Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values of the parameters to gve, for example for two parameters R(X + δx) =[R(X) + R δx + R δx 2 x x }{{ 2 } st + 2 R δx 2 x 2 2 + 2 R δx δx 2 + 2 R δx 2 2 x x 2 x 2 +...] X () 2 2 }{{} 2nd Here R s the response, x and x 2 are the parameters whose nfluences are sought, and X represents the values of these parameters at whch the terms are evaluated. The local senstvty s defned as the coeffcents of the st order effects evaluated at X.e., S x = R x X (2) Fgure depcts the response as a functon of the two parameters, x and x 2 at dfferent locatons, X. R a b c f(r) x 2 f(x ) x f(x 2 ) Fgure : Representatve Response Surface Whle local senstvtes are commonly used to nvestgate the propertes of system responses, they suffer from several serous defcences:. the computaton of 2nd and hgher order terms s not a trval undertakng.
2. they represent the behavor only at the specfc values of the parameters, X, at whch the terms of the Taylor seres are evaluated. For nonlnear problems they may vary substantally. Fgure llustrates how they may vary as X s vares. We see that at pont X = (a), the slopes are very small and the system s nsenstve to both x and x 2. At pont (b), t s senstve prmarly to x, whle at pont (c) t s senstve to both x and x 2. Clearly dfferent conclusons about the senstvty of the system would be drawn dependng upon the pont at whch the senstvtes were evaluated. 3. the conclusons are also dependent upon the values of δx and δx 2,.e. upon the drecton of nterest. For some drectons the contrbutons of some of the cross terms may vansh, whle n other drectons they may domnate. 4. the cross dervatve terms represent nteractons between the parameters and t s dffcult to understand ther effects. Addtve Models In regresson and senstvty analyses consderable mportance s attached to addtve models. These are models of system responses that are the sum of functons, each of whch s a functon of only one varable, addtve model : R(x, x 2,..., x m ) = f + f (x ) + f 2 (x 2 ) + f m (x m ) The consttuent functons, f (x ), may be complex nonlnear functons of the sngle parameter, x. Addtve models have several very desrable characterstcs,. the behavor of the model wth respect to any sngle parameter, x j, can be determned wthout specfyng any of the other parameters 2. the maxmum/mnmum response s smply the sum of the maxmum/mnmum values of each of the consttuent functons 3. when solvng nverse problems for parameter values, the sum of the resduals squared s a m dmensonal hyper ellpsod and fndng the mnmum s acheved usng standard mnmzaton technques. 4. confdence ntervals for the parameters are easly determned 5. error analyses are smple to conduct. Unfortunately, models of techncal systems are rarely addtve. Instead, the model s usually a functon of several parameters that often occur n groups, e.g. Reynolds or Nusselt numbers. When ths happens cross dervatve terms appear n the Taylor seres. As a result of these cross dervatves, parameter estmaton problems may become much more complex, dependng upon the magntude and the character of the nteractons. Because of the emphass on addtve models, much of the lterature on senstvty analyss and regresson s not applcable to engneerng models. 2) Global Senstvty Instead of usng local senstvtes, Saltell and colleagues [2] have suggested the use of global senstvty based upon varances. The dea s to evaluate the contrbuton of the dfferent parameters to the varance over the range of the parameters. Ths s best done usng Sobol s [3] concept of total senstvtes. Let the response be f(x, x 2,..., x m ). Sobol showed that a functon could be decomposed n the form m m m f(x) =f + f (x ) + f,j (x, x j ) + m m j> k>j j> m f,j,k (x, x j, x k )... (3) 2
f (x ) are termed st order effects (man effects) and f,j (x, x j ) and hgher order terms represent the nteractons. The functons f, f,j, f,j,k et seq. are of zero mean and orthogonal. Because of these propertes, evaluatng the varances s partcularly easy. Now t s not easy to decompose f(x) n the form of Eq.3. Fortunately Sobol also showed that the varances can be expanded n a smlar way and suggested the use of total senstvty defned n terms of the total varance due to x S T = sum of frst effect varance due to x + the sum of all nteractons nvolvng x Var[f (X )] = V ar [E [f(x X ]] V ar[f(x] (4) Although X can represent several parameters of nterest, we wll restrct X to be a sngle parameter x. X represents all parameters except x,.e., the complementary set of parameters. f(x X ) s the functon evaluated wth all parameters X except X consdered as known, E s the condtonal expectaton taken over X, and V ar s the varance taken over the complementary set. The senstvty of the frst order effect s defned as S = V ar [E [f(x x )]] V ar[f(x)] For a 3 parameter model S T = S + S,2 + S,3 + S,2,3 If there are no nteractons, S = S T s easy to nterpret, S has no obvous drect meanng. S T for all. Whle Haylock and O Hagan [4] gave an nterestng nterpretaton of these senstvtes by notng that from the fundamental theorem relatng the varance and the condtonal expectatons and varances so that V ar[f(x)] = E [V ar [f(x x )]] + V ar [E [f(x x )]] (5a) S = V ar[f(x)] E [V ar [f(x x )]] V ar(f(x)) (5b) In other words S s the fractonal reducton n the Var[f(X)] observed when we know x. Lkewse, S T s the fracton of the varance due to x and ts nteractons wth all of the other parameters,.e., over the range of these other parameters, X are known. The dfference between S T and S s a drect measure of the sum of the effects of x nteractng wth the rest of the parameters. An especally mportant result s that M S = of all nteractons (6) and thus M S s a drect measure of all of the nteractons between the parameters. Upon decomposng f(x), one could examne the behavor of these nteracton terms to evaluate ther effects. We defne the senstvty to the nteracton effects relatve to the frst order effects as SIE = M S M (7) S whch provdes a drect measure of the combned nteractons. 3
