Imprecise random variables, random sets, and Monte Carlo simulation - PowerPoint PPT Presentation

Imprecise random variables, random sets, and Monte Carlo simulation Thomas Fetz, Michael Oberguggenberger Unit for Engineering Mathematics University of Innsbruck, Austria Thomas.Fetz@uibk.ac.at Michael.Oberguggenberger@uibk.ac.at ISIPTA’15, Pescara, Italia Th.Fetz, M. Oberguggenberger Imprecise random variables, random sets, and Monte Carlo simulation ISIPTA’15, Pescara, Italia 1 / 10

Research team Unit of Engineering Mathematics Faculty of Engineering Sciences University of Innsbruck, Austria Research team “imprecise probabilities”: Michael Oberguggenberger Thomas Fetz (head of unit) Jelena Nedeljkovic Martin Schwarz Th.Fetz, M. Oberguggenberger Imprecise random variables, random sets, and Monte Carlo simulation ISIPTA’15, Pescara, Italia 2 / 10

Problem Given Expensive input-output map g : R n → R : x → g ( x ) . E.g. finite element computations (minutes or hours per computation). Family { X λ } λ ∈ Λ of random variables modelling the uncertainty of variable x . Aim Upper/lower probabilities that g ( x ) ∈ B . Upper/lower probabilities that g ( x ) ≤ y (upper/lower cumulative distribution functions). Upper/lower probabilities that g ( x ) ≤ 0 (upper/lower probability of failure). Two approaches Monte-Carlo simulation of { g ( X λ ) } λ ∈ Λ . Monte-Carlo simulation of the random set X generated by { g ( X λ ) } λ ∈ Λ . Th.Fetz, M. Oberguggenberger Imprecise random variables, random sets, and Monte Carlo simulation ISIPTA’15, Pescara, Italia 3 / 10

Two approaches 1 Family { X λ } λ ∈ Λ of random variables 2 Random set X based on { X λ } λ ∈ Λ Probability space ( Ω , Σ , m ) . Set-valued map X : Ω → R defined by Family { X λ } λ ∈ Λ of random variables X ( ω ) = { X λ ( ω ) : λ ∈ Λ } . X λ : Ω → R : ω → X λ ( ω ) . X is a random set, if upper/lower inverses Probability P ( X λ ∈ B ) for fixed X λ : � X − ( B ) = { ω ∈ Ω : X ( ω ) ∩ B � = ∅ } , P ( X λ ∈ B ) = Ω ✶ X λ ( ω ) ∈ B d m ( ω ) . X − ( B ) = { ω ∈ Ω : X ( ω ) ⊆ B } are measurable subsets of Ω . (for initial analysis we drop the map g ) Th.Fetz, M. Oberguggenberger Imprecise random variables, random sets, and Monte Carlo simulation ISIPTA’15, Pescara, Italia 4 / 10

Two approaches 1 Family { X λ } λ ∈ Λ of random variables 2 Random set X based on { X λ } λ ∈ Λ Probability space ( Ω , Σ , m ) . Set-valued map X : Ω → R defined by Family { X λ } λ ∈ Λ of random variables X ( ω ) = { X λ ( ω ) : λ ∈ Λ } . X λ : Ω → R : ω → X λ ( ω ) . X is a random set, if upper/lower inverses Probability P ( X λ ∈ B ) for fixed X λ : � X − ( B ) = { ω ∈ Ω : X ( ω ) ∩ B � = ∅ } , P ( X λ ∈ B ) = Ω ✶ X λ ( ω ) ∈ B d m ( ω ) . X − ( B ) = { ω ∈ Ω : X ( ω ) ⊆ B } are measurable subsets of Ω . (for initial analysis we drop the map g ) Lower/upper probabilities for { X λ } λ ∈ Λ Lower/upper probabilities for X � � P ( B ) = inf λ ∈ Λ P ( X λ ∈ B ) = inf Ω ✶ X λ ( ω ) ∈ B d m ( ω ) P ( B ) = m ( X − ( B )) = Ω ✶ X ( ω ) ⊆ B d m ( ω ) � λ ∈ Λ � � P ( B ) = m ( X − ( B )) = P ( B ) = sup P ( X λ ∈ B ) = sup Ω ✶ X λ ( ω ) ∈ B d m ( ω ) � Ω ✶ X ( ω ) ∩ B � = ∅ d m ( ω ) λ ∈ Λ λ ∈ Λ Th.Fetz, M. Oberguggenberger Imprecise random variables, random sets, and Monte Carlo simulation ISIPTA’15, Pescara, Italia 4 / 10

