# Cokriging Prediction Using as Secondary Variable a Functional Random Field with Application in Environmental Pollution

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## Abstract

**:**

## 1. Introduction and Bibliographical Review

## 2. Cokriging Using as Secondary Variable a Functional Random Field

#### 2.1. Ordinary Cokriging Predictor

- (i)
- $2{\gamma}_{lq}({\mathit{s}}_{i},{\mathit{s}}_{j})=\mathrm{Cov}({X}_{l}({\mathit{s}}_{i})-{X}_{q}({\mathit{s}}_{j}))$, for $l,q=1,\dots ,m$.
- (ii)
- ${\mathit{\gamma}}_{lq}^{\top}=({\gamma}_{lq}({\mathit{s}}_{1},{\mathit{s}}_{0}),\cdots ,{\gamma}_{lq}({\mathit{s}}_{n},{\mathit{s}}_{0}))$.
- (iii)
- ${\mathsf{\Gamma}}_{lq}=\left(\begin{array}{ccc}{\gamma}_{lq}({\mathit{s}}_{1},{\mathit{s}}_{1})& \cdots & {\gamma}_{lq}({\mathit{s}}_{1},{\mathit{s}}_{n})\\ \vdots & \ddots & \vdots \\ {\gamma}_{lq}({\mathit{s}}_{n},{\mathit{s}}_{1})& \cdots & {\gamma}_{lq}({\mathit{s}}_{n},{\mathit{s}}_{n})\end{array}\right).$

#### 2.2. Cokriging Predictor Using Functional Secondary Variables

**Remark**

**1.**

## 3. Real Data Analysis

#### 3.1. Definition of the Problem upon Study

#### 3.2. Cokriging Prediction of PM10 Using WS Curves

`R`software was used for obtaining the calculations [38] and an

`R`package named

`gstat`[39] was used to estimate the corresponding parameters. The data and

`R`codes used in this empirical application are available at [40]. The estimated range of simple and cross-variograms was 8 km (about one-third of the maximum distance between monitoring sites in Figure 1).

## 4. Concluding Remarks, Discussion, and Future Research

#### 4.1. Conclusions and Discussion

- (i)
- A cokriging predictor considering a functional secondary variable was proposed.
- (ii)
- After smoothing by using basis functions, an ordinary cokriging was defined by means of several secondary variables (as many as basis functions are used for smoothing the data).
- (iii)
- Cokriging was considered to be a better option than kriging, because including one or more secondary variables in the prediction process reduces uncertainty.
- (iv)
- It was showed how to use the proposed methodology when there are many measurements of a secondary variable over time.
- (v)
- An illustration with a real data set was considered to predict PM10 values in Bogotá city by using a cokriging predictor with wind speed as functional secondary variable.

#### 4.2. Further Work

- (i)
- A cokriging predictor with functional variables can be studied upon non-stationarity [21].
- (ii)
- Extensions to the multivariate case is also of practical relevance [41].
- (iii)
- (iv)
- (v)
- The applications of the new methodology proposed in this investigation can be of interest in diverse areas, where the functional data analysis is considered [29].
- (vi)
- (vii)
- Autoregressive model-based fuzzy clustering can be used for detecting information redundancy in air pollution monitoring networks [49].
- (viii)

