Алгоритми FLARS та виділення аномалій часових рядів
As a rule, algorithms of recognition of time series anomalies are based on time frequency or statistical analysis . This article is devoted to detailed formal description of new fuzzy set based algorithm FLARS (Fuzzy Logic Algorithm for Recognition of Signals). It recognizes time series anomalies by...
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| author | Gvishiani, A. D. Agayan, S. M. Bogoutdinov, Sh. R. Tikhotski, S. A. Hinderer, J. Bonnin, J. Diament, M. |
| author_facet | Gvishiani, A. D. Agayan, S. M. Bogoutdinov, Sh. R. Tikhotski, S. A. Hinderer, J. Bonnin, J. Diament, M. |
| author_institution_txt_mv | [
{
"author": "A. D. Gvishiani",
"institution": null
},
{
"author": "S. M. Agayan",
"institution": null
},
{
"author": "Sh. R. Bogoutdinov",
"institution": null
},
{
"author": "S. A. Tikhotski",
"institution": null
},
{
"author": "J. Hinderer",
"institution": null
},
{
"author": "J. Bonnin",
"institution": null
},
{
"author": "M. Diament",
"institution": null
}
] |
| author_sort | Gvishiani, A. D. |
| baseUrl_str | http://journal.iasa.kpi.ua/oai |
| collection | OJS |
| datestamp_date | 2019-06-27T14:30:07Z |
| description | As a rule, algorithms of recognition of time series anomalies are based on time frequency or statistical analysis . This article is devoted to detailed formal description of new fuzzy set based algorithm FLARS (Fuzzy Logic Algorithm for Recognition of Signals). It recognizes time series anomalies by means "smooth" modelling (in fuzzy mathematics sense) of interpreter's logic, which searches for anomalies at the record. |
| first_indexed | 2025-07-17T10:25:23Z |
| format | Article |
| fulltext |
© A.D. Gvishiani, S.M. Agayan, Sh.R. Bogoutdinov, S.A. Tikhotsky, J. Hinderer, J. Bonnin,
M. Diament, 2004
Системні дослідження та інформаційні технології, 2004, № 3 7
TIДC
ТЕОРЕТИЧНІ ТА ПРИКЛАДНІ ПРОБЛЕМИ І
МЕТОДИ СИСТЕМНОГО АНАЛІЗУ
УДК 519.68
ALGORITHM FLARS AND RECOGNITION
OF TIME SERIES ANOMALIES
A.D. GVISHIANI, S.M. AGAYAN, SH.R. BOGOUTDINOV, S.A. TIKHOTSKY,
J. HINDERER, J. BONNIN, M. DIAMENT
As a rule, algorithms of recognition of time series anomalies are based on time fre-
quency or statistical analysis . This article is devoted to detailed formal description
of new fuzzy set based algorithm FLARS (Fuzzy Logic Algorithm for Recognition
of Signals). It recognizes time series anomalies by means «smooth» modelling (in
fuzzy mathematics sense) of interpreter’s logic, which searches for anomalies at the
record.
INTRODUCTION
The present article extends the fuzzy set theory approach [11] to recognition of
high-frequency anomalies on time series. It may be considered as an extension of
the series «Mathematical methods of geoinformatics» [1–5]. This article is de-
voted to detailed formal description of new fuzzy set based algorithm FLARS
(Fuzzy Logic Algorithm for Recognition of Signals). Applications of FLARS to
wide spectrum of geophysical data time series along with the application of
DRAS algorithm of the same family give successful results [10].
As a rule, algorithms of recognition of time series anomalies are based on
time frequency or statistical analysis [6–8, 12]. An exclusion is constituted by the
algorithm of «probabilistic» modelling of interpreter’s logic [9], which was de-
veloped by B. Naimark already in 1966. In a certain sense, our article extends the
ideas [9].
The algorithm FLARS may be considered as a results of «smooth» model-
ling (in fuzzy mathematics sense) of interpreter’s logic, which searches for
anomalies at the record. Moreover, FLARS is really based on fuzzy mathematic
principals. We proceed from the following understanding of interpreter’s logic. At
first, interpreter glances at the record and estimates activity of its sufficiently
small fragments by positive numbers. At the same time, he puts some numeric
marks to the centres of the fragments. In this way, from initial record interpreter
necessarily proceeds to a non-negative function. It is naturally to call this function
by rectification of the initial time series. Indeed, greater values of this function
(Figs. 1a, 1b) correspond to more anomalous points (centres of fragments). In the
other words searching the anomalies on the record can be found by searching the
A.D. Gvishiani, S.M. Agayan, Sh.R. Bogoutdinov, S.A. Tikhotsky, J. Hinderer, J. Bonnin, M.Diament
ISSN 1681–6048 System Research & Information Technologies, 2004, № 3 8
uplifts on its rectification (Fig. 1b). Thus, FLARS works on two levels. The local
one is rectification of the record, and the global one is search of the uplifts on the
rectification.
