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 *N*_{obs}
→ ∞, 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__: Levy 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__: 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

*σ*_{u} =
[∑_{i}
(∂*u*/∂*x*_{i})^{2
}*σ*_{i}^{2}]^{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 3*n* data points for the fit to
be meaningful.

@ __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"].

**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.

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send feedback and suggestions to bombelli at olemiss.edu – modified 11 jul 2016