Exercise 3: Random Numbers and Monte Carlo

Transcription

1 Exercise : Random Numbers and Monte Carlo J. F. Ider Chitham Astrophysics Group, H.H. Wills Physics Laboratory, University of Bristol, Tyndall Avenue, Bristol BS8 TL Accepted March. Received March ; in original form March ABSTRACT This exercise address the use of random numbers in Monte Carlo techniques. These are often the fastest or most straightforward way of analysing complex problems in computational analysis and physical simulations. The degree of success is heavily reliant on the availability and reliability number generators. INTRODUCTION Most Monte Carlo sampling or integration techniques assume a random number generator; a routine that returns a random number drawn from a specified probability distribution. Consider the generation of uniform and statistically independent values x gen, on the open interval (, ). Effective routines should incorporate the following properties: (i) Unpredictability: if the last N values returned by the routine are known, the preceding returned values should not be easily predicted. (ii) Correct coverage: the frequency with which any x gen is generated must be proportional to the underlying probability density function (PDF) P (x). (iii) Uniqueness: True randomness is seen in many aspects of physics, for example constructing a circuit that measures the Johnson noise in a resistor, and using the measurement to generate a value would give a uniquely genuine random number, however this is idealisation is difficult to simulate computationally. Many algorithms are in fact pseudo-random number generators (PRNGs) which are unpredictable, but reproducible. These routines can be reset and the exact same sequence of random numbers can be yielded []. True randomness can however be approximated by seeding the generating algorithm with time to eliminate periods of sequential repetition. This can provide uniquely ordered values as the exact moment the generating function is called within the source code is effectively random to the precision of a computational clock.. Central Limit Theorem In probability theory, the central limit theorem (CLT) states that, the mean of a sufficiently large number of mutually independent random variables, will be approximately normally distributed (in the limit that the number of independent events tending to infinity) providing they have a well-defined expectation value and variance []. A Poisson distribution describing the probability of N discrete events occurring tends to a continuous function known as a normal distribution: lim P (n, µ) N (n, µ, σ) () as shown in Figure. n. [ ] µ n e µ lim n n! σ /σ π e (n µ) () P(n) n Figure. Poisson probabilities for means of µ = {.5, }, note orgin of gaussian resemblence emerging at N = independent events. Frequent observations of discrete distributions tending to relatively continuous ones at large numbers iterations can be seen throughout this exercise according to CLT. TASK A uniform random distribution, P (θ) which generates angles from ( < θ gen < ) must be converted to a non-uniform one, proportional to the sine of the transformed angle P (θ ) sin(θ ) with ( < θ gen < π). Providing both distributions are normalized to unit area () (N.B required factor of / ) conversion is possible via analytical inversion and accept-reject routines using contrasting methods of coordinate transformation. θ gen sin(θ )dθ = θgen n P (θ)dθ () θ gen = Q(θ req) θ req = Q (θ gen) (). Inverse Transform The inversion method is exact when an explicit form of Q is known and most convenient when the inverse function of the indefinite integral of P can be manually calculated. This is the case for some common functions such as sin(θ) (). Very commonly an analytic form for Q is unknown or too complex to work with, so