Let us proceed with exact statement.
FLARS LOCAL LEVEL. RECORD RECTIFICATION
Activity at a record (time series) is a multivalued notion, which can change from
record to record and within a record. The notion of activity doesn’t have a formal
mathematical definition like notions as «set», «element», «function», etc. Thus
we interpret «activity of a record» as intuitively clear notion. For its adequate
modelling we introduce strictly defined «rectifications» opened for replenish-
ments. Rectifying functionals serve as basis of the described below FLARS algo-
rithm.
Let us proceed with exact statement.
We consider a discrete semiaxis +
hR { , 0kh h= > , 1, 2,3, }k = … and a record
(discrete function, time series) { ( ), 1, 2,3, }ky y y kh k= = = … , which is defined
on a «coherent» subset (registration period) +⊂ hY R . At the semiaxis +
hR we in-
troduce the parameter of a local observation 0∆ > , which is divisible by «h»,
+∈∆ hR . The segment of the time series «y» with centre at a point «kh»
a
b
c
Fig. 1. The example of FLARS working: а — the initial record «y»; b — the rectification
)(kyχ ; c — the measure of anomality )(kµ
0.5
-0.5
0
Algorithm FLARS and recognition of time series anomalies
Системні дослідження та інформаційні технології, 2004, № 3 9
{ }, , , ,k
k h k k hy y y y−∆ +∆∆ = ∈… …
12
+
∆
hR
we call by a fragment of local observation.
Definition 1. A non-negative mapping :yχ ℑ→ +R defined on the set of
fragments { }k yℑ= ∆ ⊂
12
+
∆
hR we call by a functional of the time series «y».
In the set { }yχ there is a subset ( )y kΦ of the functionals, which transfer
anomalous fragments { }k y∆ into «uplifted» points on the graph of the corre-
sponding curve { }yχ .
Definition 2. We call any function ( )y kΦ by rectification of the initial time
series «y». Correspondingly, we call functional yΦ by a rectifying functional of
the given record «y».
In FLARS we use the following functionals of time series, which in many
important cases occur to be the rectifying functionals:
1. Length of the fragment
( )
1
1
k
h
k
j j
j k
h
L y y y
∆
+ −
+
∆
= −
∆ = −∑ .
2. Energy of the fragment
( ) ( )2
k
h
k
j k
j k
h
E y y y
∆
+
∆
= −
∆ = −∑ , where
2
k
h
k j
j k
h
hy y
h
∆
+
∆
= −
=
∆ + ∑ .
3. Difference of the fragment from its regression of order n
2( ) [ Regr ( )]k
k
h
k n
n j y
j k
h
R y y jk
∆
+
∆
∆
= −
∆ = −∑ ,
here as usual Regr k
n
y∆
is an optimal mean squares approximation of order n of the
fragment k y∆ . If 0=n we get the previous functional «energy of the fragment»
(see 2.):
0Regr
2k
k
h
j ky
j k
h
h y y
h
∆
+
∆
∆
= −
= =
∆ + ∑ ,
( ) ( )
2 20
0 ( ) Regr ( ) ( )k
k k
h h
k k
j j ky
j k j k
h h
R y y jh y y E y
∆ ∆
+ +
∆
∆ ∆
= − = −
∆ = − = − = ∆∑ ∑ .
A.D. Gvishiani, S.M. Agayan, Sh.R. Bogoutdinov, S.A. Tikhotsky, J. Hinderer, J. Bonnin, M.Diament
ISSN 1681–6048 System Research & Information Technologies, 2004, № 3 10
4. Oscillation of the fragment
( ) max min
k k
h hk
j j
j kj k
hh
O y y y
∆ ∆
+ +
∆∆ = −= −
∆ = − .
Illustration of rectification in the case of natural electric potential in the re-
gion of La Furnaise volcano (Reunion island, France) is given in Fig. 1,b .
