University of Texas at El Paso. Vladik Kreinovich University of Texas at El Paso,

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1 Uiversity of Texas at El Paso Departmetal Techical Reports (CS) Departmet of Computer Sciece How to Deal with Ucertaities i Computig: from Probabilistic ad Iterval Ucertaity to Combiatio of Differet Approaches, with Applicatios to Egieerig ad Bioiformatics Vladik Kreiovich Uiversity of Texas at El Paso, vladik@utep.edu Follow this ad additioal works at: Part of the Computer Scieces Commos Commets: Techical Report: UTEP-CS To appear i Proceedigs of the Eighth Iteratioal Coferece o the Applicatios of Digital Iformatio ad Web Techologies ICADIWT'2017, Ciudad Juarez, Chihuahua, Mexico, March 29-31, 2017 Recommeded Citatio Kreiovich, Vladik, "How to Deal with Ucertaities i Computig: from Probabilistic ad Iterval Ucertaity to Combiatio of Differet Approaches, with Applicatios to Egieerig ad Bioiformatics" (2017). Departmetal Techical Reports (CS) This Article is brought to you for free ad ope access by the Departmet of Computer Sciece at DigitalCommos@UTEP. It has bee accepted for iclusio i Departmetal Techical Reports (CS) by a authorized admiistrator of DigitalCommos@UTEP. For more iformatio, please cotact lweber@utep.edu.

2 How to Deal with Ucertaities i Computig: from Probabilistic ad Iterval Ucertaity to Combiatio of Differet Approaches, with Applicatios to Egieerig ad Bioiformatics Vladik KREINOVICH a,1 a Departmet of Computer Sciece, Uiversity of Texas at El Paso, USA Abstract. Most data processig techiques traditioally used i scietific ad egieerig practice are statistical. These techiques are based o the assumptio that we kow the probability distributios of measuremet errors etc. I practice, ofte, we do ot kow the distributios, we oly kow the boud o the measuremet accuracy hece, after the get the measuremet result x, the oly iformatio that we have about the actual (ukow) value x of the measured quatity is that x belogs to the iterval [ x, x + ]. Techiques for data processig uder such iterval ucertaity are called iterval computatios; these techiques have bee developed sice 1950s. I may practical problems, we have a combiatio of differet types of ucertaity, where we kow the probability distributio for some quatities, itervals for other quatities, ad expert iformatio for yet other quatities. The purpose of this paper is to describe the theoretical backgroud for iterval ad combied techiques ad to briefly describe the existig practical applicatios. Keywords. ucertaity, iterval computatios, probabilistic ucertaity 1. Data Processig uder Ucertaity: A Geeral Problem Need for idirect measuremets. I may practical situatios, we are iterested i the value of a quatity y which is difficult (or eve impossible) to measure directly, such as the amout of oil i a give area or tomorrow s temperature. Sice we caot measure y directly, a atural idea is to measure y idirectly, i.e., fid some easier-to-measure quatities x 1,...,x which are related to y by a kow depedece y = f (x 1,...,x ), ad use the results x i of measurig x i to estimate y as f ( x 1,..., x ). 1 Author: Vladik Kreiovich, Departmet of Computer Sciece, Uiversity of Texas at El Paso, 500 W. Uiversity, El Paso, TX 79968, USA; vladik@utep.edu.

3 x 1 x 2 f ỹ = f ( x 1,..., x ) x Idirect measuremets are also kow as data processig. Need to take ucertaity ito accout. Measuremets are ever 100% accurate: the measuremet result x i is, i geeral, somewhat differet from the actual (ukow) def values x i, ad we have a o-zero measuremet error x i = x i x i. Hece, the result ỹ = f ( x 1,..., x ) of idirect measuremet is, i geeral, differet, from the actual value y = f (x 1,...,x ). How big is the resultig ucertaity y def = ỹ y? x 1 x 2... f y x Probabilistic ad iterval ucertaity. Traditioal approach to ucertaity assumes that we kow the probability distributios for x i (usually, they are assumed to be Gaussia). I may practical situatios, we ca ideed determie these probability distributios by calibratig the correspodig measurig istrumet (MI), i.e., to comparig its result with the results of measurig the same quatity by a much more accurate ( stadard ) MI. However, there are two case whe the calibratio is ot possible: state-of-the-art measuremets, e.g., i fudametal sciece, whe we use the best MI available ad so, o more-accurate MIs are available for compariso, ad maufacturig, whe calibratio of each MI is possible, but would be prohibitively expesive. I such cases, all we kow is the upper boud i o the (absolute value of the) measuremet error x i, the boud provided by the maufacturer of the MI; see, e.g., [13]. I this case, oce we have the measuremet result x i, the oly thig we kow about the actual (ukow) value x i is that x i belogs to the iterval [ x i i, x i + i ]. Algorithms for processig such iterval ucertaity are kow as iterval computatio; see, e.g., [3,8].

