Estimation of a rare event probability on a complex system modeled - PowerPoint PPT Presentation

Kriging and Rare Events Estimation of a rare event probability on a complex system modeled with a kriging algorithm. R.PASTEL 1 , 2 J.MORIO 2 H.PIET-LAHANIER 2 1 ENSTA-ParisTech FRANCE 2 ONERA FRANCE Rare Events Simulation, Rennes, 2008

Kriging and Rare Events Introduction Introduction Probabilities and statistics are more and more frequently used in mathematical modeling to: structure what is unknown in a system, quantify the errors, measure a system reliability. What is, in average, the output of a complex system with random inputs?

Kriging and Rare Events Introduction Unfortunately, the problem has often no analytical solution , and/or is not analytical itself, = ⇒ we have to go through a numerical resolution. Two numerical resolution drawbacks 1 Numerical techniques require a long calculation time, 2 Random number generators are not perfectly random! = ⇒ no rare event is generated.

Kriging and Rare Events Introduction Hereby, we will conjugate two approaches to solve those issues: 1 to free ourselves from the system’s analytical expression and shorten the calculation time, we will use kriging, a statistical interpolation technique. 2 to quantify the randomness taking rare events into account, we will compare three methods: The Crude Monte-Carlo (CMC), The histogram, The Importance Sampling (IS).

Kriging and Rare Events Summary Summary 1 Kriging The problem intuition Elements of theory An example Use and limitations

Kriging and Rare Events Summary Summary 1 Kriging The problem intuition Elements of theory An example Use and limitations 2 Probabilistic methods The experience framework The Crude Monte-Carlo (CMC) Importance Sampling The Histogram Experimental results

Kriging and Rare Events Summary Summary 1 Kriging The problem intuition Elements of theory An example Use and limitations 2 Probabilistic methods The experience framework The Crude Monte-Carlo (CMC) Importance Sampling The Histogram Experimental results 3 Application to missile firing

Kriging and Rare Events Kriging Current Section 1 Kriging The problem intuition Elements of theory An example Use and limitations 2 Probabilistic methods The experience framework The Crude Monte-Carlo (CMC) Importance Sampling The Histogram Experimental results 3 Application to missile firing

Kriging and Rare Events Kriging The problem intuition Progress 1 Kriging The problem intuition Elements of theory An example Use and limitations 2 Probabilistic methods The experience framework The Crude Monte-Carlo (CMC) Importance Sampling The Histogram Experimental results 3 Application to missile firing

Kriging and Rare Events Kriging The problem intuition Kriging The problem intuition Kriging is an interpolation technique originated in Geophysics. Measurement drillings have to be in limited number to reduce costs and shorten the exploration time of a given subsoil.

Kriging and Rare Events Kriging The problem intuition Figure: Subsoil to explore to find the gold.

Kriging and Rare Events Kriging The problem intuition Figure: Some gold concentration measurements are done.

Kriging and Rare Events Kriging The problem intuition Figure: What is the gold concentration elsewhere?

Kriging and Rare Events Kriging Elements of theory Progress 1 Kriging The problem intuition Elements of theory An example Use and limitations 2 Probabilistic methods The experience framework The Crude Monte-Carlo (CMC) Importance Sampling The Histogram Experimental results 3 Application to missile firing

Kriging and Rare Events Kriging Elements of theory Kriging Formal idea One wants to know the values of the criterion c on a domain of interest D . However, it is impossible to have infinitely many exact values as: c has no analytical expression, AND/OR c is costly to evaluate numerically. Problem framework It is therefore about building up a good approximation of the criterion on the whole domain of interest, based on few exact values measured in well-chosen locations.

Kriging and Rare Events Kriging Elements of theory A classical regression approach Let us review a classical regression approach.

Kriging and Rare Events Kriging Elements of theory A classical regression approach Let us review a classical regression approach. Basic idea: Criterion = Model + Error For a given x ∈ D : c ( x ) = F ( β, x ) + ǫ ( β, x )

Kriging and Rare Events Kriging Elements of theory A classical regression approach Let us review a classical regression approach. Basic idea: Criterion = Model + Error For a given x ∈ D : c ( x ) = F ( β, x ) + ǫ ( β, x ) Usually: c ( x ) = f ( x ) β + ǫ ( β, x )

