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Two Topics in Parametric IntegrationApplied to Stochastic Simulation
in Industrial Engineering
Jeremy Staum
Department of Industrial Engineering and Management SciencesNorthwestern University
September 15th, 2014
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Outline
1 Simulation Metamodeling
introduction and overview
2 Multi-Level Monte Carlo Metamodeling
with Imry Rosenbaumhttp:
//users.iems.northwestern.edu/~staum/MLMCM.pdf
3 Generalized Integrated Brownian Fields for SimulationMetamodeling
with Peter Salemi and Barry L. Nelsonhttp://users.iems.northwestern.edu/~staumGIBF.pdf
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
MCQMC / IBC Application Domain
Industrial Engineering & Operations Research
using math to analyze systems and improve decisions
Stochastic Simulation: production, logistics, financial, . . .
integration: µ = E[Y ] =∫Y (ω) dω
parametric integration: approx. µ def. by µ(x) = E[Y (x)]
optimization: minµ(x) : x ∈ X
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
What is Stochastic Simulation Metamodeling?
Stochastic simulation model example
Fuel injector production line
System performance measure µ(x) = E[Y (x)]
x: design of production lineY (x): number of fuel injectors produced
Simulating each scenario (20 replications) takes 8 hours
Stochastic simulation metamodeling
Simulation output Y (xi ) at xi , i = 1, . . . , n
Predict µ(x) by µ(x), even without simulating at x
µ(x) is usually a weighted average of Y (x1), . . . , Y (xn)
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Overview of Multi-Level Monte Carlo (MLMC)
Error in Stochastic Simulation Metamodeling
prediction µ(x) =k∑
i=1
wi (x)Y (xi )
variance: Var[µ(x)] caused by variance of simulation output
interpolation error (bias): E[µ(x)] 6= µ(x)
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Main Idea of Multi-Level Monte Carlo
Ordinary Monte Carlo
to reduce variance: large number n of replicationsper simulation run (design point)
to reduce bias: large number k of design points (fine grid)
very large computational effort kn
Multi-Level Monte Carlo
to reduce variance: coarser grids, many replications each
to reduce bias: finer grids, few replications each
less computational effort / better convergence rate
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Our Contributions
Theoretical
mix and match or expand (from Heinrich’s papers):derive desired conclusions under desired assumptions
to suit IE goals and applications
Practical
algorithm design (based on Giles)
Experimental
show how much MLMC speeds up realistic examples in IE
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Heinrich (2001): MLMC for Parametric Integration
Assumptions
Approximate µ given by µ(x) =∫
Ω Y (x, ω) dω over x ∈ X .
X ⊆ Rd and Ω ⊆ Rd2 : bounded, open, Lipschitz boundary.
With respect to x, Y has weak derivatives up to r th order.
Y and weak derivatives are Lq-integrable in (x, ω).
Sobolev embedding condition: r/d > 1/q.
Measure error as (∫
Ω ‖µ− µ‖pq dω)1/p, where p = min2, q.
Conclusion: There is a MLMC method with optimal rate.
MLMC attains the best rate of convergence in C , the number ofevaluations of Y . The error bound is proportional to
C−r/d if r/d < 1− 1/p
C 1/p−1 logC if r/d = 1− 1/p
C 1/p−1 if r/d > 1− 1/p.
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Assumptions
Smoothness: assume r = 1
Stock option, Y (x , ω) = maxxR(ω)− K , 0Queueing: waiting time Wn+1 = maxWn + Bn − An+1, 0Inventory: Sn = minIn + Pn,Dn, In+1 = In − Sn
Parameter Domain
Assume X ⊂ Rd is compact (not open).
Heinrich and Sindambiwe (1999), Daun and Heinrich (2014)
If X were open, we would have to extrapolate.
No need to approximate unbounded µ near a boundary of X .
Domain of Integration
Ω ⊆ Rd2 is not important; d2 does not appear in theorem.
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Changing Perspective
Measure of Error
Use p = q = 2 to get Root Mean Integrated Squared Error(∫Ω
∫X (µ(x)− µ(x))2 dx dω
)1/2
Sobolev Embedding Criterion with r = 1, q = 2
r/d > 1/q becomes 1/d > 1/2, i.e. d = 1!??
Why We Don’t Need the Sobolev Embedding Condition
Assume the domain X is compact.
Assume Y (·, ω) is (almost surely) Lipschitz continuous.
Conclude Y (·, ω) is (almost surely) bounded.