4 2. Iterval Computatios Iterval computatios: a problem. We kow a algorithm y = f (x 1,...,x ) ad we kow itervals x i = [x i,x i ]. We wat to compute the rage of possible values of y: y = [y,y] = { f (x 1,...,x ) x 1 [x 1,x 1 ],...,x [x,x ]}. x 1 x 2 f y = f (x 1,...,x ) x I geeral, this problem is NP-hard eve for quadratic f ; see, e.g., [5]. So, from the practical viewpoit, we face the followig challeges: whe are feasible algorithms possible? ad whe computig y is ot feasible, how ca we fid a good approximatio Y y. Iterval computatios: a brief history. Iterval computatios ca be traced to Archimedes who used lower ad upper bouds to estimate π. I moder times, this area was revived by three 1950s pioeers: M. Warmus (Polad), T. Suaga (Japa), ad R. Moore (USA). The early 1960s witessed the first boom, icludig takig iterval ucertaity ito accout whe plaig spaceflights to the Moo. Curret applicatios rage from desigig elemetary particle colliders to checkig whether a comet will hit the Earth to robotics ad chemical egieerig; see, e.g., [3,8]. But why ot use Maximum Etropy (MaxEt) approach? Ofte i practice, may differet probability distributios are cosistet with the same observatios. Istead of cosiderig all possible distributios as iterval computatios do why ot select oe of these distributios e.g., the oe with the largest etropy. For example, if all we kow is that x belogs to a iterval, the this MaxEt approach leads to a uiform distributio o this iterval. For several variables, if we have o iformatio about their depedece, this approach cocludes that these variables are idepedet. This idea is reasoable, but it has a serious limitatio. As a example, let us take the simplest possible data proceedig algorithm y = x x. I this case, y = x x. Let us assume that all measuremet errors x i are limited to the iterval [, ]. The, the worst case situatio is whe x i = ad thus, y =. O the other had, if we use the MaxEt approach, the for large, due to Cetral Limit Theorem, y is close to ormal, with σ =. So, if we, as usual, igore 3 deviatios exceedig 6σ, we get a upper boud but it is possible that =. Coclusio: usig a sigle distributio ca be very misleadig, especially if we wat guarateed results ad this is very importat i high-risk applicatio areas such as space exploratio or uclear egieerig.

5 Iterval arithmetic: foudatios of iterval techiques. The geeral problem of iterval computatios is to compute the rage of a give fuctio f (x 1,...,x ) o give itervals x i. For arithmetic operatios f (x 1,x 2 ) (ad for elemetary fuctios), we have explicit formulas for the rage; e.g., whe x 1 x 1 = [x 1,x 1 ] ad x 2 x 2 = [x 2,x 2 ], the: The rage x 1 + x 2 for x 1 + x 2 is [x 1 + x 2,x 1 + x 2 ]. The rage x 1 x 2 for x 1 x 2 is [x 1 x 2,x 1 x 2 ]. The rage x 1 x 2 for x 1 x 2 is [y,y], where y = mi(x 1 x 2,x 1 x 2,x 1 x 2,x 1 x 2 ) ad y = max(x 1 x 2,x 1 x 2,x 1 x 2,x 1 x 2 ). The rage 1/x 1 for 1/x 1 is [1/x 1,1/x 1 ] (if 0 x 1 ). Straightforward iterval computatios: example. Iside the computer, each computatio is a sequece of arithmetic operatios. For example, computig f (x) = (x 2) (x + 2) meas that we compute r 1 := x 2, the r 2 := x + 2, ad fially, r 3 := r 1 + r 2. To fid the rage of fuctio o a give iterval, a seemigly atural idea is to perform the same operatios, but with itervals istead of umbers. For example, for x [1,2], we compute r 1 := [1,2] [2,2] = [ 1,0], r 2 := [1,2]+[2,2] = [3,4], ad r 3 := [ 1,0] [3,4] = [ 4,0]. However, the actual rage is arrower: f (x) = [ 3,0]. Thus, we eed to come up with more efficiet ways of computig a eclosure Y y. First idea: use of mootoicity. For arithmetic operatios, we had exact rages because +,, ad are mootoic i each variable. I geeral, if f (x 1,...,x ) is (ostrictly) icreasig ( f ) i each x i, the f (x 1,...,x ) = [ f (x 1,...,x ), f (x 1,...,x )]. We ca get similar simplifyig formulas if f for some x i ad f for other x j. How ca we check whether f? It is well kow that f i x i if f x i 0 for all values x = (x 1,...,x ). So, to check for mootoicity, it is sufficiet to check whether the rage [r i,r i ] of f x i o x i has r i 0. Differetiatio ca easily doe, e.g., by Automatic Differetiatio (AD) tools. Estimatig rages of f x i ca be doe, e.g., by straightforward iterval computatios. So, if the rage [r i,r i ] of each f x i o x i has r i 0, the f (x 1,...,x ) = [ f (x 1,...,x ), f (x 1,...,x )]. I the above example of f (x) = (x 2) (x + 2) o x = [1,2], we have d f dx = 1 (x+2)+(x 2) 1 = 2x. The rage of this derivative is [r,r] = [2,4], with 2 0. Thus, we get f ([1,2]) = [ f (1), f (2)] = [ 3,0], which is the exact rage. Secod idea: cetered form. Not all fuctios are mootoic. I such cases, we ca use aother idea from calculus the Itermediate Value Theorem, accordig to which

6 f (x 1,...,x ) = f ( x 1,..., x ) + f (χ) (x i x i ) x i for some χ i x i. We ca thus coclude that f (x 1,...,x ) Y, where Y = ỹ + f (x 1,...,x ) [ i, i ]. x i Here, differetiatio ca be performed by the AD tools, ad the rage of the derivatives ca be estimated, if appropriate, by mootoicity, or by straightforward iterval computatios or itself by cetered form (this will take more time but lead to a more accurate estimatio). As a example, we ca take a o-mootoic fuctio f (x) = x (1 x) o the iterval x = [0,1]. Computig this fuctio meas computig r 1 := 1 x ad r 2 := x r 1. Thus, straightforward iterval computatios lead to r 1 := [1,1] [0,1] = [0,1] ad r 2 := [0, 1] [0, 1] = [0, 1], while the actual rage is [0, 0.25]. Here, x = [ x, x + ], with x = 0.5 ad = 0.5, ad d f = 1 (1 x) + x ( 1) = dx 1 2x, thus d f (x) = 1 2 [0,1] = [ 1,1]. So, we get the estimate dx Y = f ( x) + d f (x) [, ] = 0.5 (1 0.5) + [ 1,1] [ 0.5,0.5] = dx [ 0.5,0.5] = [ 0.25,0.75]. The upper boud is better tha for straightforward iterval computatios, but the lower boud is worse. This example shows that we eed other ideas. Third idea: bisectio. The cetered form is based o the first order formula whose accuracy is O( 2 i ). Thus, if he itervals are too wide, we ca split oe of them i half decreasig the error from 2 i to 2 i /4 ad take the uio of the resultig rages. For example, for f (x) = x (1 x) o x x = [0,1], we split ito x = [0,0.5] ad x = [0.5,1]. O both subitervals, f (x) is mootoic: e.g., o x, 1 2 x = 1 2 [0.5,1] = [ 1,0] with r 0. So, we get the exact bouds f (x ) = [ f (0), f (0.5)] = [0,0.25] ad f (x ) = [ f (1), f (0.5)] = [0,0.25], ad the exact overall estimate f (x ) f (x ) = [0,0.25]. Of course, this is a toy example; i practice, we do ot always get the exact rage, but by combiig the above three ideas, we get reasoable estimates for the rage. Alterative approach: affie arithmetic. So far, we compute the rage of x (1 x) by multiplyig rages of x ad 1 x. We igored the fact that both factors deped o x ad are, thus, depedet. A atural idea is thus, for each itermediate result a, to keep a explicit depedece o x i = x i x i (at least its liear terms). I other words, we represet each itermediate result as a = a 0 + a i x i + [a,a]. We start with x i = x i x i, i.e., with

7 x i + 0 x x i 1 + ( 1) x i + 0 x i x + [0,0], where a 0 = x i, a i = 1, a j = 0 for j i, ad [a,a] = [0,0]. The, for a = a 0 + a i x i + a ad b = b 0 + b i x i + b, we get: Additio: c 0 = a 0 + b 0, c i = a i + b i, c = a + b. Subtractio: c 0 = a 0 b 0, c i = a i b i, c = a b. Multiplicatio: c 0 = a 0 b 0, c i = a 0 b i + b 0 a i, ad c = a 0 b + b 0 a + i j a i b j [ i, i ] [ j, j ] + a i b i [ i, i ] 2 + i ( ) ( ) a i [ i, i ] i b + b i [ i, i ] i a + a b. For example, for f (x) = x (1 x) o x [0,1], = 1, x = 0.5, ad = 0.5. The computer computes f (x) by computig r 1 := 1 x ad r 2 := x r 1. I affie arithmetic, we start with x = 0.5 x + [0,0], the compute r 1 := 1 (0.5 x) = x ad r 2 := (0.5 x) (0.5 + x), i.e., r 2 = x [, ] 2 = [ 2,0]. The resultig rage is y = [ 0.25,0] = [0,0.25], which is the exact rage. I our simple example, we got the exact rage. However, i geeral, rage estimatio is NP-hard. This meas that ay feasible (polyomial-time) algorithm will sometimes lead to excess width: Y y. I particular, affie arithmetic may lead to excess width. How to get more accurate estimates? Oe idea is to use bisectio. Aother idea is that, i additio to liear terms i x i, we also keep quadratic ad higher order terms i the Taylor expasio. This Taylor arithmetic has ideed bee successful, e.g., i collider desig. Iterval computatios vs. affie arithmetic: comparative aalysis. Affie arithmetic is usually more accurate, but it is also slower. Ideed, i iterval computatios, for each itermediate result a, we compute two values: edpoits a ad a of [a,a]. I affie arithmetic, for each a, we compute + 3 values: a 0, a 1,..., a, a, ad a. Thus, affie arithmetic is times slower. Solvig systems of equatios: extedig kow algorithms to situatios with iterval ucertaity. I may practical situatios, we eed to fid the ukows y i from the system of equatios g i (y 1,...,y ) = a i. May algorithms y j = f j (a 1,...,a m ) are kow for the cases whe we kow the values a i. I practice, however, we ofte a i with iterval ucertaity: we oly kow that a i [a i,a i ]. I this case, we wat to fid the correspodig rages of y j. A atural idea is to apply iterval computatios techiques to fid the rage f j ([a 1,a 1 ],...,[a,a ]). For specific equatios, we ofte already kow which ideas work best. For example, for liear equatios Ay = b, we ca use the kow fact that y is mootoic i b. Solvig systems of equatios whe o algorithm is kow. The mai idea is that we parse each equatio ito elemetary costraits, ad the use iterval computatios to improve origial rages util we get a arrow rage (= solutio).

8 First example: the equatio x x 2 = 0.5 for x [0,1] (this equatio has o solutio). Parsig leads to r 1 = x 2, ad 0.5(= r 2 ) = x r 1. From r 1 = x 2, we extract two rules: (1) x r 1 = x 2 ad (2) r 1 x = r 1. From 0.5 = x r 1, we extract two more rules: (3) x r 1 = x 0.5 ad (4) r 1 x = r We will apply these rules oe by oe, the agai, util we coverge. We start with x = [0,1], r = (, ). Rule (1) leads to r = [0,1] 2 = [0,1], so the ew rage for r is r ew = (, ) [0,1] = [0,1]. Now, Rule (2) leads to x ew = [0,1] [0,1] = [0,1] o chage. Rule (3) leads to r ew = ([0,1] 0.5) [0,1] = [ 0.5,0.5] [0,1] = [0,0.5]. Rule (4) leads to x ew = ([0,0.5] + 0.5) [0,1] = [0.5,1] [0,1] = [0.5,1]. Rule (1) lead to r ew = [0.5,1] 2 [0,0.5] = [0.25,0.5]. Rule (2) leads to x ew = [0.25,0.5] [0.5,1] = [0.5,0.71]; here, we roud a dow ad a up, to guaratee eclosure. Rule (3) leads to r ew = ([0.5,0.71] 0.5) [0.25,5] = [0.0.21] [0.25,0.5], i.e., r ew = /0. We coclude that the origial equatio has o solutios. Aother example: equatio x x 2 = 0 for x [0,1]. Parsig leads to r 1 = x 2, ad 0 = x r 1. So, we get the followig rules: (1) r = x 2 ; (2) x = r; (3) r = x; (4) x = r. We start with x = [0,1] ad r = (, ). The problem is that after Rule (1), we re stuck with x = r = [0,1]. A atural solutio is thus to bisect x = [0,1] ito [0,0.5] ad [0.5,1]. For the 1st subiterval: Rule (1) leads to r ew = [0,0.5] 2 [0,0.5] = [0,0.25]; Rule (4) leads to x ew = [0,0.25]; Rule (1) leads to r ew = [0,0.25] 2 = [0,0.0625]; Rule (4) leads to x ew = [0,0.0625]; etc. we coverge to x = 0. For the secod subiterval, we coverge to x = 1. Optimizatio: extedig kow algorithms to situatios with iterval ucertaity. I optimizatio, we eed to fid y 1,...,y m for which g(y 1,...,y m,a 1,...,a m ) max. I may practical situatios, we kow a i with iterval ucertaity: a i [a i,a i ]. We wat to fid the correspodig rages of y j. Ofte, for the case of exactly kow a i, we have a algorithm y j = f j (a 1,...,a ) for solvig the optimizatio problem. This is true, e.g., for a quadratic objective fuctio g. I this case, we ca apply iterval computatios techiques to fid the rage f j ([a 1,a 1 ],...,[a,a ]). For specific f, we ofte already kow which ideas work best, ad thus, we get a eve better solutio. Optimizatio whe o algorithm is kow. The mai idea is to divide the origial box x ito subboxes b. The, if max x b g(x) < g(x ) for a kow x, we dismiss the subbox b. As a example, let us take g(x) = x (1 x) for x = [0,1]. Let s divide [0,1] ito 10 subboxes b = [0,0.1],[0.1,0.2],... For each b, we fid g( b); the largest is = The, we compute G(b) = g( b) + (1 2 b) [, ], ad dismiss subboxes for which Y < For example, for [0.2,0.3], we have

