Abstract
Applications of statistical models to describe spatial dependence in geo-referenced data are widespread across many disciplines including the environmental sciences. Most of these applications assume that the data follow a Gaussian distribution. However, in many of them the normality assumption, and even a more general assumption of symmetry, are not appropriate. In non-spatial applications, where the data are uni-modal and positively skewed, the Birnbaum–Saunders (BS) distribution has excelled. This paper proposes a spatial log-linear model based on the BS distribution. Model parameters are estimated using the maximum likelihood method. Local influence diagnostics are derived to assess the sensitivity of the estimators to perturbations in the response variable. As illustration, the proposed model and its diagnostics are used to analyse a real-world agricultural data set, where the spatial variability of phosphorus concentration in the soil is considered—which is extremely important for agricultural management.
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References
Anderson J, Hardy E, Roach J, Witmer R (1976). A land use and land cover classification system for use with remote sensor data. Technical Report Paper 964. US Geological Survey Professional, Washington, DC
Assumpção R, Uribe-Opazo M, Galea M (2011) Local influence for spatial analysis of soil physical properties and soybean yield using Student-t distribution. Rev Bras Ciênc Solo 35:1917–1926
Assumpção R, Uribe-Opazo M, Galea M (2014) Analysis of local influence in geostatistics using Student-t distribution. J Appl Stat 41:2323–2341
Azzalini A, Capitanio A (1999) Statistical applications of the multivariate skew-normal distribution. J R Stat Soc B 61:579–602
Birnbaum Z, Saunders S (1969) A new family of life distributions. J Appl Probab 6:319–327
Borssoi J, De Bastiani F, Uribe-Opazo M, Galea M (2011) Local influence of explanatory variables in Gaussian spatial linear models. Chil J Stat 2:29–38
Cambardella C, Moorman T, Novak J, Parkin T, Karlen D, Turco R, Konopka A (1994) Field-scale variability of soil properties in central Iowa soils. Soil Sci Soc Am J 58:1501–1511
COAMO/COODETEC (2001) Soil fertility and plant nutrition. Technical report. Cooperativa Agropecuaria Mouraoense Ltda./Development Center Technological and Economic Cooperative Ltda. (COAMO/COODETEC), Cascavel, Brazil
Cook R (1987) Influence assessment. J Appl Stat 14:117–131
Davis D (1952) An analysis of some failure data. J Am Stat Assoc 47:113–150
De Bastiani F, Cysneiros A, Uribe-Opazo M, Galea M (2015) Influence diagnostics in elliptical spatial linear models. TEST 24:322–340
Diggle P, Ribeiro P (2007) Model-based geostatistics. Springer, New York
EMBRAPA (2009) Brazilian system of soil classification. Technical report. Brazilian Enterprise for National Agricultural Research/Centre of Soil Research (EMBRAPA/CPI), Rio de Janeiro, Brazil
Ferreira M, Gomes M, Leiva V (2012) On an extreme value version of the Birnbaum–Saunders distribution. Revstat Stat J 10:181–210
Galea M, Leiva V, Paula G (2004) Influence diagnostics in log-Birnbaum–Saunders regression models. J Appl Stat 31:1049–1064
Gimenez P, Galea M (2013) Influence measures on corrected score estimators in functional heteroscedastic measurement error models. J Multivar Anal 114:1–15
Grzegozewski D, Uribe-Opazo M, De Bastiani F, Galea M (2013) Local influence when fitting Gaussian spatial linear models: an agriculture application. Ciênc Investig Agrár 40:235–252
Isaaks E, Srivastava R (1989) An introduction to applied geostatistics. Oxford University Press, Oxford
Johnson N, Kotz S, Balakrishnan N (1994) Continuous univariate distributions, vol 1. Wiley, New York
Johnson N, Kotz S, Balakrishnan N (1995) Continuous univariate distributions, vol 2. Wiley, New York
Kendrick D (2002) Stochastic control for economic models. McGraw Hill, New York
Krige D (1951) A statistical approach to some basic mine valuation problems on the Witwatersrand. J Chem Metall Min Soc S Afr 52:119–139
Krippendorff K (2004) Content analysis: an introduction to its methodology. Sage, Thousand Oaks
Lange K (2001) Numerical analysis for statisticians. Springer, New York
Lange K, Little J, Taylor M (1989) Robust statistical modeling using the \(t\) distribution. J Am Stat Assoc 84:881–896
Leiva V, Barros M, Paula G, Sanhueza A (2008) Generalized Birnbaum–Saunders distribution applied to air pollutant concentration. Environmetrics 19:235–249
