\mathbf{Best}\mathbf{Weight}\mathit{a}\left(1\right)~\mathit{a}\left(12\right)$ | $\mathbf{Polynomial}\mathbf{Coefficients}{\mathit{c}}_{0},{\mathit{c}}_{1},{\mathit{c}}_{2}$ | ${\mathit{Q}}_{\mathit{T}}\left(\mathit{a}\right),{\mathit{Q}}_{\mathit{V}}\left(\mathit{a}\right)$ ^{#} |
---|---|---|---|

1-0-PPAR | −0.273, 0.226, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0.935 | 20.984, 11.002 | 253.9, 234.3 |

1-1-PPAR | 0.314, −0.845, 0.401, 0, 0, 0, 0, 0, 0, 0.1631, 0, 0 | 0.008, −3.739 | 239.3, 220.3 |

2-0-PPAR | −0.339, 0.488, −0.200, 0, 0, 0, 0, 0, 0, 0, 0, 0.779 | 20.988, 13.397, −0.048 | 246.3, 234.9 |

2-2-PPAR | −0.321, 0.687, −0.430, 0, 0, 0, 0, 0, 0, 0.490, 0, 0 | 0.276, −1.002, −4.117 | 225.6, 973.9 |

2-1-PPAR | 0.267, −0.761, 0, 0, 0, 0, 0, 0, 0, 0.591, 0, 0 | 0.010, −1.135 | 235.7, 224.4 |

^{#}${Q}_{T}\left(a\right)\mathrm{a}\mathrm{n}\mathrm{d}{Q}_{V}\left(a\right)$ represent the sum of squared error (SSE) (objective function value) of the training and validation samples, respectively. The smaller the values are, the better the model’s fitting accuracy and prediction ability (generalization ability) are. The autoregressive term with the best weight of “0” can be deleted, as shown below.

**Table 3.**Optimization results and performance comparisons of different PPAR models with three predictors (autoregressive items).

Model | $\mathbf{The}\mathbf{Optimal}\mathbf{Weights}\mathit{a}\left(1\right)-\mathit{a}\left(3\right)$ | $\mathbf{Polynomial}\mathbf{Coefficients}{\mathit{c}}_{0},{\mathit{c}}_{1},{\mathit{c}}_{2}$ | ${\mathit{Q}}_{\mathit{T}}\left(\mathit{a}\right),{\mathit{Q}}_{\mathit{V}}\left(\mathit{a}\right)$ |
---|---|---|---|

1-0-PPAR * | 0.174, −0.575, 0.800 | 20.973, 24.627 | 131.58, 178.32 |

1-1-PPAR | −0.770, 0.597, 0.227 | 0.0000, 0.0001 | 131.58, 178.32 |

1-2-PPAR | −0.208, 0.775, −0.597 | −0.0977, 2.7771, 35.0328 | 119.02, 289.15 |

2-0-PPAR | 0.176, −0.574, 0.800 | 20.916, 24.563, 0.8681 | 138.84, 244.96 |

2-2-PPAR | 0.209, −0.778, 0.592 | −0.083, −2.607, 32.529 | 120.78, 392.86 |

2-1-PPAR | 0.193, −0.790, 0.582 | 0.0006, −0.0804 | 130.83, 244.02 |

1-PPAR | 0.173, −0.545, 0.820 | 20.913, 21.674 | 93.591, 5.549 |

**Table 4.**Comparison of performance metrics between the training and verification samples of the 1-0-PPAR model with one linear PRF.

Model | Sample Subset | RMSE * | MAE | Max_AE | MAPE (%) | Max_RE (%) |
---|---|---|---|---|---|---|

1-0-PPAR | training | 0.750 | 0.525 | 3.82 | 2.63 | 11.25 |

verification | 3.855 | 3.076 | 8.44 | 5.80 | 15.70 | |

1-PPAR | training | 0.654 | 0.479 | 2.408 | 2.52 | 9.78 |

verification | 0.680 | 0.496 | 1.371 | 2.11 | 5.81 |

**Table 5.**Comparison of the optimal weights, polynomial coefficients, and objective function values of different PPR models for predicting pork prices.

Model | $\mathbf{The}\mathbf{Best}\mathbf{Weights}\mathit{a}\left(1\right)-\mathit{a}\left(12\right)$ | $\mathbf{Coefficients}{\mathit{c}}_{0},{\mathit{c}}_{1},{\mathit{c}}_{2}$ | ${\mathit{Q}}_{\mathit{T}}\left(\mathit{a}\right),{\mathit{Q}}_{\mathit{V}}\left(\mathit{a}\right)$ |
---|---|---|---|

1-PPR | 0.020, −0.056, 0.955 *, 0.195, −0.075, −0.040, 0.050, 0.072, 0.122, −0.113, −0.062, 0.007 | 20.901, 9.142 | 189.66, 156.23 |

2-PPR | −0.024, −0.048, 0.961, 0.191, −0.068, −0.037, 0.039, 0.066, 0.088, −0.126, −0.049, 0.005 | 21.008, 9.738, −0.180 | 189.22, 198.75 |

1-PPR-11 | 0.642, 0.116, 0.157, −0.012, −0.013, 0.153, −0.001, 0.673, −0.111, −0.245, −0.030 | 20.820, 8.5692 | 232.07, 252.84 |

