Statistics and Error Analysis in Physics |
In General: Data, Fluctuations and Errors
> s.a. particle statistics [spin-statistics]; probability in physics.
* Statistical uncertainties:
They vanish in general for Nobs
→ ∞, except for certain systems said to possess non-averaging
properties, as in random media.
* Epistemic uncertainty:
A kind of uncertainty whose complete probabilistic description is not
available, largely due to incomplete knowledge.
@ Books: Hacking 90;
Roe 92;
Epps 13;
Willink 13;
> s.a. statistics.
@ General references: Herbut a1512 [ensemble theory and experiment].
@ Related topics: Lévy a0804 [use of the median vs the mean in physics];
Ishikawa a1207 [quantum-linguistic formulation];
Chen et al JCP(13)
[epistemic uncertainty, flexible numerical approach for its quantification];
Vivo EJP(15)-a1507 [aspects of Extreme Value Statistics];
> s.a. Benford's Law.
Experimental Errors > s.a. physics teaching.
* Types: They can
be statistical/random or systematic; Errors in reading measuring
instruments can be either type.
* Combining uncertainties:
There is no universally accepted prescription for combining statistical
and systematic errors into one number, so they are usually given separately;
In terms of probabilities, the only way to deal with issues like this one
is to abandon the frequentist view in favor of 'degrees of belief'.
* Variance:
* Confidence interval:
* Error propagation: The rule
σu = [ ∑i (∂u/∂xi)2 σi2 ]1/2
applies to variances of random, uncorrelated variables, not to confidence intervals.
@ Error analysis: Taylor 97;
Silverman et al AJP(04)aug [error propagation];
Berendsen 11;
Nikiforov A&AT-a1306
[algorithm for the exclusion of "blunders"].
Data Analysis, Inference > s.a. Paradoxes.
* Curve fitting:
This is a minimization problem, in which one minimized an error function;
For non-linear curve fitting (non-linear regression) the most widely used
algorithm is the Levenberg-Marquardt method, an iterative one based on
computing the gradient of the error as a function of the parameters in the
fit; As a rule of thumb, if the fit involves n parameter values,
one should have at the very least 3n data points for the fit to
be meaningful.
* Statistical significance:
In particle physics the gold standard for reporting a discovery is obtaining
an experimental result that is 5 standard deviations (5 σ) away from a
theoretical prediction, corresponding to a one-in-3.5-million chance of an
observation being a fluke (3.3 σ corresponds to a one in 1,000 chance,
3.3 σ to a one in 40,000 chance).
@ General references:
Bevan 13 [II].
@ Bayesian:
Lemm 03;
Lee 04;
James 06;
Sivia & Skilling 06 [II].
@ Curve fitting: Sorkin pr(80);
Sorkin IJTP(83)ap/05 [Occam's razor and goodness of fit];
Turney BJPS(90) [balancing stability and accuracy];
Gould ap/03 [linear fits];
Transtrum et al PRL(10) [non-linear fitting process];
Banerji CP(11) [least-squares method];
> for a different, but related concept see Spline.
@ Related topics: Maltoni & Schwetz PRD(03)hp [compatibility of data sets];
Pilla et al PRL(05)phy [signal in noisy background];
Łuksza et al PRL(10) [statistical significance of structures in random data];
Cubitt et al PRL(12) ["extracting dynamical equations from experimental data is NP hard"];
Murugan & Robertson a1904 [topological data analysis, introduction].
Specific Areas and Topics > s.a. correlations;
random processes; stochastic processes.
@ In quantum mechanics: Rylov qp/01;
Rajeev MPLA(03).
@ In astrophysics / cosmology: Szapudi ap/00-proc [variances of correlations];
Hill ap/01-proc [Bayesian statistics in neutrino detection];
Feigelson & Babu ap/04-conf;
Verde a0712-ln,
LNP(10)-a0911;
Feigelson a0903-en [rev];
Heavens a0906;
Madore AJ(10)-1004;
Feigelson & Babu a1205-ch [rev];
Feigelson & Babu 12
[r CP(14)];
> s.a. observational cosmology.
main page
– abbreviations
– journals – comments
– other sites – acknowledgements
send feedback and suggestions to bombelli at olemiss.edu – modified 15 apr 2021