# Multivariate EMD-Based Modeling and Forecasting of Crude Oil Price

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## Abstract

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## 1. Introduction

## 2. Multivariate EMD Theory

- (1)
- Sample over an (n-1) dimensional sphere to obtain a point set.
- (2)
- Along the direction vector x
^{θk}, calculate the projection p^{θk}${\left(\mathrm{t}\right)}_{\mathrm{t}-1}^{\mathrm{T}}$ of the input signal v${\left(\mathrm{t}\right)}_{\mathrm{t}=1}^{\mathrm{T}}$. - (3)
- Find the time instants ${\mathrm{t}}_{\mathrm{j}}^{\mathsf{\theta}\mathrm{k}}$ corresponding to the maxima of the set of projected signals p
^{θk}${\left(\mathrm{t}\right)}_{\mathrm{k}=1}^{\mathrm{k}}$ for the whole set of direction vector k. - (4)
- Calculate the multivariate envelope curve e
^{θk}${\left(\mathrm{t}\right)}_{\mathrm{k}=1}^{\mathrm{k}}$ using the interpolation method over the interval [${\mathrm{t}}_{\mathrm{j}}^{\mathsf{\theta}\mathrm{k}}$, v(${\mathrm{t}}_{\mathrm{j}}^{\mathsf{\theta}\mathrm{k}}$)] - (5)
- Calculate the mean m(t) of the envelope curves as in $m\left(t\right)=\frac{1}{\mathrm{K}}{\displaystyle \sum}_{\mathrm{k}=1}^{\mathrm{K}}{\mathrm{e}}^{{\mathsf{\theta}}_{\mathrm{k}}}\left(\mathrm{t}\right)$.
- (6)
- Calculate the c
_{i}(t) = v(t) − m(t) for i-th order of IMF. Evaluate the c_{i}(t) using the stoppage criterion. If the stoppage criterion is satisfied, apply the above procedure to v(t) − c_{i}(t), otherwise apply it to c_{i}(t).

## 3. A Multivariate EMD Based Forecasting Model for Crude Oil Price Movement

_{i}, i = (1, 2, …, N), the m-dimensional phase space x is constructed as in Equation (1):

_{i}, r

_{i}

_{+ τ}, ···, r

_{i}

_{+ (m − 1)τ}], i = 1, 2, …, N

_{m}

_{m}= N − (m − 1)τ is the size of the vector point. x is the two-dimensional matrix that is constructed using the delayed embedding method.

_{j}is the p-variate IMF aligned with scale at time t. $\u03f5$ is the residual.

_{j}in the higher dimension back to the univariate data y

_{k}, k = (1, 2, …, N

_{m}) at a lower dimension. Different statistical tests are performed on the univariate data to help determine the appropriate model specification, i.e., the model equations and the lag orders. For example, the lag orders for time series equations are determined following the Akaike Information Criteria (AIC) and Bayesian Information Criteria (BIC) minimization principles. Then, the optimal parameters are determined for the model specification chosen, using appropriate econometric, machine learning, and optimization techniques. The parameters of Autoregressive Moving Average (ARMA) model are determined using the maximum likelihood estimation (MLE) technique as in Equation (3):

_{k}is the conditional mean of the data, y

_{k−i}is the lag m returns with parameter φ, and ε

_{k−j}is the lag n residuals in the previous period with parameter θ. δ is the constant coefficient. $\u03f5$ is the error term. This transforms the original feature extract matrix into the individual forecasting matrix.

_{1}. The model parameters, such as the weighting functions, are determined using the model tuning dataset.

_{k}

_{+1}using the rolling window method:

## 4. Empirical Results

_{i}, i = 1, 2, …, 8 refer to the proposed MEMDF with scale i assumed as the main underlying factor for forecasting. Experiment results in Table 2 show that as we assume different scales as the main underlying driving factor for the forecasting model, different performance of the proposed model would result. The performance variation suggests that the multivariate EMD-based forecasting model provides different assumptions on the market microstructure; some may be twisted and biased. One common criterion to choose the optimal model specification and parameters is to use the in-sample MSE as the optimization objective. This implies that we assume that the main driving at the particular scale chosen with the optimal model specification is the true one that would remain in the future out-of-sample data. Using optimal in-sample MSE as the optimization criterion using the model tuning data, we choose scale 7 with a Huber weight function for the WTI market with an in-sample MSE 6.7061 × 10

^{−4}and choose scale 7 with a Talwar weight function for the Brent market with an in-sample MSE 5.1590 × 10

^{−4}.

