Local and Global Sensitivity Analysis

Transcription

1 Omar 1,2 1 Duke University Department of Mechanical Engineering & Materials Science omar.knio@duke.edu 2 KAUST Division of Computer, Electrical, Mathematical Science & Engineering omar.knio@kaust.edu.sa

2 Motivation In this section, we will be mainly interested with the following, fairly straightforward but as it turns out delicate and important, question : Given f (x, y) does f vary more with x or with y? Why is this important? Planning Decision analysis and support Designing experiments or field campaigns, etc... For the discussion that follows, assume we have a minimum of information, specifying : shape of f (equation, table, spreadsheet, simulation code, etc...) range of x and y optionally a design point, say x 0, y 0

3 L 2 functions over unit-hypercubes Let L 2 (U d ) be the space of real-valued squared-integrable functions over the d-dimensional hypercube U : f : x 2 U d 7! f (x) 2 R, f 2 L 2 (U d ), f (x) 2 dx < 1. U d L 2 (U d ) is equipped with the inner product h, i, 8f, g 2 L 2 (U d ), hu, vi := f (x)g(x)dx, U d and norm k k 2, 8f 2 L 2 (U d ), kf k 2 := hf, f i 1/2.

4 L 2 functions over unit-hypercubes Let L 2 (U d ) be the space of real-valued squared-integrable functions over the d-dimensional hypercube U : f : x 2 U d 7! f (x) 2 R, f 2 L 2 (U d ), f (x) 2 dx < 1. U d NB : all subsequent developments immediately extend to product-type situations, where x 2 A = A 1 A d R d, and weighted spaces L 2 (A, ), : x 2 A 7! (x) 0, (x) = 1 (x 1 ) d (x d ). (e.g. : is a pdf of a random vector x with mutually independent components.)

5 Ensemble notations Let D = {1, 2,...,d}. Given i D, we denote i := D \ i its complement set in D, such that For instance i = {1, 2} and i = {3,...,d}, i = D and i = ;. i [ i = D, i \ i = ;.

6 Ensemble notations Let D = {1, 2,...,d}. Given i D, we denote i := D \ i its complement set in D, such that i [ i = D, i \ i = ;. Given x =(x 1,...,x d ), we denote x i the vector having for components the x i2i, that is D i = {i 1,...,i i } ) x i =(x i1,...,x i i ), where i := Card(i). For instance f (x)dx i = U i U i f (x 1,...,x d ) Y i2i dx i, and Yi /2i f (x)dx i = f (x 1,...,x d ) dx i, U d i U d i i2d

7 Sobol-Hoeffding decomposition Any f 2 L 2 (U d ) has a unique hierarchical orthogonal decomposition of the form f (x) =f (x 1,...,x d )=f 0 + dx dx f i (x i )+ i=1 dx i=1 j=i+1 k=j+1 dx i=1 j=i+1 dx f i,j (x i, x j )+ dx f i,j,k (x i, x j, x k )+ + f 1,...,d (x 1,...,x d ). Hierarchical : 1st order functionals (f i )! 2nd order functionals (f i,j )! 3rd order functionals (f i,j,l )!! d-th order functional (f 1,...,d ). Using ensemble notations : Decomposition in a sum of 2 k functionals f (x) = X i D f i (x i ).

8 Sobol-Hoeffding decomposition Any f 2 L 2 (U d ) has a unique hierarchical orthogonal decomposition of the form f (x) = X f i (x i ). i D Orthogonal : the functionals if the S-H decomposition verify the following orthogonality relations : f i (x i )dx j = 0, 8i D, j 2 i, U f i (x i )f j (x j )dx = hf i, f j i = 0, 8i, j D, i 6= j. U d It follows the hierarchical construction f ; = f (x)dx = hf i ; =D U d f {i} = f (x)dx {i} f ; = hf i D\{i} f ; i 2 D U d 1 X X f i = f (x)dx i f j = hf i i f j i 2 D, i 2. U i j(i j(i

9 Example Consider a 3D example : Y = f (X 1, X 2, X 3 ) Then SH yields : f? = E[Y ] f 1 (X 1 )=E[Y X 1 ] f? f 2 (X 2 )=E[Y X 2 ] f? f 12 (X 1, X 2 )=E[Y X 1, X 2 ] f 1 f 2 f? f 13 (X 1, X 3 )=E[Y X 1, X 3 ] f 1 f 3 f? f 23 (X 2, X 3 )=E[Y X 2, X 3 ] f 2 f 3 f? f 3 (X 3 )=E[Y X 3 ] f? f 123 (X 1, X 2, X 3 ) = E[Y X 1, X 2, X 3 ] f 1 f 2 f 3 f 12 f 13 f 23 f? = f (X 1, X 2, X 3 ) f 1 f 2 f 3 f 12 f 13 f 23 f?

