RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 10, ES1001, doi:10.2205/2007ES000278, 2008

4. Fuzzy Comparisons

[26]  In many cases, the usual difference measure of the excess of one number over another is overly rough. In particular, DMA algorithms require finer constructions for the comparison of numbers.

Definition 2.
[27]  Fuzzy comparison n(a, b) of real numbers a and b quantifies the degree of excess of b over a on a [-1, 1] scale:

eq018.gif(1)

[28]  Thus, fuzzy comparison can be realized in terms of any function

f(a, b), eq019.gif that increases with b at a fixed a and decreases with increasing a at a fixed b (in this case, the increase and decrease have usual meaning); in addition, the following boundary conditions must be fulfilled:

eq020.gif

[29]  Actually, such functions will possess properties that are naturally required for comparison of numbers.

[30]  If n(a, b) is a fuzzy comparison and y is a monotonically increasing mapping of the segment [-1, 1] into itself, the superposition ( ycirc n)(a, b) is also a fuzzy comparison called a y -induced variation of n. The choice of y allows one to strengthen or weaken the basic comparison n.

[31]  In algorithms of the DRAS and FLARS families, it is sufficient to use fuzzy comparisons defined on positive numbers. Actually, records are processed by these algorithms through their rectifications taking solely positive values. We introduce the following family of basic fuzzy comparisons nn(a, b), n > 0 and their variations of a specific type ng,n(a, b).

Definition 3.
[32]  If  eq021.gif then

(i)eq022.gif for any n > 0 and

(ii)     we set ngn(a, b) = yg(nn(a, b)) for any gin (-1, 1), where

eq023.gif

[33]  This variation is correct: n0, n(a, b) = y0(nn(a, b)) = nn(a, b), so that nn becomes larger at g > 0 and smaller at g< 0. In what follows, the comparison n(a, b) means a value of ng, n(a, b), n > 0, -1 < g< 1.

[34]  We need to extend n(a, b) to the concept of fuzzy comparisons n(a, A) and n(A, a) of an arbitrary number a ge 0 with an arbitrary weighted set of numbers

eq024.gif

is the weight of ai, i = 1,ldots, N}. Such an extension is not unambiguous, and each of its variants formalizes in a unique way the notion "large (small) as compared with A (modulus of A )''. The value n(a, A) = mes(a < A) is understood as a function meaning that eq025.gif belongs to the fuzzy notion "small compared to A modulus,'' while n(A, a) = mes(A < a) means that eq026.gif belongs to the fuzzy notion "large compared to A modulus''. In our further development of the DRAS and FLARS algorithmic constructions, we used the following three extensions.

[35]  Binary extension:

eq027.gif(2)

[36]  Gravitational extension: Let gr A be the center of gravity of the set A, i.e.

eq028.gif

then

eq029.gif(3)

[37]  σ-extension: The left moment

eq030.gif

is an argument in favor of the maximality of a compared to A modulus. Accordingly, the right moment

sr(a, A) = (sum(ai - a)wi: ai > a)

is an argument in favor of the minimality of a compared to A modulus. Then,

eq031.gif(4)

[38]  It is natural to set that, if the validity of a certain property is expressed in terms of the [-1, 1], then a value from [0.5, 1] ([0, 0.5]) means a strongly (weakly) extremal manifestation of this property. Following these lines, we formalize the notions "large'' and "small'' with respect to the weighted set A (modulus of A ).

Definition 4.
[39]  Based on a given fuzzy comparison n (for a given weighted set A ) and its extensions n(A, a) and n(a, A), a number a ge 0 is defined to be

(I) strongly (weakly) large if

n(A, a)in [0.5, 1] (n(A, a)in [0, 0.5])

and (II) strongly (weakly) small if

n(a, A) in [0.5, 1](n(a, A)in [0, 0.5]).

[40]  Example. The extremality measure m(k) in FLARS (the FLARS measure) is obtained as a result of comparison (2) of the rectification value Fy(k) with the weighted set

eq032.gif

where eq033.gif is a model of global survey on the segment [a, b] of the record y at the point k:

eq034.gif(5)

2007ES000278-fig02
Figure 2
[41]  The standard FLARS [Gvishiani et al., 2004] is obtained with the use of s -extension (4). Alternative FLARS versions can be constructed using the binary (2) and gravitational (3) extensions, which lead to more "rigid'' FLARS models. The distinctions in the "rigidity'' of decision making on the basis of these three FLARS versions are illustrated by the following synthetic example (Figure 2).

[42]  Example. The local survey parameter D in DRAS and FLARS quantifies the closeness of the record y in the recording interval T. Using fuzzy comparisons, the choice of this parameter can also be made automatic as follows. Let eq035.gif be the set of all nontrivial distances on T. Then, an element D strongly minimal with respect to mod dT (see Definition 4) is found by solving the equation

n(D, dT) = 0.5.


RJES

Citation: Gvishiani, A. D., S. M. Agayan, Sh. R. Bogoutdinov, E. M. Graeva, J. Zlotnicki, and J.  Bonnin (2008), Recognition of anomalies from time series by fuzzy logic methods, Russ. J. Earth Sci., 10, ES1001, doi:10.2205/2007ES000278.

Copyright 2008 by the Russian Journal of Earth Sciences

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