Ichikawa's wiki

**以前のリビジョンの文書です** ----

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====== 統計についてメモ ====== ===== Covariance Matrix and PDF ===== ==== Covariance Matrix ==== $ X=\pmatrix{ X_1\\ X_2\\ .\\ .\\ .\\ X_n\\} $is a set of observables. $\mu_i$ is a mean of $X_i$. Covariance matrix is defined as <jsmath> \Sigma = \pmatrix{ <(X_1-\mu_1)(X_1-\mu_1)> & & <(X_1-\mu_1)(X_2-\mu_2)> & ... & <(X_1-\mu_1)(X_n-\mu_n)> & \\ <(X_2-\mu_2)(X_1-\mu_1)> & & <(X_2-\mu_2)(X_2-\mu_2)> & ... & <(X_2-\mu_2)(X_n-\mu_n)> & \\ .\\ .\\ .\\ <(X_n-\mu_n)(X_1-\mu_1)> & & <(X_n-\mu_n)(X_2-\mu_2)> & ... & <(X_n-\mu_n)(X_n-\mu_n)> & \\ } </jsmath>\\ Note: diagonal term is equal to $\sigma_i^2$.\\ $\Sigma_{ij}>0$ : positive correlation.\\ $\Sigma_{ij}=0$ : no correlation.\\ $\Sigma_{ij}<0$ : negative correlation.\\ Correlation matrix is defined as <jsmath> \rho = (\rho_{ij}), \rho_{ij}=\frac{cov(X_i,X_j)}{\sigma_i \sigma_j} </jsmath>\\ ==== Probability Density Function w/ Covariance matrix==== <jsmath> f(x_1,x_2,...,x_n)=\frac{1}{(2\pi)^{n/2}\sqrt{\det(\Sigma))}}\exp(-\frac{1}{2}(x_1-\mu_1,...,x_n-\mu_n)\Sigma^{^1}(x_1-\mu_1,...,x_n-\mu_n)^T) </jsmath>\\ For two variable case,\\ <jsmath> f(x,y)=\frac{1}{2\pi\sqrt{1-\rho}}\exp[-\frac{1}{2(1-\rho^2)}(\frac{(x-\mu_x)^2}{\sigma_x^2}-\frac{2\rho(x-\mu_x)(y-\mu_y)}{\sigma_x\sigma_y}+\frac{(y-\mu_y)^2}{\sigma_y^2})] </jsmath>\\