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Construction scheme of DMA
Fussy comparisons onpositive numbers
Nearness in finite metrical space
Limit in finite metrical space
Density as measure oflimitness
Smoothtime series.Equilibrium
Monotonous time series
Fussy logicand geometry on
time series:geometrymeasures
Separation of dense subset.
Crystal. Monolith.
Clasterization.Rodin
Predicationof time series.
Forecast
Anomalies ontime series.
DRAS. FLARS
Extremums on
time series.
Convextime series
Search of linearstructure.Tracing
FTS anomaly recognition algorithms: DRAS, FLARS and FCARS
• DRAS (Difference Recognition Algorithm for Signals ) - 2003• FLARS (Fuzzy Logic Algorithm for Recognition of Signals) – 2005
• FCARS (Fuzzy Comparison Algorithm for Recognition of Signals) - 2007realize “smooth” modeling (in fuzzy mathematics sense introduced by L. Zade) of
interpreter’s logic, that searches for anomalies on FTS.
FTSDRAS,FLARS,FCARS local
level
Rectificationof FTS
FTSAnomalies
FLARSglobal level
DRASglobal level
FCARSglobal level
Examples of FTS rectification functionalsLength of the fragment, energy of the fragment, difference of the
fragment from its regression of order n.
Interpreter’s Logic. Illustration
Global level - searching the uplifts on rectification
Local level - rectification of the record
Record
DRAS and FLARS: local level - rectification
Discrete positive semiaxes h+={kh; k=1,2,3,…}
Record y={yk=y(kh), k=1,2,3,…}
Registration period Y h+
Parameter of local observation Δ=lh, l=1,2,…
Fragment of local observation Δk y={yk-Δ/h ,… , yk ,… , yk+Δ/h}Δh+1
Definition.
A non-negative mapping defined on the set of fragments
{Δk y}2Δ/h+1
we call by a rectifying functional of the given record “y”.
We call any function ykΔky by rectification of the record “y”.
Examples of rectifications1 Length of the fragment:
2 Energy of the fragment:
3 Difference of the fragment from its regression of order n:
here as usual is an optimal mean squares approximation of order n of the fragment . If n=0 we get the previous functional “energy of the fragment”:
4 Oscillation of the fragment:
1
1
kh
kj j
j kh
L y y y
2k
hk
j k
j kh
E y y y
2
kh
k j
j kh
hy y
h
2( ) [ Regr ( )]k
kh
k nn j y
j kh
R y y jk
0Regr2
k
kh
j ky
j kh
hy y
h
2 200 ( ) Regr ( ) ( )k
k kh h
k kj j ky
j k j kh h
R y y jh y y E y
( ) max mink k
h hk
j jj kj k
hh
O y y y
Illustration of rectification
Record
Rectification «Energy»
Rectification «Length»
-400
-350
-300
-250
-200
mV
/km
67.5 68 68.5 69
DRAS: Difference Recognition Algorithm for Signals.
Left and Right background measures
Record rectification
Record fragmentation
Potential anomaly on the record
Genuine anomaly on the record
,
Record
DRAS: global level. Recognition of potential anomalies.
( ) : ( ) , ,( )( )
( ) : ,
( ) : ( ) , ,( )( )
( ) : ,
yk
y
k
yk
y
k
kh k k k kh
L kkh k k k
h
kh k k k kh
R kkh k k k
h
Left and right background measures of silence -
β– horizontal level of background
Potential anomaly on the record y:
- vertical level of background
PA={khY : min((LαΦy)(k), (RαΦy)(k)) < β}
Regular behavior of the record y: B={khY : min((LαΦy)(k), (RαΦy)(k)) β}
DRAS: global level. Recognition of genuine anomalies.
Potential anomalies PA = UP(i), n=1,2, N. is a union of coherent components
DRAS recognizes genuine anomalies A(n) as parts of P(n) by analyzing operator DΦ(k) = LΦ(k) - RΦ(k).
The beginning of A(n) is the first positive maximum of DΦ(k) on P(n). Indeed , the difference between “calmness” from the left and anomaly behavior from the right is the biggest in this point. By the same reason, the end of A(n) is in the last negative minimum of DΦ(k).
DRAS: recognition of genuine anomaly.
( )( ) ( )( ) ( )( )y y yD k L k R k
Genuine anomalies on the record y, A = {alternating-sign decreasing segments for (DαΦy)(k)}
FLARS: Global level. recognition of genuine anomaly.
α[0,1] – vertical level of how extreme are the measure values
Regular behavior/ potential anomaly NA = { kh Y : μ(k)<α}
Genuine anomaly on the record y A = { kh Y : μ(k)α}
FLARS: global level. Recognition of potential anomaly.
( ) ( ( ))k k
( ) 0 ( )k k ( ) 0 ( )k k ( ) 0 ( )k k
We introduce the function that possesses the following properties:
One-sided background measures -
( ) ( )( ) , [ , 0]
( )k
yk
kh kk k k hk
L
( ) ( )( ) , [0, ]
( )k
yk
kh kk k k hk
R
Θ – the parameter of intermediate observation: Δ<Θ≤Δ .
β – horizontal level of background, (-1,1)
Potential anomaly on the record y PA={kh NA : min((LαΦy)(k), (RαΦy)(k)) < β}
Regular behavior of the record y B={kh NA : min((LαΦy)(k), (RαΦy)(k)) β}
FLARS: anomaly measure μ(k)
- parameter of global observation
δkk - model of global observation record at the point k( ) { [ , ] : ( ) ( )}y yk k h kh kh k k ( ) { [ , ] : ( ) ( )}y yk k h kh kh k k
( , ) ( ) ( ) ( ) : ( )y y kk k k k k h k
( , ) ( ) ( ) ( ) : ( )y y kk k k k k h k
( , ) ( , )( ) ( ) [ 1,1]
max( ( , ), ( , ))
k kk
k k
The following sum will be an “argument” for minimality (regularity) of the point “kh”
The following sum will be an “argument” for maximality (anomaly) of the point “kh”
The measure is a result of the comparison of the “arguments” and( )k
What algorithm to apply to FTS data sets?
• DRAS. Calm and anomaly points are quite well distinguished, but genuine anomalies are not evident. DRAS is useful in searching big anomalies.
• FLARS. High amplitude anomalies are quite obvious and small anomalies are not so evident on the background of noise. Useful to search very small isolated anomalies.
• FCARS. Important in searching oscillating anomalies and identification of the beginning and ends of the signals.