Leiva V, Sanhueza A, Angulo JM (2009) A length-biased version of the Birnbaum–Saunders distribution with application in water quality. Stoch Environ Res Risk Assess 23:299–307
Leiva V, Rojas E, Galea M, Sanhueza A (2014) Diagnostics in Birnbaum–Saunders accelerated life models with an application to fatigue data. Appl Stoch Models Bus Ind 30:115–131
Leiva V, Marchant C, Ruggeri F, Saulo H (2015a) A criterion for environmental assessment using Birnbaum–Saunders attribute control charts. Environmetrics 26:463–476
Leiva V, Tejo M, Guiraud P, Schmachtenberg O, Orio P, Marmolejo F (2015b) Modeling neural activity with cumulative damage distributions. Biol Cybern 109:421–433
Leiva V, Ferreira M, Gomes M, Lillo C (2016a) Extreme value Birnbaum–Saunders regression models applied to environmental data. Stoch Environ Res Risk Assess. doi:10.1007/s00477-015-1069-6
Leiva V, Liu S, Shi L, Cysneiros F (2016b) Diagnostics in elliptical regression models with stochastic restrictions applied to econometrics. J Appl Stat 43:627–642
Leiva V, Santos-Neto M, Cysneiros F, Barros M (2016c) A methodology for stochastic inventory models based on a zero-adjusted Birnbaum–Saunders distribution. Appl Stoch Models Bus Ind 32:74–89
Liu S, Leiva V, Ma T, Welsh A (2016) Influence diagnostic analysis in the possibly heteroskedastic linear model with exact restrictions. Stat Methods Appl. doi:10.1007/s10260-015-0329-4
Marchant C, Leiva V, Cavieres M, Sanhueza A (2013) Air contaminant statistical distributions with application to PM10 in Santiago, Chile. Rev Environ Contam Toxicol 223:1–31
Marchant C, Leiva V, Cysneiros F (2016) A multivariate log-linear model for Birnbaum–Saunders distributions. IEEE Trans Reliab. doi:10.1109/TR.2015.2499964
Mardia K, Marshall R (1984) Maximum likelihood estimation of models for residual covariance in spatial regression. Biometrika 71:135–146
Militino A, Palacius M, Ugarte M (2006) Outliers detection in multivariate spatial linear models. J Stat Plan Inference 136:125–146
Muirhead R (1982) Aspects of multivariate statistical theory. Wiley, New York
Müller W, Stehlík M (2009) Issues in the optimal design of computer simulation experiments. Appl Stoch Models Bus Ind 25:163–177
Müller W, Stehlík M (2010) Compound optimal spatial designs. Environmetrics 21:354–364
Nocedal J, Wright S (1999) Numerical optimization. Springer, New York
Ortega E, Bolfarine H, Paula G (2003) Influence diagnostics in generalized log-gamma regression models. Comput Stat Data Anal 42:165–186
Podlaski R (2008) Characterization of diameter distribution data in near-natural forests using the Birnbaum–Saunders distribution. Can J For Res 18:518–527
R-Team (2015) R: a language and environment for statistical computing. R Foundation for Statistical Computing, Vienna
Rieck J, Nedelman J (1991) A log-linear model for the Birnbaum–Saunders distribution. Technometrics 3:51–60
Saulo H, Leiva V, Ziegelmann F, Marchant C (2013) A nonparametric method for estimating asymmetric densities based on skewed Birnbaum–Saunders distributions applied to environmental data. Stoch Environ Res Risk Assess 27:1479–1491
Uribe-Opazo M, Borssoi J, Galea M (2012) Influence diagnostics in Gaussian spatial linear models. J Appl Stat 39:615–630
Vilca F, Sanhueza A, Leiva V, Christakos G (2010) An extended Birnbaum–Saunders model and its application in the study of environmental quality in Santiago, Chile. Stoch Environ Res Risk Assess 24:771–782
Villegas C, Paula G, Leiva V (2011) Birnbaum–Saunders mixed models for censored reliability data analysis. IEEE Trans Reliab 60:748–758
Waller L, Gotway C (2004) Applied spatial statistics for public health data. Wiley, Hoboken
Xia J, Zeephongsekul P, Packer D (2011) Spatial and temporal modelling of tourist movements using semi-Markov processes. Tour Manag 51:844–851
Zhu H, Ibrahim J, Lee S, Zhang H (2007) Perturbation selection and influence measures in local influence analysis. Ann Stat 35:2565–2588
Zhu H, Lee S (2001) Local influence for incomplete-data models. J R Stat Soc B 63:111–126
Acknowledgments
The authors thank the Editors and anonymous referees for their constructive comments on an earlier version of the manuscript, which resulted in this improved version. We are grateful to Carolina Brianezi-Melchior, who translated this work into English, from its original Portuguese. This research work was partially supported by CNPq Grants from the Brazilian Government, and by FONDECYT 1120879 Grant from the Chilean Government.
