Difference between revisions of "Documentation:Monte Carlo Equilibration"

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Revision as of 12:45, 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}) = \sigma_{\beta_1}^2 = \frac{\sigma_y^2}{\sum_i (x_i - \bar{x}_i)^2}

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


Hypothesis testing: Monte Carlo statistics has reached steady state?

\begin{array}{l} H_0 : \beta_1 = 0 \\ H_1 : \beta_1 \ne 0 \end{array}

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