9 0.25 (1 0.25) + (1 2 [0.2,0.3]) [ 0.05,0.05]. Here, Y = < , so we dismiss [0.2,0.3]. As a result, we keep oly boxes [0.3,0.7]. Further subdivisio will get us closer ad closer to x = A Example Where We Need to Combie Differet Approaches to Ucertaity: Chip Desig Formulatio of the problem. Oe of the mai problems of computer chip desig is to estimate the clock cycle o the desig stage. The clock cycle of a chip is costraied by the maximum path delay over all the circuit paths D def = max(d 1,...,D N ). The path delay D i alog the i-th path is the sum of the delays correspodig to the gates ad wires alog this path. Each of these delays, i tur, depeds o several factors such as the variatio caused by the curret desig practices, evirometal desig characteristics (e.g., variatios i temperature ad i supply voltage), etc.; see, e.g., [12]. Traditioal (iterval) approach to estimatig the clock cycle. The traditioal approach assumes that each factor takes the worst possible value. The resultig time delay correspods to the case whe all the factors are at their worst. This is ot be a very realistic case: differet factors are usually idepedet, ad thus, combiatio of may worst cases is improbable. As a result, the worst-case estimates are 30% above the observed clock time. Sice the clock time is set too high, chips are over-desiged ad uder-performig. Robust statistical methods are eeded. I the ideal case, whe we kow all the probability distributios, we ca use Mote-Carlo simulatios to estimate the correspodig ucertaity. I practice, we oly have partial iformatio about the distributios of some of the parameters. Usually, we kow the mea, ad we kow some characteristic of the deviatio from the mea e.g., the iterval that is guarateed to cotai possible values of this parameter. A possible approach would be to apply Mote-Carlo methods with several possible distributios. The problem is that there are ifiitely may possible distributios, ad so there is o guaratee that the result of usig fiitely may of them is a valid boud for all possible distributios. It is therefore desirable to provide robust bouds, i.e., bouds that work for all possible distributios. Towards a mathematical formulatio of the problem. Each gate delay d depeds o the differece x 1,...,x betwee the actual ad the omial values of the parameters. These differeces are usually small, so we ca safely igore terms which are quadratic ad higher order i x i ad thus assume that d is a liear fuctio of x i. Each path delay D i is the sum of gate delays, thus D i is also a liear fuctio of x j : D i = a i + j=1 a i j x j for some a i ad a i j. The desired maximum delay D = maxd i is thus the maximum of liear fuctios. It is kow that every maximum of liear fuctios is covex. Thus, the depedece D = F(x 1,...,x ) is covex. i

10 We kow that the factors x i are idepedet. We kow the distributios of some of the factors. For others, we kow rages [x j,x j ] ad meas E j. We are give the largest allowed probability ε > 0 of the fault. We eed to fid the smallest y 0 for which, for all possible distributios, we have y y 0 with the probability 1 ε. Additioal property: depedecy is o-degeerate. Sometimes, we lear additioal iformatio about oe of the factors x j. For example, we lear that x j actually belogs to a proper subiterval of the origial iterval [x j,x j ]. As a result, the origial class P of possible distributios is replaced with a smaller oe P P. I this case, the ew value y 0 ca oly decrease: y 0 y 0. If x j is irrelevat for y, the y 0 = y 0. Our assumptio is that irrelevat variables have bee weeded out. I this case, if we arrow dow oe of the itervals [x j,x j ], the resultig