Two approaches 1 Family { X λ } λ ∈ Λ of random variables 2 Random set X based on { X λ } λ ∈ Λ Probability space ( Ω , Σ , m ) . Set-valued map X : Ω → R defined by Family { X λ } λ ∈ Λ of random variables X ( ω ) = { X λ ( ω ) : λ ∈ Λ } . X λ : Ω → R : ω → X λ ( ω ) . X is a random set, if upper/lower inverses Probability P ( X λ ∈ B ) for fixed X λ : � X − ( B ) = { ω ∈ Ω : X ( ω ) ∩ B � = ∅ } , P ( X λ ∈ B ) = Ω ✶ X λ ( ω ) ∈ B d m ( ω ) . X − ( B ) = { ω ∈ Ω : X ( ω ) ⊆ B } are measurable subsets of Ω . (for initial analysis we drop the map g ) Lower/upper probabilities for { X λ } λ ∈ Λ Lower/upper probabilities for X � � P ( B ) = inf λ ∈ Λ P ( X λ ∈ B ) = inf Ω ✶ X λ ( ω ) ∈ B d m ( ω ) P ( B ) = m ( X − ( B )) = Ω ✶ X ( ω ) ⊆ B d m ( ω ) � λ ∈ Λ � � P ( B ) = m ( X − ( B )) = P ( B ) = sup P ( X λ ∈ B ) = sup Ω ✶ X λ ( ω ) ∈ B d m ( ω ) � Ω ✶ X ( ω ) ∩ B � = ∅ d m ( ω ) λ ∈ Λ λ ∈ Λ Theorem ≤ P ≤ P ≤ � P P X is more imprecise than { X λ } λ ∈ Λ ! � Th.Fetz, M. Oberguggenberger Imprecise random variables, random sets, and Monte Carlo simulation ISIPTA’15, Pescara, Italia 4 / 10

Example � 1 e − ω 2 / 2 d ω . Probability space: ( Ω , Σ , m ) = ( R , B ( R ) , m ) , m ( B )= √ R ✶ ω ∈ B 2 π 1 ⇒ X ( µ , σ ) ∼ N ( µ , σ 2 ) . Family { X ( µ , σ ) } ( µ , σ ) ∈ Λ : X ( µ , σ ) ( ω ) = σω + µ = Λ = [ µ , µ ] × [ σ , σ ] = [ − 0 . 5 , 2 ] × [ 1 , 2 ] , B = [ 1 , 2 . 5 ] . X ( ω ) = { X λ ( ω ) : λ ∈ Λ } = [ X ( ω ) , X ( ω )] � 4 σω + µ ω < 0 X X ( ω ) = inf X ( µ , σ ) ( ω ) = σω + µ ω ≥ 0 X ( ω = 1 ) µ ∈ [ µ , µ ] 2 X ( ω ) � σ ∈ [ σ , σ ] X ( 1 . 5 , 1 . 3 ) σω + µ ω < 0 X ( ω ) = X ( µ , σ ) ( ω ) = sup 0 σω + µ ω ≥ 0 µ ∈ [ µ , µ ] X X σ ∈ [ σ , σ ] − 1 1 0 3 P ( B ) = ( µ , σ ) ∈ Λ P ( X ( µ , σ ) ∈ B ) = P ( X ( − 0 . 5 , 1 ) ∈ B ) inf 2 , 1 ) X ( − 1 = 0 . 0655 2 . 5 P ( B ) = sup P ( X ( µ , σ ) ∈ B ) = P ( X ( 1 . 75 , 1 ) ∈ B ) B ( µ , σ ) ∈ Λ = 0 . 5467 1 X ( ω ) X ( 1 . 75 , 1 ) P ( B ) = m ( X − ( B )) = m ( ∅ ) = 0 . 0000 1 1 � X P ( B ) = m ( X − ( B )) = m ([ − 1 , 3 ]) � 1 1 − 1 0 3 4 = Φ ( 3 ) − Φ ( − 1 ) = 0 . 8400 ω Th.Fetz, M. Oberguggenberger Imprecise random variables, random sets, and Monte Carlo simulation ISIPTA’15, Pescara, Italia 5 / 10 1 1 1 1 1 1 1 1 1 1 1 1

