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## Hierarchical Modelling and Analysis for Spatial Data. Chapman and Hall/CRC, (2004)

Citations: | 442 - 45 self |

### Citations

2139 | Statistics for Spatial Data - Cressie - 1991 |

528 |
Interpolation of Spatial Data: some theory for kriging
- Stein
- 1999
(Show Context)
Citation Context ... for geostatistical models, it can be more sensitive than ML to the chosen model for the drift (see, for example, Diggle and Ribeiro Jr., 2007, p.117). Also it can have higher MSE. 7 General Issues with MLE We only consider estimation of θ since the drift estimation is generally not a challenge. Multimodality. The likelihood could have several modes for θ depending on the scheme but there seems to be always a global maximum. There has been a very close scrutiny of the behaviour of the MLE for the spatial linear model (see Christensen, 2004; Mardia and Watkins, 1989; Ripley, 1988; Smith, 2000; Stein, 1999; Warnes and Ripley, 1987). For the multimodality, there could be several reasons, for examples, the likelihood maybe flat, e.g. due to a wrong model; the scheme could be non-differentiable, e.g.,the spherical scheme with respect to the range parameter; the nugget parameter can be on the boundary; the numerical method may not give enough accuracy. There are some effective solutions. It is found that a modest increase in additional number of the model parameters can resolve some of the issues. But it is always wise to check various points: the underlying geometry, long range correlations versus... |

417 |
Directional Statistics.
- Mardia, Jupp
- 2000
(Show Context)
Citation Context ...or non-Gaussian cases the use of the Box and Cox transformation (for example, Mardia and Goodall, 1993), link between kriging and thin plate spline (for example, Kent and Mardia, 1984), use of spectral representation for covariances (for example, Kent and Mardia, 1996), kriging with derivative (for example, Mardia et al., 1996), CAR and SAR models (for example, Mardia, 1990). There are other estimation methods , e.g. minimum norm quadratic estimators (for example, Mardia and Marshall, 1985). Directional statistics is another area of great importance (for examples, Mardia, 1981a,b, 1989, 2002; Jupp and Mardia, 2000). Incidentally ,the directional statistics paper ,Mardia (2002), was presented as a plenary talk to the IAMG Conference in Berlin! We have not covered here various other key topics. Bayesian methods in geostatistics are becoming effective (see, for example, Banerjee et al., 2004; Diggle and Ribeiro Jr., 2007). Another area is of spatial temporal modelling through Kriged Kalman filter with kriging providing principal fields (Mardia et al., 1998; Sahu and Mardia, 2005). Beyond the spatial linear models, there are many other type of statistical models ,eg. for rock fractures (Mardia et al., 2007a... |

306 |
Maximum likelihood approaches to variance component estimation and to related problems.
- Harville
- 1977
(Show Context)
Citation Context ...on, so that the pdf is given by |2πPΣP |− 1 2 exp[− 1 2 XT (PΣP )−X]. (17) In the second approach, we chose a basis and the work on co-ordinates. Thus we have various specific choices. We can work on X∗2 defined by X ∗ 2 = A2X where A T 2 F = 0. Now we can write fully the distribution of X∗2 where Cov(X ∗ 2 ) is of the full rank. Also A = (A1, A2) is non-singular. In this transformation, it is clear that X∗1 does not give any information on θ. Whether the full likelihood is better or the REML, it depends except for a singular Σ (the intrinsic case) the choice is clear - you have to have REML! Harville (1977) simplified (17) when Σ−1 exists. Let X∗1 = A T 1 X, X ∗ 2 = A T 2X ; A1 = F (F TF )− 1 2 , AT2 F = 0 (18) so that A1 is an orthogonal basis of F , and the columns of A2 are orthonormal to F . Thus A = (A1, A2), Σ ∗ = AT ΣA. The REML principle says use the resulting likelihood which from (17) is given by |Σ∗22| − 1 2 exp{− 1 2 X∗T2 (Σ ∗ 22) −1X∗2} and if Σ−1 exists then Harville (1977) has shown that |Σ∗22 |= |Σ ||F T Σ−1F |/|FTF |, X∗T2 Σ ∗−1 22 X ∗ 2 = (X − F β) T Σ−1(X − F β). Hence we are back to the original quantities X, β,Σ, etc. in the likelihood. Although REML is widely recommended ... |

226 | Model-based Geostatistics - Diggle, Ribeiro - 2007 |

195 |
Statistical inference for spatial processes.
- Ripley
- 1988
(Show Context)
Citation Context ... REML is widely recommended for geostatistical models, it can be more sensitive than ML to the chosen model for the drift (see, for example, Diggle and Ribeiro Jr., 2007, p.117). Also it can have higher MSE. 7 General Issues with MLE We only consider estimation of θ since the drift estimation is generally not a challenge. Multimodality. The likelihood could have several modes for θ depending on the scheme but there seems to be always a global maximum. There has been a very close scrutiny of the behaviour of the MLE for the spatial linear model (see Christensen, 2004; Mardia and Watkins, 1989; Ripley, 1988; Smith, 2000; Stein, 1999; Warnes and Ripley, 1987). For the multimodality, there could be several reasons, for examples, the likelihood maybe flat, e.g. due to a wrong model; the scheme could be non-differentiable, e.g.,the spherical scheme with respect to the range parameter; the nugget parameter can be on the boundary; the numerical method may not give enough accuracy. There are some effective solutions. It is found that a modest increase in additional number of the model parameters can resolve some of the issues. But it is always wise to check various points: the underlying geometry, long... |

