Modern Methods of Data Analysis - WS 07/08

Transcription

1 Modern Methods of Data Analysis Lecture III ( ) Contents: Overview & Test of random number generators Random number distributions

2 Monte Carlo The terminology Monte Carlo-methods originated around 1940, related to the construction of the atomic bomb. Random processes were for the first time extensively used to theoretical predict the interaction of neutrons with matter.

3 History: Buffon's Needles (1777) Determine the value π by throwing needles (length l) on a plate width (l) 2 crossing needles k p = = --- = --all needles n π Numerical Problems can be solved approximately by statistics!

4 Monte Carlo: A method of statistical experiments Often analy. treatment of physical problem difficult/impossible Must do approximation <- or -> statistical description (MC) Application: High energy physics Numerical calculations (Integration,Differentiation coding/encoding e.g. ssh-connection Reliability tests Investment banking Earth sciences (earthquakes) Method: Find statistic model Find sequence of random numbers Calculate estimators from random quantities

5 Random Numbers First look at random numbers distributed uniform in [0,1] : U(0,1) U(0,1) random numbers shall have the correct size of fluctuations for a uniform distribution real random numbers can be extracted from physics processes radioactive decays thermal noise over a resistor alternatively on can use pseudo random numbers generation via numerical algorithms not truly random, but hopefully indistinguishable from true random numbers available any time and with a reproducible sequence important for debugging purposes

6 Historical Example Von Neumann's mid-squares generator Start with m-digit number seed. Calculate seed*seed and take the central m digits as new seed. Normalization of seed to the range [0,1] yields a pseudo-random sequence

7 Linear Congruential Generators (LCG) x(n+1) = (a * x(n) + b) mod m in [0, m-1] The properties/quality of the generator is determined by the parameters a,b and m Maximal period m is reached only for good choice of a,b and m -> distribution of single values: Example: a = 1601; b = 3456; m = 10000;

8 Properties of LCG: LCG random numbers are located on hyperplanes # of hyperplanes if function of dimension d of representation minimal requirements for good generator number of hyperplanes as large as possible period as long as possible both characteristics grow with the number of bits per integer & choice of parameters a,b,m important Mathematical max. nb. of hyperplanes:

9 RANDU Generator 1960's: RANDU (IBM, 32bit): m =, a=65539, c=0; seed: n = 1 RANDU (10000 numbers) high periodicity, however very low rate of hyperplanes many scientific studies suffered from this!

10 RANLUX Generators based on Marsaglia-Zaman algorithm mathematically equivalent to a LCG completely different implementation [hep-lat/930902] RANLUX implementation: with m = z[n] = (a * z[n-1]) mod m (prime) and a = Effectively the generator is based on 576-bit integer variables. Period: ; # of hyperplanes for d=100: However: a has only very few bits set, subsequent 576-bit states are still correlated. RANLUX-luxury-level 4, therefore discards 8760 bits of the bit stream produced by the generator, before the next 576 bits are accepted and convert them to 24 single precision floating point numbers.

11 Combining Several Generators Two standard procedures to combine two generators: combination: two random numbers are generate with two separate generators. New random number obtained by combination, e.g. via +, - or bit wise exclusive OR mixing: an array is filled with random numbers from one generator. Random number from second generator determine order in which random numbers are used. Naively one could think more complex functions yield more randomness, this is not true! [D.E. Knuth: The Art of Computer Programming]

12 Generator by L'Ecuyer combination of two LCG: x[n+1] = (a1 * x[n]) mod m1 y[n+1] = (a2 * y[n]) mod m2 z[n+1] = (x[n+1]+y[n+1]) mod m1 with following choice of variables: a1 = 40014; a2 = 40692; m1 = = * a m2 = = * a with period m1, m2: m1 = 2 * 3 * 7 * 631 * m2 = 2 * 19 * 31 * 1019 * 1789 z has period of and significantly larger nb. of hyperplanes than individual generators x, y.

13 State of the Art 1998 by Makoto Matsumoto and Takuji Nishimura Marsenne Twister algorithm based on bit-shifting bit long seed value, stored in a 624 long array with 32 bit values (31 bits are free) is a Marsenne prime number if a shift register as a size which is a Marsenne prime the testing procedure for periodicity is significantly simplified periode: in decimals: up to 623 dimension maximal # of hyperplanes it is fast! Recommended generator!

14 Marsenne Twister (MT)

15 Random Generators in ROOT TRandom (LCG,, known to have correlations!) 34 ns/call TRandom1 (RANLUX, math prooven random properties, 242 ns/call TRandom2 (based on l'ecuyer, ) 37 ns/call TRandom3 ) recommended one! (Mersenne Twister, good random properties 45 ns/call )

16 Test of Random Generators (I) Test for distance of hyperplanes of random numbers Test for distance of hyperplanes of last digits of random nb. Sequence-(up-down-) test: compare and. Set bit 0/1 according to or, count how often 0 and 1 are in a row of k numbers. For N truly uniformly distributed numbers we expect:

17 Test of Random Generators (II) Test for uniform distribution: [0,1] is split in k subintervals of same size. N random numbers are generated. Compare expected number of events (N/k) to number of events in interval i. Sum the deviation divided by the expected fluctuation (σ² =N/k) in quadrature, compare to expected distribution. Random Walk-test: choose 0 < α «1, generate large set of r.nb., count numbers below α, compare to expected number many more tests: Most try to compare generated distribution to true random distribution.

