Discrete Mathematical Analysis: theory and geophysical applications.

Discrete Mathematical Analysis:

theory and geophysical applications

DMA definition

Discrete mathematical analysis (DMA) is an approach to studying of multidimensional massifs and time series, based

on modeling of limit in a finite situation, realized in a series of algorithms.

The basis of the finite limit was formed on a more stable character, compared to a mathematic character, of human idea of

discontinuity and stochasticity. Fuzzy mathematics and fuzzy logic are sufficient for modeling of human ideas

and judgments. That was reason why they became technical foundation of DMA.

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

Fuzzy comparisons

Definition. A fuzzy comparison ( , )f a b of real numbers a and b measures in the alternative

scale of [ 1,1] segment the rate of superiority of “b” over “a ”:

( , ) ( ) [ 1,1]n a b es a b .

Example. For ( 1,1) we define

0( , ) ( ( , ))n a b n a b , где

, [ ,1]1

( )

, [ 1, ]1

tt

tt

t

, 0( , )max( , )

b an a ba b

Remark. For given pairs (5,10) and (70,75)

0

5 1(5 10) (5,10)

10 2es n and 0

5 1(70 75) (70,75)

75 15es n

Extension. Let us extend ( , )n a b to some arbitrary non-negative finite sets A and B . I f

1 20 NA a a a , 1 20 MB b b b , then

, ( , )( ) ( , ) [ 1,1]i j i jn a b

es A B n A BNM

Measures of nearness

green - 1.0 ( ) 0.5x y

sky blue - 0.5 ( ) 0.0x y

blue - 0.0 ( ) 0.5x y

red - 0.5 ( ) 1.0x y

Definition: Measure of nearness ( )x y answers in scale [0,1] or [ 1,1] with normalization ( ) 1x x

to the question: how close is point y to point x in FMS ( , )X d ? X 1

X 2 Example: 1) Let { ( , ): }dX d x y x y X set of non-trivial distances in ( , )X d . Then ( )x y is

measure of smallness ( , )d x y compared to dX :

( ( ( , ), ( , ))

( ) ( ( , ), )| | (| | 1)

x yx

n d x y d x yy n d x y dX

X X

Measures of nearness

2) Let ( ) { ( , ) : }dX x d x y y X x .

Then ( ( ( , ), ( , ))

( ) ( ( , ), ( ))| | 1

y xx

n d x y d x yy n d x y dX x

X

green - 1.0 ( ) 0.5x y

sky blue - 0.5 ( ) 0.0x y

blue - 0.0 ( ) 0.5x y

red - 0.5 ( ) 1.0x y

Nearness measures: nearness point to set

Let point x X and A is subset in X . Then

1) Fuzzy belonging ( )A x : ( )

( )| |

yy AA

xx

A

,

( ) ( )( )

( )

y xy AA

xy A

x yx

y

2) Monolithness mon ( )A x : let us 1 20 ... ( )Ar r r x is ordered set of non-trivial

distances from x to elements of A ; h – positive quantum and { , 0,1,...}h kh k .

We called quantization a correspondence ( )t h t :

, if ( )

( 1) , if ( , ( 1) )

t t nhh t

n h t nh n h

Definition. Let

1( ) ( )mon ( )

( ) : 0, ( )A

AA

h r h r xx

nh nh h r x

, where ( ) 1

( )A

nhnh

h r x h

x

X 1

x

X 2

x

X 3

1mon ( ) 1X x >

2mon ( )X x >

3mon ( )X x

3) Metrical displacement ( | )x A . Notation:

( | ) ( ), mon ( ), ( | )

( | ) ( ), mon ( )

A A

A A

S x A x x x A

P x A x x

General concept of finite limit in FMS

( , )X d - finite metrical spase: A - X -distribution: 1( , ) |ni iA A , iA X , (0,1]i

( | )S A measure of nearness to A

( | ) { ( | ), } mes( | , )i ix S x S x A x SS

AA A

SS

( | )S x A - comparision of A to x on the basis of S

( | )iS x A - result of comparison A with x at stage iA

i - weight, which has such comparison

S – integral characteristics ( | )S x A

Example: , 0 S - Kolmogorov average

1

1 ( | )mes( | , )

ni ii

i

S x Ax S

A

*

*, *

еслиargmax mes( | , )

slim slim ,argmin mes( | , )

XS

X

x x Px

x x

S

S

S

AA A

A











Crystal. Monolith.



Forecast


DRAS. FLARS

Extremums on

time series.

Convextime series


Density as measure of limitness

x – point in X

1 2 ( )0 ... n xr r r – ordered set of non-trivial distances from x to other elements from X

i – weight of ball ( )x iD r Examples: 1) Average nearness ( )x iD r to x on the basis of this or that measure of nearness x :

( ) : ( )

( )x x i

ix i

y y D r

D r

2) Degree of smallness of radius ir modulo set

( ) { ( , ) : }dX x d x y y X on the basis of fuzzy comparison n :

( , ( , )) :

| |i

i

n r d x y y Y

X

( )1( ), |n x

x x i iD D r – discrete analogue of filtration X around x .