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Appendix A

#### Appendix A.1. Cokriging System of Equations

#### Appendix A.2. Coefficients of Basis Functions

## References

- Diggle, P.; Ribeiro, P. Model-Based Geoestatistics; Springer: New York, NY, USA, 2007. [Google Scholar]
- Cressie, N. Statistics for Spatial Data; Wiley: New York, NY, USA, 1993. [Google Scholar]
- Ver Hoef, J.; Barry, R. Constructing and fitting models for cokriging and multivariable spatial prediction. J. Stat. Plan. Inference
**1998**, 69, 275–294. [Google Scholar] [CrossRef] - Chiles, J.; Delfiner, P. Geostatistics: Modeling Spatial Uncertainty; Wiley: New York, NY, USA, 1999. [Google Scholar]
- Marchant, C.; Leiva, V.; Cavieres, M.F.; Sanhueza, A. Air contaminant statistical distributions with application to PM10 in Santiago, Chile. Rev. Environ. Contam. Toxicol.
**2013**, 223, 1–31. [Google Scholar] - Cappelli, C.; D’Urso, P.; De Giovanni, L.; Massari, R. Regime change analysis of interval-valued time series with an application to PM10. Chemom. Intell. Lab. Syst.
**2015**, 146, 337–346. [Google Scholar] [CrossRef] - Leiva, V.; Marchant, C.; Ruggeri, F.; Saulo, H. Monitoring urban environmental pollution by bivariate control charts: New methodology and case study in Santiago, Chile. Environmetrics
**2015**, 30, e2551. [Google Scholar] - D’Urso, P.; Cappelli, C.; De Giovanni, L.; Massari, R. Autoregressive metric-based trimmed fuzzy clustering with an application to PM10 time series. Chemom. Intell. Lab. Syst.
**2017**, 161, 15–26. [Google Scholar] [CrossRef] - Marchant, C.; Leiva, V.; Christakos, G.; Cavieres, M.F. A criterion for environmental assessment using Birnbaum-Saunders attribute control charts. Environmetrics
**2019**, 26, 463–476. [Google Scholar] - Cavieres, M.F.; Leiva, V.; Marchant, C.; Rojas, F. A methodology for data-driven decision making in the monitoring of particulate matter environmental contamination in Santiago of Chile. Rev. Environ. Contam. Toxicol.
**2020**. [Google Scholar] [CrossRef] - Leiva, V.; Saulo, H.; Souza, R.; Aykroyd, R.G.; Vila, R. A new BISARMA time series model for forecasting mortality using weather and particulate matter data. J. Forecast.
**2020**. [Google Scholar] [CrossRef] - Le, N.; Zidek, J. Statistical Analysis of Environmental Space-Time Processes; Springer: New York, NY, USA, 2006. [Google Scholar]
- Garcia-Papani, F.; Leiva, V.; Ruggeri, F.; Uribe-Opazo, M.A. Kriging with external drift in a Birnbaum-Saunders geostatistical model. Stoch. Environ. Res. Risk Assess.
**2018**, 32, 1517–1530. [Google Scholar] [CrossRef] - Wackernagel, H. Cokriging versus kriging in regionalized multivariate data analysis. Geoderma
**1994**, 62, 83–92. [Google Scholar] [CrossRef] - Helterbrand, J.; Cressie, N. Universal cokriging under intrinsic coregionalization. Math. Geol.
**1994**, 26, 205–226. [Google Scholar] [CrossRef] - Rivoirard, J. Which models for collocated cokriging? Math. Geol.
**2001**, 33, 117–131. [Google Scholar] [CrossRef] - Pardo-Igúzquiza, E.; Dowd, P. Multiple indicator cokriging with application to optimal sampling for environmental monitoring. Comput. Geosci.
**2005**, 31, 1–13. [Google Scholar] [CrossRef] - Isaaks, E.; Srivastava, M. Applied Geostatistics; Oxford University Press: New York, NY, USA, 1989. [Google Scholar]
- Delicado, P.; Giraldo, R.; Comas, C.; Mateu, J. Geostatistics for spatial functional data: Some recent contributions. Environmetrics
**2010**, 21, 224–239. [Google Scholar] [CrossRef] - Giraldo, R. Geostatistics for Functional Data. Ph.D. Thesis, Universitat Politècnica de Catalunya, Barcelona, Spain, 2009. [Google Scholar]
- Martinez, S.; Giraldo, R.; Leiva, V. Birnbaum-Saunders functional regression models for spatial data. Stoch. Environ. Res. Risk Assess.
**2019**, 33, 1765–1780. [Google Scholar] [CrossRef] - Goulard, M.; Voltz, M. Geostatistical interpolation of curves: A case study in soil science. In Geostatistics Tróia’92; Soares, A., Ed.; Kluwer Academc Press: Dordrecht, The Netherlands, 1993; Volume 2, pp. 805–816. [Google Scholar]
- Nerini, D.; Monestiez, P.; Manté, C. Cokriging for spatial functional data. J. Multivar. Anal.