167 |
On stationary processes in the plane.”
- Whittle
- 1954
(Show Context)
Citation Context ...; θ, σ 2). Also, we then can obtain β = β(θ) and σ2 = σ2(θ) from (8) and (9). Covariance functions. One of the most popular covariance functions with statisticians is what is called Matern scheme given by {2/Γ(ν)}(r/2c)νKν(r/c), r = |h |> 0, ν > 0, c > 0, where Kν(.) is the Bessel function of the second kind and order ν, and c is a scale parameter. In fact, the parameter ν determines the differentiability of the underlying process; the special case ν = 0.5 and ν = ∞ correspond to the exponential scheme and the Gaussian scheme respectively. This scheme was introduced by Matern (1960) but Whittle (1954) proposed the use of this scheme with ν = 1 in 2 dimensions. This scheme for ν = 0.5 can be related approximately to the spherical scheme. 5 The nugget case The Model. Consider Cov(X) = σ2P + ψ2I, (10) where P is a correlation matrix with ρij = ρ(ti − tj), Cov(xi, xj) = σ 2ρij + ψ 2δij , ρ(0) = 1, and δij = 1 if i = j; = 0 if i 6= j. We will call ψ2 a nugget parameter. We could write the underlying process (with zero drift) as X(ti) = ǫ(ti) + η(ti), (11) where Cov(ǫ(ti), ǫ(tj)) = σ 2ρ(ti − tj), ρ(0) = 1, Var[η(ti)] = ψ 2, Cov{ǫ(ti), η(ti)} = 0. Hence (11) is an errors-in-variable model. It is ... |

133 |
Maximum Likelihood Estimation of Models for Residual Covariance in Spatial Regression
- Mardia, Marshall
- 1984
(Show Context)
Citation Context ...ther key topics are briefly outlined, including rock fracture modelling and risk assessment for the safe storage of hazardous wastes in underground repositories. 2 Introduction It is well known that the maximum likelihood (ML) method is a powerful statistical tool in estimation for parametric models. The method of maximum likelihood was introduced by R.A. Fisher as early as 1931 but its application in geostatistics for inference of variogram parameters has been slow and it was first introduced by Mardia (1980) - incidently, this paper was presented to the Geological Congress in Paris in 1980! Mardia and Marshall (1984) and Mardia (1990) have considered the problem in detail. In earth sciences applications, the method was studied by Kitandis (1983) and Kitandis and Lane (1985). Since then, the number of applications has been on increase. Additionally, the computer simulations have allowed numerical studies of problems that are difficult to handle theoretically: small sample theory, the effect of drift on bias of the covariance parameter estimates, robustness of the Gaussian assumption, effect of transformations of the data and transformation on the parameters. There is no doubt that the availability of cheap... |

58 | Multi-dimensional multivariate Gaussian Markov random fields. - Mardia - 1988 |

47 |
Estimation and model identification for continuous spatial processes.
- Vecchia
- 1988
(Show Context)
Citation Context ...ether to fit a higher order drift and white noise vs simply coloured error with long range correlation: expert opinion can help but the drift can be poor in extrapolation. Computational Procedures. With increase in the computational power, the ML method is becoming easier to implement for even large data sets. Now inverting even a 500 × 500 matrix is feasible. Various methods are available - methods using derivatives of likelihood versus methods not using the derivatives but using the likelihood itself. For small numbers of parameters, grid method seems to be adequate. For very large data set Vecchia (1988) type selection of neighbourhood (a pseudo-likelihood type approach) is found to be useful even when n >> 500! 8 Generalized Linear Geostatistical Models (GLGM) We now describe a versatile model of Diggle et al. (1998). Let S(t) be a stationary Gaussian process with mean zero and variance σ2, and correlation function ρ(h). Suppose that the observations xi|S(.), i = 1, . . . , n are conditionally independent with E{xi|S(.)} = g(α+ Si) = µi say, Var{xi|S(.)} = v(µi). The function of g(.) is the analytic inverse of the link function of Generalized Linear Model. We can now obtain the semi-variogra... |

46 |
Spatial-temporal analysis of multivariate environmental monitoring data.
- Mardia, Goodall
- 1993
(Show Context)
Citation Context ...omponent rather than the covariance parameters and his comments on p.156 are worth noting (the equation (4-2-7) below refers to the universal kriging equations and S to the Domain): “Our optimal estimator is more general than the maximum likelihood estimator, for it is not related to a Gaussian hypothesis and is generalized to the case where S is infinite. In applications, the systems (4-2-7) is easier to solve, as it requires only one matrix inversion.” Within Geostatistical models, there are many other key ideas: for non-Gaussian cases the use of the Box and Cox transformation (for example, Mardia and Goodall, 1993), link between kriging and thin plate spline (for example, Kent and Mardia, 1984), use of spectral representation for covariances (for example, Kent and Mardia, 1996), kriging with derivative (for example, Mardia et al., 1996), CAR and SAR models (for example, Mardia, 1990). There are other estimation methods , e.g. minimum norm quadratic estimators (for example, Mardia and Marshall, 1985). Directional statistics is another area of great importance (for examples, Mardia, 1981a,b, 1989, 2002; Jupp and Mardia, 2000). Incidentally ,the directional statistics paper ,Mardia (2002), was presented as... |