Simulation of a family { X λ } λ ∈ Λ of random variables 1 Basic sample x 1 ,..., x N samp Generate a sample x 1 ,..., x N samp which is distributed as a basic random variable X ∗ . Distribution of X ∗ should cover a greater range than a distribution of a single X λ does. 2 N samp function evaluations g ( x k ) , k = 1 ,..., N samp We compute g ( x k ) either using g directly or a cost saving surrogate model ˜ g . 3 Approximation of P ( g ( X λ ) ≤ y ) Probability P ( g ( X λ ) ≤ y ) for fixed λ is computed by reweighting the original sample. Weights w k ( λ ) depending on parameters λ for reweighting the sample x 1 ,..., x N samp according to the distribution of X λ : w k ( λ ) = f X λ ( x k ) = f new ( x k ) 1 1 f X ∗ ( x k ) N samp f old ( x k ) N samp where f X λ and f X ∗ are strictly positive densities. P ( g ( X λ ) ≤ y ) for different X λ without additional function evaluations of g : � N samp N samp ∑ P ( g ( X λ ) ≤ y ) = Ω ✶ g ( X λ ( ω )) ≤ y d m ( ω ) ≈ ∑ ✶ g ( X λ ( ω k )) ≤ y · w k ( λ ) = ✶ g ( x k ) ≤ y · w k ( λ ) . k = 1 k = 1 Th.Fetz, M. Oberguggenberger Imprecise random variables, random sets, and Monte Carlo simulation ISIPTA’15, Pescara, Italia 6 / 10

Simulation of a family { X λ } λ ∈ Λ of random variables 4 Approximation of P ( g ≤ y ) and P ( g ≤ y ) For the computation of the upper/lower probabilities P ( g ≤ y ) and P ( g ≤ y ) we use a grid of representative parameter values λ i , estimate the probabilities P ( g ( X λ i ) ≤ y ) at the grid points λ i by means of MC simulation and take the maximum/minimum value: N samp ∑ P ( g ≤ y ) = sup P ( g ( X λ ) ≤ y ) ≈ max P ( g ( X λ i ) ≤ y ) ≈ max ✶ g ( x k ) ≤ y · w k ( λ i ) , i = 1 ,..., N grid i = 1 ,..., N grid λ ∈ Λ k = 1 N samp ∑ P ( g ≤ y ) ≈ ✶ g ( x k ) ≤ y · w k ( λ i ) . min i = 1 ,..., N grid k = 1 Effort: N grid · N samp reweightings, N samp expensive function evaluations of g . Th.Fetz, M. Oberguggenberger Imprecise random variables, random sets, and Monte Carlo simulation ISIPTA’15, Pescara, Italia 7 / 10

Simulation of a random set X 1 Propagation of a random set through g G ( ω ) = g ( X ( ω )) = { g ( X λ ( ω ))) : λ ∈ Λ } G ( ω ) = [ G ( ω ) , G ( ω )] random interval G ( ω ) = min g ( X ( ω )) , G ( ω ) = max g ( X ( ω )) 2 Cumulative distribution functions F ( y ) = � P ( g ≤ y ) , F ( y ) = P ( g ≤ y ) � � � � � F ( y )= P ( − ∞ , y ] ∩ [ G , G ] � = ∅ = P G ≤ y = F G ( y ) � � � � F ( y )= P [ G , G ] ⊂ ( − ∞ , y ] = P G ≤ y = F G ( y ) 3 Algorithm for computing F ( y ) Generate ω 1 ,..., ω N samp distributed as m. For each ω n , estimate G ( ω n ) ≈ min i g ( X λ i ( ω n )) using grid points λ 1 ,..., λ N grid on Λ . N samp 1 F ( y ) ≈ ∑ ✶ G ( ω k ) ≤ 0 · N samp . k = 1 Effort: N grid · N samp expensive evaluations of g . Th.Fetz, M. Oberguggenberger Imprecise random variables, random sets, and Monte Carlo simulation ISIPTA’15, Pescara, Italia 8 / 10