2.) Example Consder the smple functon defned over x, x 2. The decomposton s f(x, x 2 ) = x 2 + x x 2 + x 2 (8) f(x, x 2 ) = 3 2 + (x2 + x 2 7 2 ) + (3x 2 2 3 4 ) + (x x 2 x 2 x 2 2 + 4 ) (9) Fgure 2 depcts f(x, x 2 ) andf,2 (x, x 2 ). From the fgure t appears that there s a strong nteracton but t s dffcult to estmate the effects of the nteracton. Whle t may be possble to analytcally or numercally evaluate the nteracton effects, there are a total of 2 M terms n the expanson, meanng a computatonally ntractable approach for a large number of parameters, and even unrealstc for M as small as 4..25 3 2.5 2.5.2..5.5..2.8.25.6.4.2 X2 X.2.4.6.8 X2 X Fgure 2: left: Response Surface for f(x, x 2 ) = x 2 + x x 2 + x 2 rght: nteracton From Fgure 2b t appears that there s consderable nteracton. However, S =.4982, S 2 =.4839, and S,2 =.79. Thus, contrary to the mpresson ganed from Fg. 2b, the nteracton s seen to be neglgble. 2.2) Example 2 Consder the functon shown n Fgure 3a f(x, x 2 ) = x x 2 () From Eq. the functon s nothng but an nteracton and one mght thnk that the frst order effects would be zero. However, ts expanson n the form of Eq.3 s f(x, x 2 ) = 4 2 (x 2 ) 2 (x 2 2 ) + (x x 2 + x 2 + x 2 2 3 4 ) () wth the nteracton shown n Fgure 3b. From Eq., a drect evaluaton of the varances gves and var(f) = 7 44, var x = 3 44, var x 2 = 3 44, var x x 2 = 44 S x = 3 7, S x 2 = 3 7, S x x 2 = 7 (2) 4
F(x,y)=xy Fxy.8.2..6.4 -..2 -.2.6.4.2.8 -.3.8.6.4.2 y x 8-Feb-28 2:9:37 Fgure 3: left: Response Surface for f(x, x 2 ) = x x 2 rght: nteracton It s not always easy or even possble to express a nonlnear functon n terms of Sobol s expanson, Eq. 3. However, one can always evaluate the senstvtes usng Eq. 5b. For ths functon, assumng unform and ndependent dstrbutons for x, x 2 we have var x2 (f x 2 ) = x 2 2, E x = 36 and usng Eq.5b we easly obtan the senstvtes. Note that the total senstvtes are S T x = 4/7, S T x 2 = 4/7 and that the nteractons are /3th of the frst order senstvtes. 2.3) Example 3 In general, the response of engneerng systems cannot be descrbed by equatons such as treated n Examples and 2. Instead the response s commonly determned through the soluton of partal dfferental equatons or ther dscrete equvalents, fnte elements or fnte volumes. In these cases, t s not possble to represent them n the form of Eq.3 and the senstvtes can only be found usng Eq.5b. Consder one-dmensonal heat transfer n a slab of thckness L. We examned the effect of the dfferent thermal parameters on the temperature measured at x = when a heat flux, Q, s prescrbed at x = L. Snce the tme at whch the thermal flux reaches x = s a functon of the dffusvty, κ = k/ρc, we expected to see consderable nteracton between these two propertes. As shown on Fgure 3a, we see a hgh senstvty to k and ρc wth both parameters showng substantal nteractons. The ntal response at x= s a very weak functon of Q and h. As steady state s approached, the senstvty to k and ρc vanshes. Wth the steady state temperature gven by (3) T(x = ) T = Q/h It s clear that t wll be dffcult to accurately estmate k and ρc smultaneously because of: a) the rapd reducton of senstvtes, b) the strong nteracton between these two parameters. The cause of ths nteracton s obvous, but the evaluatng ts effect s not trval. The estmaton of Q and h at longer tmes s smlarly affected. However, ts magntude s easly estmated snce t occurs because of the exstence of the second order dervatve wth respect to Q and h. Note that the second dervatve wth respect to Q vanshes whle that to h exsts. Thus the senstvty to Q s lnear whle that to h s nonlnear. 5
.7. Senstvtes.6.4.3.2. k ρc h Q S S T Sum of S.98.96.94.92.9....5 2. Fo.88...5 2. Fgure 3a Frst Effect and Total Senstvtes Fgure 3b Sum of Frst Effect Senstvtes for a prescrbed heat flux Quantfyng the effect can only be done by examnng the sum of the senstvtes, Eq. Fgure 4b llustrates the total nteracton nvolved n the model. At early tmes, the nteractons are substantal, approxmately %, at steady state they are reduced to the order of 2%. One precauton must be taken. Even though the nteractons may be large, they are of lttle concern f the total varance s neglgble. The star on Fgure 4b marks the pont where the varance frst exceeds 5% of the steady state value; nteractons pror to ths tme of lttle nterest. The maxmum sum of the nteractons are less than 5% and show the same reducton as steady state s approached for the other two problems. References. Beck. J. V. and Arnold, K. J., 977, Parameter Estmaton n Engneerng and Scence, J. Wley and Sons, N.Y., NY 2. Saltell, A., Tarantola, S., Campolongo,F. and Ratto, M., 24, Senstvty Analyss n Practce, J. Wley and Sons, N.Y., NY 3. Sobol, I. M., 993, Senstvty Estmates for Nonlnear Mathematcal Models, MMCE, Vol., No. 4, pp. 47-44 4. Haylock, R. G. and O Hagan, A., 996, On Inference for Outputs of Computatonally Expensve Algorthms wth Uncertanty on The Inputs, Bayesan Statstcs 5, Oxford Unversty Press, pp. 629-637 Fo 6