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Statistical tools documentation.

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tayral 2014-02-13 17:37:42 +00:00 committed by Olivier Parcollet
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mctools/intro mctools/intro
det_manip/contents det_manip/contents
parameters/parameters parameters/parameters
statistics/contents
utilities/contents utilities/contents
using_the_lib/profiling using_the_lib/profiling

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Autocorrelation time
=====================
Synopsis
---------
Example
--------

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Binning
==========
Synopsis
----------
`make_binned_series(T series, int bin_size)`
- series: object with **TimeSeries** concept
- bin_size: size of the bin
returns the binned time series.
Example
--------
.. compileblock::
#include <triqs/statistics.hpp>
using namespace triqs::statistics;
int main(){
observable<double> A;
A<<1.;A<<1.5;A<<.2;A<<1.1;
auto A_b = make_binned_series(A,2);
std::cout << A_b << std::endl;
return 0;
}

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Tools for statistical analysis: binning and jackknife
======================================================
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 <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
---------
`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
--------
.. compileblock::
#include <triqs/clef.hpp>
#include <triqs/statistics.hpp>
using namespace triqs::statistics;
int main(){
observable<double> X;
X<<1.0;
X<<-1.0;
X<<.5;
X<<.0;
std::cout << average_and_error(X) << std::endl;
std::cout << average_and_error(X*X) << std::endl;
return 0;
}
.. toctree::
binning
jackknife
autocorrelation_time
ising2d

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.. highlight:: c
Full example: Monte-Carlo simulation of the 2D Ising model
===========================================================
.. literalinclude:: src/ising2d.cpp
The output is
.. literalinclude:: src/ising2d.output

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.. highlight:: c
Jackknife
============
Synopsis
---------
`make_jackknife(T series)`
- series: object with **TimeSeries** concept
returns the jackknifed time series.
Example
---------
.. compileblock::
#include <triqs/statistics.hpp>
using namespace triqs::statistics;
int main(){
observable<double> A;
A<<1.;A<<1.5;A<<.2;A<<1.1;
auto A_j = make_jackknife(A);
std::cout << A_j << std::endl;
return 0;
}

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../../../../test/triqs/statistics

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* *
******************************************************************************/ ******************************************************************************/
#pragma once #pragma once
#include "./clef.hpp"
#include "./statistics/statistics.hpp" #include "./statistics/statistics.hpp"