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G. Cowan Lectures on Statistical Data Analysis Lecture 2 page 1
Statistical Data Analysis: Lecture 4
1 Probability, Bayes’ theorem, random variables, pdfs 2 Functions of r.v.s, expectation values, error propagation 3 Catalogue of pdfs 4 The Monte Carlo method 5 Statistical tests: general concepts 6 Test statistics, multivariate methods 7 Goodness-of-fit tests 8 Parameter estimation, maximum likelihood 9 More maximum likelihood 10 Method of least squares 11 Interval estimation, setting limits 12 Nuisance parameters, systematic uncertainties 13 Examples of Bayesian approach 14 tba 15 tba
What it is: a numerical technique for calculating probabilities and related quantities using sequences of random numbers. The usual steps: (1) Generate sequence r 1 , r 2 , ..., r m uniform in [0, 1]. (2) Use this to produce another sequence x 1 , x 2 , ..., x n distributed according to some pdf f ( x ) in which we’re interested ( x can be a vector). (3) Use the x values to estimate some property of f ( x ), e.g., fraction of x values with a < x < b gives → MC calculation = integration (at least formally) MC generated values = ‘simulated data’ → use for testing statistical procedures
The Monte Carlo method
Random number generators (2)
The sequence is (unfortunately) periodic! Example (see Brandt Ch 4): a = 3, m = 7, n 0
← sequence repeats Choose a , m to obtain long period (maximum = m 1); m usually close to the largest integer that can represented in the computer. Only use a subset of a single period of the sequence.
Random number generators (3)
are in [0, 1] but are they ‘random’? Choose a , m so that the r i pass various tests of randomness: uniform distribution in [0, 1], all values independent (no correlations between pairs), e.g. L’Ecuyer, Commun. ACM 31 (1988) 742 suggests a = 40692 m = 2147483399 Far better generators available, e.g. TRandom3 , based on Mersenne twister algorithm, period = 2 19937 1 (a “Mersenne prime”). See F. James, Comp. Phys. Comm. 60 (1990) 111; Brandt Ch. 4
Example of the transformation method
Exponential pdf: Set (^) and solve for x ( r ). → works too.)
The acceptance-rejection method
Enclose the pdf in a box: (1) Generate a random number x , uniform in [ x min , x max ], i.e. r 1 is uniform in [0,1]. (2) Generate a 2nd independent random number u uniformly distributed between 0 and f max , i.e. (3) If u < f ( x ), then accept x. If not, reject x and repeat.
Improving efficiency of the
acceptance-rejection method
The fraction of accepted points is equal to the fraction of the box’s area under the curve. For very peaked distributions, this may be very low and thus the algorithm may be slow. Improve by enclosing the pdf f ( x ) in a curve C h ( x ) that conforms to f ( x ) more closely, where h ( x ) is a pdf from which we can generate random values and C is a constant. Generate points uniformly over C h ( x ). If point is below f ( x ), accept x.
Monte Carlo event generators
Simple example: e
e → μ
μ
Generate cos θ and φ:
Less simple: ‘event generators’ for a variety of reactions: e
e
- → μ + μ , hadrons, ... pp → hadrons, D-Y, SUSY,... e.g. PYTHIA, HERWIG, ISAJET... Output = ‘events’, i.e., for each event we get a list of generated particles and their momentum vectors, types, etc.
Monte Carlo detector simulation
Takes as input the particle list and momenta from generator. Simulates detector response: multiple Coulomb scattering (generate scattering angle), particle decays (generate lifetime),
ionization energy loss (generate Δ),
electromagnetic, hadronic showers, production of signals, electronics response, ... Output = simulated raw data → input to reconstruction software: track finding, fitting, etc. Predict what you should see at ‘detector level’ given a certain hypothesis for ‘generator level’. Compare with the real data. Estimate ‘efficiencies’ = #events found / # events generated. Programming package: GEANT
Wrapping up lecture 4
We’ve now seen the Monte Carlo method: calculations based on sequences of random numbers, used to simulate particle collisions, detector response. So far, we’ve mainly been talking about probability. But suppose now we are faced with experimental data. We want to infer something about the (probabilistic) processes that produced the data. This is statistics, the main subject of the following lectures.
“True” random numbers
1955 the RAND Corporation published a book of random numbers generated with an “electronic roulette wheel”, based on random frequency electronic pulses. You can download all 1,000,000 of them (and buy the book) from www.rand.org.