**2010**, 101, 409–418. [Google Scholar] [CrossRef] - Giraldo, R.; Delicado, P.; Mateu, J. Ordinary kriging for function-valued spatial data. Environ. Ecol. Stat.
**2011**, 18, 411–426. [Google Scholar] [CrossRef] [Green Version] - Menafoglio, A.; Petris, G. Kriging for Hilbert-space valued random fields. The operational point of view. J. Multivar. Anal.
**2016**, 146, 84–94. [Google Scholar] [CrossRef] - Caballero, W.; Giraldo, R.; Mateu, J. A universal kriging approach for spatial functional data. Stoch. Environ. Res. Risk Assess.
**2013**, 27, 1553–1563. [Google Scholar] [CrossRef] - Ignaccolo, R.; Mateu, J.; Giraldo, R. Kriging with external drift for functional data for air quality monitoring. Stoch. Environ. Res. Risk Assess.
**2014**, 28, 1171–1186. [Google Scholar] [CrossRef] [Green Version] - Reyes, A.; Giraldo, R.; Mateu, J. Residual kriging for functional prediction of salinity curves. Commun. Stat. Theory Methods
**2005**, 44, 798–809. [Google Scholar] [CrossRef] [Green Version] - Ramsay, J.; Silverman, B. Functional Data Analysis; Springer: New York, NY, USA, 2005. [Google Scholar]
- Hooyberghsa, J.; Mensinka, C.; Dumontb, G.; Fierensb, F.; Brasseurc, O. A neural network forecast for daily average PM10 concentrations in Belgium. Atmos. Environ.
**2005**, 39, 3279–3289. [Google Scholar] [CrossRef] - Pérez, P.; Reyes, J. Prediction of maximum of 24-h average of PM10 concentrations 30 h in advance in Santiago, Chile. Atmos. Environ.
**2002**, 36, 4555–4561. [Google Scholar] [CrossRef] - Giri, D.; Krishna-Murthy, V.; Adhiraky, P. The influence of meteorological conditions on PM10 concentrations in Kathmandu valley. Int. J. Environ. Res.
**2008**, 2, 49–60. [Google Scholar] - Emery, X. Iterative algorithms for fitting a linear model of coregionalization. Comput. Geosci.
**2010**, 36, 1150–1160. [Google Scholar] [CrossRef] - Giraldo, R. Propuesta de un indicador como variable auxiliar en el análisis cokriging. Rev. Colomb. Estadística
**2001**, 24, 1–12. [Google Scholar] - Rodríguez-Camargo, L.; Sierra-Parada, R.; Blanco-Becerra, L. Análisis espacial de las concentraciones de PM2.5 en Bogotá según los valores de las guías de la calidad del aire de la Organización Mundial de la Salud para enfermedades cardiopulmonares, 2014–2015. Biomédica
**2020**, 40, 137–152. [Google Scholar] [CrossRef] [Green Version] - Myers, D. Matrix formulation of cokriging. Math. Geol.
**1982**, 14, 249–257. [Google Scholar] [CrossRef] [Green Version] - Bogaert, P. Comparison of kriging techniques in a space-time context. Math. Geol.
**1996**, 28, 73–86. [Google Scholar] [CrossRef] - R Core Team. R: A Language and Environment for Statistical Computing; R Foundation for Statistical Computing: Vienna, Austria, 2018. [Google Scholar]
- Pebesma, E. Multivariable geostatistics in S: The gstat package. Comput. Geosci.
**2004**, 30, 683–691. [Google Scholar] [CrossRef] - R Code. Available online: https://sites.google.com/a/unal.edu.co/ramon-giraldo-webpage/r-code (accessed on 20 July 2020).
- Sánchez, L.; Leiva, V.; Galea, M.; Saulo, H. Birnbaum-Saunders quantile regression models with application to spatial data. Mathematics
**2020**, 8, 1000. [Google Scholar] [CrossRef] - Huerta, M.; Leiva, V.; Rodriguez, M.; Villegas, D. On a partial least squares regression model for asymmetric data with a chemical application in mining. Chemom. Intell. Lab. Syst.
**2019**, 190, 55–68. [Google Scholar] [CrossRef] - Saulo, H.; Leão, J.; Leiva, V.; Aykroyd, R.G. Birnbaum-Saunders autoregressive conditional duration models applied to high-frequency financial data. Stat. Pap.
**2019**, 60, 1605–1629. [Google Scholar] [CrossRef] [Green Version] - Carrasco, J.M.F.; Figueroa-Zuniga, J.I.; Leiva, V.; Riquelme, M.; Aykroyd, R.G. An errors-in-variables model based on the Birnbaum-Saunders and its diagnostics with an application to earthquake data. Stoch. Environ. Res. Risk Assess.
**2020**, 34, 369–380. [Google Scholar] [CrossRef] - Sánchez, L.; Leiva, V.; Galea, M.; Saulo, H. Birnbaum-Saunders quantile regression and its diagnostics with application to economic data. Appl. Stoch. Model. Bus. Ind.
**2020**. [Google Scholar] [CrossRef] - Garcia-Papani, F.; Leiva, V.; Uribe-Opazo, M.A.; Aykroyd, R.G. Birnbaum-Saunders spatial regression models: Diagnostics and application to chemical data. Chemom. Intell. Lab. Syst.
**2018**, 177, 114–128. [Google Scholar] [CrossRef] [Green Version] - Liu, Y.; Mao, G.; Leiva, V.; Liu, S.; Tapia, A. Diagnostic analytics for an autoregressive model under the skew-normal distribution. Mathematics
**2020**, 8, 693. [Google Scholar] [CrossRef] - Genton, M.G.; Zhang, H. Identifiability problems in some non-Gaussian spatial random fields. Chilean J. Stat.
**2012**, 3, 171–179. [Google Scholar] - D’Urso, P.; Di Lallo, D.; Maharaj, E.A. Autoregressive model-based fuzzy clustering and its application for detecting information redundancy in air pollution monitoring networks. Soft Comput.
**2013**, 17, 83–131. [Google Scholar] [CrossRef] - D’Urso, P.; De Giovanni, L.; Massari, R. Time series clustering by a robust autoregressive metric with application to air pollution. Chemom. Intell. Lab. Syst.
**2015**, 141, 107–124. [Google Scholar] [CrossRef] - Velasco, H.; Laniado, H.; Toro, M.; Leiva, V.; Lio, Y. Robust three-step regression based on comedian and its performance in cell-wise and case-wise outliers. Mathematics
**2020**, 8, 1259. [Google Scholar] [CrossRef]

