# Functional Location-Scale Model to Forecast Bivariate Pollution Episodes

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## Abstract

**:**

## 1. Introduction

## 2. Methodology

#### 2.1. Mathematical Model

#### 2.2. Estimation Algorithm

**Step 1:**Perform a decomposition of each covariate ${X}^{j}\left(t\right)$ in basis functions of the form ${X}^{j}\left(t\right)\approx {\sum}_{k=1}^{K}{\xi}_{k}^{j}{\varphi}_{k}\left(t\right)$, where ${\varphi}_{k}$ ($k=1,\dots ,K$) are K basis functions (i.e., B-splines, wavelets), and ${\xi}_{il}$ are either the coefficients of an expansion in fixed basis or the principal component scores of the Karhunen-Loève expansion [44,45]. As a result, we obtain the transformed covariates

**Step 2:**For $r=1,2$, fit an additive model to the sample ${\{{\tilde{\mathbf{X}}}_{i},{Y}_{i1},{Y}_{i2}\}}_{i=1}^{n}$ and obtain an estimation of the means

**Step 3:**Compute the standardized residuals

**Step 4:**Obtain the kernel estimation of the bivariate density $\widehat{f}({\epsilon}_{1},{\epsilon}_{2})$ given by

## 3. Case Study: Joint Forecasting of $\left(\right)$ Pollution Episodes

## 4. Discussion

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Five curves of the historical matrix for pollutants $\left(\mathrm{NO}\right(t),\mathrm{SO}(t)$) and their corresponding first derivatives $({\mathrm{NO}}^{\prime}\left(t\right),{\mathrm{SO}}^{\prime}\left(t\right))$, observed in a period of time ${t}_{lag}=20$.

**Figure 2.**Observed and forecasted concentrations of SO${}_{2}$ and NO${}_{x}$ for two pollution episodes.

**Figure 3.**Observations (solid black line), mean (solid gray line) and $0.95th$ quantile estimations (discontinuous line) for both pollutants, SO${}_{2}$ and NO${}_{x}$ and for two pollution episodes.

**Figure 4.**Example of a pollution incident showing SO${}_{2}$ and NO${}_{x}$ concentrations versus time. Notice that there is an advance in the first pollutant compared to the second.

**Table 1.**Selected models from equation in (11). Cross X indicates the covariates included in each of the four considered models. The derivatives of the functions are indicated with a single quote.

${\mathit{X}}^{1}$ | ${\mathit{X}}^{2}$ | |||||||
---|---|---|---|---|---|---|---|---|

Model | $\mathrm{SO}\left(\mathit{t}\right)$ | $\mathrm{NO}\left(\mathit{t}\right)$ | ${\mathrm{SO}}^{\prime}\left(\mathit{t}\right)$ | ${\mathrm{NO}}^{\prime}\left(\mathit{t}\right)$ | $\mathrm{SO}\left(\mathit{t}\right)$ | $\mathrm{NO}\left(\mathit{t}\right)$ | ${\mathrm{SO}}^{\prime}\left(\mathit{t}\right)$ | ${\mathrm{NO}}^{\prime}\left(\mathit{t}\right)$ |

M${}_{1}$ | X | X | ||||||

M${}_{2}$ | X | X | X | X | ||||

M${}_{3}$ | X | X | X | X | ||||

M${}_{4}$ | X | X | X | X | X | X | X |

**Table 2.**Nominal $\tau $ and estimated $\widehat{\tau}$ coverages for each of the four models under study. Two time lags, ${t}_{lag}=10$, ${t}_{lag}=20$, two sizes of the training sample ${n}_{train}=10,000$ and ${n}_{train}=4900$, and two numbers of principal components, K = 3 and K = 5, were considered. Results correspond to the test sample.

