# Analysis of East Asia Wind Vectors Using Space–Time Cross-Covariance Models

^{1}

^{2}

^{3}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Data

^{−1}), are the eastward (the horizontal speed of the air moving toward the east) and the northward components of the wind, respectively. We can obtain the wind speed (ms

^{−1}) and direction (in radians) of the horizontal wind using the following equations:

## 3. Method

#### 3.1. Model

#### 3.2. Inference Scheme

- Step 1. We fit the TGH-AR(p) at each spatial location and each variable. Then, we obtained the residuals by filtering out the fitted values based on their estimated parameters of TGH-AR(p), i.e., ${Y}_{t}-{\widehat{Y}}_{t}={Y}_{t}-\widehat{\xi}+\widehat{\omega}\xb7{\tau}_{\widehat{g},\widehat{h}}\left({\widehat{Z}}_{t}\right)$. Here, we could select either the univariate TGH-AR(p) or the bivariate Tukey g-and-h vector autoregressive model of order p as in [23]. In any case, we could perform estimation independently at each site, i.e., the computation could be easily parallelized.
- Step 2. We fit the space–time cross-covariance model to the residuals and estimated the remaining parameters. Here, if the computation costs matter, we could sub-sample in each partition and use them for parameter estimation. In this study, we split the whole dataset into $16\times 5$ partitions (16 spatial pieces × 5 temporal pieces), and then sub-sampled approximately 15% and 15% of observations from each spatial and temporal piece, respectively. As a result, the number of data points was 78,400, which was still large enough for the parameter estimation. We tried different partition numbers (from $16\times 4$ to $25\times 6$) and subsampling proportions (from $10\%$ to $25\%$) and did not find notable changes in inference performance.

## 4. Results

#### 4.1. Parameter Estimation

- Model 1: A parsimonious version of the space–time Gneiting-Matérn of (1); i.e., we assumed the same spatial range r for both variables.
- Model 2: This model was the same as Model 1, but the variances ${\sigma}_{1}$ and ${\sigma}_{2}$ had different values over the land and the ocean.
- Model 3: This model was the same as Model 1, but the variances ${\sigma}_{1}$ and ${\sigma}_{2}$ had varying values depending on the latitude over the land and the ocean. Here, we fixed the variances to their estimated values from the second-order polynomial regression in Figure 5.

#### 4.2. Prediction

^{−2}), which is often used as a metric to evaluate the availability of wind energy for power generation [44,45]. The WPD for conversion from a wind turbine can be computed as

^{−3}, which is the typically used air density value, as in [6].

## 5. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 4.**Plot of the estimated parameters for the Tukey g-and-h transformation, with (

**a**) $\xi $, (

**b**) $\omega $, (

**c**) g, and (

**d**) h for U.

**Figure 8.**Absolute prediction errors of the U component for (

**a**) Model 1, (

**b**) Model 2, and (

**c**) Model 3 on 3 July 2021 (for 1 case out of the 30 repetitions).

**Figure 9.**Absolute prediction errors of the V component for (

**a**) Model 1, (

**b**) Model 2, and (

**c**) Model 3 on 3 July 2021 (for 1 case out of the 30 repetitions).

**Figure 10.**(

**a**) Predicted wind speed on 4 July 2021, and (

**b**) scatter plot of the predicted and original wind speeds from Model 3.

Number of Parameters | Log-Likelihood | AIC | BIC | |
---|---|---|---|---|

Model 1 | 11 | 519.433 | −1016.865 | −914.900 |

Model 2 | 13 | 650.794 | −1275.589 | −1155.084 |

Model 3 | 9 | 525.153 | −1032.306 | −948.880 |

${\mathit{\rho}}_{12}$ | $\mathit{\alpha}$ | a | b | r | ${\mathit{\nu}}_{1}$ | ${\mathit{\nu}}_{2}$ | |
---|---|---|---|---|---|---|---|

Model 1 | 0.259 | 0.930 | 0.999 | 0.999 | 131.059 | 4.685 | 0.853 |

Model 2 | 0.065 | 1296.877 | $0.931$ | 0.745 | 507.661 | 0.513 | 0.524 |

Model 3 | 0.155 | 4392.535 | <0.001 | <0.001 | 429.204 | 0.394 | 0.393 |

${\mathit{\sigma}}_{\mathbf{1}}^{\mathrm{land}}$ | ${\mathit{\sigma}}_{\mathbf{1}}^{\mathrm{ocean}}$ | ${\mathit{\sigma}}_{\mathbf{2}}^{\mathrm{land}}$ | ${\mathit{\sigma}}_{\mathbf{2}}^{\mathrm{ocean}}$ | ${\mathit{\delta}}_{\mathbf{1}}$ | ${\mathit{\delta}}_{\mathbf{2}}$ | ||

Model 1 | 1.427 | 1.427 | 0.700 | 0.700 | $0.058$ | <0.001 | |

Model 2 | 0.907 | 0.668 | 0.905 | 0.692 | <0.001 | <0.001 | |

Model 3 | <0.001 | <0.001 |

U | Model 1 | Model 2 | Model 3 | |||
---|---|---|---|---|---|---|

MAE(Land/Ocean) | RMSE(Land/Ocean) | MAE(Land/Ocean) | RMSE(Land/Ocean) | MAE(Land/Ocean) | RMSE(Land/Ocean) | |

Min | 0.135 (10.174/0.103) | 0.187 (0.229/0.146) | 0.051 (0.085/0.030) | 0.089 (0.127/0.059) | 0.044 (0.066/0.026) | 0.072 (0.093/0.045) |