FLARS GLOBAL LEVEL
Having the results of the rectification procedure (local level operations) FLARS
starts to search uplifts on the corresponding graph. As it is shown at the Fig.1b the
relief of the rectification may be enough complicated. Therefore, generally speak-
ing analysis of only vertical level of the curve ( )y kΦ is insufficient to define of
the uplifts. Anomalies may not possess constant high intensity. Often, the anoma-
lies they are heterogeneous. There is a majority of active fragments, divided by
short non-anomalous segments. In this case the corresponding fragments of recti-
fication occur to be oscillating uplifts. Analysis of rectifications ( )y kΦ shows
that its domain of definition Υ consists of three types of points: anomalous, which
belonging to uplifts, background (calm), located far enough from the uplifts, and
potentially anomalous (not calm), which have intermediate character. The latter
formally not belong to uplifts, but locate close enough to them to experience their
influence.
FLARS makes such division of Υ into above three subsets in two steps. At
the one the algorithm constructs an alternating-sign measure of extremality
( ) [ 1,1]kµ ∈ − , which characterises yΦ in a point «kh». Possibility of growth of
( )kµ from –1 to +1 corresponds to growth of «maximality of yΦ in the point
« kh » and in this way to growth of anomality of the initial record in that point.
Apparently, ( ) 1kµ = corresponds to maximum anomality and when ( ) 1kµ = − —
to maximum calmness in the point «kh». Being based on ( )kµ , FLARS chooses
in Υ the segment A, that consists of certainly anomalous points. On the next step
the complement \A Y A= of non-anomalous points, is divided by FLARS into
the set of calm (background) points S ′ and the set A′ of not calm (potentially
anomalous) points. One-sided background measures αL ( )y kΦ and αR ( )y kΦ ,
which characterise the rate of calmness of the rectification yΦ from left and right
hand sides of the point «kh», play key role in the division A S A′ ′= ∪ . It should
be noticed here that FLARS principally differs from DRAS [11], which is based
on the same principle of rectification. Indeed, FLARS at first recognizes the
fragments of a certain anomality which form the set A . On the next stage frag-
ments of a potential anomality A′ are added to the set A . On the contrary, DRAS
at the first stage recognises the set A A′∪ of potential anomalies and at the sec-
ond stage establishes certainly anomalous zones A A A′⊂ ∪ .
Let us proceed with exact statements.
Algorithm FLARS and recognition of time series anomalies
Системні дослідження та інформаційні технології, 2004, № 3 11
First we introduce the FLARS free parameter Λ >> ∆ , +∈Λ hR , which we
call by parameter of a global observation,. The following set we call as a fragment
of a global observation
( ), , ( ), , ( )k
y y y yk k k
h h
Λ Λ⎧ ⎫Λ Φ = Φ − Φ Φ + ∈⎨ ⎬
⎩ ⎭
… …
12
+
Λ
hR . (1)
We denote by ℵ the set of fragments (1).
In FLARS we will use the following weight function on the segment
[ , ]kh kh− Λ + Λ
h
kkhh
kk +Λ
−−+Λ
=)(δ . (2)
It is easy to see that the graph of this weight function (2) looks like an isos-
celes triangle with the base 2 ⋅ Λ and the height 1.
The function kδ represents another model of global observation of the frag-
ment k
yΛ Φ . In this model we, in a certain sense, «look at the fragment» giving
bigger weight to the points ( )kh k∈Λ , which are located closer to the centre «kh».
We symbolize
( ) { [ , ] : ( ) ( )}y yk k h kh kh k k+ + +Λ = ∈ −Λ + Λ Φ ≥ Φ ,
( ) { [ , ] : ( ) ( )}y yk k h kh kh k k− − −Λ = ∈ −Λ + Λ Φ <Φ .
The following sum will be an «argument» for minimality (backgroundness)
of the point «kh»
( )( , ) ( ) ( ) ( ) : ( )y y kk k k k k h kσ δ+ + + + +Λ = Φ −Φ ∈Λ∑ . (3)
At the same time the following sum will be an «argument» for maximality
(anomality) of the point «kh»
( )( , ) ( ) ( ) ( ) : ( )y y kk k k k k h kσ δ− − − − −Λ = Φ −Φ ∈Λ∑ . (4)
The measure ( )kµ is a result of the comparison of the «arguments» (3) and (4)
( , ) ( , )( ) ( ) [ 1,1]
max ( ( , ), ( , ))
k kk
k k
σ σµ µ σ σ
σ σ
− +
− +
+ −
Λ − Λ
= < = ∈ −
Λ Λ
.