22 |
On multimodality of the likelihood in the spatial linear model.
- Mardia, Watkins
- 1989
(Show Context)
Citation Context ...n the likelihood. Although REML is widely recommended for geostatistical models, it can be more sensitive than ML to the chosen model for the drift (see, for example, Diggle and Ribeiro Jr., 2007, p.117). Also it can have higher MSE. 7 General Issues with MLE We only consider estimation of θ since the drift estimation is generally not a challenge. Multimodality. The likelihood could have several modes for θ depending on the scheme but there seems to be always a global maximum. There has been a very close scrutiny of the behaviour of the MLE for the spatial linear model (see Christensen, 2004; Mardia and Watkins, 1989; Ripley, 1988; Smith, 2000; Stein, 1999; Warnes and Ripley, 1987). For the multimodality, there could be several reasons, for examples, the likelihood maybe flat, e.g. due to a wrong model; the scheme could be non-differentiable, e.g.,the spherical scheme with respect to the range parameter; the nugget parameter can be on the boundary; the numerical method may not give enough accuracy. There are some effective solutions. It is found that a modest increase in additional number of the model parameters can resolve some of the issues. But it is always wise to check various points: the underlying ... |

22 |
Problems with likelihood estimation of covariance functions of spatial Gaussian processes.
- Warnes, Ripley
- 1987
(Show Context)
Citation Context ...stical models, it can be more sensitive than ML to the chosen model for the drift (see, for example, Diggle and Ribeiro Jr., 2007, p.117). Also it can have higher MSE. 7 General Issues with MLE We only consider estimation of θ since the drift estimation is generally not a challenge. Multimodality. The likelihood could have several modes for θ depending on the scheme but there seems to be always a global maximum. There has been a very close scrutiny of the behaviour of the MLE for the spatial linear model (see Christensen, 2004; Mardia and Watkins, 1989; Ripley, 1988; Smith, 2000; Stein, 1999; Warnes and Ripley, 1987). For the multimodality, there could be several reasons, for examples, the likelihood maybe flat, e.g. due to a wrong model; the scheme could be non-differentiable, e.g.,the spherical scheme with respect to the range parameter; the nugget parameter can be on the boundary; the numerical method may not give enough accuracy. There are some effective solutions. It is found that a modest increase in additional number of the model parameters can resolve some of the issues. But it is always wise to check various points: the underlying geometry, long range correlations versus trend; stability of the m... |

19 |
Kriging and splines with derivative information.
- Mardia, Kent, et al.
- 1996
(Show Context)
Citation Context ...aximum likelihood estimator, for it is not related to a Gaussian hypothesis and is generalized to the case where S is infinite. In applications, the systems (4-2-7) is easier to solve, as it requires only one matrix inversion.” Within Geostatistical models, there are many other key ideas: for non-Gaussian cases the use of the Box and Cox transformation (for example, Mardia and Goodall, 1993), link between kriging and thin plate spline (for example, Kent and Mardia, 1984), use of spectral representation for covariances (for example, Kent and Mardia, 1996), kriging with derivative (for example, Mardia et al., 1996), CAR and SAR models (for example, Mardia, 1990). There are other estimation methods , e.g. minimum norm quadratic estimators (for example, Mardia and Marshall, 1985). Directional statistics is another area of great importance (for examples, Mardia, 1981a,b, 1989, 2002; Jupp and Mardia, 2000). Incidentally ,the directional statistics paper ,Mardia (2002), was presented as a plenary talk to the IAMG Conference in Berlin! We have not covered here various other key topics. Bayesian methods in geostatistics are becoming effective (see, for example, Banerjee et al., 2004; Diggle and Ribeiro Jr., 20... |

14 |
Monte Carlo maximum likelihood in model-based geostatistics.
- Christensen
- 2004
(Show Context)
Citation Context ...ties X, β,Σ, etc. in the likelihood. Although REML is widely recommended for geostatistical models, it can be more sensitive than ML to the chosen model for the drift (see, for example, Diggle and Ribeiro Jr., 2007, p.117). Also it can have higher MSE. 7 General Issues with MLE We only consider estimation of θ since the drift estimation is generally not a challenge. Multimodality. The likelihood could have several modes for θ depending on the scheme but there seems to be always a global maximum. There has been a very close scrutiny of the behaviour of the MLE for the spatial linear model (see Christensen, 2004; Mardia and Watkins, 1989; Ripley, 1988; Smith, 2000; Stein, 1999; Warnes and Ripley, 1987). For the multimodality, there could be several reasons, for examples, the likelihood maybe flat, e.g. due to a wrong model; the scheme could be non-differentiable, e.g.,the spherical scheme with respect to the range parameter; the nugget parameter can be on the boundary; the numerical method may not give enough accuracy. There are some effective solutions. It is found that a modest increase in additional number of the model parameters can resolve some of the issues. But it is always wise to check vario... |