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Changing Perspective
Measure of Error
Use p = q = 2 to get Root Mean Integrated Squared Error(∫Ω
∫X (µ(x)− µ(x))2 dx dω
)1/2
Sobolev Embedding Criterion with r = 1, q = 2
r/d > 1/q becomes 1/d > 1/2, i.e. d = 1!??
Why We Don’t Need the Sobolev Embedding Condition
Assume the domain X is compact.
Assume Y (·, ω) is (almost surely) Lipschitz continuous.
Conclude Y (·, ω) is (almost surely) bounded.
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Changing Perspective
Measure of Error
Use p = q = 2 to get Root Mean Integrated Squared Error(∫Ω
∫X (µ(x)− µ(x))2 dx dω
)1/2
Sobolev Embedding Criterion with r = 1, q = 2
r/d > 1/q becomes 1/d > 1/2, i.e. d = 1!??
Why We Don’t Need the Sobolev Embedding Condition
Assume the domain X is compact.
Assume Y (·, ω) is (almost surely) Lipschitz continuous.
Conclude Y (·, ω) is (almost surely) bounded.
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Our Assumptions
On the Stochastic Simulation Metamodeling Problem
X ⊂ Rd is compact
Y (x) has finite variance for all x ∈ X|Y (x′, ω)− Y (x, ω)| ≤ κ(ω)‖x′ − x‖, a.s., and E[κ2] <∞.
On the Approximation Method and MLMC Design
µ(x) =∑N
i=1 wi (x)Y (xi ) where each wi (x) ≥ 0 and
Total weight on points xi far from x gets close to 0.
Total weight on points xi near x gets close to 1.
Thresholds for “far”/“‘near” and “close to”are O(N−1/2φ) as number N of points increases.
Examples: piecewise linear interpolation on a grid;nearest-neighbors, Shepard’s method, kernel smoothing
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Approximation Method Used in Examples
Kernel Smoothing
µ(x) =N∑i=1
wi (x)Y (xi )
weight wi (x) is
0 if xi is outside the cellcontaining xotherwise, proportional toexp(−‖x− xi‖)
weights are normalized tosum to 1
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Our Conclusions
MLMC Performance
As number N of points used in a level increases,
Errors due to bias and refinement variance are like O(N−1/φ).
Example: nearest-neighbor approximation on grid, φ = d/2
Computational Complexity (based on Giles 2013)
To attain RMISE < ε, the required number of evaluations of Y isO(ε−2(1+φ)) for standard Monte Carlo and for MLMC it is
O(ε−2φ) if φ > 1
O((ε−1(log ε−1))2) if φ = 1
O(ε−2) if φ < 1.
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Sketch of Algorithm (based on Giles 2008)
Goal: add levels until target RMISE < ε is achieved.
1 INITIALIZE level ` = 0.2 SIMULATE at level `:
1 Run level ` simulation experiment with M0 replications.2 Observe sample variance of simulation output.3 Choose number of replications M` to control variance; run
more replications if needed.
3 TEST CONVERGENCE:1 Use Monte Carlo to estimate the size of the refinement ∆µ`,∫
X (∆µ`(x))2 dx.2 If refinements are too large compared to target RMISE,
increment ` and return to step 2.
4 CLEAN UP: Finalize number of replications M0, . . . ,M` tocontrol variance; run more replications at each level if needed.
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Asian Option Example, d = 3
MLMC up to 150 times better than standard Monte Carlo
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Inventory System Example, d = 2
MLMC was 130-8900 times better than standard Monte Carlo
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Conclusion on Multi-Level Monte Carlo
Celebration
Multi-Level Monte Carlo worksfor typical IE stochastic simulation metamodeling too!
Future Research
Handle discontinuities in simulation output.
Combine with good experiment designs.Grids are not good in high dimension.
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Introduction: Generalized Integrated Brownian Field
Kriging / Interpolating Splines
Pretend µ is a realization of a Gaussian random field M withmean function m and covariance function σ2.
