RUSSIAN JOURNAL OF EARTH SCIENCES VOL. 10, ES1001, doi:10.2205/2007ES000278, 2008
[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.
![]() | (1) |
[28] Thus, fuzzy comparison can be realized in terms of any function
f(a, b),
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:
![]() |
[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 ( y 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).
(i) | ![]() | for any n > 0 and |
(ii) we set
ngn(a, b) = yg(nn(a, b)) for any
g (-1, 1), where
![]() |
[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 0 with an arbitrary weighted set of numbers
![]() |
[35] Binary extension:
![]() | (2) |
[36] Gravitational extension: Let gr A be the center of gravity of the set A, i.e.
![]() |
then
![]() | (3) |
[37] σ-extension: The left moment
![]() |
is an argument in favor of the maximality of a compared to A modulus. Accordingly, the right moment
is an argument in favor of the minimality of a compared to A modulus. Then,
![]() | (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 ).
(I) strongly (weakly) large if
[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
![]() |
where
is a model of
global survey on the segment
[a, b] of the record
y at the point
k:
![]() | (5) |
![]() |
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
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
Citation: 2008), Recognition of anomalies from time series by fuzzy logic methods, Russ. J. Earth Sci., 10, ES1001, doi:10.2205/2007ES000278.
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