2 J. F. Ider Chitham Table. Comparative summary of goodness of fit analysis Number Of Random Numbers, N χ inv χ a/r that obtaining an inverse analogously is impractical. θ gen = cos(θ gen). Acceptance-Rejection θ gen = arccos( θ gen) Suppose that for any given value of θ, we can enclose P (θ) entirely inside a shape which is π times R(θ), a uniform random distribution with ( < θ gen < ). To generate P (θ), θ gen is generated according to R(θ) then P (θ) is calculated along with the height of the envelope; another random angle, ( < ϕ < ) is generated and then tested if ϕπr(θ) P (θ) to determine whether or not the angle will contribute to the distribution. If so, accept θ; if not reject until points populate the entire area πr(θ) in a smooth manner []. () P (θ ) P (θ ) Inverse Reject-Accept fiti fit ra π/ π/ π/ π θ (rad) COMPARATIVE ANALYSIS Each technique is successful to a consistently similar extent, with the fitted curves being visually indistinguishable from each other at every logarithmic interval of N shown in Figure.. This is confirmed via analysis of fit quality in terms of Pearson s χ whereby a more reliable (mean) value is deduced by averaging over randomly generated P (θ ) distributions for each value of N, a sample of results is summarised in Table.. Despite this each χ will still change slightly due to random variation making comparative analysis difficult although the general trend for both appeared to be χ as N as expected. The goodness of fit however, is of varying quality across each bin in the distribution as the angles in each bin will be normally distributed within their respective bins and the standard error of the mean will vary across the distribution as shown in Figure.. This was further investigated by calculating the deviation from the ideal fit χ as a function of N as illustrated in Figure. over the range 5 N. The deviation from χ = for the analytical inversion transformation method appeared relatively superior on average, indicated by the greater degree of clustering approaching the horizontal axis. This was unexpected however can be explained via the contrasting computational times required to generate N random angles (Figure. 5) as faster algorithms will have a higher degree of similarity between consecutively (time) generated seeds. The linear fit present in the computational time analysis suggests an exponential increase in cumulative generation time as N increases, the relative gradients indicate the generation rate via the reject-accept method is slower. This is to be expected as not every generated angle will contribute to the distribution. The ratio of the total number of generations to the number that successfully contribute to the distribution, η should increasingly resemble the ratio of integrated areas under the distribution and the total, η = /π as the number of random numbers is increased (5). η = N N i N dis i lim N η η = η = π (5) P (θ ). Inverse Reject-Accept fiti fit ra π/ π/ π/ π θ (rad). Inverse Reject-Accept fiti fit ra π/ π/ π/ π θ (rad) Figure. Histograms of P (θ ) (normalized frequency) for inverse transform (red) and reject-accept (blue) methods with fitted sine profiles for N = {,,5,6} (goodness of fit summarised in Table..) The deviation between discrete and integral solutions η = η η, was investigated via a percentage difference as a function of N as summarised by Figure. 6. The difference between each run emphasises the true randomness of the generator whilst the tendency towards continuity outlines the basis of the central limit theorem as mentioned in.. Despite more in depth analysis suggesting the analytical transform method is superior in terms of precision, computational time and number of iterations, this is to be expected considering the simplistic form of P (θ ). If the PDF is more complex inversion may not be a viable option meaning each method is appropriate for dif-

3 Exercise : Random Numbers and Monte Carlo. Standard Error. e-5 η / η (%) e Bin N Figure. Standard error σ N = σ/ N, accross each bin in a N = 5 analytical transformation distribution as shown in Figure.. χ Inverse Transform Reject-Accept Figure. Deviation of mean χ (each averaged over random P (θ ) distributions) from its ideal value as a function of N for for inverse transform (red) and reject-accept (blue) methods. N Figure 6. Percentage difference between discrete and integral ratios of areas, η of P (θ ) for the accept reject method (5 random runs). Note logarithmic axis and approximate exponential drop off as N. P (x) x N= N= N= 5 N= 6 Figure 7. Histograms of Gaussian transformed distribution (via reject accept see ) P (x ) (mean µ =.5, standard deviation σ =.5) for N = {,,5,6}, N bins =. Computational Time (ms).. Inverse Reject-Accept ferent situations (reject-accept proved more applicable for a Gaussian PDF, Figure. 7). Occasionally a combination of the two is required for example, how would you generate random numbers from the distribution: f(θ) = + cos(θ)/ θ. Accept-reject method is not possible due to the asymptote at θ = (see Figure. 8) nor is the transform method because the integral is not easily calculated, a hybrid of the two is required log (N) Figure 5. Computational time to generate log N random angles within the normalized distribution P (θ ) for inverse transform (red) and rejectaccept (blue), note logarmic time axis, note more significant contrast at large N and approximatley constant gradient at N. PHYSICS PROBLEM A beam of unstable, unpolarised nuclei are initially injected into a m cavity perpendicular to an electromagnetic calorimeter array at ms (relativistic effects can be considered negligible as β << ). The radioactive decay rate is well represented by an exponential random probability distribution with a mean lifetime of τ = 5µs (mean path length of l =.m) producing isotropic gamma ray emission. In terms of spherical polar coordinates with respect to the x-y plane of the detector array and the z-axis defined by the path of the nuclei, this translates to a random distribution

