physics:stat
====== 差分 ====== この文書の現在のバージョンと選択したバージョンの差分を表示します。
| 次のリビジョン | 前のリビジョン | ||
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physics:stat [2011/08/22 04:45] ichikawa 作成 |
physics:stat [2013/11/20 05:05] (現在) |
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| ライン 1: | ライン 1: | ||
| - | ====== 統計についてもメモ ====== | + | ====== 統計についてメモ ====== |
| - | ===== Covariance Matrix ===== | + | ===== 良く使う数字 ===== |
| - | <jsmath> | + | {{:physics:gauss.png?200|}} |
| - | X=pmatrix{ | + | {{:physics:dchi2.png?200|}} |
| + | |||
| + | ===== 役に立つリンク ===== | ||
| + | * http://www-cdf.fnal.gov/physics/statistics/statistics_recommendations.html | ||
| + | * [[http://www.vibrationdata.com/math/int_pdf.pdf|INTEGRATION OF THE NORMAL DISTRIBUTION CURVE]] | ||
| + | |||
| + | ===== Covariance Matarix and PDF ===== | ||
| + | ==== Covariance Matrix ==== | ||
| + | |||
| + | $ | ||
| + | X=\pmatrix{ | ||
| X_1\\ | X_1\\ | ||
| X_2\\ | X_2\\ | ||
| ライン 9: | ライン 19: | ||
| .\\ | .\\ | ||
| X_n\\} | X_n\\} | ||
| - | </jsmath> | + | $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\sigma_x\sigma_y\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>\\ | ||
| + | |||
| + | ==== simulation ==== | ||
| + | * http://approximity.com/papers/ptfopt/node19.html | ||
| + | |||
physics/stat.1313988338.txt.gz · 最終更新: 2013/11/20 05:05 (外部編集)
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