mirror of
https://github.com/triqs/dft_tools
synced 2024-12-27 06:43:40 +01:00
edd1ff4529
A first general restructuration of the doc according to the pattern [tour|tutorial|reference]. In the reference part, objects are documented per topic. In each topic, [definition|c++|python|hdf5] (not yet implemented)
84 lines
3.2 KiB
ReStructuredText
84 lines
3.2 KiB
ReStructuredText
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
|
|
--------
|
|
|
|
.. triqs_example:: ./contents_0.cpp
|
|
.. toctree::
|
|
binning
|
|
jackknife
|
|
autocorrelation_time
|
|
ising2d
|