Difference between revisions of "Documentation:Monte Carlo Equilibration"

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(Theory)
(Theory)
Line 39: Line 39:
 
\right) =
 
\right) =
 
\frac{1}{N\sum_i x_i^2 - (\sum_i x_i)^2}
 
\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)
 
</math>
 
</math>

Revision as of 12:03, 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  :


\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)

Therefore, 
\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)