Forskjell mellom versjoner av «CorrelationMatrix»
Fra mn/fys/epf
(Ny side: The covariance matrix is <br> <math>C_{ij} = <(x_i-\bar{x_i})(x_j-\bar{x_j})></math>.<br> Suppose <math>\delta_1,\ \delta_2,\ \delta_3</math> are three independent sources of normally-distr...) |
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| (2 mellomliggende revisjoner av samme bruker vises ikke) | |||
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| − | The covariance matrix is <br> | + | The covariance matrix is <br> <math>C_{ij} = <(x_i-\bar{x_i})(x_j-\bar{x_j})></math>.<br> Suppose <math>\delta_1,\ \delta_2,\ \delta_3</math> are three independent sources of normally-distributed unit fluctuations (with <span class="texhtml"> < δ<sub>''i''</sub> > = 0</span> and <span class="texhtml"> < δ<sub>''i''</sub> * δ<sub>''j''</sub> > = δ<sub>ij</sub></span> |
| − | <math>C_{ij} = <(x_i-\bar{x_i})(x_j-\bar{x_j})></math>.<br> | + | and where <math>\delta_{ij}</math> is the kronikker delta function (1 for <math>i=j</math> and zero for <math>i\neq j</math>). |
| − | Suppose <math>\delta_1,\ \delta_2,\ \delta_3</math> are three independent sources of normally-distributed unit fluctuations (with < | ||
| − | and < | ||
| + | <br> Pseudo-measurements of <span class="texhtml">(''x''<sub>1</sub>,''x''<sub>2</sub>)</span> can be generated from the expressions <span class="texhtml">(''x''<sub>1</sub>,''x''<sub>2</sub>) = (''x''<sub>10</sub> + α * δ<sub>1</sub> + β * δ<sub>2</sub>,''x''<sub>20</sub> + γ * δ<sub>3</sub> + λ * δ<sub>2</sub>)</span>. | ||
| − | + | Expanding the covariance matrix one finds <math>C_{ij}=<x_i*x_j>-\bar{x_i}\bar{x_j}</math>.<br> | |
| − | + | ||
| − | Expanding the covariance matrix one finds | + | Using the properties of <math><\delta_i*\delta_j></math> above and <math><x_i>=\bar{x_i}</math>, one finds <math><x_1*x_2> = x_{10}*x_{20}+\beta*\lambda</math>; so that <span class="texhtml">''C''<sub>12</sub> = ''C''<sub>21</sub> = β * λ</span>. Similar substitution gives for the diagonal matrix elements |
| − | <math>C_{ij}=<x_i*x_j>-\bar{x_i}\bar{x_j}< | + | <br> |
| − | Using the properties of <math><\delta_i*\delta_j></math> above and <math><x_i>=\bar{x_i}</math>, one finds <x_1*x_2> = x_{10}*x_{20}+\beta*\lambda</math> so that < | + | <math>C_{11}=x_{10}^2+\alpha^2+\beta^2-x_{10}^2</math><br> and <br> <math>C_{22}=x_{20}^2+\gamma^2+\lambda^2-x_{20}^2</math> |
| − | <math>C_{11}=x_{10}^2+\alpha^2+\beta^2-x_{10}^2</math> and | + | <br> |
| − | <math>C_{22}=x_{20}^2+\gamma^2+\lambda^2-x_{ | + | so that |
| + | C= | ||
| + | {| width="200" cellspacing="1" cellpadding="1" border="1" | ||
| + | |- | ||
| + | | <span class="texhtml">α<sup>2</sup> + β<sup>2</sup></span> | ||
| + | | <span class="texhtml">β * λ</span> | ||
| + | |- | ||
| + | | <math>\beta*\lambda</math> | ||
| + | | <math>\gamma^2+\lambda^2</math> | ||
| + | |} | ||
Nåværende revisjon fra 25. mar. 2011 kl. 23:34
The covariance matrix is
.
Suppose are three independent sources of normally-distributed unit fluctuations (with < δi > = 0 and < δi * δj > = δij
and where is the kronikker delta function (1 for and zero for ).
Pseudo-measurements of (x1,x2) can be generated from the expressions (x1,x2) = (x10 + α * δ1 + β * δ2,x20 + γ * δ3 + λ * δ2).
Expanding the covariance matrix one finds .
Using the properties of above and , one finds ; so that C12 = C21 = β * λ. Similar substitution gives for the diagonal matrix elements
and
so that
C=
| α2 + β2 | β * λ |
