Documentation:Monte Carlo Equilibration

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Equilibration in Monte Carlo simulations

Description

Synposis

pyalps.checkSteadyState(sets=None, outfile=None, observable=None, confidenceInterval=0.6827, simplified=False, includeLog=False)

Description

argument default type remark
sets None Pyalps Dataset usually returned by pyalps.loadMeasurements
outfile None Python str ALPS hdf5 output file(name)
observable None Python str measurement observable
confidenceInterval 0.6827 Python float confidence interval in which steady state has been reached
simplified False Python bool shall we combine the checks of all observables as 1 final boolean answer?
includeLog False Python bool shall we print the detailed log?

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


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


Using the standard z-test, we reject H_0 at confidence interval \alpha if


\left| \frac{\beta_1 - 0 }{ \sigma_{\beta_1} } \right| > z_{\frac{1-\alpha}{2}}


© 2013 by Matthias Troyer, Ping Nang Ma