10 Parametric sensitivity analysis Consider x as a set of d independent random parameters uniformly distributed on U d, and f (x) a model-output depending on these random parameters. It is assumes that f is a 2nd order random variable : f 2 L 2 (U d ). Thus, f has a unique S-H decomposition f (x) = X i D f i (x i ). Further, the integrals of f with respect to i are in this context conditional expectations, E[f x i ]= f (x)dx i = g(x i ) 8i D, U i so the S-H decomposition follows the hierarchical structure f ; = E[f ] f {i} = E f x {i} E[f ] i 2 D X f i = E[f x i ] f j i D, i 2. j(i

11 Variance decomposition Because of the orthogonality of the S-H decomposition the variance V [f ] of the model-output can be decomposed as i6=; X V [f ]= V [f i ], V [f i ]=hf i, f i i. i D V [f i ] is interpreted as the contribution to the total variance V [f ] of the interaction between parameters x i2i. The S-H decomposition thus provide a rich mean of analyzing the respective contributions of individual or sets of parameters to model-output variability. However, as there are 2 d 1 contributions, so one needs more "abstract" characterizations.

12 Sensitivity indices To facilitate the hierarchization of the respective influence of each parameter x i, the partial variances V [f i ] are normalized by V [f ] to obtain the sensitivity indices : S i (f )= V [f i6=; i] X apple 1, S i (f )=1. V [f ] i D The order of the sensitivity indices S i is equal to i = Card(i). 1st order sensitivity indices. The d first order indices S {i}2d characterize the fraction of the variance due the parameter x i only, i.e. without any interaction with others. Therefore, dx 1 S {i} (f ) 0, i=1 measures globally the effect on the variability of all interactions between parameters. If P d i=1 S {i} = 1, the model is said additive, because its S-H decomposition is dx f (x i,...,x d )=f 0 + f i (x i ), and the impact of the parameters can be studied separately. i=1

13 Sensitivity indices To facilitate the hierarchization of the respective influence of each parameter x i, the partial variances V [f i ] are normalized by V [f ] to obtain the sensitivity indices : S i (f )= V [f i6=; i] X apple 1, S i (f )=1. V [f ] i D The order of the sensitivity indices S i is equal to i = Card(i). Total sensitivity indices. The first order SI S {i} measures the variability due to parameter x i alone. The total SI T {i} measures the variability due to the parameter x i, including all its interactions with other parameters : T {i} := X i3i S i S {i}. Important point : for x i to be deemed non-important or non-influent on the model-output, S {i} and T {i} have to be negligible. Observe that P i2d T {i} 1, the excess from 1 characterizes the presence of interactions in the model-output.

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15 Sensitivity indices In many uncertainty problem, the set of uncertain parameters can be naturally grouped into subsets depending on the process each parameter accounts for. For instance, boundary conditions BC, material property ', external forcing F, and D is the union of these distinct subsets : D = D BC [ D ' [ D F. The notion of first order and total sensitivity indices can be extended to characterize the influence of the subsets of parameters. For instance, S D' = X i D ' S i, measures the fraction of variance induced by the material uncertainty alone, while T DF = X S i. i\d F 6=; measures the fraction of variance due to the external forcing uncertainty and all its interactions.

16 Recap Sensitivity index S u, total sensitivity index T u it is easy to verify that S u + T u = S u + T u = 1 P it immediately follows that i S {i} apple 1 ) P i T {i} 1 S u high means that X u has substantial contribution to V (Y ) T u low means that X u has weak contribution to V (Y ) (and so it is a candidate to be dropped)

17 S-H decomposition from PC expansions. Consider the model-output f : 2 R d 7! R, where =( 1,..., d ) are independent real-valued r.v. with joint-probability density function p (x 1,...,x d )= dy p i (x i ). Let { } be the set of d-variate orthogonal polynomials, i=1 ( ) = dy i=1 (i) i ( i ), with (i) l 0 2 l the uni-variate polynomials mutually orthogonal with respect to the density p i. If f 2 L 2 (, p ), it has a convergent PC expansion f ( ) = lim X No!1 appleno ( )f, = dx i. i=1

18 S-H decomposition from PC expansions. Given the truncated PC expansion of f, X ˆf ( ) = ( )f, applea one can readily obtain the PC approximation of the S-H functionals through ˆfi ( i )= X ( i )f, applea(i) where the multi-index set A(i) is given by A(i) ={ 2 A; i > 0 for i 2 i, i = 0 for i /2 i} ( A. For the sensitivity indices it comes P 2A(i) S i (ˆf )= f 2 P h, i P 2A f 2 h, i, T 2T (i) {i}(ˆf )= f 2 h, i P 2A f, 2 h, i where T (i) ={ 2 A; i > 0 for i 2 i}