13 |
Maximum likelihood parameter estimation of hydrologic spatial processes by the Gauss-Newton methods.
- Kitandis, Lane
- 1985
(Show Context)
Citation Context ...2 Introduction It is well known that the maximum likelihood (ML) method is a powerful statistical tool in estimation for parametric models. The method of maximum likelihood was introduced by R.A. Fisher as early as 1931 but its application in geostatistics for inference of variogram parameters has been slow and it was first introduced by Mardia (1980) - incidently, this paper was presented to the Geological Congress in Paris in 1980! Mardia and Marshall (1984) and Mardia (1990) have considered the problem in detail. In earth sciences applications, the method was studied by Kitandis (1983) and Kitandis and Lane (1985). Since then, the number of applications has been on increase. Additionally, the computer simulations have allowed numerical studies of problems that are difficult to handle theoretically: small sample theory, the effect of drift on bias of the covariance parameter estimates, robustness of the Gaussian assumption, effect of transformations of the data and transformation on the parameters. There is no doubt that the availability of cheap and powerful computers has increased the interest in ML applications. With spatially correlated data, the maximization of the likelihood requires considerable ... |

12 | Link between kriging and thin plate splines. In: Kelly F.P. (Ed.) Festschrift Volume to - Kent, Mardia - 1994 |

10 |
Spectral and circulant approximations to the likelihood for stationary Gaussian random fields.
- Kent, Mardia
- 1996
(Show Context)
Citation Context ...the Domain): “Our optimal estimator is more general than the maximum likelihood estimator, for it is not related to a Gaussian hypothesis and is generalized to the case where S is infinite. In applications, the systems (4-2-7) is easier to solve, as it requires only one matrix inversion.” Within Geostatistical models, there are many other key ideas: for non-Gaussian cases the use of the Box and Cox transformation (for example, Mardia and Goodall, 1993), link between kriging and thin plate spline (for example, Kent and Mardia, 1984), use of spectral representation for covariances (for example, Kent and Mardia, 1996), kriging with derivative (for example, Mardia et al., 1996), CAR and SAR models (for example, Mardia, 1990). There are other estimation methods , e.g. minimum norm quadratic estimators (for example, Mardia and Marshall, 1985). Directional statistics is another area of great importance (for examples, Mardia, 1981a,b, 1989, 2002; Jupp and Mardia, 2000). Incidentally ,the directional statistics paper ,Mardia (2002), was presented as a plenary talk to the IAMG Conference in Berlin! We have not covered here various other key topics. Bayesian methods in geostatistics are becoming effective (see, fo... |

7 | Meeting the statistical needs of 21st-century science. - Mardia, Gilks - 2005 |

6 |
Model based geostatistics (with discussion).
- Diggle, Tawn, et al.
- 1998
(Show Context)
Citation Context ...in the computational power, the ML method is becoming easier to implement for even large data sets. Now inverting even a 500 × 500 matrix is feasible. Various methods are available - methods using derivatives of likelihood versus methods not using the derivatives but using the likelihood itself. For small numbers of parameters, grid method seems to be adequate. For very large data set Vecchia (1988) type selection of neighbourhood (a pseudo-likelihood type approach) is found to be useful even when n >> 500! 8 Generalized Linear Geostatistical Models (GLGM) We now describe a versatile model of Diggle et al. (1998). Let S(t) be a stationary Gaussian process with mean zero and variance σ2, and correlation function ρ(h). Suppose that the observations xi|S(.), i = 1, . . . , n are conditionally independent with E{xi|S(.)} = g(α+ Si) = µi say, Var{xi|S(.)} = v(µi). The function of g(.) is the analytic inverse of the link function of Generalized Linear Model. We can now obtain the semi-variogram of the process S as (Diggle and Ribeiro Jr., 2007; Diggle et al., 1998): γs(h) = 1 2 [ES{g(α+ Si) − g(α+ Sj)} + 2ES{v(g(µ+ Si))]. All this looks daunting but the model can be implemented even for non-Gaussian cases s... |

6 | Spatial statistics in environmental science.
- Smith
- 2000
(Show Context)
Citation Context ...y recommended for geostatistical models, it can be more sensitive than ML to the chosen model for the drift (see, for example, Diggle and Ribeiro Jr., 2007, p.117). Also it can have higher MSE. 7 General Issues with MLE We only consider estimation of θ since the drift estimation is generally not a challenge. Multimodality. The likelihood could have several modes for θ depending on the scheme but there seems to be always a global maximum. There has been a very close scrutiny of the behaviour of the MLE for the spatial linear model (see Christensen, 2004; Mardia and Watkins, 1989; Ripley, 1988; Smith, 2000; Stein, 1999; Warnes and Ripley, 1987). For the multimodality, there could be several reasons, for examples, the likelihood maybe flat, e.g. due to a wrong model; the scheme could be non-differentiable, e.g.,the spherical scheme with respect to the range parameter; the nugget parameter can be on the boundary; the numerical method may not give enough accuracy. There are some effective solutions. It is found that a modest increase in additional number of the model parameters can resolve some of the issues. But it is always wise to check various points: the underlying geometry, long range correl... |