2-PPR-11 | 0.688, 0.031, 0.148, −0.058, −0.026, 0.173, 0.030, 0.650, −0.072, −0.204, 0.015 | 20.357, 8.3245, 0.5764 | 219.68, 309.95 |

1-PPR-9 | 0.656, −0.372, 0.219, 0.188, 0.335, −0.152, 0.245, 0.092, −0.379 | 20.653, 11.428 | 423.70, 810.91 |

2-PPR-9 | 0.651, −0.367, 0.201, 0.189, 0.346, −0.154, 0.253, 0.103, −0.384 | 20.542, 11.327, 0.2637 | 423.04, 717.64 |

H-PPR-13 | 0.015, −0.024, 0.968, 0.163, −0.055, −0.033, 0.058, 0.071, 0.077, −0.113, −0.007, 0.012, −0.060 | 21.168, 9.067 | 188.00, 162.70 |

H-PPR-8 | 0.003, −0.023, 0.983, 0.129, 0, 0, 0, 0.075, 0.036, −0.088, 0, 0, −0.043 | 21.188, 9.093 | 189.80, 155.70 |

H-PPR-7 | 0.049, −0.031, 0.939, 0.195, 0, 0, 0, 0, 0.127, −0.189, 0, 0, 0.158 | 21.215, 8.031 | 196.70, 162.00 |

H-PPR-6 | −0.0474, −0.0549, 0.9787, 0.1305, 0, 0, 0, 0, −0.0482, 0, 0, 0, 0.132 | 21.170, −8.796 | 201.50, 165.00 |

H-PPR-5 | −0.105, −0.045, 0.633, 0, 0, 0, 0, 0, −0.044, 0, 0, 0, 0.764 | 21.131, 7.982 | 237.40, 198.30 |

H-PPR-6a | 0.028, 0.001, 0.992, 0.077, 0, 0, 0, 0, 0.050, 0, 0, 0, 0.076 | 20.990, 7.789 | 118.20, 10.69 |

H-PPR-12b | 0.417, −0.064, 0, 0.247, −0.099, −0.039, 0.050, 0.050, 0.451, −0.201, −0.111, 0.025, 0.698 | 21.105, 7.553 | 209.88, 194.19 |

H-PPR-6b | 0.377, −0.059, 0, 0.191, 0, 0, 0, 0, 0.394, −0.292, 0, 0, 0.760 | 21.147, 7.921 | 214.49, 191.39 |

Model | Sample | RMSE | MAE | Max_AE | MAPE (%) | Max_RE (%) | Bias |
---|---|---|---|---|---|---|---|

H-PPR-6 | Training | 0.936 | 0.628 | 6.323 | 3.03 | 15.05 | 0 |

Verification | 3.873 | 3.462 | 7.535 | 6.62 | 14.92 | 0.872 | |

H-PPR-6a | Training | 0.742 | 0.559 | 2.640 | 2.88 | 12.71 | 0 |

Verification | 0.986 | 0.788 | 2.159 | 3.40 | 9.24 | 0.706 | |

Equation (7) | Training | 0.910 | 0.628 | 6.214 | 3.13 | 14.79 | 0 |

Verification | 3.701 | 3.064 | 7.568 | 5.85 | 14.99 | 1.256 | |

Equation (8) | Training | 0.706 | 0.537 | 2.121 | 2.84 | 12.69 | 0 |

Verification | 1.000 | 0.751 | 2.502 | 3.21 | 10.60 | 0.741 |

Model | Univariate Time Series | Multivariate Pork Price Times Series | ||
---|---|---|---|---|

RMSE | SMAPE | RMSE | SMAPE | |

DMA | / | / | 0.541 */ | 3.387 / |

Dynamic model selection | / | / | 0.557 / | 3.391 / |

Bayes model averaging | / | / | 0.664 / | 3.906 / |

PPAR/H-PPR | 0.650/0.680 | 2.509/2.493 | 0.743/0.986 | 2.887/3.212 |

SVR * | 0.503/1.195 | 1.752/3.415 | 0.517/1.0216 | 2.044/3.155 |

BPNN | 0.655/0.657 | 2.561/2.127 | 0.684/1.420 | 2.819/5.125 |

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**MDPI and ACS Style**

Yu, X.; Liu, B.; Lai, Y.
Monthly Pork Price Prediction Applying Projection Pursuit Regression: Modeling, Empirical Research, Comparison, and Sustainability Implications. *Sustainability* **2024**, *16*, 1466.
https://doi.org/10.3390/su16041466

**AMA Style**

Yu X, Liu B, Lai Y.
Monthly Pork Price Prediction Applying Projection Pursuit Regression: Modeling, Empirical Research, Comparison, and Sustainability Implications. *Sustainability*. 2024; 16(4):1466.
https://doi.org/10.3390/su16041466

**Chicago/Turabian Style**

Yu, Xiaohong, Bin Liu, and Yongzeng Lai.
2024. "Monthly Pork Price Prediction Applying Projection Pursuit Regression: Modeling, Empirical Research, Comparison, and Sustainability Implications" *Sustainability* 16, no. 4: 1466.
https://doi.org/10.3390/su16041466