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Appendices
Appendix 1: The score vector
For the BS log-linear spatial model, the score vector is defined by
Using the log-likelihood function defined in (20), we have
where \({\varvec{W}}=(W_{1},\ldots ,W_{n})^{\top},\) with \(W_{i}=\cosh ({(y_{i}-\mu )}/{2}),\) and \({\varvec{V}}=(V_{1},\ldots ,V_{n})^{\top}\) such as given in (19) with \(V_{i} =\sinh({(y_{i}-\mu )}/{2}),\) for \(i=1,\ldots ,n.\) Thus, considering that \({\varvec{\Sigma} }=\varphi _{1}{\varvec{I}}_{n}+\varphi _{2}{\varvec{R}},\) we have
Therefore,
Considering the Matérn model to describe the spatial variability given in (11), we have that \({\partial {\varvec{R}}}/{\partial \varphi _{3}}=({\partial r_{ij}}/{\partial \varphi _{3}}),\) where, for \(K^{\prime}_{\delta }(u)={\partial K_{\delta} (u)}/{\partial u}=- ({1}/{2})(K_{\delta -1}(u)+ K_{\delta +1}(u)),\)
Appendix 2: The observed information matrix
The observed Fisher information matrix for the BS log-linear spatial model is defined by \(-{\ddot{\varvec{\ell }}}({\varvec{\theta} })\) evaluated at \({\varvec{\theta} }=\widehat{\varvec{\theta }},\) where \({\ddot{\varvec{\ell }}}({\varvec{\theta} })\) is the Hessian matrix given by
with, for \({\ddot{\ell }}_{\alpha \mu }={\ddot{\ell} }_{\mu \alpha },\)
Note that \({\ddot{\ell} }_{\varphi \alpha }={\ddot{\ell} }_{\alpha \varphi} ^{\top}\) is a \(3\times 1\) vector with elements given by
where \({\partial {\varvec{\Sigma} }}/{\partial \varphi _{i}}\) is given in Appendix 1. Then, with \({\ddot{\ell} }_{{\alpha }{\varphi }_{1}}={\ddot{\ell} }_{{\varphi }_{1}{\alpha }},\) \({\ddot{\ell} }_{{\alpha }{\varphi }_{2}}={\ddot{\ell} }_{{\varphi }_{2}{\alpha }}\) and \({\ddot{\ell} }_{{\alpha }{\varphi }_{3}}={\ddot{\ell} }_{{\varphi }_{3}{\alpha }},\) we have
Furthermore, \({\ddot{\ell} }_{\varphi \mu }={\ddot{\ell} }_{\mu \varphi} ^{\top}\) is a \(3\times 1\) vector with elements
Then, with \({\ddot{\ell} }_{{\mu }{\varphi }_{1}}={\ddot{\ell} }_{{\varphi }_{1}{\mu }},\, {\ddot{\ell} }_{{\mu }{\varphi }_{2}}={\ddot{\ell} }_{{\varphi }_{2}{\mu }}\) and \({\ddot{\ell} }_{{\mu }{\varphi }_{3}}={\ddot{\ell} }_{{\varphi }_{3}{\mu }},\) we have
Moreover, \({\varvec{\ddot{\ell }}}_{{\varvec{\varphi} }{\varvec{\varphi }}} =({\ddot{\ell }}_{\varphi _{j}\varphi _{k}})\) is a \(3\times 3\) symmetric matrix with elements given by
Then, with \({\ddot{\ell} }_{{\varphi }_{1}{\varphi }_{2}}={\ddot{\ell} }_{{\varphi }_{2}{\varphi }_{1}},\,{\ddot{\ell} }_{{\varphi }_{1}{\varphi }_{3}}={\ddot{\ell} }_{{\varphi }_{3}{\varphi }_{1}}\) and \({\ddot{\ell} }_{{\varphi }_{2}{\varphi }_{3}}={\ddot{\ell} }_{{\varphi }_{3}{\varphi }_{2}},\) we have
Appendix 3: Expected information matrix
The expected Fisher information matrix is \({\varvec{I}}({\varvec{\theta} }) = {\text {E}}(-{\varvec{\ddot{\ell} }}({\varvec{\theta} })).\) For the BS log-linear spatial model, this matrix is given by