Kriging and Rare Events Kriging Elements of theory A classical regression approach Let us review a classical regression approach. Basic idea: Criterion = Model + Error For a given x ∈ D : c ( x ) = F ( β, x ) + ǫ ( β, x ) Usually: c ( x ) = f ( x ) β + ǫ ( β, x ) ⇓ ⇓ ⇓ Measured data matrices: C = F β + E

Kriging and Rare Events Kriging Elements of theory A classical regression approach Let us review a classical regression approach. Basic idea: Criterion = Model + Error For a given x ∈ D : c ( x ) = F ( β, x ) + ǫ ( β, x ) Usually: c ( x ) = f ( x ) β + ǫ ( β, x ) ⇓ ⇓ ⇓ Measured data matrices: C = F β + E = ⇒ β is defined

Kriging and Rare Events Kriging Elements of theory A classical regression approach Let us review a classical regression approach. Basic idea: Criterion = Model + Error For a given x ∈ D : c ( x ) = F ( β, x ) + ǫ ( β, x ) Usually: c ( x ) = f ( x ) β + ǫ ( β, x ) ⇓ ⇓ ⇓ Measured data matrices: C = F β + E ⇒ β is defined as the solution of min β � E � 2 = 2 .

Kriging and Rare Events Kriging Elements of theory Two steps further ˆ c ,the kriging estimator The two extra hypothesis behind kriging are

Kriging and Rare Events Kriging Elements of theory Two steps further ˆ c ,the kriging estimator The two extra hypothesis behind kriging are 1 ǫ ( β, x )

Kriging and Rare Events Kriging Elements of theory Two steps further ˆ c ,the kriging estimator The two extra hypothesis behind kriging are 1 ǫ ( β, x ) is the trajectory of a random process.

Kriging and Rare Events Kriging Elements of theory Two steps further ˆ c ,the kriging estimator The two extra hypothesis behind kriging are 1 ǫ ( β, x ) is the trajectory of a random process. 2 ˆ c ( x )

Kriging and Rare Events Kriging Elements of theory Two steps further ˆ c ,the kriging estimator The two extra hypothesis behind kriging are 1 ǫ ( β, x ) is the trajectory of a random process. c ( x ) = κ T ( x ) C with κ : R n �→ R m . 2 ˆ

Kriging and Rare Events Kriging Elements of theory Two steps further ˆ c ,the kriging estimator The two extra hypothesis behind kriging are 1 ǫ ( β, x ) is the trajectory of a random process. c ( x ) = κ T ( x ) C with κ : R n �→ R m . 2 ˆ = ⇒ ∀ x ∈ D , ( β, κ ) is defined

Kriging and Rare Events Kriging Elements of theory Two steps further ˆ c ,the kriging estimator The two extra hypothesis behind kriging are 1 ǫ ( β, x ) is the trajectory of a random process. c ( x ) = κ T ( x ) C with κ : R n �→ R m . 2 ˆ = ⇒ ∀ x ∈ D , ( β, κ ) is defined as the solution of c ( x ) − c ( x ) � 2 � min β,κ E [ � ˆ 2 ] Under the no bias constraint F T c = f ( x )

Kriging and Rare Events Kriging An example Progress 1 Kriging The problem intuition Elements of theory An example Use and limitations 2 Probabilistic methods The experience framework The Crude Monte-Carlo (CMC) Importance Sampling The Histogram Experimental results 3 Application to missile firing

Kriging and Rare Events Kriging An example

Kriging and Rare Events Kriging An example

Kriging and Rare Events Kriging An example

Kriging and Rare Events Kriging An example

Kriging and Rare Events Kriging Use and limitations Progress 1 Kriging The problem intuition Elements of theory An example Use and limitations 2 Probabilistic methods The experience framework The Crude Monte-Carlo (CMC) Importance Sampling The Histogram Experimental results 3 Application to missile firing

Kriging and Rare Events Kriging Use and limitations Kriging Use and limitations Kriging in a nutshell Kriging enables to interpolate any given criterion, analytical or not, on a restrained domain thanks to sufficiently many well-spread exact values. Error is nil at the measure sites and increases when moving away from them. In particular: Kriging interpolates but does not forecast → the proxy is bad in subdomains with no measure. The proxy quality is good in a limited area around the measurement site → a good proxy requires a sufficient measurement site density.

Kriging and Rare Events Probabilistic methods Current Section 1 Kriging The problem intuition Elements of theory An example Use and limitations 2 Probabilistic methods The experience framework The Crude Monte-Carlo (CMC) Importance Sampling The Histogram Experimental results 3 Application to missile firing