**Figure 1.**Spatial location of ten air quality monitoring stations in the Bogotá area. Circles are proportional to maximum PM10 values (maximum at each station was calculated based on data that were collected hourly between 26 January 2011 at 12:00 p.m. to 3 February 2011 at 2:00 p.m.).

**Figure 2.**WS values (in m/s) of ten air quality monitoring stations in the Bogotá area (data at each monitoring station were collected each two hours between 26 January 2011 at 10:00 a.m. to 3 February 2011 at 12:00 p.m.). In total, WS data were collected in 98 time periods (time period 1 corresponds to 10:00 a.m. of 26 January 2011 and time period 98 to 12:00 p.m. 3 February 2011).

**Figure 3.**WS curves obtained by smoothing the data set of each station using a Fourier basis (of dimension $k=7$). Time period 1 corresponds to 10:00 a.m. 26 January 2011 and time period 98 to 12:00 p.m. 3 February 2011.

**Figure 4.**Map of PM10 predictions in the Bogotá area obtained with cokriging and a secondary variable corresponding to WS curves.

**Figure 5.**Map of PM10 prediction variances with the highest magnitudes (dark grey) corresponding to zones distant from the sampling sites.

**Table 1.**PM10 values and coefficients ${a}_{ij}$, for $i=1,\dots ,n$ and $j=1,\dots ,7$, of Fourier basis functions fitted to SW data (collected each two hours between 26 January 2011 and 3 February 2011 at each of ten environmental monitoring stations in Bogotá, Colombia).

Station ID | Monitoring Station | PM10 | ${\mathit{a}}_{\mathit{i}1}$ | ${\mathit{a}}_{\mathit{i}2}$ | ${\mathit{a}}_{\mathit{i}3}$ | ${\mathit{a}}_{\mathit{i}4}$ | ${\mathit{a}}_{\mathit{i}5}$ | ${\mathit{a}}_{\mathit{i}6}$ | ${\mathit{a}}_{\mathit{i}7}$ |
---|---|---|---|---|---|---|---|---|---|

1 | Carvajal | 242 | 4.35 | 2.31 | 1.87 | 1.40 | –0.40 | –0.10 | –0.67 |

2 | Fontibon | 174 | 11.24 | 3.97 | 2.12 | 2.17 | –1.05 | –0.37 | –0.83 |

3 | Guaymaral | 179 | 3.49 | 1.67 | 1.26 | 0.85 | –0.48 | –0.51 | –0.50 |

4 | Kennedy | 265 | 9.76 | 3.52 | 1.72 | 1.54 | –0.55 | –0.05 | –0.60 |

5 | Las Ferias | 116 | 6.90 | 2.17 | 1.37 | 0.91 | –0.69 | –0.26 | –0.56 |

6 | Simón Bolivar | 130 | 5.77 | 2.13 | 1.52 | 0.88 | –0.27 | –0.48 | –0.24 |

7 | Puente Aranda | 237 | 9.80 | 3.00 | 1.49 | 1.77 | –0.37 | –0.48 | –0.63 |

8 | Suba | 157 | 6.41 | 0.76 | 1.63 | 0.51 | –0.13 | –0.83 | –0.23 |

9 | Tunal | 162 | 4.64 | 1.94 | 1.27 | 1.07 | –0.33 | –0.01 | –0.35 |

10 | Usaquen | 118 | 5.39 | 1.12 | -0.22 | 0.19 | –0.07 | –0.17 | –0.59 |

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**MDPI and ACS Style**

Giraldo, R.; Herrera, L.; Leiva, V.
Cokriging Prediction Using as Secondary Variable a Functional Random Field with Application in Environmental Pollution. *Mathematics* **2020**, *8*, 1305.
https://doi.org/10.3390/math8081305

**AMA Style**

Giraldo R, Herrera L, Leiva V.
Cokriging Prediction Using as Secondary Variable a Functional Random Field with Application in Environmental Pollution. *Mathematics*. 2020; 8(8):1305.
https://doi.org/10.3390/math8081305

**Chicago/Turabian Style**

Giraldo, Ramón, Luis Herrera, and Víctor Leiva.
2020. "Cokriging Prediction Using as Secondary Variable a Functional Random Field with Application in Environmental Pollution" *Mathematics* 8, no. 8: 1305.
https://doi.org/10.3390/math8081305