CC BY
25
Mathematical Structures and Modeling 2013. N. 1(27). PP. 38-41
UDC 519.6
FOR DESCRIBING UNCERTAINTY, ELLIPSOIDS ARE BETTER THAN GENERIC POLYHEDRA AND PROBABLY BETTER THAN BOXES: A REMARK
O. Kosheleva, V. Kreinovich
For a single quantity, the set of all possible values is usually an interval. An interval is easy to represent in a computer: e.g., we can store its two endpoints. For several quantities, the set of possible values may have an arbitrary shape. An exact description of this shape requires infinitely many parameters, so in a computer, we have to use a finite-parametric approximation family of sets. One of the widely used methods for selecting such a family is to pick a symmetric convex set and to use its images under all linear transformations. If we pick a unit ball, we end up with ellipsoids; if we pick a unit cube, we end up with boxes and parallelepipeds; we can also pick a polyhedron. In this paper, we show that ellipsoids lead to better approximations of actual sets than generic polyhedra; we also show that, under a reasonable conjecture, ellipsoids are better approximators than boxes.
1. Formulation of the Problem
Need for describing sets of possible values. Measurement and estimates are never 100% accurate. As a result, we usually do not know the exact value of a physical quantity; we usually know the set of possible values of this quantity. For a single quantity, this set is usually an interval. Representing an interval in a computer is easy: e.g., we can represent an interval by its endpoints; see, e.g., [7,10].
For several quantities xx,... , xn, in addition to interval bounds on each of these quantities, we often have additional restrictions on their combinations; as a result, the set of possible values of x = (xi,... ,xn) can have different shapes. The space of all possible sets is infinite-dimensional, meaning that we need infinitely many real-valued parameters to represent a generic set. In a computer, at any given time, we can only store finitely many parameters; so, we cannot represent generic sets exactly, we need to approximate them by sets from a finite-parametric family.
Copyright © 2013 O. Kosheleva, V. Kreinovich
University of Texas at El Paso (USA)
E-mail: [email protected], [email protected]
Mathematical Structures and Modeling. 2013. N 1(27).
39
Convex set-based representation of sets of possible values. In many practical situations, e.g., when x are spatial coordinates, the selection of the quantities is rather arbitrary: we can use a different coordinate system in which, instead of
m
the original quantities x,, we use linear combinations y = Tx, i.e., y = tj • Xj.
j=i
In view of this, a reasonable way to select a finite-parametric set is to pick a bounded symmetric convex set S0 with non-empty interior, and to use images TS0 of this set S0 under arbitrary linear transformations T.
If we start with a Euclidean unit ball S0 = B =f |x : ^ xf < l|, we get
the family of ellipsoids (see, e.g., [1-4,11-14,16]); if we start with a unit cube S0 = C = {x : |x,| < l for all i}, we get the family of all boxes (plus the corresponding parallelepipeds); alternatively, we can also start with a symmetric convex polyhedron P.
Which set S0 should we choose? Once we pick a set S0, we can (precisely) represent sets S of the type TS0. If we start with such a set S, we enclose it into a set TS0 = S, and then, if we want to enclose TS0 in a set A • S corresponding to the original S-based representations, we get the same original set S = TS0 back, with A = l.
For sets S which are different from TS0, the S0-based representation is only approximate. We start with a set S, and we enclose it in a set TS0 D S for an appropriate linear transformation T. If we then try to enclose TS0 in a set of the type A • S, then we inevitably get A > l.
The smaller A, the better the approximation. It is therefore reasonable, as a measure d(S0,S) of accuracy of approximating S by S0, to use the smallest A corresponding to all possible T:
d(S0, S) = inf{A : 3T (S c TS0 C A • S)}.
This quantity is known as a Banach-Mazur distance between the convex sets S and S0; see, e.g., [15,17].
For each "standard" set S0, we get different values A(S0,S) for different sets S. As a measure of quality Q(S0) of choosing S0, it is reasonable to select the worst-case approximation accuracy
Q(S0) =f sup d(S0,S),
S
where the supremum is taken over all possible bounded symmetric convex sets S with non-empty interior.
40 O. Kosheleva, V. Kreinovich. For Describing Uncertainty, Ellipsoids.
2. Main Results
Main conclusion: ellipsoids are better than generic polyhedra. According to the well-known John's Theorem [8,15,17], for the Euclidean unit ball B, we have d(B,S) < jn for all symmetric convex sets S. Thus, we have Q(B) < jn.
On the other hand, according to Gluskin's theorem [6,15,17], there exists a constant c > 0 such that for each dimension n, there exist polyhedra P and P' for which d(P, P') > c■ n and for which, therefore, d(P) > c■ n. Moreover, if we take a convex hull P of 2n points randomly selected from a unit Euclidean sphere, then, with high probability, we get Q(P) > c ■ n. Since for large n, we have c ■ n > jn and therefore, Q(B) c Q(P), this shows that for large dimensions, ellipsoids are indeed better than generic polyhedra.
