Basics of probability in statistical simulation and stochastic programming

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AACIMP 2010 Summer School lecture by Leonidas Sakalauskas. "Applied Mathematics" stream. "Stochastic Programming and Applications" course. Part 2.
More info at http://summerschool.ssa.org.ua

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Basics of probability in statistical simulation and stochastic programming

  1. 1. Lecture 2 Basics of probability in statistical simulation and stochastic programming Leonidas Sakalauskas Institute of Mathematics and Informatics Vilnius, Lithuania <sakal@ktl.mii.lt> EURO Working Group on Continuous Optimization
  2. 2. Content  Random variables and random functions  Law of Large numbers  Central Limit Theorem  Computer simulation of random numbers  Estimation of multivariate integrals by the Monte-Carlo method
  3. 3. Simple remark  Probability theory displays the library of mathematical probabilistic models  Statistics gives us the manual how to choose the probabilistic model coherent with collected data  Statistical simulation (Monte-Carlo method) gives us knowledge how to simulate random environment by computer
  4. 4. Random variable Random variable is described by  Set of support SUPP(X )  Probability measure Probability measure is described by distribution function: F ( x) Pr ob( X x)
  5. 5. Probabilistic measure  Probabilistic measure has three components:  Continuous;  Discrete (integer);  Singular.
  6. 6. Continuous r.v. Continuous r.v. is described by probability density function f (x) Thus: x F ( x) f ( y )dy
  7. 7. Continuous variable If probability measure is absolutely continuous, the expected value of random function: Ef ( X ) f ( x) p( x)dx
  8. 8. Discrete variable Discrete r.v. is described by mass probabilities: x1 , x2 ,..., xn p1 , p2 ,..., pn
  9. 9. Discrete variable If probability measure is discrete, the expected value of random function is sum or series: n Ef ( X ) f ( xi ) p i i 1
  10. 10. Singular variable Singular r.v. probabilistic measure is concentrated on the set having zero Borel measure (say, Kantor set).
  11. 11. Law of Large Numbers (Chebyshev, Kolmogorov) N i 1 zi lim z, N N here z1 , z2 ,..., z N are independent copies of r. v. , z E
  12. 12. What did we learn ? The integral f ( x, z ) p( z )dz is approximated by the sampling average N i 1 i N if the sample size N is large, here j f ( x, z j ), j 1,..., N , z1 , z 2 ,..., z N is the sample of copies of r.v. , distributed with the density p(z ) .
  13. 13. Central limit theorem (Gauss, Lindeberg, ...) xN lim P x ( x), N / N x y2 here 1 2 ( x) e dy, 2 N xi 2 xN i 1 , EX , D2 X E(X )2 N
  14. 14. Beri-Essen theorem 3 EX sup FN ( x) ( x) 0.41 3 x N where FN ( x) Pr ob x N x
  15. 15. What did we learn ? According to the LLN: N N N ( xi xN ) 2 3 1 xi xN x xi , 2 i 1 , EX EX 3 i 1 Ni 1 N N Thus, apply CLT to evaluate the statistical error of approximation and its validity.
  16. 16. Example Let some event occurred n times repeating N independent experiments. Then confidence interval of probability of event : 1.96 p (1 p) 1.96 p (1 p) p ,p N N here n (1,96 – 0,975 quantile of normal distribution, p , confidence interval – 5% ) N If the Beri-Esseen condition is valid: N p (1 p) 6 !!!
  17. 17. Statistical integrating … b I f ( x)dx ??? a Main idea – to use the gaming of a large number of random events
  18. 18. Statistical integration Ef ( X ) f ( x) p( x)dx N f ( xi ) i 1 , xi  p( ) N
  19. 19. Statistical simulation and Monte-Carlo method F ( x) f ( x, z ) p( z )dz min x N f ( x, z i ) i 1 min , zi  p ( ) N x (Stochastic Analytical Approximation (SAA), Shapiro, (1985), etc)
  20. 20. Simulation of random variables There is a lot of techniques and methods to simulate r.v. Let r.v. be uniformly distributed in the interval (0,1] Then, the random variable U , where F (U ) , is distributed with the cumulative distribution function F ( )
  21. 21. F (a) x cos( x sin( a x) ) e x dx 0 f ( x) x cos( x sin( a x) ) N=100, 1000
  22. 22. Wrap-Up and conclusions o the expectations of random functions, defined by the multivariate integrals, can be approximated by sampling averages according to the LLN, if the sample size is sufficiently large; o the CLT can be applied to evaluate the reliability and statistical error of this approximation

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