Geostatistics

「An Introduction to Applied Geostatistics」+「Applied Geostatistics with SGeMS」+Wikiより、備忘録兼ねて。
BloggerはTeXコマンドを図に変換してくれるんですね。便利です。


Measures of Spread
Variance
(標本分散） $$\begin{align*}\sigma ^{2}=\frac{1}{n}\sum_{i=1}^{n}{(xi-m)^{2}}\end{align*}$$
(不偏分散） $$\begin{align*}\sigma ^{2}=\frac{1}{n-1}\sum_{i=1}^{n}{(xi-m)^{2}}\end{align*}$$
Standard deviation
(Standard Deviation of the Sample) $$\begin{align*}\sigma =\sqrt{\frac{1}{n}\sum_{i=1}^{n}{(xi-m)^{2}}}\end{align*}$$
(Sample Standard Deviation) $$\begin{align*}\sigma =\sqrt{\frac{1}{n-1}\sum_{i=1}^{n}{(xi-m)^{2}}}\end{align*}$$
Interquartile Range
$$\begin{align*}IQR=Q_{3}-Q_{1}\end{align*}$$

Measures of Shape
Coefficient of Skewness
$$\begin{align*}CS=\frac{\frac{1}{n}\sum_{i=1}^{n}{(xi-m)^{3}}}{\sigma ^{3}}\end{align*}$$
Coefficient of Variation
$$\begin{align*}CV=\frac{\sigma }{m}\end{align*}$$

Comparing Two Distributions
q-q plot
Scatterplot

Correlation
Correlation Coefficient
$$\begin{align*}\rho=\frac{\frac{1}{n}\sum_{i=1}^{n}{(x_{i}-m_{x})(y_{i}-m_{y})}}{\sigma _{x}\sigma _{y}}\end{align*}$$
Covariance
$$\begin{align*}C=\frac{1}{n}\sum_{i=1}^{n}{(x_{i}-m_{x})(y_{i}-m_{y})}\end{align*}$$
Moment of Inertia
$$\begin{align*}\gamma=\frac{1}{2n}\sum_{i=1}^{n}{(x_{i}-y_{i})^{2}}\end{align*}$$

Linear Regression
$$\begin{align*}y=ax+b\end{align*}$$
$$\begin{align*}a=\rho \frac{\sigma _{y}}{\sigma _{x}}\end{align*}$$
$$\begin{align*}b=m_{y}-am_{x}\end{align*}$$

Spatial Description
h-Scatterplot　　　　　  V(t)とV(t+h)の分布
x=V(t)= vi、y=V(t+h)= vjとすると、Moment of Inertia = Semivariogramとなる。（Semiは1/2の意）

Semivariogram
$$\begin{align*}\gamma (h)=\frac{1}{2N(h)}\sum_{(i,j)| h_{ij}=h}{(v_{i}-v_{j})^{2}}\end{align*}$$

Kriging
Simple kriging
$$\begin{align*}Z_{SK}^{}(u)-m=\sum_{\alpha =1}^{n(u)}{\lambda _{\alpha }^{SK}}(u)[ Z(u_{\alpha })-m]\end{align*}$$
Ordinary kriging
$$\begin{align*}Z_{OK}^{}(u)=\sum_{\alpha =1}^{n(u)}{\lambda _{\alpha }^{OK}}(u)Z(u_{\alpha })\end{align*}$$

Indicator Kriging 
The ordinary kriging of indicators at several cutoffs, using a separatevariogram model for each cutoff, is usually referred to simply asindicator kriging.

Median Indicator Kriging 
There is an approximation to indicator kriging that, in many situations,produces very good results. This approximation consists of using thesame variogram model for the estimation at all cutoffs. The variogrammodel chosen for all cutoffs is most commonly developed from theindicator data at a cutoff close to the median. This procedure isusually referred to as median indicator kriging.

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20190727
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