5 |
Maximum likelihood estimation for spatial models. In:
- Mardia
- 1990
(Show Context)
Citation Context ...lined, including rock fracture modelling and risk assessment for the safe storage of hazardous wastes in underground repositories. 2 Introduction It is well known that the maximum likelihood (ML) method is a powerful statistical tool in estimation for parametric models. The method of maximum likelihood was introduced by R.A. Fisher as early as 1931 but its application in geostatistics for inference of variogram parameters has been slow and it was first introduced by Mardia (1980) - incidently, this paper was presented to the Geological Congress in Paris in 1980! Mardia and Marshall (1984) and Mardia (1990) have considered the problem in detail. In earth sciences applications, the method was studied by Kitandis (1983) and Kitandis and Lane (1985). Since then, the number of applications has been on increase. Additionally, the computer simulations have allowed numerical studies of problems that are difficult to handle theoretically: small sample theory, the effect of drift on bias of the covariance parameter estimates, robustness of the Gaussian assumption, effect of transformations of the data and transformation on the parameters. There is no doubt that the availability of cheap and powerful comp... |

5 |
Some minimum norm quadratic estimators of the components of spatial covariance.
- Mardia, Marshall
- 1985
(Show Context)
Citation Context ...2-7) is easier to solve, as it requires only one matrix inversion.” Within Geostatistical models, there are many other key ideas: for non-Gaussian cases the use of the Box and Cox transformation (for example, Mardia and Goodall, 1993), link between kriging and thin plate spline (for example, Kent and Mardia, 1984), use of spectral representation for covariances (for example, Kent and Mardia, 1996), kriging with derivative (for example, Mardia et al., 1996), CAR and SAR models (for example, Mardia, 1990). There are other estimation methods , e.g. minimum norm quadratic estimators (for example, Mardia and Marshall, 1985). Directional statistics is another area of great importance (for examples, Mardia, 1981a,b, 1989, 2002; Jupp and Mardia, 2000). Incidentally ,the directional statistics paper ,Mardia (2002), was presented as a plenary talk to the IAMG Conference in Berlin! We have not covered here various other key topics. Bayesian methods in geostatistics are becoming effective (see, for example, Banerjee et al., 2004; Diggle and Ribeiro Jr., 2007). Another area is of spatial temporal modelling through Kriged Kalman filter with kriging providing principal fields (Mardia et al., 1998; Sahu and Mardia, 2005). ... |

5 | A Bayesian kriged Kalman model for short-term forecasting of air pollution levels.
- Sahu, Mardia
- 2005
(Show Context)
Citation Context ...rdia and Marshall, 1985). Directional statistics is another area of great importance (for examples, Mardia, 1981a,b, 1989, 2002; Jupp and Mardia, 2000). Incidentally ,the directional statistics paper ,Mardia (2002), was presented as a plenary talk to the IAMG Conference in Berlin! We have not covered here various other key topics. Bayesian methods in geostatistics are becoming effective (see, for example, Banerjee et al., 2004; Diggle and Ribeiro Jr., 2007). Another area is of spatial temporal modelling through Kriged Kalman filter with kriging providing principal fields (Mardia et al., 1998; Sahu and Mardia, 2005). Beyond the spatial linear models, there are many other type of statistical models ,eg. for rock fractures (Mardia et al., 2007a; Xu et al., 2006) to assess risk for the safe storage of hazardous wastes in underground repositories. All in all, the interplay between “statisticians” and “geostatisticians” has really become stronger, and we end with a few quotations on modelling (Speed, 2007): George Box: “All models are wrong, some models are useful”. John Tukey: “Our focus should be on questions, not models . . .Models can - and will - get us in deep trouble if we expect them to tell us what t... |

3 |
Spatial Variation, 2nd edn.,
- Matern
- 1960
(Show Context)
Citation Context ...) rather than l(X ; θ, σ 2). Also, we then can obtain β = β(θ) and σ2 = σ2(θ) from (8) and (9). Covariance functions. One of the most popular covariance functions with statisticians is what is called Matern scheme given by {2/Γ(ν)}(r/2c)νKν(r/c), r = |h |> 0, ν > 0, c > 0, where Kν(.) is the Bessel function of the second kind and order ν, and c is a scale parameter. In fact, the parameter ν determines the differentiability of the underlying process; the special case ν = 0.5 and ν = ∞ correspond to the exponential scheme and the Gaussian scheme respectively. This scheme was introduced by Matern (1960) but Whittle (1954) proposed the use of this scheme with ν = 1 in 2 dimensions. This scheme for ν = 0.5 can be related approximately to the spherical scheme. 5 The nugget case The Model. Consider Cov(X) = σ2P + ψ2I, (10) where P is a correlation matrix with ρij = ρ(ti − tj), Cov(xi, xj) = σ 2ρij + ψ 2δij , ρ(0) = 1, and δij = 1 if i = j; = 0 if i 6= j. We will call ψ2 a nugget parameter. We could write the underlying process (with zero drift) as X(ti) = ǫ(ti) + η(ti), (11) where Cov(ǫ(ti), ǫ(tj)) = σ 2ρ(ti − tj), ρ(0) = 1, Var[η(ti)] = ψ 2, Cov{ǫ(ti), η(ti)} = 0. Hence (11) is an errors-in-var... |

