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Random Number Generation Using Low Discrepancy Points

Learn about low discrepancy point generators, including the Faure method and other alternatives, for efficient and systematic random number generation. Explore the use of LDPs in various scenarios and potential concerns.

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Random Number Generation Using Low Discrepancy Points

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  1. Random Number GenerationUsing Low Discrepancy Points Donald Mango, FCAS, MAAA Centre Solutions June 7, 1999 1999 CAS/CARe Reinsurance Seminar Baltimore, Maryland

  2. What is Discrepancy? • Large # of points inside a unit hypercube :n-dimensional hypercube of length 1 on each side • For any “sub-volume” of the hypercube, Discrepancy = the difference between the proportion of points inside the volumeand the volume itself

  3. Low Discrepancy Point Generator: • Method to generate a set of points which fills out a given n-dimensional unit hypercube, with as little discrepancy as possible • Attempt to be systematic and efficient in filling a space, given the number of points • My paper discusses “Faure” Points, just one of many alternatives • Faure method relies on prime numbers

  4. Other Low Discrepancy Point Generators: • Named after number theorists: Sobol’, Neiderreiter, Halton, Hammersley, ... • More advanced methods use “irreducible polynomials” -- polynomial equivalents of prime numbers (cannot be factored) • More complex algorithms • Less flexible than Faure

  5. Linear Congruential Generator: • Xn+1 = (aXn + c) mod m • Used in spreadsheets -- RAND() in Excel, @RAND in Lotus • Sequential • Cyclical, with a long cycle length or “period” • “Randomized” in spreadsheets by using a random seedvalue ( X0 ) = the system clock

  6. LDPMAKER Excel 97 Workbook: • Available in the 1999 Spring Forum section of the CAS Website:www.casact.org/pubs/forum/99spforum/99spftoc.htm • Includes both: • A spreadsheet-only calculation (recalc-driven), and • A Visual Basic for Applications (VBA) macro-driven generator (run with a button)

  7. LDPMAKER Excel 97 Workbook: • “Example” sheet is spreadsheet-only calculation • Demonstrates formulas • Not very flexible

  8. Example: 4 Dimensions, 24 Iterations • Dimension #1: • First, convert each iteration number N to base Prime (= 5) • Iteration 1 = 01base5Iteration 10 = 20base5 • F(N, 1) = Faure point (Iteration N, Dimension 1)F(1,1) = 0/52 + 1/5 = 0.20F(10,1) = 2/52 + 0/5 = 0.08

  9. Example: 4 Dimensions, 24 Iterations • Dimension #2: • Start with the base Prime digits from Dimension #1 and “shuffle” them • Using combinations, sum of digits and MOD operator • First digit in Dimension #2 = [ Sum (first digit, second digit) from Dimension #1 ] MODPrime • Dimension #1, Iteration 10 = 20base5Dimension #2, Iteration 10 = 22base5 • Formula for F(N,2) is the same

  10. Example: 4 Dimensions, 24 Iterations • Dimensions #3 and higher: • Start with the base Prime digits from the previous dimension and “shuffle” them • Formula for F(N,3) ... is the same

  11. Loops in the Faure Algorithm: • Fills out the space in ever-larger loops of ever-smaller spacing • Fills out the space sequentially • There MAY be an issue with ending the iterations in the middle of one of these loops • Examples later in the test results...

  12. Visual Basic for Applications (VBA) Version: • VBA = real programming language • Recursive algorithm using “dynamic arrays” - arrays which are dimensioned (sized) at run-time • Generalization of spreadsheet-only calculations • FAST

  13. Performance Test #1:Sum of Limited Paretos Table 2 (from Paper) - Pareto Parameters

  14. Performance Test #1:Sum of Limited Paretos Table 3: Sum of 2 Limited Paretos

  15. Performance Test #1:Sum of Limited Paretos Table 4: Sum of 5 Limited Paretos

  16. Performance Test #2:Sum of Poissons Table 5: Sum of 2 Poissons (l = 8)

  17. Performance Test #2:Sum of Poissons Table 6: Sum of 5 Poissons (l = 8)

  18. Performance Test #3:Low Frequency Events Table 7 - Pareto Parameters used for Severity

  19. Performance Test #3:Low Frequency Events Table 8: One Event, 5% Prob of Occurrence

  20. Performance Test #3:Low Frequency Events Table 9: Two Events, each with 5% Prob of Occurrence

  21. Performance Test #4:99th Percentile of Sum of Normals Table 10 - Normal Parameters

  22. Performance Test #4:99th Percentile of Sum of Normals Table 11 - 99th Pctle of Sum of 2 Normals

  23. Performance Test #4:99th Percentile of Sum of Normals Table 12 - 99th Pctle of Sum of 5 Normals

  24. Performance Test #5:Mixed Bag • Sum of 5 each from: • LogNormal • Pareto • Uniform • Normal • Testing variability of estimates over 10 runs

  25. Performance Test #5:Mixed Bag Table 14 - Avg % Error and Std Dev of % Error over 10 runs

  26. Possible Concerns in Using LDPs • Unused Dimensions: • Example: modeling Excess Claims • # of Excess claims between 0 and 30 • requires 30 dimensions • If # claims < 30, are the “used” dimensions still filled out with low discrepancy? • Dr. Tom?

  27. Possible Concerns in Using LDPs • Time Series: • Example: Probability of 2 consecutive years of loss ratio exceeding 75% • How many dimensions is this problem? • Can’t use a single dimension of LDPs, because they are sequentially dependent • Need to know “over how many years”, then set dimensions

  28. Possible Concerns in Using LDPs • Correlation: • If two variables are • 100% correlated ==> 1 dimension • 0% correlated ==> 2 dimensions • x% correlated ==> ? dimensions • Is promise of “low discrepancy” still fulfilled? • How to implement?

  29. Possible Concerns in Using LDPs • Loop Boundaries: • Faure algorithm fills out space sequentially in ever-expanding loops of ever-finer granularity • If iteration count does not finish on a loop boundary (depends on Prime), there may be potential bias... • See Appendix B of paper

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