value y 0 decreases: y 0 < y 0. Formulatio of the problem i precise terms. GIVEN:, k, ε > 0; a covex fuctio y = F(x 1,...,x ) 0; k cdfs F j (x), k + 1 j ; itervals x 1,...,x k, values E 1,...,E k, TAKE: all joit probability distributios o R for which: all x i are idepedet, x j x j, E[x j ] = E j for j k, ad x j have distributio F j (x) for j > k. FIND: the smallest y 0 s.t. for all such distributios, F(x 1,...,x ) y 0 with probability 1 ε. WHEN: the problem is o-degeerate if we arrow dow oe of the itervals x j, y 0 decreases. Mai result ad how we ca use it. Our mai result is that y 0 is attaied whe for def each j from 1 to k, we take x j = x j with probability p j = x j E j, ad x j = x j with the x j x j remaiig probability p j def = E j x j x j x j. (The proof is give i the Appedix.) Thus, we arrive at the followig algorithm: simulate these distributios for x j, j < k; simulate kow distributios for j > k; use the simulated values x (s) j sort N values y (s) : y (1) y (2)... y (Ni ); take y (Ni (1 ε)) as y 0. to fid y (s) = F(x (s) 1,...,x(s) ); Commet about Mote-Carlo techiques. It is usually believed that Mote-Carlo methods are iferior to aalytical: they are approximate, they require large computatio time; ad simulatios for several distributios, may mis-calculate the (desired) maxi-

11 mum over all distributios. I our case, however, the value correspodig to the selected distributios ideed provide the desired maximum value y 0. This is a particular case of the geeral fact that justified Mote-Carlo methods ofte lead to faster computatios tha aalytical techiques a good example is multi-d itegratio, where Mote-Carlo methods were origially iveted. 4. Combiig Iterval ad Probabilistic Ucertaity: Geeral Case Need for such a combiatio. Chip desig is just oe of the examples of the practical situatios i which some iformatio comes i probabilistic form ad some i iterval form. So, we eed to lear how to combie these two types of ucertaity. How to represet a probability distributio. There are may ways to represet a probability distributio. A atural idea is to look for a objective. The objective of processig ucertai data is to make decisios, ad it is kow that ratioal decisio makig is equivalet to maximizig the expected values of a appropriate fuctio u(x, a) called utility: E x [u(x,a)] max; see, e.g., [2,7,9,14]. a I may practical situatios, the fuctio u(x) is smooth. We ca therefore expad u(x) i Taylor series ad keep oly the first few terms: we have u(x) = u(x 0 )+ (x x 0 ) u (x 0 ) +... The, to fid the expected value, it is eough to kow the momets E[(x x 0 ) k ]. If we do ot kow the exact momets, the we should kow (iterval) bouds o momets. Sometimes, we have a threshold-type u(x): e.g., whe a fie is imposed if the cocetratio of a certai pollutats exceeds a certai threshold. I this case, to compute E[u], we eed to kow cdf F(x) = Prob(ξ x). I the case of ucertaity, we have bouds [F(x),F(x)]; such bouds are kow as p-boxes. Extesio of iterval arithmetic to probabilistic case: successes ad challeges. To process the correspodig ucertaity, it is reasoable to parse the origial algorithm ito a sequece of elemetary operatios +,,, 1/x, max, mi, ad the use the formulas for these elemetary operatios. Explicit formulas are kow for itervals, for p-boxes, ad for itervals + 1st momets E i. I some cases, these formulas are easy to get, e.g., for +,, ad for product of idepedet x i ; see, e.g., [1,6,10]. For multiplicatio y = x 1 x 2, whe we have o iformatio about the correlatio, we get o-trivial formulas: E = max(p 1 + p 2 1,0) x 1 x 2 + mi(p 1,1 p 2 ) x 1 x 2 + mi(1 p 1, p 2 ) x 1 x 2 + max(1 p 1 p 2,0) x 1 x 2 ; E = mi(p 1, p 2 ) x 1 x 2 + max(p 1 p 2,0) x 1 x 2 + max(p 2 p 1,0) x 1 x 2 + mi(1 p 1,1 p 2 ) x 1 x 2, where p i def = (E i x i )/(x i x i ). The mai remaiig challeges are to deal with situatios whe we kow itervals + 1st ad 2d momets, or whe we ow momets + p-boxes. I some cases, such formulas are kow. Let us describe oe of these cases.