2 |
Statistical estimation of polynomial generalized covariance functions and hydrologic applications.
- Kitandis
- 1983
(Show Context)
Citation Context ...round repositories. 2 Introduction It is well known that the maximum likelihood (ML) method is a powerful statistical tool in estimation for parametric models. The method of maximum likelihood was introduced by R.A. Fisher as early as 1931 but its application in geostatistics for inference of variogram parameters has been slow and it was first introduced by Mardia (1980) - incidently, this paper was presented to the Geological Congress in Paris in 1980! Mardia and Marshall (1984) and Mardia (1990) have considered the problem in detail. In earth sciences applications, the method was studied by Kitandis (1983) and Kitandis and Lane (1985). Since then, the number of applications has been on increase. Additionally, the computer simulations have allowed numerical studies of problems that are difficult to handle theoretically: small sample theory, the effect of drift on bias of the covariance parameter estimates, robustness of the Gaussian assumption, effect of transformations of the data and transformation on the parameters. There is no doubt that the availability of cheap and powerful computers has increased the interest in ML applications. With spatially correlated data, the maximization of the like... |

2 |
Some statistical inference problems in Kriging II: Theory. In:
- Mardia
- 1980
(Show Context)
Citation Context ...ainly discuss here in detail maximum likelihood methods for spatial linear model (kriging). Some other key topics are briefly outlined, including rock fracture modelling and risk assessment for the safe storage of hazardous wastes in underground repositories. 2 Introduction It is well known that the maximum likelihood (ML) method is a powerful statistical tool in estimation for parametric models. The method of maximum likelihood was introduced by R.A. Fisher as early as 1931 but its application in geostatistics for inference of variogram parameters has been slow and it was first introduced by Mardia (1980) - incidently, this paper was presented to the Geological Congress in Paris in 1980! Mardia and Marshall (1984) and Mardia (1990) have considered the problem in detail. In earth sciences applications, the method was studied by Kitandis (1983) and Kitandis and Lane (1985). Since then, the number of applications has been on increase. Additionally, the computer simulations have allowed numerical studies of problems that are difficult to handle theoretically: small sample theory, the effect of drift on bias of the covariance parameter estimates, robustness of the Gaussian assumption, effect of tra... |

2 | Intrinsic random fields and image deformations.
- Mardia, Bookstein, et al.
- 2006
(Show Context)
Citation Context ...n the model X = Fβ + error, but now the error process may be only increment stationary, ie. we are dealing with intrinsic random function (IRF). The main principle of the REML is to project the data in a hyperplane perpendicular to F, and work on the marginal likelihood in that plane, namely on X∗ = PX,P = I − F (FTF )−1FT (16) where PF = 0 is the hyperplane. The ML can be biased so REML is used for estimating θ as above. For implementation, we can take an ortho-normal basis in this hyperplane. Many distribution takes a form independent of basis but not in the intrinsic case (see, for example Mardia et al., 2006). There are two main approaches: (1) Basis free method (ie. implicit basis only) and (2) Basis explicitly specified. In the first approach, we have the marginal distribution as singular normal distribution, so that the pdf is given by |2πPΣP |− 1 2 exp[− 1 2 XT (PΣP )−X]. (17) In the second approach, we chose a basis and the work on co-ordinates. Thus we have various specific choices. We can work on X∗2 defined by X ∗ 2 = A2X where A T 2 F = 0. Now we can write fully the distribution of X∗2 where Cov(X ∗ 2 ) is of the full rank. Also A = (A1, A2) is non-singular. In this transformation, it is ... |

1 |
Are statisticians helping earth sciences enough?.
- Mardia
- 1981
(Show Context)
Citation Context ...e are many other key ideas: for non-Gaussian cases the use of the Box and Cox transformation (for example, Mardia and Goodall, 1993), link between kriging and thin plate spline (for example, Kent and Mardia, 1984), use of spectral representation for covariances (for example, Kent and Mardia, 1996), kriging with derivative (for example, Mardia et al., 1996), CAR and SAR models (for example, Mardia, 1990). There are other estimation methods , e.g. minimum norm quadratic estimators (for example, Mardia and Marshall, 1985). Directional statistics is another area of great importance (for examples, Mardia, 1981a,b, 1989, 2002; Jupp and Mardia, 2000). Incidentally ,the directional statistics paper ,Mardia (2002), was presented as a plenary talk to the IAMG Conference in Berlin! We have not covered here various other key topics. Bayesian methods in geostatistics are becoming effective (see, for example, Banerjee et al., 2004; Diggle and Ribeiro Jr., 2007). Another area is of spatial temporal modelling through Kriged Kalman filter with kriging providing principal fields (Mardia et al., 1998; Sahu and Mardia, 2005). Beyond the spatial linear models, there are many other type of statistical models ,eg. f... |

