Physics 509C --- Theory of Measurement

(Also known as: how to get the most from your data without making a fool of yourself.)



Term: September 2011
Lecturer: Scott Oser
Class coordinates: Tuesdays/Thursdays, 9:30-11:00 in Hennings 302
Office Hours: Tuesdays 12-1pm, or by appointment
TA:
Shimpei Tobayama

Topics covered: Interpretation of probability; basic descriptive statistics; common probability distributions; Monte Carlo methods; Bayesian analysis; methods of error propagation; systematic uncertainties; parameter estimation; hypothesis testing and statistical significance; confidence intervals; blind analyses; methods of multivariate analysis; non-parametric tests; periodicity searches; "robust" statistics; deconvolution and unfolding

Prerequisites: Officially, none. However, you will be expected to have some facility with computational techniques and programming in a high-level language, or at least a willingness to learn very quickly. Quite simply, it's not possible to do much data analysis or statistics without being able to program. Almost all homework assignments will have a large computational component, although this class will not teach programming per se. If you don't already know basic computational physics, your time might be better spent taking Physics 210 or Physics 410 instead.

Textbooks: There are two textbooks for this course:

  1. Statistics: A Guide to the Use of Statistical Methods in the Physical Sciences, by Roger Barlow

  2. Bayesian Logical Data Analysis for the Physical Sciences, by P.C. Gregory

Each has a different focus with different strengths and weaknesses, and I'll draw on material from both.

Supplemental material: You may also find these books enlightening:


Your grade will be determined by:


Homework

60%

Midterm

20%

Final Exam

20%


Homework:
There will be approximately five lengthy homework assignments. They will generally include analytic calculations, essay-type questions, and computational problems that will require you to analyze data sets and most usually to write some computer code to do so. You are welcome to discuss problems informally with your classmates. However, you must complete the assignment yourself, and if you hand in obviously copied homework, you should expect a mark of zero on that assignment, or worse. Assignments are due by the end of class on the day they are due. I will give more guidelines in class for how to submit completed assignments.


Useful software: This course will require some computational facility on your part. The entire course can be done using free software, and you're not required to buy anything. The most important things you'll need are access to a good plotting package and a library of scientific routines (capable of random number generation, non-linear fitting, and matrix operations at a minimum). I encourage you to use whatever tools your field uses or that you already know, but if you want some recommendations, you may find the following useful:

  1. ROOT: a combined plotting/analysis package developed by the high energy physics community (but of general utility), based on a C++ interpreter. Extremely powerful, with decent tutorials available. Free. Includes most numerical routines you might want, and since it's based on C++ it can work with other libraries or code as well.

  2. gnuplot: a free plotting package with some basic fitting capability (although not enough to do every HW problem). This might be a good option if you're writing standalone code in C/C++/FORTRAN and just need a way of plotting the output.

  3. Mathematica: an integrated plotting and mathematical analysis package. Quite expensive (prohibitively so if you're not a student). A single user license for an old, slow version of Mathematica 4 is available on the main physics server.

  4. GNU Scientific Library: a free library of computational routines. To some extent it is a freeware equivalent of the routines in Numerical Recipes. Available in C and C++.

  5. Numerical Recipes: Very commonly used. Although the text of the book is available online for free, the routines are proprietary, and you're supposed to buy the book if you use any of the routines. Available in C, C++, and FORTRAN.


Programming languages: It's up to you to choose what programming language you feel most comfortable with. These days I'd generally recommend C++, as it is quickly coming to dominate many areas of the physical sciences, and most libraries of scientific routines are available in that language. But I will confess that I am personally still much more fluent in FORTRAN and C. If you want to use something besides C++, C, or FORTRAN, please be my guest, but I won't be able to offer you much specific advice on coding issues. Not that I will anyway.


Missed exams: There will be one in-class midterm exam. If you miss the exam with a legitimate excuse (proof of illness, family emergency, etc), see me to discuss make-up options.