Kriging predictor:
µ(x) = m(x) + σ2(x)Σ−1(Y −m) = m(x) +∑i
βiσ2(x, xi )
σ2(x) is a vector with ith element σ2(x, xi )Σ is a matrix with i , jth element σ2(xi , xj)Y −m is a vector with ith element Y (xi )−m(xi )
Stochastic Kriging / Smoothing Splines
µ(x) = m(x) + σ2(x)(Σ + C )−1(Y −m) = m(x) +∑i
βiσ2(x, xi )
C = covariance matrix of noise, estimated from replications
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Radial Basis Functions vs. Integrated Brownian Field
Radial Basis Functions
Basis Functions from r -Fold Integrated Brownian Field
(d) r = 0 (e) r = 1 (f) r = 2
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Response Surfaces in IE Stochastic Simulation
(g) Credit Risk (h) Inventory
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
r -Integrated Brownian Field Br
Covariance function / reproducing kernel
σ2(x, y) =d∏
i=1
1
(r !)2
∫ 1
0(xi − ui )
r+(yi − ui )
r+ dui
Inner product
〈f , g〉 =
∫(0,1)d
(f ([r ···r ])(u))(g ([r ···r ])(u))du
Space
Tensor product of Sobolev Hilbert space H r (0, 1)with boundary conditions f (j)(0) = 0 for j = 0, . . . , r
What’s missing? polynomials of degree r
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Removing Boundary Conditions: d = 1
Generalized integrated Brownian motion
Xr (x) =r∑
k=0
√θkZk
xk
k!+√θr+1Br (x)
Covariance function / reproducing kernel
σ2(x , y) =r∑
k=0
θkxkyk
(k!)2+ θr+1
∫ 1
0
(x − u)r+(y − u)r+(r !)2
du
Sobolev space H r (0, 1) , no boundary conditions
Inner product
〈f , g〉 =r∑
k=0
1
θk(f (k)(u))(g (k)(u)) +
1
θr+1
∫ 1
0(f (r)(u))(g (r)(u))du
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Multidimensional, Without Boundary Conditions
Tensor-Product RKHS with Weights
Example of reproducing kernel for d = 2, r = 1
K (x, y) = θ00 + θ10x1y1 + θ20(x1 ∧ y1) + θ01x2y2 + θ02(x2 ∧ y2)
+θ11x1x2y1y2 + θ12x1y1(x2 ∧ y2)
+θ21(x1 ∧ y1)x2y2 + θ22(x1 ∧ y1)(x2 ∧ y2)
In general, one weight for each of∏d
i=1(ri + 2) subspaces.
Generalized Integrated Brownian Field
Covariance function / reproducing kernel
σ2(x, y) =d∏
i=1
(ri∑
k=0
θi ,kxki y
ki
(k!)2+ θi ,ri+1
∫ 1
0
(xi − ui )ri+(yi − ui )
ri+
(ri !)2dui
)
In general, number of weights is∑d
i=1(ri + 2).
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Multidimensional, Without Boundary Conditions
Tensor-Product RKHS with Weights
Example of reproducing kernel for d = 2, r = 1
K (x, y) = θ00 + θ10x1y1 + θ20(x1 ∧ y1) + θ01x2y2 + θ02(x2 ∧ y2)
+θ11x1x2y1y2 + θ12x1y1(x2 ∧ y2)
+θ21(x1 ∧ y1)x2y2 + θ22(x1 ∧ y1)(x2 ∧ y2)
In general, one weight for each of∏d
i=1(ri + 2) subspaces.
Generalized Integrated Brownian Field
Covariance function / reproducing kernel
σ2(x, y) =d∏
i=1
(ri∑
k=0
θi ,kxki y
ki
(k!)2+ θi ,ri+1
∫ 1
0
(xi − ui )ri+(yi − ui )
ri+
(ri !)2dui
)
In general, number of weights is∑d
i=1(ri + 2).
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Our Contributions
more parsimonious parametrization
makes maximum likelihood estimation easierand MLE search for r1, . . . , rd
GIBF has Markov property
d = 1: proofd > 1: conjecture
IE simulation examples
stochastic and deterministic simulationstandard and nonstandard information
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Credit Risk Example, d = 2
Experiment design: 63 Sobol’ points,predictions in a smaller square
Factor by which MISE decreased using (1,1)-GIBF
Number of replications ∞ 100 25Noise level none low medium
Without gradient estimates 94 111 120With gradient estimates 81 83 69
(i) Credit risk surface (j) Gaussian (k) (1, 1)-GIBF
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Conclusion on Generalized Integrated Brownian Field
Emancipating Simulation Metamodeling from Geostatistics
a new covariance function for kriging,designed for simulation metamodeling in engineering
Superior Practical Performance
4-120 times better than Gaussian covariance functionin 2-6 dimensional exampleswith or without gradient information
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering
Thank You!
Jeremy Staum Topics in Simulation Metamodeling in Industrial Engineering