Difference between revisions of "Documentation:Monte Carlo Equilibration"

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(Theory)
(Theory)
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<math>
 
<math>
\rightarrow \left(  
+
\Rightarrow \left(  
 
\begin{array}{cc}  
 
\begin{array}{cc}  
 
N              & \sum_i x_i    \\
 
N              & \sum_i x_i    \\
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<math>
 
<math>
\left(
+
\Rightarrow \left(
 
\begin{array}{c}
 
\begin{array}{c}
 
\beta_0 \\  
 
\beta_0 \\  

Revision as of 12:05, 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)