Difference between revisions of "Documentation:Monte Carlo Equilibration"

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\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)
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= \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)
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= \frac{1}{(\sum_i (x_i - \bar{x}_i)^2)^2}
 
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Revision as of 12:18, 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{1}{(\sum_i (x_i - \bar{x}_i)^2)^2}

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