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 potential anomaly.

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

FLARS: recognition of anomaly on the record.

Anomality measure

Rectification

Record

-0.5

0

0.5

FLARS: application to the Superconducting Gravimeters data preprocessing (Strasbourg, France)

DRAS

FLARS

DRAS and FLARS recognition comparison

FCARS

FCARS anomaly recognition

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.