12 5. Case Study: Bioiformatics Practical problem. It is importat to fid geetic differece betwee cacer cells ad healthy cells. Ideal solutio to this problem is to directly measure cocetratios c of the gee i cacer cells ad h i healthy cells. I reality, such cells are difficult to separate. Thus, what we really measure is ot c, but y i x i c+(1 x i ) h, where x i is the percetage of cacer cells i i-th sample. This amout ca be described as a x i + h y i, where a def = c h. Ideal case. If we kow x i exactly, we ca use the Least Squares Method (a x i +h y i ) 2 C(x,y) mi. We the fid a = ad h = E(y) a E(x), where E(x) = a,h V (x) 1 x i, V (x) = 1 1 (x i E(x)) 2, ad C(x,y) = 1 1 (x i E(x)) (y i E(y)). Real-life case. I practice, experts maually cout x i, ad oly provide iterval bouds x i, e.g., x i [0.7,0.8]. We the eed to fid the rage of a ad h correspodig to all possible values x i [x i,x i ]. Solvig the problem. I geeral, we kow itervals x 1 = [x 1,x 1 ],..., x = [x,x ], ad we eed to compute the rage of E(x), V (x), etc. I geeral, this problem is NP-hard eve for the populatio variace V (x) [10]. However, there are efficiet algorithms for V ad C(x,y) for reasoable situatios; see, e.g., [10]. So, to solve our problem, we use these algorithms to fid itervals for C(x,y) ad for V (x) ad the divide these itervals. 6. Aother Case Study: Detectig Outliers Outlier detectio is importat. I may applicatio areas, it is importat to detect outliers, i.e., uusual, abormal values: i medicie, uusual values may idicate disease; i geophysics, abormal values may idicate a mieral deposit (or a erroeous measuremet result); i structural itegrity testig, abormal values may idicate faults i a structure, etc. The traditioal egieerig approach is that a ew measuremet result x is classified as a outlier if x [L,U], where L def = E k 0 σ ad U def = E + k 0 σ, ad k 0 > 1 is preselected. Most frequetly, k 0 = 2, 3, or 6. Outlier detectio uder iterval ucertaity. I may practical situatios, we oly have itervals x i = [x i,x i ]. Differet values x i x i lead to differet itervals [L,U]. We ca therefore distiguish betwee possible ad guarateed outliers. A possible outlier is whe x is outside some k 0 -sigma iterval. Fidig possible outliers is importat i structural itegrity so as ot to miss a fault. A guarateed outlier is a value which is outside all k 0 -sigma itervals. For example, before a breast surgery, we wat to make sure that there is a micro-calcificatio. Oe ca easily check that x is a possible outlier if x [L,U], ad x is a guarateed outlier if x [L,U]. So, to detect outliers, we must fid the rages [L,L] ad [U,U] of L = E k 0 σ ad U = E + k 0 σ. Algorithms for computig such rages have ideed bee proposed; see, e.g., [10].

13 7. Fuzzy Ucertaity Need for fuzzy ucertaity. Kowledge ofte comes from experts, ad expert usually describe their kowledge by usig imprecise ( fuzzy ) words from atural laguage such as small. To describe this kowledge i precise terms, a special techique called fuzzy was iveted; see, e.g., [4,11,15]. I this techiques, for each possible value x of the correspodig quatity, we ask the expert to estimate the degree µ(x) to which this value satisfies the appropriate property (i.e., to what extet this value is small). The expert ca do it, e.g., by markig a value o a scale from 0 to 10; if the expert marks 7, we take µ(x) = 7/10. The resultig fuctio µ(x) is kow as a membership fuctio or a fuzzy set. Data processig uder fuzzy ucertaity. We have a data processig algorithm y = f (x 1,...,x ) ad fuzzy sets µ i (x i ). We eed to come up with a fuzzy set µ(y) that describes the resultig ucertaity about y. To come up with the appropriate formulas, let us take ito accout that y is a possible value of the possible value of the correspodig quatity Y if ad oly if there exist umbers x 1,...,x such that each x i is a possible value of X i ad f (x 1,...,x ) = y. We kow the degrees µ i (x i ) to which each x i is a possible value of X