_{i}refers to the calculated MSE for different models in the market i, CW

_{i,j}refers to the calculated Clark West test of predictive accuracy for different models against the benchmark model j in the market i. Experiment results in Table 3 show that the performance of the proposed MEMD-based algorithm is significantly better than the benchmark RW and ARMA models for WTI, in terms of the level of predictive accuracy. Overall, the proposed model achieves lower MSE than the benchmark RW and ARMA models. The forecasting performance gap is statistically significant at 95% confidence level for the WTI market. For the Brent market, it is statistically significant at 80% against the ARMA model and 70% confidence level against the RW model. For both markets, the ARMA model is inferior to the RW model in terms of forecasting accuracy. This suggests that the market contains nonlinear dynamics instead of the linear features captured by simple linear model, such as the ARMA model. The slight inferior performance of the proposed model in the Brent market may indicate that the Brent market may be less efficient than the WTI market., the dominating factor in the Brent market, unlike the WTI market, is less linear, which results in a larger estimation bias for the ARMA model. More accurate nonlinear modeling of the underlying DGPs than the currently-employed ARMA model are needed. As our out-of-sample test results confirm that the model specification chosen in the model tuning phase leads to the improved forecasting results, scale 7 is supposed to be the optimal scale where the underlying data components are extracted. Since different weighting functions are chosen for the robust regression model in different markets, this suggests that the market has different levels of noise, which would result in different levels of disruption on the main driving factors in different markets.

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

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Statistics | Mean | Standard Deviation | Skewness | Kurtosis | p_{J}_{B} | p_{B}_{D}_{S} |
---|---|---|---|---|---|---|

r_{WTI} | 0.0001 | 0.0229 | −0.4367 | 6.4688 | 0.001 | 0.0013 |

r_{B}_{r}_{ent} | 0.0001 | 0.0169 | 0.0664 | 6.3167 | 0.001 | 0 |

Models | RW | ARMA | MEMDF | |||||||
---|---|---|---|---|---|---|---|---|---|---|

1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | |||

WTI_{MSE,×}_{10}^{−4} | 6.7194 | 6.7133 | 9.9698 | 7.6461 | 8.2590 | 7.5943 | 7.0964 | 7.0676 | 6.8689 | 6.9802 |

Brent_{MSE,×}_{10}^{−4} | 5.1652 | 5.1837 | 7.6798 | 6.2551 | 6.3701 | 5.9294 | 5.5196 | 5.3251 | 5.2520 | 5.6616 |

Models | RW | ARMA | MEMDF |
---|---|---|---|

MSE_{WTI,}_{10}^{−}^{4} | 3.7891 | 3.8271 | 3.7856 |

CW_{WTI,ARMA} | 0.0061 | N/A | 0.0036 |

CW_{WTI,RW} | N/A | 0.6897 | 0.0828 |

MSE_{B}_{r}_{ent,}_{10}^{−}^{4} | 4.7533 | 4.7565 | 4.7529 |

CW_{B}_{r}_{ent,AR}_{M}_{A} | 0.1177 | N/A | 0.1748 |

CW_{B}_{r}_{ent,RW} | N/A | 0.7460 | 0.3006 |

© 2016 by the authors; licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC-BY) license (http://creativecommons.org/licenses/by/4.0/).

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**MDPI and ACS Style**

He, K.; Zha, R.; Wu, J.; Lai, K.K.
Multivariate EMD-Based Modeling and Forecasting of Crude Oil Price. *Sustainability* **2016**, *8*, 387.
https://doi.org/10.3390/su8040387

**AMA Style**

He K, Zha R, Wu J, Lai KK.
Multivariate EMD-Based Modeling and Forecasting of Crude Oil Price. *Sustainability*. 2016; 8(4):387.
https://doi.org/10.3390/su8040387

**Chicago/Turabian Style**

He, Kaijian, Rui Zha, Jun Wu, and Kin Keung Lai.
2016. "Multivariate EMD-Based Modeling and Forecasting of Crude Oil Price" *Sustainability* 8, no. 4: 387.
https://doi.org/10.3390/su8040387