Θ is the stratified sampling estimator of θ. Note: y i s subdivide sampling space into k strata ;

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1 1 STRATIFIED SAMPLING Stratified Sampling Background: suppose θ = E[X]. Simple MC simulation would use θ X. If X simulation depends on some discrete Y with pmf P {Y = y i } = p i, i = 1,..., k, and X can be simulated given Y = y i, E[X] = p i E[X Y = y i ] p i Xi = Θ. Θ is the stratified sampling estimator of θ. Note: y i s subdivide sampling space into k strata ; Analysis: If np i samples are used to determine X i, then V ar( X i ) = V ar(x Y = y i), and np i V ar(θ) = p 2 i V ar( X i ) = 1 n = 1 E[V ar(x Y )]. n p i V ar(x Y = y i ) Now V ar(x) = E[V ar(x Y )] + V ar(e[x Y ]), and V ar( X) = 1 nv ar(x), so V ar(θ) = V ar( X) 1 n V ar(e[x Y ]). Stratified sampling simulation can reduce variance significantly. Note: if S 2 i /(np i) is the sample variance for X i with np i samples, V ar(θ) = k p2 i V ar( X i ) 1 n k p is 2 i.

2 2 Application: one-dimensional integration, where θ = E[h(U)] = 1 h(x)dx. Strata are subintervals [ i 1 k, i k ], with p i = 1 k. For subinterval i, generate U ij = i 1 k + 1 k U, j = 1,..., n i, i = 1,..., k. Example: π 4 = E[ 1 U 2 ] = 1 1 x2 dx Try n = 5: simple MC ( raw simulation ), N = 5; U = rand(1,n); X = sqrt(1-u.^2); disp( [mean(x) 2*std(X)/sqrt(N)]) With stratified sampling there are different choices for k. a) k = n, n i = 1, so use U ij = U i = i 1 n + 1 nu, i = 1,..., n. US = [:N-1]/N + U/N; XS = sqrt(1-us.^2); disp( [mean(xs) 2*std(XS)/sqrt(N)] ) b) k = 5, n i = 1, p i = 1 5, so use U ij = i U, j = 1,..., 1, i = 1,..., 5. V ar( Θ) 5 1 S 2 1(5) 2 i = 1 5 Si K = 5; Ni = N/K; for i = 1 : K XS = sqrt(1-((i-1+rand(1,ni))/k).^2); XSB(i) = mean(xs); SS(i) = var(xs); end, SST = mean(ss/n); disp( [mean(xsb) 2*sqrt(SST)] ) e-5

3 3 Optimal Distribution of Samples : to reduce V ar more, assume n i samples for stratum i, with k n i = n. If Xi E[X Y = y i ] using n i samples, let Θ = p i Xi, with V ar( Θ) = p 2 i V ar( X i ). Given estimate s 2 i for each V ar( X i ) s2 i n i, choose n i s to minimize k p2 i s2 i /n i, subject to k n i = n; Lagrange multiplier solution is p i s i n i = n k j=1 p. js j Algorithm: use small n to estimate s i s, larger n for Θ; V ar( Θ) p 2 i Si 2, n i is estimated using sample variances S 2 i from strata.

4 4 Continued Example: π 4 = E[ 1 U 2 ] = 1 1 x2 dx c) k = 5, p i = 1 5, optimal n i = 5s i /( 5 s i). U ij = i U,j = 1,..., n i, i = 1,..., 5, V ar( Θ) 1 5 Si 2 5 5n i. ON = max( 3, round(n*sqrt(ss)/sum(sqrt(ss))) ); for i = 1 : K XS = sqrt(1-((i-1+rand(1,on(i)))/k).^2); XSB(i) = mean(xs); SS(i) = var(xs); end, SST = mean(ss./(5*on)); disp( [mean(xsb) 2*sqrt(SST)] ) e-5 More Stratified Sampling Examples Estimate P = cos(t2 ) e t2 /2 2π dt. For simulation convert to x [, 1] using x = Φ(t): so dx = e t2 /2 2π dt, P = 1 cos(φ 1 (x) 2 )dx. a) Simple MC f N = 5; U = rand(1,n); X = f(u); disp( [mean(x) 2*std(X)/sqrt(N)])

5 b) k = 5, n i = 1, p i = 1 5, so use U ij = i U, j = 1,..., 1, i = 1,..., 5. V ar( Θ) 5 1 S 2 1(5) 2 i = 1 5 Si K = 5; Ni = N/K; for i = 1 : K XS = f((i-1+rand(1,ni))/k); XSB(i) = mean(xs); SS(i) = var(xs); end, SST = mean(ss/n); disp( [mean(xsb) 2*sqrt(SST)] ) c) k = 5, p i = 1 5, optimal n i = 5s i /( 5 s i). U ij = i U,j = 1,..., n i, i = 1,..., 5, V ar( Θ) 1 5 Si 2 5 5n i. ON = max( 3, round(n*sqrt(ss)/sum(sqrt(ss))) ); for i = 1 : K XS = f((i-1+rand(1,on(i)))/k); XSB(i) = mean(xs); SS(i) = var(xs); end, SST = mean(ss./(5*on)); disp( [mean(xsb) 2*sqrt(SST)] )

6 Video Poker Example: 5 cards dealt, 1-5 discards are replaced. Let C = ( 52 5 ) 1, and assume payoff table is Hand Payoff Probability RF: Royal Flush 8 4C SF: Straight Flush 5 36C 4K: Four-of-a-kind C FH: Full House C FL: Flush 5 518C ST: Straight 4 12C 3K: Three-of-a-kind C 2P: Two Pair C HP: High Pair(> 1) C LP: Low Pair(<= 1) 7632C OT: Other 1- p i Note: 1- p i Textbook Strategy: no replacement cards dealt if > 3K; otherwise keep pairs and triplets but replace other cards. Raw simulation: for each run a) use random permutation of [1:52] (p.51) to shuffle cards; b) pick top 5 cards, discard, replace using text strategy; c) compute X from table. Compute X using many runs. 6