Definition: Density ( )XP x of point x in FMS ( , )X d is one of measures of limitness xD to x

on the basis of nearness ( ,mon)P :

( ) ( | , )X xP x es D x P

Algorithm “Choosing of foundation”Let ( , )X d be FMS and ( )XP x – chosen model of density, x X , ( ) { ( ), }XP X P x x X .

Definition: Let us call point x strong (weak) n – foundation in X , if for measure of

superiority ( )XP x over ( )P X , induced by fuzzy comparison n , inequality is corrected

μes( ( ) ( )) ( ( ), ( )) 0.5X XP X P x n P X P x ( 0) ( )

“Choosing of foundation” – algorithm of recognition in dense areas

in X by correlation ( )

CRYSTAL description

• CRYSTAL goal:

to identify -dense subsets against a general background

• Definition 1. Subset AX is dense against the background X if

• Definition 2. Subset AX is -dense against the background X if

CRYSTAL description

CRYSTAL block-scheme:

CRYSTAL description

“Growth I” block:• – the current version of i-th crystal Ki

• for

• , =1 (1,n) 1

“Struggle” between and in :

• If then has “won”, and we proceed to the “Foundation” block

• If then has “won”, but Kin+1={Kn

i, x} should

remain dense against the background X in all its points (the “Growth II” block)

CRYSTAL description

“Growth II” block (1):• Statement. If the “Growth I” condition is fulfilled then Kn+1={Kn, xn+1}

will be -dense only if the “Growth II” condition is fulfilled:

=1

• If there is no such xn+1, we proceed to the “Foundation” block for

choosing another foundation for the next crystallization

• If there are several such xn+1 ( ), then

CRYSTAL description

“Growth II” block (2):

• Recalculation of the densities and in for their further “struggle” in the “Growth I” block:

CRYSTAL description

“End” block:

• Identification of the final crystal versions K1, …, KI,

I = I(F,).

Examples

=1.0

=1.2

=2.3

Examples

=2.5 =4.0

blue – foundations; red – crystallised points

Examples

blue – foundations; red – crystallised points

(1) (2)

=2.6

Geophysical applications


N=3

Географич. к-ты региона

70ВД, 210СШ

TN 38 500 нТл

DN -3.50

IN 250

Real data – region Hoggar (Algeria):definition of magnetization

T

Geophysical applicationsReal data – region Hoggar (Algeria):

definition of magnetization

RODIN overview

• The cluster definition:

cluster in X, if

A – cluster in X, if

RODIN overview

Let A be a cluster in X and xAX:• The cluster quality:

• Measure of separability of x in A:

where

The cluster separability:

A1 A

2

X

A1

A2X

RODIN overview

Block-scheme of the Global RODIN: – given level of quality, – given level of separability (clusterness), K0 – initial version of cluster, Kn – current version of cluster

RODIN overview

Algorithms of RODIN family:

RODIN overviewThe examples of Global RODIN clustering with different levels of quality:

RODIN overview


C artographied area

L a n n io n

G u in g a m p

Golfe deSaint-Malo

J ersey

MCCPC

Zone

Shear Armorican

North 0 10 20 km

S a i n t B r i e u c U n i t

ortho-gneisses

brioverian meta-volcanics

brioveriansediments

gabbroic & tonaliticintrusions

Guingamp Unit

metagabbrosmigmatitesleuco-granites

Saint Malo Unit

migmatites granite(Lamballe)

sediments

granitoid (Ploufragan-Saint brieuc)

post cadomian intrusions

paleozoic and mezosoic sediments

variscan granitic intrusions

doleriticdikes

Saint Malo Region

Geophysical applicationsSaint Malo Region

Algorithm “Monolith”

X – multi-dimensional massif, xX, A – subset in X

Crossing AA (1) =MonA X – “multi-dimensional topological” smoothing A in X

MonA X – subsep points in X with large A-monolithness (A-foundations)

radius of monolithness r weight of monolithness parameter of choosing number of iterations i

Algorithm “Monolith”: parameters

monA x – monolithness A in x measure of limitness A in x

Algorithm “Monolith”: block-scheme

A m o n A (. ) A (1 ) m o n A (1 )(. ) A (2 ) A ( i)m o n A ( i-1 )(. )...

r (X )

...

Etna. Interferrogramm

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0102030405060708090100110120130140150160170180190200210220230240250

Smoothness of displacement

Let be a set of pixels, ( ) п - a shift of pixel п measured by satellite, ( )D п - its

neighborhood (it could be neighboring points or a sphere of radius r around pixel п).