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The evolution of directional models in geosciences since Fisher. In: Merriam, D.F. Down-To-Earth Statistics: Solution Targeting Geological Problems, Syracuse Univ.
- Mardia
- 1981
(Show Context)
Citation Context ...e are many other key ideas: for non-Gaussian cases the use of the Box and Cox transformation (for example, Mardia and Goodall, 1993), link between kriging and thin plate spline (for example, Kent and Mardia, 1984), use of spectral representation for covariances (for example, Kent and Mardia, 1996), kriging with derivative (for example, Mardia et al., 1996), CAR and SAR models (for example, Mardia, 1990). There are other estimation methods , e.g. minimum norm quadratic estimators (for example, Mardia and Marshall, 1985). Directional statistics is another area of great importance (for examples, Mardia, 1981a,b, 1989, 2002; Jupp and Mardia, 2000). Incidentally ,the directional statistics paper ,Mardia (2002), was presented as a plenary talk to the IAMG Conference in Berlin! We have not covered here various other key topics. Bayesian methods in geostatistics are becoming effective (see, for example, Banerjee et al., 2004; Diggle and Ribeiro Jr., 2007). Another area is of spatial temporal modelling through Kriged Kalman filter with kriging providing principal fields (Mardia et al., 1998; Sahu and Mardia, 2005). Beyond the spatial linear models, there are many other type of statistical models ,eg. f... |

1 | Statistics in Earth Sciences. - Mardia - 1989 |

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Why is directional statistics pivotal to geosciences?.
- Mardia
- 2002
(Show Context)
Citation Context ...mple, Mardia and Goodall, 1993), link between kriging and thin plate spline (for example, Kent and Mardia, 1984), use of spectral representation for covariances (for example, Kent and Mardia, 1996), kriging with derivative (for example, Mardia et al., 1996), CAR and SAR models (for example, Mardia, 1990). There are other estimation methods , e.g. minimum norm quadratic estimators (for example, Mardia and Marshall, 1985). Directional statistics is another area of great importance (for examples, Mardia, 1981a,b, 1989, 2002; Jupp and Mardia, 2000). Incidentally ,the directional statistics paper ,Mardia (2002), was presented as a plenary talk to the IAMG Conference in Berlin! We have not covered here various other key topics. Bayesian methods in geostatistics are becoming effective (see, for example, Banerjee et al., 2004; Diggle and Ribeiro Jr., 2007). Another area is of spatial temporal modelling through Kriged Kalman filter with kriging providing principal fields (Mardia et al., 1998; Sahu and Mardia, 2005). Beyond the spatial linear models, there are many other type of statistical models ,eg. for rock fractures (Mardia et al., 2007a; Xu et al., 2006) to assess risk for the safe storage of hazar... |

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Some statistical inference problems in kriging I: numerical applications.
- Mardia, Gill
- 1982
(Show Context)
Citation Context ...(θ)] −2. (13) The ML equations for δ and β do no depend on σ2, and are β(θ, δ) = [FTC(Λ + δI)−1CTF ]−1CT (Λ + δI)−1CX, (14) and [ n ∑ i=1 ui(θ, β) 2[δ + λi(θ)] −1 ] [ n ∑ i=1 [δ + λi(θ)] −2 ] = n n ∑ i=1 ui(θ, β) 2[δ + λi(θ)] −2. (15) Now in substituting β from (14), for a given θ, a solution can be obtained in terms of δ(θ). In turn, we substitute δ into (14) to obtain β(θ, δ), and then from (13) we get σ2 = σ2(δ, θ, β). Thus the profile likelihood with respect to θ can be obtained from (12). Hence, the ML estimate of θ can be derived, which in turn gives the MLEs of δ, β, and σ2. Mardia and Gill (1982) gave some early interpretations. 6 Restricted Maximum Likelihood (REML) Assume again the model X = Fβ + error, but now the error process may be only increment stationary, ie. we are dealing with intrinsic random function (IRF). The main principle of the REML is to project the data in a hyperplane perpendicular to F, and work on the marginal likelihood in that plane, namely on X∗ = PX,P = I − F (FTF )−1FT (16) where PF = 0 is the hyperplane. The ML can be biased so REML is used for estimating θ as above. For implementation, we can take an ortho-normal basis in this hyperplane. Many distributio... |

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The Kriged Kalman filter. Discussion paper. Presented to the Spanish Statistical Society.
- Mardia, Goodall, et al.
- 1998
(Show Context)
Citation Context ...tors (for example, Mardia and Marshall, 1985). Directional statistics is another area of great importance (for examples, Mardia, 1981a,b, 1989, 2002; Jupp and Mardia, 2000). Incidentally ,the directional statistics paper ,Mardia (2002), was presented as a plenary talk to the IAMG Conference in Berlin! We have not covered here various other key topics. Bayesian methods in geostatistics are becoming effective (see, for example, Banerjee et al., 2004; Diggle and Ribeiro Jr., 2007). Another area is of spatial temporal modelling through Kriged Kalman filter with kriging providing principal fields (Mardia et al., 1998; Sahu and Mardia, 2005). Beyond the spatial linear models, there are many other type of statistical models ,eg. for rock fractures (Mardia et al., 2007a; Xu et al., 2006) to assess risk for the safe storage of hazardous wastes in underground repositories. All in all, the interplay between “statisticians” and “geostatisticians” has really become stronger, and we end with a few quotations on modelling (Speed, 2007): George Box: “All models are wrong, some models are useful”. John Tukey: “Our focus should be on questions, not models . . .Models can - and will - get us in deep trouble if we expec... |

