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

3. Detection of Anomalies by Fuzzy Logic Methods

[10]  The DMA-based algorithms DRAS and FLARS [Gvishiani et al., 2003, 2004] are an alternative approach (with respect to the methods presented in section 2) to modeling of human reasoning and actions in search for anomalies. DRAS and FLARS are an attempt to model the logic of a researcher recognizing an anomaly from visual inspection of a record for an automated use of this model in analysis of large sets of data that cannot be manually processed. These algorithms yield estimates for boundaries of sought anomalies and subdivide them morphologically into initial, central, and final stages, identifying strong and weak phases in the central stage [Gvishiani et al., 2003]. The algorithms are rather versatile due to a wide set of "rectifications'' [Gvishiani et al., 2003, 2004] arising in modeling interpreter's work.

[11]  In a simplified form, work of an interpreter detecting an anomaly by visual inspection of a record is understood here as follows. Initially he looks over the record, estimating activity of its fragments in terms of positive numbers and mentally assigns the inferred numerical estimates to the fragments or their centers. Thus, the interpreter passes from the initial record to a nonnegative function that can be naturally called "rectification'' of a record. Actually, larger values of this function (rectification) will correspond to record points that are more active from the standpoint of sought signals. Further, the interpreter searches for rises in the record rectification that correspond to the most active record fragments. Thus, the interpreter works at two levels, local (rectification of a record) and global (search for rises in the rectification).

[12]  Naturally, the proposed simplified model of interpreter's logic cannot be regarded as unique and/or universal. Moreover, interpreter's reasoning is largely determined by the concrete type of anomalies (data) in question. However, in our opinion, the rectification process functions, one way or other, in any case.

3.1. Local level: Construction of record rectification.
[13]  An anomaly in a record (time series) is an ambiguous notion changing its form both from one record to another and within one record. Similarly to other intuitively clear mathematical notions (e.g., an element of a set), we do not attempt to give its strict definition. Anomalous nature is clear from examples given by experts. In terms of the DMA approach, a set of rectifications open for updating is applied for adequate modeling of "anomalies'' (higher activity zones). Now we pass to exact constructions.

[14]  Let a discrete positive semiaxis be eq001.gif eq002.gif and let y = {yk = y(kh)} be a finite time series (FTS) defined in the interval (recording period) eq003.gif We introduce a local survey parameter D > 0 multiple of eq004.gif and define a local survey fragment of the record y with a center at khin T as the interval

eq005.gif

Definition 1.
[15]  If J = Dk y is a set of local survey fragments of the record y and if eq006.gif where eq007.gif is the set of positive real numbers, we define a rectification of y on the basis of F as the superposition eq008.gif The mapping F is called here a rectifying functional.

[16]  A rectification determination can be regarded as successful if anomalies identified by an interpreter are mapped onto rises in the rectification. Accordingly, the presence of training data (i.e. results obtained by an interpreter from processing of a sufficiently long record fragment) is beneficial to the construction of a rectification. Examples of rectifications:

(1) survey fragment length,

eq009.gif

(2) survey fragment energy,

eq010.gif

where

eq011.gif

[17]  Many other types of rectification were used in [Gvishiani et al., 2003, 2004; Zlotnicki et al., 2005]. At a local level common to algorithms of the DRAS and FLARS families, the rectification Fy is constructed and specified for a record y. This transformation of a record is the first stage of visual analysis performed by an interpreter.

2007ES000278-fig01
Figure 1
3.2. Global level: Search for rises in a rectification.
[18]  Examples show that a rectification topography can be rather complex (Figure 1). The activity of anomalies cannot be invariably high and they can be inhomogeneities (activity intervals are several and they are divided by "quiet'' points). The corresponding rectification intervals are oscillating rises. It is natural to seek a "platform'', i.e. a connected base of such a rise, and to detect sought "spikes'' against this base. A procedure required for the determination of rises in the curve Fy(k) does not reduce to simple selection of points in accordance with their heights. This procedure should combine a union process (search for platforms) and a subdivision process (extraction of spikes within platforms). The algorithmic implementation of this logic at a global level divides algorithms into the DRAS and FLARS families and enables differentiation between concrete implementations within each family.