${Q}_{1}$ | 0.136 (0.189/0.109) | 0.192 (0.248/0.155) | 0.056 (0.097/0.034) | 0.104 (0.149/0.067) | 0.046 (0.078/0.028) | 0.079 (0.115,0.047) |

${Q}_{2}$ | 0.139 (0.192/0.112) | 0.196 (0.253/0.158) | 0.060 (0.101/0.037) | 0.113 (0.158/0.075) | 0.047 (0.082/0.029) | 0.083 (0.122/0.051) |

Mean | 0.139 (0.192/0.111) | 0.196 (0.251/0.158) | 0.060 (0.104/0.037) | 0.117 (0.166/0.078) | 0.047 (0.081/0.029) | 0.084 (0.121/0.053) |

${Q}_{3}$ | 0.142 (0.195/0.113) | 0.200 (0.255/0.163) | 0.062 (0.109/0.039) | 0.120 (0.171/0.080) | 0.048 (0.084/0.030) | 0.086 (0.127/0.056) |

Max | 0.144 (0.205/0.117) | 0.207 (0.268/0.170) | 0.094 (0.158/0.059) | 0.256 (0.367/0.166) | 0.052 (0.091/0.033) | 0.095 (0.141/0.069) |

V | Model 1 | Model 2 | Model 3 | |||
---|---|---|---|---|---|---|

MAE(land/Ocean) | RMSE(Land/Ocean) | MAE(Land/Ocean) | RMSE(Land/Ocean) | MAE(Land/Ocean) | RMSE(Land/Ocean) | |

Min | 0.042 (0.073/0.022) | 0.076 (0.115/0.038) | 0.057 (0.099/0.030) | 0.104 (0.148/0.056) | 0.050 (0.084/0.026) | 0.090 (0.132/0.042) |

${Q}_{1}$ | 0.045 (0.078/0.025) | 0.089 (0.129/0.043) | 0.061 (0.105/0.036) | 0.110 (0.165/0.069) | 0.051 (0.090/0.030) | 0.097 (0.142/0.054) |

${Q}_{2}$ | 0.046 (0.083/0.026) | 0.091 (0.140/0.049) | 0.064 (0.112/0.038) | 0.120 (0.172/0.077) | 0.054 (0.096/0.031) | 0.101 (0.150/0.058) |

Mean | 0.046 (0.082/0.027) | 0.094 (0.140/0.054) | 0.065 (0.113/0.038) | 0.125 (0.181/0.079) | 0.054 (0.096/0.032) | 0.103 (0.152/0.062) |

${Q}_{3}$ | 0.047 (0.085/0.029) | 0.096 (0.144/0.062) | 0.066 (0.119/0.040) | 0.128 (0.188/0.083) | 0.056 (0.100/0.033) | 0.106 (0.161/0.067) |

Max | 0.056 (0.101/0.034) | 0.133 (0.175/0.108) | 0.097 (0.165/0.059) | 0.254 (0.366/0.162) | 0.064 (0.110/0.040) | 0.131 (0.180/0.107) |

**Table 5.**Summary statistics of prediction measures of wind speed for each model based on the 30 repetitions.

Speed | Model 1 | Model 2 | Model 3 | |||
---|---|---|---|---|---|---|

MAE | RMSE | MAE | RMSE | MAE | RMSE | |

Min | 0.102 | 0.150 | 0.054 | 0.093 | 0.049 | 0.083 |

${Q}_{1}$ | 0.104 | 0.157 | 0.055 | 0.099 | 0.051 | 0.090 |

${Q}_{2}$ | 0.106 | 0.159 | 0.059 | 0.106 | 0.053 | 0.092 |

Mean | 0.107 | 0.159 | 0.059 | 0.111 | 0.053 | 0.093 |

${Q}_{3}$ | 0.109 | 0.162 | 0.062 | 0.114 | 0.054 | 0.096 |

Max | 0.115 | 0.174 | 0.076 | 0.205 | 0.062 | 0.112 |

WPD | Model 1 | Model 2 | Model 3 | |||
---|---|---|---|---|---|---|

MAE | RMSE | MAE | RMSE | MAE | RMSE | |

Min | 3.937 | 7.064 | 1.724 | 3.339 | 1.612 | 3.206 |

${Q}_{1}$ | 4.179 | 7.676 | 1.834 | 3.868 | 1.757 | 3.386 |

${Q}_{2}$ | 4.261 | 7.885 | 1.993 | 4.276 | 1.866 | 3.749 |

Mean | 4.282 | 7.909 | 1.984 | 4.416 | 1.863 | 3.755 |

${Q}_{3}$ | 4.405 | 8.165 | 2.086 | 4.561 | 1.949 | 3.945 |

Max | 4.600 | 8.827 | 2.427 | 7.317 | 2.169 | 4.787 |

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**MDPI and ACS Style**

Jeong, J.; Chang, W.
Analysis of East Asia Wind Vectors Using Space–Time Cross-Covariance Models. *Remote Sens.* **2023**, *15*, 2860.
https://doi.org/10.3390/rs15112860

**AMA Style**

Jeong J, Chang W.
Analysis of East Asia Wind Vectors Using Space–Time Cross-Covariance Models. *Remote Sensing*. 2023; 15(11):2860.
https://doi.org/10.3390/rs15112860

**Chicago/Turabian Style**

Jeong, Jaehong, and Won Chang.
2023. "Analysis of East Asia Wind Vectors Using Space–Time Cross-Covariance Models" *Remote Sensing* 15, no. 11: 2860.
https://doi.org/10.3390/rs15112860