Additional conclusion: ellipsoids are probably better than boxes. A Euclidean unit ball B (corresponding to ellipsoids) and a unit cube C (corresponding to boxes) can be viewed as particular cases of unit balls Bp =f {x : ||x||p < 1}
def ( n \ 1/p
in the €p-metric ||x||p = I |x^|M : B is a unit ball in the ¿2-metric while
C is a unit ball in the ¿^-metric: B = B2 and C = B^. The exact values of d(Bp, Bq) are known only when both p and q are on the same side of 2; in this case, d(Bp, Bq) = n|1/p-1/q|. In particular, for p =1 and q = 2, we get d(B1, B2) = jn.
These values have the property that when p < q, then d(Bp,Bq) strictly increases when p decreases or when q increases; in other words, the larger the difference between p and q, the larger the value d(Bp,Bq). For values p and q on different sides of 2, this monotonicity does not hold for n = 2, since in this case, B1 (rhombus) and B^ (square) are linearly equivalent and thus, d(B1, B^) = 0. However, for n > 3, we do not have this anomaly and therefore, it is reasonable to conjecture that for n > 3, this monotonicity holds. Under this hypothesis, d(B^,B1) > d(B2,B1) = jn, and thus, Q(B^) > d(B^,B1) > jn. Since Q(B2) = jn, we therefore conclude that Q(B2) < Q(B^) and thus, ellipsoids are better than boxes.
Comment. These results are in line with a general result according to which, under certain conditions, ellipsoids are the best approximators [5,9].
Acknowledgments. This work was supported in part by the National Science Foundation grants HRD-0734825 and HRD-1242122 (Cyber-ShARE Center of Excellence) and DUE-0926721, by Grants 1 T36 GM078000-01 and 1R43TR000173-01 from the National Institutes of Health, and by a grant on F-transforms from the Office of Naval Research.
References
1. Belforte G., Bona B. An improved parameter identification algorithm for signal with unknown-but-bounded errors. Proceeding of the 7th IFAC Symposium on Identification and Parameter Estimation, York, U.K., 1985.
2. Chernousko F.L. Estimation of the Phase Space of Dynamic Systems. Moscow : Nauka publ., 1988 (in Russian).
3. Chernousko F.L. State Estimation for Dynamic Systems. Boca Raton, FL : CRC Press, 1994.
4. Filippov A.F. Ellipsoidal estimates for a solution of a system of differential equations // Interval Computations. 1992. N. 2(4). P. 6-17.
5. Finkelstein A., Kosheleva O., Kreinovich V. Astrogeometry, error estimation, and other applications of set-valued analysis // ACM SIGNUM Newsletter. 1996. V. 31, N. 4. P. 3-25.
6. Gluskin E.D. The diameter of the Minkowski compactum is approximately equal to n // Functional Analysis and Its Applications. 1981. V. 15. P. 72-73.
7. Jaulin L., Kieffer M., Didrit O., Walter E. Applied Interval Analysis, with Examples in Parameter and State Estimation, Robust Control and Robotics. London : SpringerVerlag, 2001.
8. John F. Extremum problems with inequalities as subsidiary conditions. In: Studies and Essays Presented to R. Courant on his 60th Birthday. New York : Interscience Publishers, Inc., 1948. P. 187-204.
9. Li S., Ogura Y., Kreinovich V. Limit Theorems and Applications of Set Valued and Fuzzy Valued Random Variables. Dordrecht : Kluwer Academic Publishers, 2002.
10. Moore R.E., Kearfott R.B., Cloud M.J. Introduction to Interval Analysis. Philadelphia, Pennsylviania : SIAM Press, 2009.
11. Norton J.P. Identification and application of bounded parameter models. Proceeding of the 7th IFAC Symposium on Identification and Parameter Estimation. York, U.K., 1985.
12. Schweppe F.C. Recursive state estimation: unknown but bounded errors and system inputs // IEEE Transactions on Automatic Control. 1968. V. 13. P. 22.
13. Schweppe F.C. Uncertain Dynamic Systems. Englewood Cliffs, New jersey : Prentice Hall, 1973.
14. Soltanov S.T. Asymptotic of the function of the outer estimation ellipsoid for a linear singularly perturbed controlled system / Shary S.P., Shokin Yu.I. (eds.) Interval Analysis. Krasnoyarsk, Academy of Sciences Computing Center, Technical Report N. 17, 1990. P. 35-40 (in Russian).
15. Tomczak-Jaegermann N. Banach-Mazur Distance and Finite-Dimensional Operator Ideals. Pitman Monographs and Surveys in Pure and Applied Mathematics. V. 38. Harlow, UK : Longman Sci. Tech., 1989.
16. Utyubaev G.S. On the ellipsoid method for a system of linear differential equations / Shary S.P. (eds.) Interval Analysis. Krasnoyarsk, Academy of Sciences Computing Center, Technical Report N. 16, 1990. P. 29-32 (in Russian).
17. Vershynin R. Lectures in Geometric Functional Analysis. Ann Arbor, Michigan : University of Michigan, 2013. URL: http://www-personal.umich.edu/~romanv/ (the date of circulation: 31.03.2013).
