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\title{Stat 992: Lecture 37\\
Anisotropic kernel smoothing II.}
\author{Moo K. Chung \tt{mchung@stat.wisc.edu}}
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Consider following random field model on manifolds $Y_i(x) =
\mu_i(x) +\epsilon(x), x \in \partial \Omega$ for subject $j$. We
assume i.i.d. zero mean Gaussian random field $\epsilon_j$ with
isotropic covariance $R(x,y) = \sigma^2f(d(x,y))$ where $d$ is a
geodesic distance and $f$ some function.

Suppose there are other measurement $Z_j \sim N(\mu_Z
,\sigma_Z^2)$. The sample correlation between two measurements
$Y_j$ and $Z_j$ at fixed point $x$ is given by
$$\rho(x) = \frac{\mbox{Cov}(Y(x),Z)}{$$


\item {\em Covariance function estimation.} On manifolds $\partial
\Omega$, we have model
$$Y(x) = \mu(x) + \epsilon(x)$$
where $\epsilon$ is unknown random field. We may estimate $\mu$ by
kernel smoothing so that $\widehat{\mu} = K_{\sigma}*Y$. Now we
estimate the noise with the residual $\widehat{\epsilon} = Y -
K_{\sigma}*Y$.