18 Warning Every Random generator failing those tests is a bad one Truly comprehensive tests are impossible. Use professional random generators! Use nontrivial seeds! However, keep a little suspect in mind.

19 Random numbers for any distribution given distribution f(x), F(x) = F(x) = u is analytically invertible: x = if u[i] uniform in [0,1], then follows pdf f(x) Method is applicable, if F(x) and analytical solvable

20 Example: Exponential Distribution Simulate random numbers according to: x>0 f(x,λ) = 0 x 0 Computation of inverse pdf: thus random numbers x[i] = - ln (u[i])/λ with u[i] uniform distribution in [0,1] follow f(x,λ)!

21 Hit - & Miss Method (von Neumann) If is not computable, then use Hit & Miss: f(x) pdf in [a,b] with maximum fmax generate uniform distributed random numbers x[i] in [a,b] generate second uniform distributed random numbers y[i] in [0, fmax] accept x[i] if y[i] < f(x[i]) else reject x[i] fmax f(x) a b

22 Modified Hit & Miss Especially with extreme functions, the method is inefficient. Improve by selecting invertible function f'(x) with according F(u) and f'(x) f(x). generate x[i] according f'[x] generate y[i] in [0,1] accept x[i] if y[j]*f'(x[j]) < f(x[j]) else reject For often used pdf (e.g. Gaussian distribution), there are optimized algorithms

23 Box-Muller-Transformation (I) not analytical solvable... x,y two independent Gaussian distributions

24 Box-Muller-Transformation (II) simulate uniform distribution u1 in [0,1] and u2 [0,2π]] x,y independent Gaussian distributed random variables

25 MC Methods: Integration (I) f(x) = 1/(a-b) a < x < b 0 elsewhere xi: uniformly distributed random numbers in [a,b]

26 Monte Carlo Methods: Integration (II)

27 Reduction of V[g(x)] (I) Stratification: c = 0.5 (a+b)

28 Reduction of V[g(x)] (II) Importance Sampling: f(x) normalized in [a,b], integratable & invertible for probability density f(x) x random numbers in [a,b] according to f(x) small if g(x)/f(x) almost constant alternative g(x) = g'(x) + f(x); F(x) analytical solvable;

29 Example in two dimensions Easy to get caught in a trap x uniform in [0,1] y uniform in [0,x] Accept only if y<x Alternative efficient algorithm: u uniform in [0,1], v uniform in [0,1] x = max(u,v); y = min(u,v)

30 Comparison to numerical Integration Trapezoidal rule: Error: 1/n² Simpson's rule: Error: 1/n4 In 1-D, conventional methods are always better than MC ( In d-d, trapezoidal rule: Simpson's rule: )

31 Quasi Random Numbers smaller variance, better equally distributed exploit stratification convergence [however not yet fully understood]

32 MC Methods: Example - Ks Mesons A source of Ks mesons, with mass M=0.498 GeV, produces a narrow monoenergetic beam of particles with energy E = GeV. Most Ks decay with an average lifetime s, or cm, into pairs. In the rest system of the Ks the decay is isotropic. Located at a distance of 14 cm behind the source is a silicon detector to register the decay pions. The detector is a circular disk with radius R=7cm, centered on the Ks beam. Determine the probability that both pions from the decay hit the detector.

33 Analytical Calculation parametrize the decay in the Ks rest frame perform Lorentz boost of decay pions into the lab-system boost parameter is γ = E/M describe the flight distance z by an exponential distribution uniform in φ the beam direction uniform cosine of polar angle of the beam uniform 0 < cosθ < 1 probability density ρ(z) = 1/(<z>) exp(z/<z>) in the lab system one finds <z> = <ct> 4.07 cm determine as a function of the decay configuration the allowed z range, from which both pions reach the detector

34 Direct Simulation (I) generate decay position according to ρ(z) generate isotropic decays in the Ks rest frame perform Lorentz transformation to the lab system analyze the events, check weather pions hit detector count as success cases where both pions hit the detector count as failure if at least one particle misses Repeat N times (p = n/n): e.g. given N= generated events & n= successes: A = ± 405 versus analytical result A =

35 Direct Simulation (II): easy to adapt to more complex cases... other Ks energies an energy spectrum rectangular detectors introducing Φ dependence finite efficiency of detector direct simulation allows detailed studies albeit with finite statistical precision

36 More complex in real life (I) modern high energy experiments consist of several detectors-system CMS (LHC): silicon strip detectors with about read out channels Simulation of physics processes: [Pythia, Sherpa, Herwig, EvtGen, Bgen] all produced particles, expected energy, momentum, angular distribution. Interaction with material: [Geant] energy deposit, multiple scattering, secondaries detection efficiency for each detector component [experiment specific] position and energy resolution of each component [experiment specific]

37 More complex in real life (II) End of simulation (1-20 minutes per event!), list of digitalized signals from the various detector components. No difference to real data! Simulation used to: - develop reconstruction algorithms. (particle trajectories in the tracking system, shower in the calorimeters) - optimize trigger selection - optimize reconstruction of signal signature (combination of all detector components) - determine detector acceptance Simulation crutial for RND (research and development planing phase) of experiments, for preparation of data taking, for optimizing analysis and for evaluating significance of result.

38 More complex in real life (III) CDF: proton-antiproton accelerator at the Tevatron (Fermilab/USA) In average 100 observed particles. 3-Jet event