Population Estimation: Using High-Performance Computing in Statistical Research. Craig Finch Zia Rehman

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1 Population Estimation: Using High-Performance Computing in Statistical Research Craig Finch Zia Rehman

2 Statistical estimation Estimated Value Confidence Interval Actual Value Estimator: a rule for finding an estimate of a statistical parameter (such as mean or variance), based on limited measurements with a random component

3 Population estimation How many fish are in the lake? We can t drain the lake and count the dead fish We can t catch all the fish Estimate the population based upon limited information

4 Capture-recapture method Catch a fish If the fish is unmarked, mark the fish and release it Increment number of marked fish X i If the fish is already marked, release the fish Record a recapture event at trial x i = 1 Repeat until estimate is good enough The result is a series of values: X i x i X 0 = 0 X 1 = 1 X 2 = 1 X 3 = 2 X 4 = 2 x 0 = 0 x 1 = 0 x 2 = 1 x 3 = 0 x 4 = 1 X n x n

5 Ratio estimator for closed system No fish enter or leave the lake (or breed or die ) Population: Variance: τ = X i x i VVV τ = n i τ 2 + 2τ n i + 1 p i 1 p i p i 2 n j j=1 i=1 p i p j 1 p j τ 2 + n i + 1 n j τ n j + 1 p 2 i p i = Probability of re-capture at trial i p i = X i τ

6 Non-parametric ratio estimator Non-parametric Makes no assumption about the distribution from which the data are drawn Ratio estimator Ratio of two quantities Can be extended to Open systems Open systems with cluster sampling

7 Open systems More common than closed systems Units may enter, leave, or re-enter the system

8 Cluster sampling vs More efficient to capture multiple units in one sample Fishing with a net vs. a pole

9 Monte Carlo simulations Simulation implementation is straightforward Use a pseudorandom number generator to generate random variates Algorithm: Calculate probability of recapture at trial i Randomly determine whether or not a recapture occurred Update arrays for x i and X i Repeat until desired number of recaptures is reached Run ensemble of simulations

10 Simulation of open system Ensemble of 250 runs 5% sample Population Trial (i)

11 Convergence of estimator Results shown for closed system Estimator converges to actual population as number of trials increases % 5% sampling 10% sampling 2% sampling Number of trials (n)

12 How did STOKES get involved? Simulations can actually run on a desktop PC but estimating variance is computationally intensive STOKES is a commodity cluster Any one node of STOKES is just a dual-processor Intel Xeon server STOKES accelerates calculations by taking advantage of parallel computing

13 Estimating variance Review the formula for estimating variance of a closed system VVV τ = n i=1 n i τ 2 + 2τ n i + 1 p i 1 p i p 2 i n j j=1 i=1 p i p j 1 p j τ 2 + n i + 1 n j τ n j + 1 p 2 i Note the nested summations in the second term Variance formula for open systems is even worse! Four nested summations

14 Exploiting parallelism Trivial parallelization: analyze one Monte Carlo run on one core 250 runs on 250 cores cuts run time by factor of 250 compared to sequential analysis Simple to program: no inter-process communication Data 1 Data 2 Data 3 Data k Analysis 1 Analysis 2 Analysis 3 Analysis k Result 1 Result 2 Result 3 Result k

15 More sophisticated parallelism Calculating variance for ONE run of an open system is still too slow for large numbers of trials Investigating additional ways to leverage parallelism to accelerate variance calculation Vectorize loops to maximize FLOPs per core Analyze formula to determine which calculations can be done in parallel

16 Summary Using STOKES to evaluate a novel statistical estimator High-performance computing (HPC) can benefit areas of research that have not traditionally used HPC Problems must be re-formulated to exploit parallel computing

17 Authors Zia Rehman PhD candidate Modeling and Simulation (statistics track), University of Central Florida Masters Degree in Mathematics from the University of Louisville Fellow of Casualty Actuarial Society (FCAS) Over 10 years of professional actuarial experience Publishes in statistical/actuarial journals Craig Finch Postdoctoral Research Associate in the STOKES Advanced Research Computing Center PhD in Modeling and Simulation from the University of Central Florida BS in Electrical Engineering from the University of Illinois at Urbana-Champaign

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