The measure ( )kµ gives «more delicate assessment» of difference between
positive numbers than just a simple difference | |x y− . Indeed, the difference in
age between five-year-old and ten-year-old children is more considerable than the
same difference between 70-year-old and 75-year-old people. Apparently, it is
reflected in the values of the measure µ : ( 1(5 10)
2
µ < = and 1(70 75)
15
µ < = );
while 10 5 5 75 70− = = − .
A.D. Gvishiani, S.M. Agayan, Sh.R. Bogoutdinov, S.A. Tikhotsky, J. Hinderer, J. Bonnin, M.Diament
ISSN 1681–6048 System Research & Information Technologies, 2004, № 3 12
Decision on the presence of an anomaly is taken in FLARS by analysis of
the vertical level.
Definition 3. Let [ 1,1]α∈ − . A point kh Y∈ is α -anomalous, if ( )kµ α> .
We denote by Aα the whole set of the anomalous points.
The set Aα , unfortunately, does not give a complete picture of the anomality
of the record «y». Indeed, as it is seen on the Fig.1, c, there can be narrow zones
between the fragments of anomality located closely with each other. These zones
should not be considered as calm zones since they are too small. To take it into
account we transform the initial measure ( )kµ and construct above-mentioned
one-sided background measures αL ( )y kΦ and αR ( )y kΦ . For that we need «the
helping function» ( )xαψ (Fig. 2).
, 1
1( )
, 1
1
x x
x
x x
α
α α
αψ
α α
α
−⎧ ≤ ≤⎪⎪ −= ⎨ −⎪ − ≤ ≤
⎪ +⎩
, where 1 1α− ≤ ≤ .
It’s easy to see that ( ) ( ( ))k kα αµ ψ µ= possesses the following properties:
( ) 0 ( )k kαµ µ α> ⇔ > ,
( ) 0 ( )k kαµ µ α= ⇔ = , (5)
( ) 0 ( )k kαµ µ α< ⇔ < .
From (5) we can simply state.
Statement 1. If ( ) 0kαµ > , the point kh Y∈ is α -anomalous.
Let us introduce left and right background measures. We assign Θ — the
parameter of intermediate observation: ∆ < Θ ≤ Λ . Thus,
-0.8 -0.4 0 0.4 0.8
-0.8
-0.4
0
0.4
0.8
α=0.3
Fig. 2. The helping function )(xαψ
Algorithm FLARS and recognition of time series anomalies
Системні дослідження та інформаційні технології, 2004, № 3 13
αL
( ) ( )
( ) , [ , 0]
( )
k
y
k
kh k
k k k hk
αµ δ
δ
ΘΦ = ∈ −∑
∑
, (6)
αR
( ) ( )
( ) , [0, ]
( )
k
y
k
kh k
k k k hk
αµ δ
δ
ΘΦ = ∈ +∑
∑
. (7)
The parameter Θ in FLARS plays similar role to the parameter of a global
observation in the algorithm DRAS [11]. It is necessary to emphasise that formu-
las (6) and (7), which define left and right background measures in our case, dif-
fer from functions )(kL yΦα and )(kR yΦα , which define background measures
for the algorithm DRAS [11], although both are introduced for the same purpose.
Definition 4. We call the point kh A∈ α-not calm (potentially anomalous),
if ( ) 0)(),(max >ΦΦ kk yy αα RL .
Definition 5. We call the point kh A∈ is α-calm (background), if
( ) 0)(),(max ≤ΦΦ kk yy αα RL .
Informaly the constructed measures can be interpreted as follows. The less
are values of )(kyΦαL and )(kyΦαR — the calmer (α-calmer) is the record «y»
in the point «kh» (correspondently from the left and from the right). Contrariwise,
the greater are values of the membership function the «more anomalous» (more α-
anomalous) is the initial record in the given point. Thus, )(kyΦαL and
)(kyΦαR may be interpreted as alternating-sign membership functions of the
fuzzy set of not calm points kh Y∈ .
We denote by S ′ the set of the background points, and A′ the set of poten-
tially anomalous points. Thus, the searched division of the set Y is constructed:
Y A A S′ ′= ∨ ∨ . (8)
FLARS FORMAL ALGORITHMICAL MODEL
FLARS has the following free parameters to be chosen by user:
a) +∈∆ hR — parameter of local observation;
b) +∈Λ hR , ∆>>Λ — parameter of global observation;
c) [ 1,1]α∈ − — vertical level of extremality;
d) +∈Θ hR , ∆ < Θ ≤ Λ — parameter of intermediate observation.
