# [triqs/stat] Statistical Analysis¶

## Introduction¶

This library provides statistical tools to analyze samples that are generated during a simulation. In particular, it provides routines for:

1. accumulating and binning correlated measurements, arising from a Monte Carlo simulation
2. calculating averages and errors for data and functions of data
3. constructing histograms

## Processing Correlated Data¶

In general, the output samples produced in Monte Carlo type simulations are not independent of one another, but correlated over a characteristic number of steps – the correlation time $$\tau$$. This depends on the observable as well as on the details of the Monte Carlo simulation. In order to proceed with a conventional statistical analysis, we have to obtain independent samples from by binning the raw samples. The central object in the code for this is the accumulator class. This provides functions to both to estimate $$\tau$$ and to actually bin and store data.

## Averages and Standard Errors¶

For independent samples $$\lbrace x_i\rbrace _{i=0\dots N-1}$$ of the elementary observable $$X$$, estimates of the mean and standard error are:

$\begin{split}\bar{X} \approx \frac{1}{N} \sum_{i=0}^{N-1} x_{i} \\ (\Delta \bar{X})^2 \approx \frac{N-1}{N} \frac{ \sigma^2(x)}{N}\end{split}$

Functions to perform these calculations (including mpi versions), are implemented in the library.

We often want to computing the mean and error for quantities which are derived from our measured and binned data. Consider a general function $$f$$ that depend on several observables $$X,Y,\ldots$$:

$f(\bar{X}, \bar{Y}, \dots)$

Unless $$f$$ is linear, calculating a reliable estimate the error $$\Delta f$$ is a difficult problem, which can be addressed using resampling methods – the simplest of these is the jackknife. We implement the jacknife function (and a MPI version), which allows for general user-specified functions $$f$$ and input data $$X,Y,\ldots$$.

C++ API Reference: