Surveillance Points in High Dimensional Spaces

Transcription

1 Société de Calcul Mathématique SA Tools fo decision help since 995 Suveillance Points in High Dimensional Spaces by Benad Beauzamy Januay 06 Abstact Let us conside any compute softwae, elying upon a lage numbe of paametes, typically, 40 o moe, used fo any type of simulation. Such a tool will cetainly be a vey complicated softwae, theefoe lengthy to execute, and the numbe of uns one may pefom will be limited. One wants to pefom these uns "at best", that is at specific places, called "Suveillance Points". These investigations will then be consideed as epesentative of any situation. These Suveillance Points may be intepeted as the cente of balls, coveing an hypecube which (afte nomalization) descibes all the possible configuations of paametes. Ou fist esult shows that these balls must be extemely lage. In dimension 40, fo 300 uns, thei adius must be at least 3, which means that the esult obtained at a Suveillance Point should not be adically diffeent fom a esult obtained at anothe point at this distance. In othe wods, if one wants to monito the esults given by the computational code, using a small numbe of Suveillance Points, the computational code must be quite "stable" : its esults should not be vey diffeent at points which ae not too fa. If, on the contay, the computational code may take vey diffeent values unde athe simila conditions (which is the case, fo instance, if some kind of discontinuity occus), then such a small numbe of Suveillance Points will not suffice. Siège social et bueaux :, Fauboug Saint Honoé, Pais. Tel : Fax : Société Anonyme au capital de Euos. RCS : Pais B SIRET : APE : 79Z

2 In a second pat, we give an explicit constuction of the Suveillance Points. Basically, thei numbe must be a powe of, and the pecision inceases vey slowly with the numbe of balls. A compaison between the theoetical esult and the pactical constuction shows that the odes of magnitude ae coect and satisfactoy. Fo instance, in dimension 40, if we use 56 balls in ode to cove the hypecube, they must have a adius at least On the othe hand, we can constuct explicitly 56 balls of adius which cove the hypecube. The oveall conclusion is that people who use computational codes in high dimensional spaces, that is depending on a lage numbe of paametes, should be vey cautious when they claim that a small numbe of uns suffices in ode to evaluate the outputs of the code. Acknowledgements ***** The pesent pape comes fom specific needs expessed oiginally by Famatome-AP (003), then by the Institut de Radiopotection et de Sûeté ucléaie (since 005) and moe ecently by EDF (05). Using a small numbe of uns o obsevations in ode to econstuct a global infomation was the topic of Olga Zeydina's thesis ; see the book "Pobabilistic Infomation Tansfe". The pesent pape is a pat of the geneal eseach topic "Malfunctions in Sensos etwoks", developed ointly by IRS and SCM. I. Geneal pesentation Let, in the sequel, y C x x eal valued paametes *****,..., be a simulation softwae, etuning a eal value y fom x,..., x. We may assume without loss of geneality that each paamete takes its values between 0 and, eplacing if necessay x by x x x x x min and x max ae espectively the smallest and the lagest value of the paamete. max H 0,. Theefoe, ou function is defined on the dimensional hypecube min min, whee BB Suveillance Points, 06/0

3 A Suveillance Point A is a point in H at which we will ty the code. Let a,..., a be the coodinates of A. Let be the numbe of uns we want to execute, and let coesponding Suveillance Points. The points A,..., A be the A n ae not distibuted at andom. On the contay, we want them to be distibuted as egulaly as possible, so that any abitay point A in the hypecube should be at minimal distance fom one of the A 's. In mathematical tems, it means that the euclidean balls centeed at the n A n 's, with some adius (same fo all balls) should cove the hypecube Hee is a pictue fo ; the hypecube is ust a squae, the balls ae disks, and we cove the unit squae with 9 balls, all having adius 6 : H. Figue : coveing the unit squae with 9 equal disks Hee, ou Suveillance Points will be the centes of the small cicles. We note that thee is always an ovelap between the zones, which may be used in ode to detect some malfunction o stange behavio of the sensos. In dimension 3, we cove the same way the unit cube by balls, and the desciption is the same in high dimensional spaces. Howeve, in such situations, the esults become highly counteintuitive, as we will see. Fo a geneal intoduction to the geomety of high dimensional spaces, see the book [BB]. Assume that a set of balls B,..., B, with centes A,..., A and with same adius, coves the hypecube. Then these balls must cetainly contain in paticula the summits of the hypecube. A summit is a point whose coodinates ae 0 o : they ae the exteme points of the hypecube. Fo the squae in dimension, the summits ae simply the fou cones of the squae. In dimension, thee ae coodinate may take the values 0 o. summits, since thee ae coodinates and each BB Suveillance Points, 06/0 3

