Kriging回歸

given set $\mathbf{s}=[s_1,s_2,\dots ,s_m{]}^T$, corresponding response $\mathbf{g}=[g_1,g_2,\dots ,g_m{]}^T$,

predictor:
$$\hat{g}(t)=f^T(t)\beta +Z(t)$$
$f(t)$ is a regression function, the stochastic process $Z(t)$ is assumed to have zero mean and covariance is :
$$E[Z(\mathbf{W})Z(\mathbf{Q})]={\sigma }^{2}R(\theta ;\mathbf{W},\mathbf{Q})$$
$R$ is the correlation function defined by parameter $\theta$, the most widely used is gaussian correalaton function:
$$R(\theta ;\mathbf{W},\mathbf{Q})=\prod _{j=1}^n\left[?\theta (Q_j?W_i{)}^2\right]$$
the response $G(u)$ of a given test point $\mathbf{u}$ obey a gaussian distribution, denote as
$$G(\mathbf{u})～N\left({\mu }_G(\mathbf{u}),{\sigma }_G^2(\mathbf{u})\right)$$
the mean and standard devision given as :
$${\mu }_G(\mathbf{u})=f(\mathbf{u}{)}^T{\beta }^{?}+r(\mathbf{u}{)}^T{\mathbf{R}}^{?1}(\mathbf{g}?\mathbf{F}{\beta }^{?})$$

$${\sigma }_G^2(\mathbf{u})={\sigma }^2\left[1+{\mathbf{v}}^T({\mathbf{F}}^T{\mathbf{R}}^T\mathbf{F}{)}^{?1}\mathbf{v}?r(\mathbf{u}{)}^T{\mathbf{R}}^{?1}r(\mathbf{u})\right]$$

where
$$\mathbf{v}={\mathbf{F}}^T{\mathbf{R}}^{?1}r(\mathbf{u})?f(\mathbf{u})$$

$${\sigma }^2=\frac{1}{m}(\mathbf{Y}?\mathbf{F}{\beta }^{?}{)}^T{\mathbf{R}}^{?1}(\mathbf{g}?\mathbf{F}{\beta }^{?})$$

$${\beta }^{?}=({\mathbf{F}}^T{\mathbf{R}}^{?1}\mathbf{F}{)}^{?1}{\mathbf{F}}^T{\mathbf{R}}^{?1}\mathbf{g}$$

$r(\mathbf{u})$ is correlation vector between $\mathbf{S}$ and $\mathbf{u}$
$$r(\mathbf{u})=[R(\theta ;s_1,\mathbf{u}),\dots ,R(\theta ;s_m,\mathbf{u}){]}^T$$
$F$ is the regression matrix
$$F=[f(s_1),\dots ,f(s_m){]}^T$$
the optimal coefficients ${\theta }^{?}$ of the correlation function solves
$${\theta }^{?}=\underset{\theta }{\min }\{|\mathbf{R}{|}^{\frac{1}{m}}{\sigma }^2\}$$

總結

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