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The Theory of Regionalzed Variables and its Applications.Ecole Nationale Superieure des Mines de Paris.
- Matheron
- 1971
(Show Context)
Citation Context ...) = 1 2 [ES{g(α+ Si) − g(α+ Sj)} + 2ES{v(g(µ+ Si))]. All this looks daunting but the model can be implemented even for non-Gaussian cases such as Poisson, Binomial, etc. Indeed, there is a supporting library in R called GeoR. For the Poisson case, we have g(y) = exp{α+ y} = v(y). Further, we have cov(xi, xj) = e α+ 1 2 σ2 + e2α+σ 2 (eσ 2 − eσ 2ρ(h)), h = ‖ti − tj‖. The aim is to explore the latent process S(.) as for dynamic linear model. In general, the expressions are not straightforward so one has to resort to numerical work. 9 Discussion Of course, Matheron was aware of the ML method (see Matheron, 1971, pp.155-156). But he was more concerned with the drift component rather than the covariance parameters and his comments on p.156 are worth noting (the equation (4-2-7) below refers to the universal kriging equations and S to the Domain): “Our optimal estimator is more general than the maximum likelihood estimator, for it is not related to a Gaussian hypothesis and is generalized to the case where S is infinite. In applications, the systems (4-2-7) is easier to solve, as it requires only one matrix inversion.” Within Geostatistical models, there are many other key ideas: for non-Gaussian cases... |

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Terence’s stuff: model skeptics.
- Speed
- 2007
(Show Context)
Citation Context ...mple, Banerjee et al., 2004; Diggle and Ribeiro Jr., 2007). Another area is of spatial temporal modelling through Kriged Kalman filter with kriging providing principal fields (Mardia et al., 1998; Sahu and Mardia, 2005). Beyond the spatial linear models, there are many other type of statistical models ,eg. for rock fractures (Mardia et al., 2007a; Xu et al., 2006) to assess risk for the safe storage of hazardous wastes in underground repositories. All in all, the interplay between “statisticians” and “geostatisticians” has really become stronger, and we end with a few quotations on modelling (Speed, 2007): George Box: “All models are wrong, some models are useful”. John Tukey: “Our focus should be on questions, not models . . .Models can - and will - get us in deep trouble if we expect them to tell us what the unique proper questions are”. Basil Rennie: “All thought and all communication is modelling, and most misunderstandings arise by someone confusing either a model with reality or one model with another. Every model embodies a half-truth, and as one of our wiser politicians once remarked, half-truths are like half-bricks, they are better because they carry further”. Answer to all these que... |

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A boy from the bush. In:
- Watson
- 1986
(Show Context)
Citation Context ...y accurate or if interval estimation (giving a region of suitable parameters) is more adequate. We discuss various practical and theoretical problems with MLE (The talk will have several illustrated examples). The conclusion is that the ML method is feasible for geostatistics; it can be implemented efficiently and provides a powerful tool for geostatistical inference. It provides a complete approach to variogram inference offering methodology for model selection, statistical inference and providing measures of uncertainty of the estimated parameters. Historically, the following observation of Watson (1986) is a key in understanding the development of statistical geostatistics: “In the mid 1970s the work of Georges Matheron and Jean Serra of the Center for Mathematical Morphology at the Paris School of Mines, attracted my attention. They seemed to be breathing new life into the application of statistics to geology and mining. As a result, I spent a lot of time persuading Englishspeaking geologists and statisticians that this was so, while trying to persuade the Fontainebleau School to integrate their writings with that of the anglophones! Our geologists receive very little mathematical training ... |

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A connectivity index for discrete fracture networks.
- Xu, Dowd, et al.
- 2006
(Show Context)
Citation Context ...Incidentally ,the directional statistics paper ,Mardia (2002), was presented as a plenary talk to the IAMG Conference in Berlin! We have not covered here various other key topics. Bayesian methods in geostatistics are becoming effective (see, for example, Banerjee et al., 2004; Diggle and Ribeiro Jr., 2007). Another area is of spatial temporal modelling through Kriged Kalman filter with kriging providing principal fields (Mardia et al., 1998; Sahu and Mardia, 2005). Beyond the spatial linear models, there are many other type of statistical models ,eg. for rock fractures (Mardia et al., 2007a; Xu et al., 2006) to assess risk for the safe storage of hazardous wastes in underground repositories. All in all, the interplay between “statisticians” and “geostatisticians” has really become stronger, and we end with a few quotations on modelling (Speed, 2007): George Box: “All models are wrong, some models are useful”. John Tukey: “Our focus should be on questions, not models . . .Models can - and will - get us in deep trouble if we expect them to tell us what the unique proper questions are”. Basil Rennie: “All thought and all communication is modelling, and most misunderstandings arise by someone confusi... |