i. So, if use mi(µ(a), µ(b)) to describe our degree of belief i A& B, ad max(d(a),d(b)) for A B, we coclude that µ(y) = max mi(µ 1(x 1 ),..., µ (x )). x 1,...,x : f (x 1,...,x )=y This formula is kow as Zadeh s extesio priciple. Reductio to iterval computatios. How ca we compute µ(y)? A direct optimizatio would be computatioally expesive. Good ew is that such computatios ca be simplified if we represet each fuzzy set µ i by its α-cuts X i (α) def = {x i : µ i (x i ) α}. It turs out that for cotiuous f, for every α, the α-cut Y (α) is equal to the rage of f o α-cuts of X i : Y (α) = f (X 1 (α),...,x (α)). Thus, to compute µ(y), we ca simply, for α = 0,0.1,0.2,...,1 apply iterval computatios techiques to compute Y (α). Refereces [1] S. Ferso, V. Kreiovich, L. Gizburg, D.S. Myers, ad K. Setz, Costructig Probability Boxes ad Dempster-Shafer Structures, Sadia Natioal Laboratories, Report SAND , Jauary [2] P.C. Fishbur, Utility Theory for Decisio Makig, Joh Wiley & Sos Ic., New York, [3] L. Jauli, M. Kiefer, O. Dicrit, ad E. Walter, Applied Iterval Aalysis, Spriger, Lodo, [4] G. Klir ad B. Yua, Fuzzy Sets ad Fuzzy Logic, Pretice Hall, Upper Saddle River, New Jersey, [5] V. Kreiovich, A. Lakeyev, J. Roh, ad P. Kahl, Computatioal Complexity ad Feasibility of Data Processig ad Iterval Computatios, Kluwer, Dordrecht, [6] V. Kreiovich, Iterval computatios ad iterval-related statistical techiques: tools for estimatig ucertaity of the results of data processig ad idirect measuremets, I: F. Pavese ad A.B. Forbes (eds.), Data Modelig for Metrology ad Testig i Measuremet Sciece, Birkhauser-Spriger, Bosto, 2009, pp

14 [7] R.D. Luce ad R. Raiffa, Games ad Decisios: Itroductio ad Critical Survey, Dover, New York, [8] R.E. Moore, R.B. Kearfott, ad M.J. Cloud, Itroductio to Iterval Aalysis, SIAM, Philadelphia, [9] H.T. Nguye, O. Kosheleva, ad V. Kreiovich, Decisio makig beyod Arrow s impossibility theorem, with the aalysis of effects of collusio ad mutual attractio, Iteratioal Joural of Itelliget Systems 24(1) (2009) [10] H.T. Nguye, V. Kreiovich, B. Wu, ad G. Xiag, Computig Statistics uder Iterval ad Fuzzy Ucertaity, Spriger Verlag, Berli, Heidelberg, [11] H.T. Nguye ad E.A. Walker, A First Course i Fuzzy Logic, Chapma ad Hall/CRC, Boca Rato, Florida, [12] M. Orshasky, W.-S. Wag, G. Xiag, ad V. Kreiovich, Iterval-based robust statistical techiques for o-egative covex fuctios, with applicatio to timig aalysis of computer chips, Proceedigs of the Secod Iteratioal Workshop o Reliable Egieerig Computig REC 2006, Savaah, Georgia, February 22-24, 2006, pp [13] S.G. Rabiovich, Measuremet Errors ad Ucertaity: Theory ad Practice, Spriger Verlag, Berli, [14] H. Raiffa, Decisio Aalysis, Addiso-Wesley, Readig, Massachusetts, [15] L.A. Zadeh, Fuzzy sets, Iformatio ad Cotrol 8(1965) A. Proof of the Result about Chips: Mai Idea Let us fix the optimal distributios for x 2,...,x ; the, Prob(D y 0 ) = p 1 (x 1 ) p 2 (x 2 )... (x 1,...,x ):D(x 1,...,x ) y 0 So, Prob(D y 0 ) = N c i q i, where v i are possible values of x 1, p i is the probability of v i, i=0 ad c i do ot deped o x 1. The restrictios o q i are: q i 0, N i=0 q i = 1, ad N q i v i = E 1. Thus, the worst-case distributio for x 1 is a solutio to the followig liear programmig (LP) problem: Miimize N c i q i uder the costraits N q i = 1 ad N i=0 i=0 q i 0, i = 0,1,2,...,N. i=0 i=0 q i v i = E 1, It is kow that i LP with N +1 ukows q 0,q 1,...,q N, N +1 costraits are equalities. I our case, we have 2 equalities, so at least N 1 costraits q i 0 are equalities. Hece, o more tha 2 values q i are o-0. If the correspodig values v or v of x 1 for which the probability is o-zero are i (x 1,x 1 ), the for [v,v ] x 1 we get the same y 0 i cotradictio to o-degeeracy. Thus, the worst-case distributio is located at x 1 ad x 1. The coditio that the mea of x 1 is E 1 leads to the desired formulas for p 1 ad p 1. Similarly, we get formulas for all other values p i ad p i. The statemet is prove.