Documentation:Monte Carlo Equilibration

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 $\beta_1$.

$\frac{\partial S}{\partial \beta_0 } = 0$ and $\frac{\partial S}{\partial \beta_1 } = 0$