In more «flexible» implementation of the algorithm additional parameter
1 1β− ≤ ≤ is introduced. It is horizontal background level. By substituting the
condition ( ) 0)(),(max >ΦΦ kk yy αα RL by the condition ( )(max kyΦαL ,
) βα >Φ )(kyR in the definition 2 and the in equation ( )(max kyΦαL ,
) 0)( ≤Φ kyαR by the condition ( ) βαα ≤ΦΦ )(),(max kk yy RL in definition 3 for
[ 1,1]β∈ − we obtain another, «more flexible» version on FLARS. In other words,
upon proper «tuning» of the algorithm it pays attention only to those outputs
A.D. Gvishiani, S.M. Agayan, Sh.R. Bogoutdinov, S.A. Tikhotsky, J. Hinderer, J. Bonnin, M.Diament
ISSN 1681–6048 System Research & Information Technologies, 2004, № 3 14
above level «α», which are massive enough by time: they will be surrounded by
an aureole of not calm points. Such «tuning» is achieved with the help of a
free parameter [ 1,1]β∈ − and conditions βα ≤Φ )(kyL , ) βα ≤Φ )(kyR .
For each «β» a corresponding dichotomy of non-anomalous points of A
arises:
[ ] { }βαα ≤ΦΦ∈=′ )(),(max: kkAhkS yy RL ,
[ ] { }βαα >ΦΦ∈=′ )(),(max: kkAhkS yy RL .
It is natural to call parameter «β» a horizontal level of a background, since a
massiveness of anomaly by time is its horizontal extension. The bigger is «β» the
bigger segment of a record is related to a background and the smaller aureole will
correspond to each anomalous fragment.
Block-scheme of the algorithm FLARS is given in Fig.3.
THE APPLICATION OF FLARS FOR THE SUPERCONDUCTING
GRAVIMETER DATA PREPROCESSING
The network of the Superconducting Gravimeters (SG) was started in the course
of Global Geodynamic Project at the 1st of July 1997. Now it consists of about 20
instruments deployed all around the globe that continuously measure the time
variations of gravity with the accuracy reaching the 29 m/s10− and sampling in-
terval less or equal to 1 minute. Thus it is clear that the total amount of SG data is
Record rectification
yΦ
Anomality measure
)(kµ
Record fragmentation
)( hkyyk =
Anomaly on the
record
Background
on the record
Potential anomaly
on the record
Background measures
)(),( kRkL yy ΦΦ αα
No anomaly on
the record
>0 <0∆
Λ
Fig. 3. The block-scheme of the algorithm FLARS
Algorithm FLARS and recognition of time series anomalies
Системні дослідження та інформаційні технології, 2004, № 3 15
now really huge and the processing and analysis of this data possess a real scien-
tific and technical challenge.
The main purpose of the GGP is to reveal the information about the Earth’s
inner structure and dynamics by interpreting the SG data in terms of Earth’s tides
as well as non-tidal gravity variations that are related to the Earth axis precession,
core nutations and core modes. Some processes that could manifest themselves in
very weak periodical gravity variations were theoretically predicted, thus the
problem arises of proving or rejecting these hypotheses on the basis of compre-
hensive analysis of SG data.
The scientific approaches that are used to reveal the periodical variations are
generally based on the harmonic (or Fourier) analysis. A process of revealing the
weak peaks in the Fourier power spectra could be very unstable with respect to
noise in the data. On the other hand it is clear that SG time series always contain a
significant noise associated with a wide spectra of sources. Even the small rain at
the site of the SG deployment would produce a substantial signal of tens of mi-
crogals. Other noise sources are the earthquakes, atmospheric pressure variations,
soil humidity variations, snow cover and technical sources of external (autos,
trains and nearby factories) as well of internal (electrical power instabilities, ma-
nipulations with instrument, etc.) origin. It is important that all these sources pro-
duce the noise of a wide variability of features such as amplitudes, frequencies
and specific shapes that may include spikes, gaps, offsets and others. Thus it is
difficult to remove all these noise signals on the base of convenient noise reduc-
tion approaches such as frequency filtering or cutting by the signal amplitude or
amplitude of signal time derivative. But FLARS algorithm by its fuzzy logic na-
ture provides a powerful tool for the solution of this problem.
