Revision as of 12:31, 9 September 2013

Monte Carlo equilibration

Theory

We have a timeseries of N measurements obtained from a Monte Carlo simulation, i.e. $y_0,y_1,\cdots,y_{N-1}$ .

Suppose $\bar{y}_i = \beta_0 + \beta_1 x_i$ (s.t. $i = 0, 1, \cdots, N-1$ ) is the least-squares best fitted line, we attempt to minimize $S = \sum_i (y_i - \bar{y}_i)^2$ w.r.t. $\beta_0$ and $\beta_1$ .

$\frac{\partial S}{\partial \beta_0 } = 0$ , $\frac{\partial S}{\partial \beta_1 } = 0$ :

$\Rightarrow \left( \begin{array}{cc} N & \sum_i x_i \\ \sum_i x_i & \sum_i x_i^2 \end{array} \right) \left( \begin{array}{c} \beta_0 \\ \beta_1 \end{array} \right) = \left( \begin{array}{c} \sum_i y_i \\ \sum_i x_i y_i \end{array} \right)$

$\Rightarrow \left( \begin{array}{c} \beta_0 \\ \beta_1 \end{array} \right) = \frac{1}{N\sum_i x_i^2 - (\sum_i x_i)^2} \left( \begin{array}{cc} \sum_i x_i^2 & -\sum_i x_i \\ -\sum_i x_i & N \end{array} \right) \left( \begin{array}{c} \sum_i y_i \\ \sum_i x_i y_i \end{array} \right)$

$\Rightarrow \beta_1 = \frac{N \sum_i x_i y_i - \sum_i x_i \sum_i y_i}{N\sum_i x_i^2 - (\sum_i x_i)^2}$

Slope of best-fitted line

$\Rightarrow \beta_1 = \frac{\sum_i (x_i - \bar{x}_i)( y_i - \bar{y}_i) }{\sum_i (x_i - \bar{x}_i)^2} \,\,\,\,\, \left( = \frac{s_{xy}}{s_{xx}} \right)$

Error in slope of best-fitted line

$\mathrm{Var}({\beta_1}) = \mathrm{Var} \left( \frac{\sum_i (x_i - \bar{x}_i)( y_i - \bar{y}_i) }{\sum_i (x_i - \bar{x}_i)^2} \right) = \frac{\sum_i (x_i - \bar{x}_i)^2 \mathrm{Var}(y_i) }{(\sum_i (x_i - \bar{x}_i)^2)^2}$

Denoting $\mathrm{Var}(y_i) = \sigma_{y}^2$ , we have:

$\Rightarrow mathrm{Var}({\beta_1}) = \frac{\sigma_y^2}{\sum_i (x_i - \bar{x}_i)^2}$

$\Rightarrow mathrm{Var}({\beta_1}) = \frac{ 12 \sigma_y^2 }{ N(N^2 - 1) }$

@@ Line 75: / Line 75: @@
 Denoting <math>\mathrm{Var}(y_i) = \sigma_{y}^2 </math>, we have:
-<math> \mathrm{Var}({\beta_1}) = \frac{\sigma_y^2}{\sum_i (x_i - \bar{x}_i)^2} </math>
+<math> \Rightarrow mathrm{Var}({\beta_1}) = \frac{\sigma_y^2}{\sum_i (x_i - \bar{x}_i)^2} </math>
+<math> \Rightarrow mathrm{Var}({\beta_1}) = \frac{ 12 \sigma_y^2 }{ N(N^2 - 1) }  </math>

Difference between revisions of "Documentation:Monte Carlo Equilibration"

Revision as of 12:31, 9 September 2013

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Monte Carlo equilibration

Theory

Slope of best-fitted line

Error in slope of best-fitted line

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