$\widehat{\mathit{\tau}}$ | |||||||
---|---|---|---|---|---|---|---|

$\mathbf{\tau}$ | ${\mathit{t}}_{\mathit{lag}}$ | ${\mathit{n}}_{\mathit{train}}^{\u2022}$ | K | M${}_{1}$ | M${}_{2}$ | M${}_{3}$ | M${}_{4}$ |

0.50 | 10 | 20 | 3 | 0.43 | 0.47 | 0.45 | 0.51 |

5 | 0.42 | 0.48 | 0.47 | 0.49 | |||

10 | 49 | 3 | 0.51 | 0.52 | 0.52 | 0.52 | |

5 | 0.51 | 0.50 | 0.50 | 0.50 | |||

20 | 20 | 3 | 0.45 | 0.49 | 0.44 | 0.49 | |

5 | 0.48 | 0.46 | 0.43 | 0.46 | |||

20 | 49 | 3 | 0.50 | 0.54 | 0.51 | 0.50 | |

5 | 0.49 | 0.49 | 0.49 | 0.48 | |||

0.75 | 10 | 20 | 3 | 0.70 | 0.73 | 0.72 | 0.75 |

5 | 0.69 | 0.74 | 0.73 | 0.74 | |||

10 | 49 | 3 | 0.76 | 0.78 | 0.78 | 0.78 | |

5 | 0.77 | 0.76 | 0.76 | 0.76 | |||

20 | 20 | 3 | 0.70 | 0.72 | 0.70 | 0.72 | |

5 | 0.70 | 0.71 | 0.69 | 0.69 | |||

20 | 49 | 3 | 0.77 | 0.78 | 0.75 | 0.73 | |

5 | 0.75 | 0.74 | 0.72 | 0.72 | |||

0.90 | 10 | 20 | 3 | 0.88 | 0.87 | 0.87 | 0.89 |

5 | 0.86 | 0.88 | 0.87 | 0.88 | |||

10 | 49 | 3 | 0.91 | 0.90 | 0.90 | 0.90 | |

5 | 0.90 | 0.90 | 0.90 | 0.90 | |||

20 | 20 | 3 | 0.87 | 0.87 | 0.85 | 0.86 | |

5 | 0.87 | 0.86 | 0.86 | 0.84 | |||

20 | 49 | 3 | 0.90 | 0.89 | 0.90 | 0.86 | |

5 | 0.87 | 0.87 | 0.88 | 0.85 | |||

0.95 | 10 | 20 | 3 | 0.93 | 0.93 | 0.93 | 0.93 |

5 | 0.93 | 0.93 | 0.93 | 0.93 | |||

10 | 49 | 3 | 0.96 | 0.93 | 0.93 | 0.93 | |

5 | 0.95 | 0.94 | 0.94 | 0.94 | |||

20 | 20 | 3 | 0.93 | 0.93 | 0.92 | 0.92 | |

5 | 0.92 | 0.92 | 0.93 | 0.90 | |||

20 | 49 | 3 | 0.95 | 0.94 | 0.95 | 0.92 | |

5 | 0.92 | 0.92 | 0.93 | 0.90 |

**Table 3.**RMSE values for two pollution episodes and the four models tested, considering curves with two different time lags, size of the training samples and number of principal components.

Episode 1 | Episode 2 | ||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|

Repsonse | ${\mathit{t}}_{\mathit{lag}}$ | ${\mathit{n}}_{\mathit{train}}^{\u2022}$ | K | M${}_{\mathbf{1}}$ | M${}_{\mathbf{2}}$ | M${}_{\mathbf{3}}$ | M${}_{\mathbf{4}}$ | M${}_{\mathbf{1}}$ | M${}_{\mathbf{2}}$ | M${}_{\mathbf{3}}$ | M${}_{\mathbf{4}}$ |