Since the model error \({\varvec{\varepsilon} }\sim {\text {log}}{\text{-BS}}_{n}(\alpha {\mathbf{1}},\,{\mathbf{0}},\,{\varvec{\Sigma} }),\) we have that \(({2}/{\alpha }){\varvec{V}}\sim {\text {N}}_{n}({\mathbf{0}},\,{\varvec{\Sigma} }),\) with \({\varvec{V}}\) given in (19). Then (see Muirhead 1982), \(W = ({2}/{\alpha }){\varvec{V}}^{\top} {\varvec{\Sigma} }^{-1}({2}/{\alpha }){\varvec{V}}\sim \chi _{n}^{2},\) where \(\chi _{n}^{2}\) denotes the chi-squared distribution with n degrees of freedom, and then, \({\text {E}}(W)=n.\) Thus,
Moreover, by using \({\text {E}}\left( {\varvec{X}}^{\top} {\varvec{A}}{\varvec{X}}\right) =( {\text {E}}( {\varvec{X}})) ^{\top} {\varvec{A}}( {\text {E}}( {\varvec{X}})) +{\text {tr}}( {\varvec{A}}{\varvec{C}}),\) where \({\varvec{C}}\) is the covariance matrix of \({\varvec{X}}\) (see Kendrick 2002), \({\varvec{I}}_{\varvec{\varphi }{\varvec{\varphi} }} = (I_{\varphi _{j} \varphi _{k}})\) is a symmetric \(3\times 3\) matrix with elements
In addition, \({\varvec{I}}_{\varvec{\varphi }\alpha } = {\varvec{I}}_{\alpha {\varvec{\varphi} }}^{\top} = (I_{\varphi _{1} \alpha },\, I_{\varphi _{2} \alpha },\, I_{\varphi _{3} \alpha })^{\top}\) is a \(3\times 1\) vector with elements given by
where \({\text {tr}}({\varvec{A}})\) denotes the trace of the matrix \({\varvec{A}}.\) To obtain the elements \(I_{\alpha \mu },\, I_{\mu \mu }\) and \(I_{\mu {\varvec{\varphi} }},\) for \({(y_{i}-\mu )}/{2}\) expected to be small enough with \(\cosh (\cdot )\approx 1\), using expansion in Taylor series for \(\cosh (\cdot )\) and \(\alpha\) to be small enough, we have
Furthermore, \({\varvec{I}}_{\varvec{\varphi }\mu } = {\varvec{I}}_{\mu {\varvec{\varphi} }}^{\top} = (I_{\varphi _{1}\mu },\, I_{\varphi _{2}\mu },\, I_{\varphi _{3}\mu })^{\top}\) is a \(3\times 1\) vector with elements given by
Appendix 4: Score vector \({\varvec{U}}({\varvec{\omega} })\) and matrix \({\varvec{G}}({\varvec{\omega} })\)
Score vector used in the local influence method is given by \({\varvec{U}}({\varvec{\omega} })=\left( {\partial {\varvec{V}}_{\varvec{\omega }}}/{\partial {\varvec{\omega} }^{\top} }\right) ^{\top} \left( 2{\varvec{\Sigma} }^{-1}{\varvec{V}_{\omega }}\right) +({1}/{2}){\varvec{A}}{\varvec{T}_{\omega }},\)where \({\varvec{T}_{\omega }}=({\varvec{T}}_{\omega _{1}},\ldots ,{\varvec{T}}_{\omega _{n}}),\) with \({\varvec{T}}_{\omega _{i}}=\tanh({(y_{i}+{\varvec{A}}_{i}{\varvec{\omega} }-\mu )}/{2}),\) and \({\varvec{A}}_{i}\) is the ith row of the matrix \({\varvec{A}}.\) For \(\cosh ({(y_{i}+{\varvec{a}}_{i}{\varvec{\omega} }-\mu )}/{2})\approx 1,\) with \(i=1,\ldots ,n,\) we get
