Given the statistical samples :math:`\lbrace x_i\rbrace _{i=0\dots N-1}` and :math:`\lbrace y_i\rbrace _{i=0\dots N-1}` of random variables :math:`X` and :math:`Y`, one often wants to compute the estimate of the following observables:
:math:`\langle X \rangle`, :math:`\langle X\rangle/\langle Y \rangle`, :math:`\langle X \rangle^2`, or in general :math:`f(\langle X \rangle , \langle Y \rangle, \dots)`
as well as the estimate of the errors:
:math:`\Delta\langle X \rangle`, :math:`\Delta\langle X\rangle /\langle Y \rangle`, :math:`\Delta\langle X\rangle ^2` or :math:`\Delta f(\langle X \rangle , \langle Y \rangle, \dots)`
The estimate of the expectation values is the empirical average :
:math:`\langle X \rangle \approx \frac{1}{N} \sum_{i=0}^{N-1} x_i`
If the samples are independent from each other and :math:`f` is a linear function of its variables (e.g :math:`f=Id`):
:math:`(\Delta \langle X \rangle)^2 \approx \frac{\frac{N-1}{N} \sigma^2({x})}{N}`
where :math:`\sigma^2({x})` is the empirical variance of the sample.
In the general case, however,
- the samples are correlated (with a characteristic correlation time): one needs to :doc:`bin <binning>` the series to obtain a reliable estimate of the error bar
-:math:`f` is non-linear in its arguments: one needs to :doc:`jackknife <jackknife>` the series
This library allows one to reliably compute the estimates of :math:`f(\langle X \rangle , \langle Y \rangle, \dots)` and its error bar :math:`\Delta f(\langle X \rangle , \langle Y \rangle, \dots)` in the general case.
Synopsis
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`average_and_error` takes an object with the **Observable** concept (see below) and returns a struct with two members `val` and `error`:
-`val` is the estimate of the expectation value of the random variable for a given sample of it
-`error` is the estimate of the error on this expectation value for the given sample
Concepts
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TimeSeries
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An object has the concept of a TimeSeries if it has the following member functions:
