**[triqs/statistics]** Tools for statistical analysis: binning, jackknife and autocorrelation time
====================================================================================================
Introduction
------------
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 ` the series to obtain a reliable estimate of the error bar
- :math:`f` is non-linear in its arguments: one needs to :doc:`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
---------
`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
---------
TimeSeries
~~~~~~~~~~
An object has the concept of a TimeSeries if it has the following member functions:
+-------------+-------------------+
| Return type | Name |
+=============+===================+
| value_type | operator[](int i) |
+-------------+-------------------+
| int | size() |
+-------------+-------------------+
and the following member type:
+-------------+--------------------------------------------+
| Name | Property |
+=============+============================================+
| value_type | belong to an algebra (has +,-,* operators) |
+-------------+--------------------------------------------+
Observable
~~~~~~~~~~
An object has the concept of an observable if it is a TimeSeries and has, additionally, the following member function:
+-------------+-----------------+
| Return type | Name |
+=============+=================+
| observable& | operator<<(T x) |
+-------------+-----------------+
where `T` belongs to an algebra.
Example
-------
.. literalinclude:: ./contents_0.cpp
.. toctree::
binning
jackknife
autocorrelation_time
autocorrelation_function
green_function
ising2d
Histogram
---------
`histogram` is a lightweight object used to represent and to accumulate a histogram of a real random variable.
.. literalinclude:: ./histogram_example.cpp
.. toctree::
/documentation/cpp_api/triqs/statistics/histogram