3.3. DRAS: The global level
[19]  [Gvishiani et al., 2003]. A record is first divided into background (quiet) and potentially anomalous (disturbed) parts. Connected regions in the disturbed part serve as bases (platforms) of rises. Farther, DRAS identifies undoubtedly anomalous fragments on the platforms.

[20]  To implement this procedure, the algorithm uses one-side measures LaFy(k) and RaFy(k) that quantify, on the [0, 1] scale, the quietness of the rectification Fy left and right of the point kh, detecting points whose ordinates exceed a level a [Gvishiani et al., 2003]. The latter is a free parameter of the algorithm called the vertical level of background. In other words, the quietness to the left (right) of the point kh in the record y is modeled in DRAS as a fuzzy subset on the recording interval T with the measures LaFy(k) (RaFy(k)). Using the conjunction min(LaFy(k), RaFy(k)), provides for the possibility of versatile treatment of Fy excesses over the level a. With the so-called horizontal level bin[0, 1] being properly adjusted, DRAS extracts only sufficiently dense (in time) excesses and takes no account of insignificant fragments considering them as background. This is attained by dividing the recording interval T into background (quiet) and potentially anomalous (disturbed) parts:

T = Bcup P,

B = {khin T: min (LaFy(k), RaFy(k))geb},

P = {khin T: min(LaFy(k), RaFy(k))< b}.

[21]  The set P is the union of the connected components eq012.gif It is these components that are processed by DRAS at the second stage of the global level. Identification of significantly anomalous intervals An in Pn is based on monitoring of the difference DaFy(k) = LaFy(k) - RaFy(k), which is reflected in the name of the algorithm (difference recognition algorithm for signals). The beginning of the anomaly An coincides with the first maximum of DaFy(k) in Pn. Actually, the difference between the quiet level to the left and the disturbed level to the right is most pronounced for the first time precisely at this point. For the same reason, the end of the anomaly An coincides in time with the last negative minimum of DaFy(k) in Pn. This procedure of identification of anomalies An is described in detail in [Gvishiani et al., 2003].

[22]  Free parameters of DRAS are the rectifying functional F and the following positive values: the local survey window Dll |T|, the vertical level of background a, the global survey window L > D, and the horizontal level of background bin [0.5, 1]. Accordingly, the algorithm can also be written as DRAS (F, D, a, L, b).

3.4. FLARS: The global level
[23]  [Gvishiani et al., 2004]. As distinct from DRAS, the FLARS algorithm first identifies significantly anomalous intervals and then the set of these intervals is supplemented with potentially anomalous intervals, thereby forming an "aureole'' of an identified anomaly. Thus, FLARS divides, in two stages, the recording interval into three subsets (T = A cup P cup B) denoted as follows: A, anomalous points; B, quiet background points located sufficiently far from the anomaly; and P, potentially anomalous, disturbed points covering a rather long interval (formally, they are not anomalous but lie sufficiently close to the points A and therefore "feel'' the influence of the latter).

[24]  We remind the reader that the DRAS choice of extreme points is based on analysis of the vertical level a immediately in the rectification Fy. FLARS forms indirectly the set of anomalous points A, using the search for extreme values on a Fy topography with the help of a fuzzy extremality measure m(k) taking values from the interval -1 le m(k)le 1 [Gvishiani et al., 2004]. The measure is constructed on Fy on the basis of fuzzy comparisons (5) and the vertical extremality level tin [-1, 1]. The T interval is divided into a set of significantly anomalous points A and its complement eq013.gif

eq014.gif

Like DRAS, the FLARS algorithm divides the set of nonanomalous points is subdivided into background and potentially anomalous components with the help of the alternating one-sided measures eq015.gif and eq016.gif and the horizontal background level

eq017.gif

Note that, due to the normalization m(k)in [-1, 1], the FLARS choice of the extremality level t is somewhat simpler compared to DRAS: t is usually set equal to 0, 0.5, or 0.75. A detailed description of FLARS is given in [Gvishiani et al., 2004].

[25]  Free parameters of FLARS are the rectifying functional F and the following positive values: the local survey window Dll |T|, global survey window L > D, and vertical level of extremality t. Accordingly, the algorithm can be designated as FLARS (F, D, L, t).


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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