4 J. F. Ider Chitham Original Smeared f(θ) θ (rads) Figure 8. f(θ) = + cos(θ)/ θ of azimuth ϕ, and polar θ angles (see.). For a sample of N nuclei at t =, the predicted number decaying in the interval from t to t + dt is given by dn(t) = N τ [exp( t/τ)]dt. To more realistically simulate an electromagnetic calorimeter, a quantum efficiency ɛ, was implemented via the inclusion Bernoulli random distribution (6). This is either or depending on the relative success or failure of photon detection on incidence as determined by a the sample mean probability p (equivalent to the quantum efficiency of each pixel). { f(count, p = ɛ) = p count = p count = This allows the number of detected gamma rays dn γ in the same time interval to be predicted (7), which can be alternately written in terms of displacement (t z, τ l).. Gaussian Smearing (6) dn γ(t) = ɛn τ e t/τ dt (7) When a photon is incident on a (finitely sized) pixel its exact location of incidence across the pixel can not be determined with certainty, therefore an additional probability distribution is required to simulate this limitation of precision. This was achieved via Gaussian smearing with the resolution of the detector array, x res = cm and y res = cm taken as the respective one dimensional full width at half maximum (FWHM) components of the two dimensional Gaussian spread. Combining the FWHM and the mathematical definition of a Gaussian distribution () an expression for the standard deviation was derived, σ = F W HM/ ln. The original Cartesian coordinates of the incident γ-ray (8) were altered via the addition of a Gaussian GSL RNG centred at µ =. x s = x + N x(x, σ x, µ x = ) y s = y + N y(y, σ y, µ y = ) A preliminary investigation was carried out to ensure that the influence of Gaussian smearing was fully functional. Figure. 9 shows a definite alteration in apparent position post smearing, pairs of slightly differing points can easily be identified around the periphery although difficult to distinguish in the central vicinity. There appears to be more significant smearing in the y direction, this is to be expected to be on average a factor more than in the x direction due to the respective ratio of FWHM s. The sample data (8) Figure 9. Sample of the cartesian coordinates corresponding to detected photons (ɛ =.) over a restricted range to highlight the effect of Gaussian smearing, orginal poistions shown in (red) and smeared in (blue) Table. Sample of the mean x, y alterations < x >, < y > and thier respective ratios post Gaussian smearing for a range of decay numbers, N. N.B. Expections: [< y >/< x >] [y res/x res] ( ) N < x > ( m) < y > ( m) < y >/< x > in Table. confirms this tendency to the expected ratio as N.. Analysis An isotropic distribution requires the frequency of events to be proportional to solid angle Ω, the differential element of which is dω = sin(θ)dϕ = d(cosθ)dϕ, resulting in a uniformly spherical distribution. Ω frequencies are shown in Figure. and appear approximately uniform across all bins. In theory an infinite detector array should measure 5% of detections (discounting those with beyond the detector z > m) at a perfect quantum efficiency as statically half of the photons will propagate in directions with θ π/ for a sufficiently large number of decays ( O ). This is shown in Figure. as the detection fraction.5 as the detector becomes increasingly extensive in the x-y plane and ɛ. providing additional confirmation of isotropy. The distributions in Figure. resemble expectations and are consistent with Figures. and, depicticing increasingly extensive and isotropic photon detection as N increases. Symmetrical uniformity and isocontour definition (at base of each each D surface plot) illustrate the convergence toward continuity. The effect

5 Exercise : Random Numbers and Monte Carlo logn Ntotal = logn Ntotal = logn logn Ntotal = 5 Ntotal = logn Ntotal = logn Ntotal = logn logn Ntotal = 7 Ntotal = Figure. D (left) and D (right) spacial distributions of logarithmic photon detection frequency log (N ) across a m m detector array consisting of a square pixels (each with =.7, xres = cm and yres = cm) for a range of total number of decays Ntot = {,5,6,7}. Isocontours depict central elliptical aberation of the form (x/a) + (y/b) =, with eccentricity of e /.

6 6 J. F. Ider Chitham N=x 5 N=x 5 N=x 5 N=x 5 N=5x 5 Frequency π π π π Ω (sr) Figure. Histograms of solid angle Ω = πsin (θ/), illustrating an approximate increasingly isotropic distribution for N = {,,,, 5} 5. Detection Fraction N decays = Detector Size (N pixels ) ε Figure. Detection fraction as a function of detector size for a range of quantum efficiencies of the form a(n pixels b) c + d. of Gaussian smearing can be identified via the presence of an central elliptical aberration confirming the preliminary predictions of an asymmetry about reflections in x = y (.). ACKNOWLEDGEMENTS I thank my collaborators for assistance and C. Lucas for guidance throughout the duration of the course. This work made extensive use of GNU s GSL library. REFERENCES () Press, W.H., et al, Numerical Recipes in C, rd edn. Cambridge Univ. Press, Cambridge, 7 () Grimmett, G., Stirzaker, D., Probability & Random Processes, OUP, 99 () Devroye, L., Non-uniform random variate generation, st edn. Springer, 986 This paper has been typeset from a TEX/ L A TEX file prepared by the author.