Everything You Wanted to Know About Data Analysis and Fitting but Were Afraid to Ask

88 Pages·2015·1.43 MB·English

by  Peter Young (auth.)

SPRINGER BRIEFS IN PHYSICS
Peter Young
Everything You
Wanted to Know
About Data
Analysis and
Fitting but Were
Afraid to Ask
SpringerBriefs in Physics
Editorial Board
Egor Babaev, University of Massachusetts, Massachusetts, USA
Malcolm Bremer, University of Bristol, Bristol, UK
Xavier Calmet, University of Sussex, Sussex, UK
Francesca Di Lodovico, Queen Mary University of London, London, UK
Maarten Hoogerland, University of Auckland, Auckland, New Zealand
Eric Le Ru, Victoria University of Wellington, Wellington, New Zealand
Hans-Joachim Lewerenz, California Institute of Technology, Pasadena, USA
James Overduin, Towson University, Towson, USA
Vesselin Petkov, Concordia University, Concordia, Canada
Charles H.-T. Wang, University of Aberdeen, Aberdeen, UK
Andrew Whitaker, Queen’s University Belfast, Belfast, UK
More information about this series at http://www.springer.com/series/8902
Peter Young
Everything You Wanted
to Know About Data
Analysis and Fitting
but Were Afraid to Ask
123
PeterYoung
Physics Department
University of California
SantaCruz, CA
USA
ISSN 2191-5423 ISSN 2191-5431 (electronic)
SpringerBriefs inPhysics
ISBN978-3-319-19050-1 ISBN978-3-319-19051-8 (eBook)
DOI 10.1007/978-3-319-19051-8
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These notes discuss, in a style intended for physicists,
how to average data and fit it to some functional form.
I try to make clear what is being calculated, what
assumptions are being made, and to give a derivation
of results rather than just quote them. The aim is put
a lot of useful pedagogical material together in a
convenient place.
Acknowledgments
I am grateful to Alexander Hartmann for inviting me to give a lecture at the Bad
Honnef School on “Efficient Algorithms in Computational Physics” in September
2012, which provided the motivation to write up a first version of these notes, and
also for several very helpful discussions.
In addition, I would like to thank Matt Wittmann for helpful discussions on
fitting and data analysis using python and for permission to include his python
codes. I am also grateful to Wittmann and Christoph Norrenbrock for helpful
comments on an earlier version of the manuscript. This work is supported by the
Humboldt Foundation, and by the NSF through grant DMR-1207036.
vii
Contents
1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
Reference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Averages and Error Bars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 Basic Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Advanced Analysis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.2.1 Traditional Method . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2.2 Jackknife . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.3 Bootstrap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.2.4 Jackknife or Bootstrap? . . . . . . . . . . . . . . . . . . . . . . . . 19
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3 Fitting Data to a Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.1 Fitting to a Straight Line. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
3.2 Fitting to a Polynomial. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.3 Error Bars. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3.4 Interpolating . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
3.5 Fitting to a Non-linear Model . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.6 Confidence Limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.7 Confidence Limits by Resampling the Data. . . . . . . . . . . . . . . . 36
3.8 A Tale of Two Probabilities. When Can One Rule Out a Fit? . . . 37
3.9 Model Selection (i.e. How to Avoid Over-Fitting):
Maximum Likelihood Versus Bayes. . . . . . . . . . . . . . . . . . . . . 39
3.9.1 Maximum Likelihood. . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.9.2 A Bayesian Approach. . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.9.3 Conclusions for Model Selection . . . . . . . . . . . . . . . . . . 52
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
ix
x Contents
Appendix A: Central Limit Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 55
Appendix B: The Number of Degrees of Freedom. . . . . . . . . . . . . . . . 61
Appendix C: The Chi-squared Distribution and the Goodness
of Fit Parameter Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
Appendix D: Asymptotic Standard Error and How to Get
Correct Error Bars from Gnuplot . . . . . . . . . . . . . . . . . 67
Appendix E: The Distribution of Fitted Parameters
Determined from Simulated Datasets . . . . . . . . . . . . . . . 71
Appendix F: The Distribution of Fitted Parameters
from Repeated Sets of Measurements . . . . . . . . . . . . . . . 73
Appendix G: Fitting Correlated Data. . . . . . . . . . . . . . . . . . . . . . . . . 75
Appendix H: Scripts for Some Data Analysis
and Fitting Tasks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Chapter 1
Introduction
Thesenotesdescribehowtoaverageandfitnumericaldatathatyouhaveobtained,
presumablybysomesimulation.
Typically you will generate a set of values x , y ,..., i = 1,...N, where N
i i
is the number of measurements. The first thing you will want to do is to estimate
various average values, and determine error bars on those estimates. As we shall
see, this is straightforward if one wants to compute a single average, e.g. (cid:2)x(cid:3), but
notquitesoeasyformorecomplicatedaveragessuchasfluctuationsinaquantity,
(cid:2)x2(cid:3)−(cid:2)x(cid:3)2,orcombinationsofmeasuredvaluessuchas(cid:2)y(cid:3)/(cid:2)x(cid:3)2.Averagingofdata
willbediscussedinChap.2.
Havingobtainedseveralgooddatapointswitherrorbars,youmightwanttofit
thisdatatosomemodel.Techniquesforfittingdatawillbedescribedinthesecond
partofthesenotesinChap.3.
I find that the books on these topics usually fall into one of two camps. At one
extreme,thebooksforphysicistsdon’tdiscussallthatisneededandrarelyprovethe
resultsthattheyquote.Attheotherextreme,thebooksformathematicianspresum-
ablyproveeverythingbutarewritteninastyleoflemmas,proofs,(cid:2)’sandδ’s,and
unfamiliarnotation,whichisintimidatingtophysicists.Oneexception,whichfinds
agoodmiddleground,isNumericalRecipes[1]andthediscussionoffittinggiven
hereiscertainlyinfluencedbyChap.15ofthatbook.InthesenotesIaimtobefairly
complete and also to derive the results I use, while the style is that of a physicist
writingforphysicists.Ialsoincludescriptsinpython,perl,andgnuplottoperform
certaintasksindataanalysisandfitting.Forthesereasons,thesenotesareperhaps
ratherlengthy.Nonetheless,Ihope,thattheywillprovideausefulreference.
Reference
1. W.H.Press,S.A.Teukolsky,W.T.Vetterling,B.P.Flannery,NumericalRecipesinC,2ndedn.
(CambridgeUniversityPress,Cambridge,1992)
©TheAuthor(s)2015 1
P.Young,EverythingYouWantedtoKnowAboutDataAnalysis
andFittingbutWereAfraidtoAsk,SpringerBriefsinPhysics,
DOI10.1007/978-3-319-19051-8_1