4 II. Coveing the Hypecube A. A geneal esult We will pove : Theoem. Let be the dimension of the space and let H be the dimensional hypecube 0,. Let be any numbe of balls of same adius, coveing the hypecube. Let 0 be the lagest intege satisfying : () Then the adius of the balls satisfies : 0 In the case 40 and 300, we find Poof of Theoem Assume we have constucted balls which cove the hypecube. We have balls and summits. Since the balls contain all the summits, one of the balls must contain a numbe of summits which is Let S. Let us denote by B 0 this ball.,..., be any summit, with 0 o. We say that anothe summit S,..., is a neighbou of S of ode if the coodinates of S and the coodinates of S diffe by one item only. Fo instance, if S 0,...,0, its neighbous of ode ae the points,0,...,0, 0,,0,...,0,, k 0,...,0,. Each summit has neighbous of ode. The same way, S is a neighbou of S of ode if thei coodinates diffe by items only. The neighbous of 0,...,0 of ode ae the points with coodinates equal to, the othes being 0. Thee ae!!! neighbous of type. Similaly again, S is a neighbou of S of ode if thei coodinates diffe by items only. The neighbous of 0,...,0 of ode ae the points with coodinates equal to, the othes being 0. Thee ae neighbous of type. BB Suveillance Points, 06/0 4

5 Take the ball B 0 defined above : the one with lagest numbe of summits. Assume. Cetainly, this ball cannot be such that it contains simply a point and its neighbous of ode, and no othe summit, because in this case the numbe of points it contains is, which is smalle than the numbe of points it should contain, namely. Fo instance, if 40, 300, 4 and The same way, if and ae such that, this ball cannot be such that it contains simply a point, its neighbous of ode, its neighbous of ode, and no othe summit. If 40, 300, 8. Moe geneally, this ball cannot be such that it contains simply a point, its neighbous of ode,, its neighbous of ode, and no othe summit, as long as : () Fo 40, 300,, the left hand side of () takes the value fo, the value has the value < and So the lagest fo which () holds, denoted by 0, Fom () follows that the ball B 0 must contain a point, which is a neighbou of ode 0, of its cente. But in this case, thei euclidean distance is at least d 0 and the adius of the ball is at least 0. Fo 40, 300, they have at least coodinates which diffe. Ou Theoem is poved. In pactice, evaluations of sums of binomial coefficients ae needed: BB Suveillance Points, 06/0 5

6 Coollay. The theoetical adius th can be found fom the fomula: th whee is the invese function of the Gaussian epatition function, that is H v u, with: H x x e t / dt u v if Poof of Coollay The condition: can be ewitten: i0 i () with. Let X be a andom vaiable with binomial law, of paametes ewitten as:,. Condition () may be P X () But such a andom vaiable has expectation equal to / and vaiance equal to /4. Condition () may be ewitten: X / / P / 4 / 4 (3) Using the appoximation of the binomial law by a nomal law (which is legitimate hee), we may wite (3) unde the fom : PZ / /4 (4) BB Suveillance Points, 06/0 6

7 whee Z is a nomalized Gaussian andom vaiable, that is EZ 0, va Z. Using the epatition function of the nomal law, (4) is equivalent to : H / /4 (5) / (6) /4 That is: (7) The value of is given by tables of the nomal law, and we take: 0 int (8) which gives a theoetical adius with: th (9) which poves Coollay. We deduce fom Chenoff's bound (see [Chenoff]) an explicit estimate: Coollay 3. The theoetical adius satisfies: Poof of Coollay 3 th Log We have, fom Chenoff's inequality: i0 exp So the condition: BB Suveillance Points, 06/0 7

8 is satisfied if : exp which is equivalent to: This gives a theoetical adius with: Log th Log which poves Coollay 3. B. Simple cases. Coveing the hypecube with a single ball The Theoem asks fo the lagest such that 0. Then the adius given by the theoem is th 0., which is obviously In pactice, we may cove the hypecube with a single ball, centeed at we have : C,...,. Then, obs 4 (whee the subscipt "obs" stands fo "obseved"). So the pevision of the Theoem is pessimistic, but the ode of magnitude is coect.. Case of two balls Using the Theoem, we need to find the lagest such that : BB Suveillance Points, 06/0 8

9 which gives 0 if is even, and 0 appoximately : if is odd. So, in both cases, we obtain th ow, in pactice, we can cove the hypecube with the two balls of centes: with:,,..., 4, 3,,..., 4 obs 6 4 so again the ode of magnitude given by the Theoem is coect. C. Volume consideations The volume of the hypecube is obviously, no matte what the dimension is. The volume of a ball of adius in dimension ( even) is given by the fomula : V / /! If 40, 3, we get : 3 0 V This means that thee is a consideable loss in volume : we need 300 balls of adius in ode to cove something of volume. We may wonde about the volume of the ball, centeed at the middle point of the hypecube (that is above, volume,..., ), containing all summits. This ball has adius and, by the fomula So it is extemely lage. Recall that, in dimension, the length of the diagonal of the hypecube is. The volume of the hypecube is small (equal to ), but the diagonal is lage, which is vey counte-intuitive. In fact, the hypecube extends in many diections (o, moe exactly, in many dimensions). BB Suveillance Points, 06/0 9

10 D. Chosing the numbe of balls Assume now that we fix the adius of all balls ; how many balls do we need? This is clea fom fomula (). If is fixed, we define by : 0 0 and the numbe is given by : int 0, whee int x denotes the integal pat of x. This means that : () 0 0 whee we wite : (3) Any change of in the inteval () is useless, so one should take the which is as small as possible, that is the left bound of the inteval. Fo 40 and 0, we find the inteval : 0 3 which means that we can achieve the same esult with 0 balls as with 300 balls : this is impotant in pactice. We have obtained: Theoem 4.- Fo a given value of, and 0 0 0, all values of in the inteval (whee is defined by (3)) povide the same coveing. Theefoe, one should use the value int 0 which epesents the smallest numbe of balls which ae equied in ode to obtain this coveing. Putting moe balls is a waste of time. BB Suveillance Points, 06/0 0

11 III. Choosing the centes of the balls We now indicate how to chose the balls. In this section, we give a constuctive appoach which, theoetically speaking, might not be best possible. Howeve, the ode of magnitude is coect, as Theoem shows. In what follows, the paametes ae teated one by one. If we know nothing about thei espective impotance, the ode is abitay. But if, fo some pactical easons, we have a anking upon the paametes, we should of couse stat with the most impotant one. This is often the case in pactice. Coveing with and balls has been descibed above. A. Thee balls A coveing by 3 balls is obtained by dividing the inteval fo the fist paamete into 3 subintevals, which gives the centes:,,..., 6, 3,,..., 6, 5,,..., 6 The adius is 6 4 B. Fou balls A coveing by 4 balls is obtained by dividing the intevals fo the fist and second paametes into, which gives the centes:,,,..., 4 4, 3,,,..., 4 4, 3,,,..., 4 4, 3 3,,,..., 4 4 The adius is 4 4 C. Geneal patten. Desciption Assume that the inteval fo the fist paamete is divided into n sub-intevals,, the inteval fo the th paamete is divided into n sub-intevals, then the centes ae all the points of the fom : BB Suveillance Points, 06/0,..., n n, with 0,..., n,, 0,..., n. ()

12 and the adius satisfies: 4 n () k nk k k The numbe of balls in this case is n n.. Obtaining the smallest adius Given a numbe of balls, we want the adius to be as small as possible. Fom fomula () follows that we should divide as many paametes as possible. Fo instance, if 4, dividing the fist inteval into 4 gives and dividing the fist two intevals into 4 6 gives, which is obviously smalle Consequences An obvious consequence is that a lage does not necessaily povide a bette solution. Fo instance, the value 7 allows the division of the fist inteval into 7 sub-intevals, giving: 4 7, wheeas the value 6 allows the division of 4 intevals into pieces, giving: 4 4 4, which is much smalle. 4. Pactical ule Assume that the paametes ae witten in deceasing ode of impotance. Then the best stategy is to use a division by, as long as possible. This means that the numbe of balls should be chosen in the sequence 3,,,,.... If we use k balls, we divide the inteval of vaiation of k paametes into, which means that the centes ae of the fom ( 0,) fo the fist k and of the fom fo the last k. The value of the adius is: k k k 44 (3) BB Suveillance Points, 06/0

13 5. Compaison between two binay steps Let us see what happens if we use and theefoe : k balls instead of k. Then, by the paagaph above: k k k k k 6 So, the multiplication by of the numbe of balls gives a decease on of a constant quantity, namely 3. Using any intemediate numbe of balls is athe useless Reaching a given theshold Assume we want to find the numbe of balls necessay to have (3), which gives : 3k 4A 4 A, fo a given A. Then, by 4 k 4A (4) 3 This conclusion is vey stong. It shows that, unless we accept a theshold which is popotional to, the numbe of necessay balls is exponential : 4 /3. IV. Compaison with the theoetical bounds We now compae the adius obtained in this constuction with the adius indicated by Theoem. A. umeical compaison It is easy to do when numeical values ae chosen. So let us take 40, We find 0 in the definition given in Theoem ; theefoe, the theoetical adius satisfies : th BB Suveillance Points, 06/0 3

14 Fo the pactical adius, we use fomula () above, and we get: obs k nk So the ode of magnitude is coect, once again. B. Theoetical compaison A geneal compaison between the estimates given by Theoem and the pactical constuction is hade to obtain, because it equies the evaluation of patial sums of binomial coefficients. We use Coollay 3 above, with such that : k balls, and we want to find the lagest i0 i k If we take k popotional to, that is: Coollay 3 gives: th k Log On the othe hand, the pactical constuction gives: that is: k 3k k obs 4 4 So thee is oughly a facto between the squae of the adii, which is a coect ode of magnitude. BB Suveillance Points, 06/0 4

15 V. Refeences [BB] Benad Beauzamy : Intoduction to Banach Spaces and thei Geomety. oth Holland, Collection "otas de Matematica", vol. 68. Pemièe édition : 98, seconde édition : 985. [Chenoff] [PIT] Olga Zeydina and Benad Beauzamy : Pobabilistic Infomation Tansfe. SCM SA, 03. ISB: , ISS : BB Suveillance Points, 06/0 5