Pooled variance

From PBDN

This page contains a mathematical description of pooled variance. For a simpler overview in the context of our results, see Experiments and Graphs.

The pooled variance is used to measure how well we have approximated the PDF. The set of trajectories are divided into subsets, and the mean of each subset is calculated. We can estimate how good the approximation is by calculating the variance of these means.

If we allow ourselves to be more mathematical, N is the number of trajectories in total, J is the number of pools and $x i j$ is the final state of the i:th trajectory in the j:th pool we can write the pool means as:

$\mu_j = \frac{1}{N/J} \sum_{i=1}^{N/J} x_{ij}$

and the pooled variance is given as the variance of the pool means:

$s^2_p = \frac{1}{J-1} \sum_{j=1}^{J} (\overline{\mu} - \mu_j)^2$

where the global mean $\bar{\mu}$ is given by

$\overline{\mu} = \frac{1}{J} \sum_{j=1}^{J} \mu_j$

Here, $x$ is a scalar, i.e. only one species exists. For a multi-dimensional system, the pooled variance must be calculated for each species.

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