Asymptotic standard errors of MLE Suppose, in the previous example of Carbon and Nitrogen in soil data, that we get the parameter estimates For maximum likelihood estimation, we can use Hessian matrix of the loglikelihood function to get the asymptotic standard errors of the maximum likelihood estimates Hessian matrix is the matrix of the the second-order partial derivatives of a function The observed information matrix is the negative of the Hessian matrix of the loglikelihood function evaluated at the maximum likelihood estimators Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, 2009 1 / 9
Asymptotic standard errors of MLE It is known in statistics theory that maximum likelihood estimators are asymptotically normal with the mean being the true parameter values and the covariance matrix being the inverse of the observed information matrix In particular, the square root of the diagonal entries of the inverse of the observed information matrix are asymptotic standard errors of the parameter estimates Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, 2009 2 / 9
Asymptotic standard errors of MLE In practice, how do we get the asymptotic standard errors of the parameter estimates? Let us start by the Carbon-Nitrogen example Suppose we are doing maximum likelihood estimation to estimate the parameters In R, you can ask nlm or optim functions to return Hessian matrix Once you get the Hessian matrix, you need to be careful because of the parameter transformation For Carbon-Nitrogen data, we get ˆα = 0.040(0.021), ˆβ = 50.73(33.10), ˆν = 0.52(0.25), ˆδ = 0.029(0.0038), ˆβ 0 = 3.12(0.14), ˆβ 1 = 0.80(0.019) Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, 2009 3 / 9
Spatial autoregressive models In time series, autoregressive models express the data at time t as a linear combination of the values in the past For example, if Z(t) is a time series of interest, AR(p) model is in the form p Z(t) = c + φ i Z(t i) + ɛ(t) i=1 We can do similar things with spatial data We will see SAR models and CAR models today Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, 2009 4 / 9
Simultaneous Autoregressive (SAR) Model Consider a spatial regression problem with Gaussian data If B is a matrix of spatial dependence parameters with b i i = 0, Z(s) = m(s)β + e(s) e(s) = B e(s) + v The residuals v i, i = 1,, n, have mean zero and a diagonal covariance matrix Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, 2009 5 / 9
Simultaneous Autoregressive (SAR) Model We can also express this as (I B)(Z(s) m(s)β) = v Then the covariance matrix of Z(s), Σ SAR = (I B) 1 Σ v (I B T ) 1 if Σ v is the diagonal covariance matrix of v The covariance structure of Z(s) is completely determined by B and Σ v Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, 2009 6 / 9
Simultaneous Autoregressive (SAR) Model In practice, we may need to model B using a parametric model for b ij We may let B = ρw where W is a proximity matrix that consists of 0 s and 1 s Under this setting, we can easily see how SAR model reduces to a spatial model with uncorrelated errors Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, 2009 7 / 9
Markov property In time series, if the conditional distribution of Z(t + 1) given Z(s), s = 1,, t is the same as that of Z(t + 1) given Z(t), we say the process has Markov property We can extend this to spatial data We may say a spatial random field Z(s) is a Markov random field if Z(s i ) only depends on its neighbors N i Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, 2009 8 / 9
Conditional Autoregressive (CAR) Model This model uses the concept of Markov property We consider f (Z(s i ) Z(s) i ) where Z(s) i denotes the vector of all the data except Z(s i ) Specifically we assume each of the conditional distributions is Gaussian and we let E(Z(s i ) Z(s) i ) = m(s i )β + n c ij (Z(s j ) m(s i )β) j=1 Var(Z(s i ) Z(s) i ) = σ 2 i, i = 1,, n Here c ij is nonzero only if s i N i and c ii = 0 Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, 2009 9 / 9
Conditional Autoregressive (CAR) Model Given conditional distributions, it is not easy to construct joint distributions to do estimation and inference SAR model is only defined for multivariate Gaussian distribution while CAR model may not Mikyoung Jun (Texas A&M) stat647 lecture 11 October 8, 2009 10 / 9