7 7 Video Poker Example with stratification: if payoff is RV X, E[X] = E[X RF ]P {RF } + E[X SF ]P {SF } + E[X 4K]P {4K} + E[X F H]P {F H} + E[X F L]P {F L} + E[X ST ]P {ST } + E[X 3K]P {3K} + E[X 2P ]P {2P } + E[X HP ]P {HP } + E[X LP ]P {LP } + E[X OT ]P {OT } = ( ) + E[X 2P ]P {2P } + E[X HP ]P {HP } + E[X LP ]P {LP } + E[X OT ]P {OT } E[X 2P ], E[X HP ], E[X LP ], also computed analytically. Stratified sampling: for each run a) use random permutation of [1:52] (p.51) to shuffle cards; b) pick top 5 cards, if LP, start new run; c) pick next 5 cards; compute X OT and X using X = ( ) + E[X 2P ]P {2P } +E[X HP ]P {HP } + E[X LP ]P {LP } + P {OT }X OT Compute X using many runs.

8 8 Multdimensional Integrals: consider the 2-dimensional θ = 1 1 g(x, y)dxdy Simple stratified sampling. If subdivision of [, 1] [, 1] is same for x and y, strata are squares s ij = [ i 1 k, i k ] [j 1 k, j k ], for i = 1 : k, j = 1 : k, so θ = p ij E[X (x, y) s ij ], p ij = 1 k 2, Θ = j=1 p ij Xij, with X ij E[X (x, y) s ij ], j=1 X ij = 1 n ij Note: if n ij = n/k 2, n ij l=1 ( i 1 g k V ar( Θ) V ar( Θ) 1 k 2 + U l k, j 1 k j=1 p 2 ij n ij S 2 ij. j=1 S 2 ij n. + V ) l, k

9 Two-dimensional example: θ = 1 1 e(x+y)2 dxdy. g N = 1; X = g(rand(2,n)); % Simple MC disp([mean(x) 2*std(X)/sqrt(N)]) K = 2; Nij = N/K^2; % Stratified for i = 1:K for j = 1:K XS = g([i-1+rand(1,nij);j-1+rand(1,nij)]/k); XSb(i,j) = mean(xs); SS(i,j) = var(xs); end end, SST = mean(mean(ss/n)); disp([mean(mean(xsb)) 2*sqrt(SST) ]) Note: for n dimensional integrals the growth in the number of terms k n in the sum for Θ, X i1,...,i Θ = n, k n i 1 =1 i n =1 results in a method that is infeasible for large n; however, the terms in the sum could be sampled with m X I Θ m, l=1 where I = (I 1, I 2,..., I n ), with I j Uniform(1 : k). Other strategies, e.g. with nonuniform n ij s, different ks for each variable, have been considered. 9

10 1 Transformation Method for Monotone Functions θ = 1 1 g(x 1, x 2,..., x n )dx 1 dx n. If x 1 = e y ny 1, x 2 = e y n(y 2 y 1 ),... x n = e y n(1 y n 1 ), θ = 1 y2 yn 1 yn 2 g(x 1 (y),..., x n (y))y n 1 n e y n dy 1 dy n. The pdf y n 1 n e y n is a Gamma(n, 1) pdf. For simulation, first generate Y n Gamma(n, 1), then U 1,..., U n 1 Uni(, 1), with Y i = U (i) (sorted). If Y n Gamma 1 (n, 1, U n ) (inversion), stratification can be used with U n.

11 Example 2-d continued; let x 1 = e y 2y 1, x 2 = e y 2(1 y 1 ), then θ = = e (x 1+x 2 ) 2 dx 1 dx 2 e (x 1(y 1,y 2 )+x 2 (y 1,y 2 )) 2 y 2 e y 2 dy 1 dy 2, N = 1; U = rand(2,n); % Simple MC Y(1,:) = U(1,:); Y(2,:) = gaminv(u(2,:),2,1); X(1,:) = exp(-y(1,:).*y(2,:)); X(2,:) = exp(-y(2,:).*(1-y(1,:))); Z = g(x); disp([mean(z) 2*std(Z)/sqrt(N)]) K = 1; Ni = N/K; % Stratified for i = 1:K, U=rand(2,Ni); Y(2,:) = gaminv((i-1+u(2,:))/k,2,1); X(1,:) = exp(-u(1,:).*y(2,:)); X(2,:) = exp(-y(2,:).*(1-u(1,:))); XS = g(x); XSB(i) = mean(xs); SS(i) = var(xs); end, SST = mean(ss/n); disp([mean(xsb) 2*sqrt(SST)])

12 12 Compound RVs: assume iid RVs X i and RV N: let S n (X) = n α ix i ; estimate P {S N (X) > c}. E.g. X i is insurance claim i, N is # claims by time T. In many cases, distribution for N is known, e.g. Poisson(λ), with p i = e λ (λ) i i!. Raw simulation : for K runs, generate N and N X i s; count proportion with N X i > c. Stratified simulation: given pmf for N, let g n (x) = I(S n (x) > c).. θ = m E[g n (X)]p n + E[g N (X) N > m](1 n= m p n ) n= Simulation run: a) compute m = N N > m from conditional pmf; b) generate X 1,..., X m and set Θ = m g n (X)p n + g m (X)(1 n= Then Θ P {S N (X) > c}. E.g. Poisson(λ), has conditional pmf with m p n ). n= with α = 1 m n= p n. (λ)m+i p m+i = e λ α(m + i)!,