Bayesian residual analysis for spatially correlated data.

*(English)*Zbl 07290031Summary: This work considers residual analysis and predictive techniques for the identification of individual and multiple outliers in geostatistical data. The standardized Bayesian spatial residual is proposed and computed for three competing models: the Gaussian, Student-t and Gaussian-log-Gaussian spatial processes. In this context, the spatial models are investigated regarding their plausibility for datasets contaminated with outliers. The posterior probability of an outlying observation is computed based on the standardized residuals and different thresholds for outlier discrimination are tested. From a predictive point of view, methods such as the conditional predictive ordinate, the predictive concordance and the Savage-Dickey density ratio for hypothesis testing are investigated for identification of outliers in the spatial setting. For illustration, contaminated datasets are considered to assess the performance of the three spatial models for identification of outliers in spatial data. Furthermore, an application to wind speed modelling is presented to illustrate the usefulness of the proposed tools to detect regions with large wind speeds.

##### MSC:

62 | Statistics |

##### Keywords:

residual analysis; spatial statistics; outlier detection; predictive performance; Bayesian inference; non-Gaussian process##### Software:

CODA
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\textit{V. G. R. Lobo} and \textit{T. C. O. Fonseca}, Stat. Model. 20, No. 2, 171--194 (2020; Zbl 07290031)

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##### References:

[1] | Albert, J, Chib, S (1995) Bayesian residual analysis for binary response regression models. Biometrika, 82, 747-59. · Zbl 0861.62022 |

[2] | Alqallaf, F, Gustafson, P (2001) On cross-validation of Bayesian models. The Canadian Journal of Statistics, 29, 333-40. · Zbl 0974.62019 |

[3] | Bastos, LS, O’Hagan, A (2008) Diagnostics for Gaussian process emulators. Technometrics, 51, 425-38. |

[4] | Breusch, TS, Robertson, JC, Welsh, AH (1997) The emperor’s new clothes: A critique of the multivariate t regression model. Statistica Neerlandica, 51, 269-86. · Zbl 0929.62062 |

[5] | Chaloner, K, Brant, R (1988) A Bayesian approach to outlier detection and residual analysis. Biometrika, 75, 651-59. · Zbl 0659.62037 |

[6] | Chilès, J-P, Delfiner, P (1999) Modeling Spatial Uncertainty. New York, NY: Wiley. |

[7] | Dickey, J (1971) The weighted likelihood ratio, linear hypotheses on normal location parameters. The Annals of Statistics, 42, 204-23. · Zbl 0274.62020 |

[8] | Escalante, CS (2007) Bivariate estimation of extreme wind speeds. Structural Safety, 30, 481-92. |

[9] | Fonseca, TCO, Ferreira, MAR, Migon, HS (2008) Objective Bayesian analysis for the Student-t regression model. Biometrika, 95, 325-33. · Zbl 1400.62260 |

[10] | Fonseca, TCO, Steel, MFJ (2011) Non-Gaussian spatiotemporal modelling through scale mixing. Biometrika, 98, 761-74. · Zbl 1228.62117 |

[11] | Fournier, B, Furrer, R (2005) Automatic mapping in the presence of substitutive errors: A robust kriging approach. Applied GIS, 1, 12-1-12-16. |

[12] | Fraccaro, R, Hyndman, RJ, Veevers, A (2000) Residual diagnostic plots for checking for model mis-specification in time series regression. Australian & New Zealand Journal of Statistics, 42, 463-477. · Zbl 0992.62081 |

[13] | Freeman, PR (1980) On the Number of Outliers in Data from a Linear Model, pp. 349-65. Valencia: University Press. |

[14] | Gamerman, D, Lopes, H (2006) Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference. Texts in Statistical Science. Boca Raton, FL: Taylor & Francis. · Zbl 1137.62011 |

[15] | Gelfand, A (1996) Model Determination Using Samplings Based Methods. Boca Raton, FL: Chapman & Hall. |

[16] | Hasllet, J (1999) A simple derivation of deletion diagnostic results for the general linear model with correlated errors. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 61, 603-09. · Zbl 0924.62076 |

[17] | Houseman, EA, Ryan, L, Coull, B (2004) Cholesky residuals for assessing normal errors in a linear model with correlated outcomes. Journal of the American Statistical Association, 99, 383-94. · Zbl 1117.62356 |

[18] | Juna, KM, Katzfuss, M, Hub, J, Johnson, VE (2014) Assessing fit in Bayesian models for spatial processes. Environmetrics, 25, 584-95. URL http://dx.doi.org/[10.1002/env.2315] (last accessed 12 December 2018). |

[19] | Kass, R, Raftery, AE (1995) Bayes factor. Journal of the American Statistical Association, 90, 773-95. · Zbl 0846.62028 |

[20] | Klein, N, Klein, T, Lang, S, Sohn, A (2015) Bayesian structured additive distributional regression with an application to regional income inequality in Germany. The Annals of Applied Statistics, 2, 1024-52. · Zbl 1454.62485 |

[21] | Marshall, EC, Spiegelhalter, DJ (2003) Approximate cross-validatory predictive checks in disease mapping models. Statistics in Medicine, 22, 1649-60. |

[22] | Montgomery Montgomery, DC, Peck, EA, Vining, GG (2006) Introduction to Linear Regression Analysis, 4th edition. Hoboken, NJ: Wiley & Sons. ISBN 0471754951. |

[23] | Palacios, MB, Steel, MFJ (2006) Non-Gaussian Bayesian geostatistical modeling. Journal of the American Statistical Association, 101, 604-18. · Zbl 1119.62321 |

[24] | Petit, L (1990) The conditional predictive ordinate for the normal distribution. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 52, 175-84. |

[25] | Plummer, M, Best, N, Cowles, K, Vines, K (2006) Coda: Convergence diagnosis and output analysis for MCMC. R News, 6, 7-11. URL https://journal.r-project.org/ [archive/] (last accessed 12 December 2018) |

[26] | Reich, BJ, Fuentes, M, Dunson, DB (2011) Bayesian spatial quantile regression. Journal of the American Statistical Association, 106, 6-20. · Zbl 1396.62263 |

[27] | Robert, CP (2007) The Bayesian Choice, 2nd edition. New York, NY: Springer. |

[28] | Roislien, J, Omre, H (2006) T-distributed random fields: A parametric model for heavy-tailed well-log data. Mathematical Geology, 38, 821-49. |

[29] | Sarkar, A, Singh, S, Mitra, D (2011) Wind climate modeling using Weibull and extreme value distribution. International Journal of Engineering, Science and Technology, 3, 100-106. |

[30] | Souza, ADP, Migon, HS (2010) Bayesian outlier analysis in binary regression. Journal of Applied Statistics, 37, 1355-68. · Zbl 07252511 |

[31] | Stern, HS, Cressie, N (2000) Posterior predictive model checks for disease mapping models. Statistics in Medicine, 19, 2377-97. |

[32] | Vehtari, A, Lampinen, J (2002) Bayesian model assessment and comparison using cross-validation predictive densities. Neural Computation, 14, 2439-68. · Zbl 1002.62029 |

[33] | Vehtari, A, Gelman, A, Gabry, J (2017) Practical Bayesian model evaluation using leave-one-out cross-validation and WAIC. Statistics and Computing, 27, 1413-32. · Zbl 06737719 |

[34] | West, M (1984) Outlier models and prior distributions in Bayesian linear regression. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 48, 431-39. · Zbl 0567.62022 |

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