where \(\sigma _{ij}\) is the (i, j) element of the matrix \({\varvec{A}},\) for \(i,\,j=1,\ldots ,n.\) Furthermore, for \(\cosh ({(y_{i}+{\varvec{a}}_{i}{\varvec{\omega} }-\mu )}/{2})\approx 1,\) with \(i=1,\ldots ,n,\) \({\varvec{T}_{\omega }}\) can be approximated by \({\varvec{V}_{\omega }},\) from which it follows that \({\varvec{U}}({\varvec{\omega} })=({1}/{2}){\varvec{A}}{\varvec{V}_{\omega }}-({2}/{\alpha ^{2}}){\varvec{A}}{\varvec{\Sigma} }^{-1}{\varvec{V}_{\omega }}.\) Thus,
Therefore, \({\varvec{G}}({\varvec{\omega} })={\text {E}}\left( {\varvec{U}}({\varvec{\omega} }){\varvec{U}}^{\top} ({\varvec{\omega} })\right) = {\varvec{A}}\left( \frac{\alpha }{4}{\varvec{\Sigma} }^{{\frac{1}{2}}}-{\frac{1}{\alpha}}{\varvec{\Sigma} }^{{-\frac{1}{2}}}\right) ^{2}{\varvec{A}}.\) To find the appropriate perturbation, according to the methodology proposed by Zhu et al. (2007), it is necessary to find \({\varvec{A}},\) such that \({\varvec{G}}({\varvec{\omega} })=c{\varvec{I}}_{n},\) for \(c>0.\) Considering \(c=1,\) then \({\varvec{A}}\) must satisfy \((({\alpha }/{4}){\varvec{\Sigma} }^{({1}/{2})}-({1}/{\alpha }){\varvec{\Sigma} }^{-\frac{1}{2}})^2=({\varvec{A}}^{-1})^2.\) A solution of the equation above is given by \({\varvec{A}}=(({\alpha }/{4}){\varvec{\Sigma} }^{\frac{1}{2}}-({1}/{\alpha }){\varvec{\Sigma} }^{-\frac{1}{2}})^{-1}.\) Then, \(\tilde{\varvec{\omega }}=(({\alpha }/{4}){\varvec{\Sigma} }^{\frac{1}{2}}-({1}/{\alpha }){\varvec{\Sigma} }^{-\frac{1}{2}})^{-1}{\varvec{\omega }}\) is an appropriate perturbation for the BS log-linear spatial model.
Appendix 5: The perturbation matrix
The perturbation matrix for the BS log-linear spatial model obtained from (20) is given by
where \({\varvec{U}_{\omega }}=({\varvec{U}}_{\omega _{1}},\ldots ,{\varvec{U}}_{\varvec \omega _{n}}),\) with \({\varvec{U}}_{\omega _{i}}=\cosh (({y_{i}+{\varvec{a}_{i}{\varvec{\omega}} }-\mu )}/{2}),\) for \(i=1,\ldots ,n.\) Thus, the results presented in (32), (33) and (34) are obtained, for \(\cosh ({(y_{i}+{\varvec{a}_{i}\varvec\omega }-\mu )}/{2})\approx 1,\) as
where \({\varvec{D}}=( D_{1},\ldots ,D_{n}) ^{\top},\) with \(D_{i}={\varvec{l}}_{i}\varvec\omega\) and \({\varvec{l}}_{i}\) being the ith row of the matrix
In addition,
where \({\varvec{M}}=( m_{1},\ldots ,m_{n}) ^{\top},\) with \(m_{i}={\varvec{l}}_{i}{\varvec{\omega} }\) and \({\varvec{l}}_{i}\) being the ith row of matrix
Details about \({\partial {\varvec{\Sigma} }^{\frac{1}{2}}}/{\partial \varphi _{j}}\) can be found in De Bastiani et al. (2015).
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Garcia-Papani, F., Uribe-Opazo, M.A., Leiva, V. et al. Birnbaum–Saunders spatial modelling and diagnostics applied to agricultural engineering data. Stoch Environ Res Risk Assess 31, 105–124 (2017). https://doi.org/10.1007/s00477-015-1204-4
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DOI: https://doi.org/10.1007/s00477-015-1204-4