# Difference between revisions of "Documentation:Monte Carlo Equilibration"

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\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} | \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} | ||

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+ | === Slope of best-fitted line === | ||

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\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) | \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) | ||

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+ | === Error in slope of best-fitted line === |

## Revision as of 12:15, 9 September 2013

## Contents

# Monte Carlo equilibration

## Theory

We have a timeseries of N measurements obtained from a Monte Carlo simulation, i.e. .

Suppose (s.t. ) is the least-squares best fitted line, we attempt to minimize w.r.t. and .

, :

### Slope of best-fitted line