We have successfully applied the FLARS algorithm for the processing of
data recorded by the SG deployed in Strasbourg, France, in the period from April
to December 1997 (Fig. 4). These records were processed previously by different
authors [13] and the detailed comparison of different approaches could be per-
formed. It has been demonstrated, in particular, that algorithm FLARS is able to
find almost all noise events on SG records.
Рис. 4. An example of noise (interval marked black) detection in SG data. Note the stable
detection of different noise patterns: first four signals are the seismic waves distant earth-
quakes, the fifth disturbance of unknown origin and the sixth is the offset associated with
the instrument operation
A.D. Gvishiani, S.M. Agayan, Sh.R. Bogoutdinov, S.A. Tikhotsky, J. Hinderer, J. Bonnin, M.Diament
ISSN 1681–6048 System Research & Information Technologies, 2004, № 3 16
Depending on choice of free parameters it could act in a more «strong» or
«soft» rejection approaches used by different authors. The present version of the
FLARS algorithm software adopt the «TSoft» data format which is the standard
for the SG community.
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Received 30.04.2004
From the Editorial Board: The article corresponds completely to submitted manu-
script.
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| id | journaliasakpiua-article-171679 |
| institution | System research and information technologies |
| keywords_txt_mv | keywords |
| language | English |
| last_indexed | 2025-07-17T10:25:23Z |
| publishDate | 2019 |
| publisher | The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" |
| record_format | ojs |
| resource_txt_mv | journaliasakpiua/73/72b24cdb5a4026c3b59b2c77b3a75d73.pdf |
| spelling | journaliasakpiua-article-1716792019-06-27T14:30:07Z Algorithm FLARS and recognition of time series anomalies Алгоритмы FLARS и выделение аномалий временних рядов Алгоритми FLARS та виділення аномалій часових рядів Gvishiani, A. D. Agayan, S. M. Bogoutdinov, Sh. R. Tikhotski, S. A. Hinderer, J. Bonnin, J. Diament, M. As a rule, algorithms of recognition of time series anomalies are based on time frequency or statistical analysis . This article is devoted to detailed formal description of new fuzzy set based algorithm FLARS (Fuzzy Logic Algorithm for Recognition of Signals). It recognizes time series anomalies by means "smooth" modelling (in fuzzy mathematics sense) of interpreter's logic, which searches for anomalies at the record. Общепринятые алгоритмы выделения аномалий временных рядов основываются, в основном, на частотно временном или статистическом анализе. Статья посвящена строгому построению нового алгоритма FLARS. Его можно рассматривать как результат "мягкого" (на основе нечеткой математики) моделирования логики интерпретатора, ищущего аномалии на записи. Загальноприйняті алгоритми виділення аномалій часових рядів базуються, як правило, на частотно часовому або статистичному аналізі. Стаття присвячена строгій побудові нового алгоритму FLARS. Його можна розглядати як результат "м’якого" (на основі нечіткої математики) моделювання логіки інтерпретатора, який шукає аномалії на запису. The National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute" 2019-06-27 Article Article application/pdf https://journal.iasa.kpi.ua/article/view/171679 System research and information technologies; No. 3 (2004); 7-16 Системные исследования и информационные технологии; № 3 (2004); 7-16 Системні дослідження та інформаційні технології; № 3 (2004); 7-16 2308-8893 1681-6048 en https://journal.iasa.kpi.ua/article/view/171679/171376 Copyright (c) 2021 System research and information technologies |
| spellingShingle | Gvishiani, A. D. Agayan, S. M. Bogoutdinov, Sh. R. Tikhotski, S. A. Hinderer, J. Bonnin, J. Diament, M. Алгоритми FLARS та виділення аномалій часових рядів |
| title | Алгоритми FLARS та виділення аномалій часових рядів |
| title_alt | Algorithm FLARS and recognition of time series anomalies Алгоритмы FLARS и выделение аномалий временних рядов |
| title_full | Алгоритми FLARS та виділення аномалій часових рядів |
| title_fullStr | Алгоритми FLARS та виділення аномалій часових рядів |
| title_full_unstemmed | Алгоритми FLARS та виділення аномалій часових рядів |
| title_short | Алгоритми FLARS та виділення аномалій часових рядів |
| title_sort | алгоритми flars та виділення аномалій часових рядів |
| url | https://journal.iasa.kpi.ua/article/view/171679 |
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