NO${}_{x}$ | 10 | 20 | 3 | 20.7 | 25.2 | 20.0 | 23.0 | 1.9 | 1.7 | 1.4 | 1.4 |

5 | 19.6 | 18.6 | 20.6 | 16.4 | 0.9 | 0.8 | 0.8 | 0.6 | |||

49 | 3 | 20.9 | 24.7 | 18.2 | 23.8 | 1.8 | 1.9 | 1.4 | 1.4 | ||

5 | 19.2 | 17.5 | 19.3 | 16.7 | 0.9 | 0.9 | 0.8 | 0.7 | |||

20 | 100 | 3 | 34.0 | 24.7 | 19.9 | 19.6 | 3.3 | 5.5 | 1.7 | 3.1 | |

5 | 40.9 | 23.3 | 46.5 | 29.2 | 1.5 | 2.5 | 1.0 | 1.8 | |||

49 | 3 | 30.2 | 18.2 | 18.7 | 20.4 | 3.4 | 5.3 | 1.7 | 2.8 | ||

5 | 36.2 | 27.6 | 39.8 | 35.6 | 1.5 | 2.5 | 1.0 | 1.8 | |||

SO${}_{2}$ | 10 | 100 | 3 | 505.0 | 841.0 | 407.5 | 837.3 | 544.8 | 531.9 | 419.8 | 419.1 |

5 | 914.9 | 868.6 | 669.4 | 516.3 | 215.6 | 230.3 | 184.5 | 199.1 | |||

49 | 3 | 686.0 | 685.3 | 515.8 | 518.3 | 481.4 | 484.4 | 338.0 | 361.1 | ||

5 | 991.1 | 925.5 | 846.3 | 682.9 | 199.9 | 214.7 | 170.6 | 179.9 | |||

20 | 100 | 3 | 1463.4 | 2172.7 | 825.4 | 1199.9 | 1154.5 | 1133.1 | 709.5 | 659.0 | |

5 | 1470.6 | 2531.2 | 1002.9 | 1428.5 | 525.5 | 482.1 | 352.0 | 341.3 | |||

49 | 3 | 1.458.7 | 2485.4 | 768.4 | 698.4 | 1162.6 | 1125.8 | 644.8 | 628.7 | ||

5 | 1787.6 | 2811.0 | 1111.2 | 951.5 | 548.1 | 492.0 | 352.7 | 359.3 |

**Table 4.**Maximum memory consumption (MB) and computation time (seconds) for the four models tested, ${M}_{i}$, following different strategies concerning the time lag, ${t}_{lag}$, the size of the training sample, ${n}_{train}$, and the number of basis functions, K.

Memory (MB) | Runtime (seconds) | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

${\mathit{t}}_{\mathit{lag}}$ | ${\mathbf{n}}_{\mathbf{train}}$ | K | M${}_{\mathbf{1}}$ | M${}_{\mathbf{2}}$ | M${}_{\mathbf{3}}$ | M${}_{\mathbf{4}}$ | M${}_{\mathbf{1}}$ | M${}_{\mathbf{2}}$ | M${}_{\mathbf{3}}$ | M${}_{\mathbf{4}}$ |

10 | 20 | 3 | 652.70 | 1083.03 | 1319.57 | 2266.34 | 17.98 | 29.66 | 35.99 | 61.62 |

5 | 1090.78 | 1855.39 | 2299.14 | 4075.88 | 35.55 | 55.29 | 63.15 | 124.7 | ||

49 | 3 | 329.94 | 548.58 | 669.42 | 1153.38 | 10.53 | 17.42 | 21.38 | 37.01 | |

5 | 552.74 | 942.68 | 1171.67 | 2089.53 | 19.79 | 31.28 | 46.62 | 88.20 | ||

20 | 20 | 3 | 653.34 | 1084.15 | 1320.66 | 2268.16 | 18.11 | 29.63 | 35.97 | 61.21 |

5 | 1091.90 | 1857.26 | 2300.99 | 4078.95 | 32.14 | 49.51 | 72.45 | 124.81 | ||

49 | 3 | 330.10 | 548.84 | 669.69 | 1153.82 | 10.51 | 17.56 | 21.49 | 37.82 | |

5 | 553.01 | 943.13 | 1172.20 | 2090.36 | 17.80 | 36.44 | 39.03 | 76.11 |

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**MDPI and ACS Style**

Oviedo-de La Fuente, M.; Ordóñez, C.; Roca-Pardiñas, J.
Functional Location-Scale Model to Forecast Bivariate Pollution Episodes. *Mathematics* **2020**, *8*, 941.
https://doi.org/10.3390/math8060941

**AMA Style**

Oviedo-de La Fuente M, Ordóñez C, Roca-Pardiñas J.
Functional Location-Scale Model to Forecast Bivariate Pollution Episodes. *Mathematics*. 2020; 8(6):941.
https://doi.org/10.3390/math8060941

**Chicago/Turabian Style**

Oviedo-de La Fuente, Manuel, Celestino Ordóñez, and Javier Roca-Pardiñas.
2020. "Functional Location-Scale Model to Forecast Bivariate Pollution Episodes" *Mathematics* 8, no. 6: 941.
https://doi.org/10.3390/math8060941