Religious holidays: Students are entitled to request an alternate test date if a scheduled test date falls on one of their holy days. If you think this may apply to you, please contact me as soon as possible to make an alternate arrangement. Please don't put this off until the last minute---you must give at least two week's notice.

FINAL EXAM:
The final exam is not yet scheduled. A take-home exam is a likely possibility.


Syllabus
: A tentative lecture schedule follows. It will almost certainly be adjusted as the course proceeds.


Lecture #

Date

Topics Covered

Reading Material
(Textbook Sections)

Assignment Due (tentative)

1

9/6

First day of class. Introduction; Interpretations of probability

B7.1; G1.1-1.4



9/8

NO CLASS --- TA training day



2

9/13

Basic descriptive statistics; random variables; Gaussian and binomial distributions

B2.1-2.6; B3.1-3.2


3

9/15

Poisson, exponential, and chi^2 distributions; mathematics of manipulating probability distributions

B3.3-3.5


4

9/20

Monte Carlo and basic computational methods: random number generation, minimization routines, coding hints

B10.1-10.4

HW1

5

9/22

Intro to Bayesian analysis: general principles, basic applications, contrast with frequentist approach, nuisance parameters and systematic uncertainties

G Ch 3-4



9/27

NO CLASS - INSTRUCTOR TO BE ABDUCTED BY ALIENS




9/29

NO CLASS - INSTRUCTOR TO BE ABDUCTED BY ALIENS



6

10/4

Bayesian analysis: choice of priors, maximum entropy principles

G Ch 4, 8


7

10/6

The central limit theorem; the Chebyshev limit; covariance matrices and multidimensional Gaussian distributions

B4.1-4.4


8

10/11

Estimators I: introduction & maximum likelihood method

B5.1-5.4

HW2

9

10/13

Estimators II: least squares methods

B5.5-5.6, B6.1-6.7


10

10/18

Error propagation methods: meaning and interpretation of error bars; the error propagation equation; dealing with correlations; handling asymmetric and non-Gaussian errors

B4.1-4.4



10/20

IN-CLASS MIDTERM EXAM



11

10/25

Systematic Uncertainties I: distinction or lack thereof between statistical and systematic uncertainties; Monte Carlo evaluation; covariance matrix approach

B4.1-4.4


12

10/27

Systematic Uncertainties II: the pull method/"floating systematics", how to evaluate systematics; common mistakes in systematic error propagation

B4.1-4.4


13

11/01

Hypothesis/significance testing I: introduction, interpretation, significance and power, Neyman-Pearson lemma; trials factors

B8.1-8.2.2


14

11/03

Hypothesis/significance testing II: likelihood ratio test, goodness of fit, Kolmogorov-Smirnov tests, the two-sample problem and the t-test

B8.2.3-8.4

HW3


11/08

DISCUSSION DAY



15

11/10

Bayesian analysis: Numerical methods---Laplace's approximation, methods of marginalizing over nuisance parameters, numerical integration, Markov Chain Monte Carlo and the Metropolis-Hastings algorithm

G11-12


16

11/15

Confidence regions: Bayesian and frequentist interpretations; non-physical regions; Feldman-Cousins confidence intervals

B7.2, this paper


17

11/17

Multivariate analysis: linear Fisher discriminants; likelihood ratio approximations; decision trees; machine learning

class notes

HW4

18

11/22

DISCUSSION DAY. Also, please read the attached notes and paper on blind analyses.

supplemental


19

11/24

Non-parametric tests: sign test for the median; the Mann-Whitney test; matched pairs; Spearman's correlation coefficient; run tests

B8.3.2-8.3.3, B9.1-9.3


20

11/29

Robust methods of parameter estimation; bootstrap method

Numerical Recipes 15.7; class notes


21

12/01

Deconvolution and unfolding

LAST DAY OF CLASS

class notes; see also supplemental text Cowan, Ch 11.

HW5

22


Periodicity studies

BONUS NOTES


G Appendix B, G Ch 13, + this paper

Bayesian analysis essay, with critics' comments attached




Scott Oser (email me) June 22, 2011