There are different variants of smoothness ( )G п of shift in point п . Let’s cite

several examples:

1. ( )G п - average deviation

( ) ( )( )

( )G

D

п п

пп

, ( )Dп п .

A palette was used in our calculations ( )D п - 3х3 and

ï

ï

( ) ( )( )

8G

п п

п

Etna. Smooth points

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Etna. Monolith (1-st iteration)

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Etna. Monolith (4-th iteration)

470 475 480 485 490 495 500 505 510 515 520

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Etna. Monolith (boundary)

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Etna. General solution

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0102030405060708090100110120130140150160170180190200210220

Tracing











Crystal. Monolith.



Forecast


DRAS. FLARS

Extremums on

time series.

Convextime series


Gravitation limits of time series

Initial data: time series :[ , ]x a b , points ( 1)it a i h , 1a t , nb t , b a

hn ,

1( ) ~ ( ) |n hi i ix x t x x . M is fixed and [ , ]it a b .

( ) max(| |,| |)i ir i t a t b h , ( , ) ( )(1 ), 0, ,m

r i m r i m MM

, 0( , ) { , }|Mi i i m mx t A A A - gravitational distribution for x in it

Quantitative component iA : , { ( ) :| | ( , )}i m j i jA x t t t r i m

Nearness ,( | )i mS z A - square deviation from weighted gravity centre ( ),( ; ( ))m

i m igr A j :

2( )

( ),( )

( )( | )

mj i

mi

i mx j

zj

S z A

,

where ( )

| |1 , | | ( , )

( , )

0 , | | ( , )

( )i j

i j

i j

mi

t tt t r i m

r i m

t t r i m

j

and ( )( )Mi ijj - symbol of Kroneker

Gravitation limits of time series

Dynamic component iA : 11m

mM

( ) 2,0

0

|| ( ; ( )) ||mes( | , )

mMm i m i

i Mm

z gr A jz S

A ,

( ),0

,0

( ; ( ))lim ( ) slim

i

mMm i m i

i MSt t m

gr A jgr x t

A

Algorithm “Equilibrium” Task: for time series y on [ , ]a b to construct smooth skeleton ( )x Sm y

Logic: “x - smooth skeleton for y ” ( x - smooth time series) ( x - approximation y )

x - smooth time series deviation of gravitational smoothness ( )CGr x

x - approximation y functional of scanning ( | )Sc x y

Deviation of gravitational smoothness: zero level (0)( )CGr x

(0) 21( ) | lim ( ) |

i

nii

t txCGr x gr x t

Deviation of gravitational smoothness: s -valued level ( )( )sCGr x

(0)( )( ) ( )s sCGr x CGr x

sx - s -valued derivation x : :[ , ]s sa b b shx

0( 1)( ) , [ , ]

s s l ls ssl i l

is

C xx i t a b

h

Algorithm “Equilibrium”

Deviation of gravitational smoothness: final level ( )CGr x

( )10( ) ( )sn

ssCGr x w CGr x , sw - non-negative weights

Approximation 2( | ) || ||Sc x y y x

Final functional of smoothness ( | )Sm x y

( | ) ( ) (1 ) ( | )Sm x y CGr x Sc x y , [0,1]

* *( ) Grad ( | ) 0x Sm y Sm x y


0.00.9999990.999990.99990.9990.990.9

Dependence from


( ) ( ) (1 ) ( | )Sm x CGr x Sc x y , [0,1] *x Sm y *Grad ( | ) 0Sm x y

10( ) ( )n s

s sCGr x CGr x

Let us denote through 0G non-negative operator on [ , ]ВР a b , associated with derivation CGr :

( ) ( , )CGr x Gx x , [ , ]x ВР a b

[ , ]s a bP – subspace [ , ]ВР a b , induced polynomials of degree s .

Statement. *

*

1Ker Ker Ker [ , ]s

sG CGr D a b P , where *s is number of the first non-trivial

weight s in derivation CGr .

[ . ] Ker ImВР a b G G , *dimIm 1G n s , *dimKer 1G s .

1( (1 ) ) (1 )Sm G E , so than smoothing y Sm y leaves invariant subspaces

KerG and ImG , and acts identically on KerG . I f ker y - projection y on KerG , then kern

nSm y

.

Component ker y – square regression for y of degree *s .


Definition: For [ , ]t a b and z let’s denote through ztx series on [ , ]a b : ( )z

t ix t z , if

it t , ( )zt i ix t x , if it t . Then x Gr is smooth, if

( ) ( )i

ztCGr x CGr x , [ , ]it a b , :| |iz z x

Statement. At 0 for any [ , ]y ВР a b there always exists Gr -smooth iteration

nSm y , ( , , )n n y .

Example. Our experience work with algorithm “Equilibrium” has shown that at h ,

0.9999 the number of iteration, as a rule, didn’t exceed 4. The smaller is , the more

is the number of iterations.


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-30

-20

-10

0

10

20

30

-2 -1.95 -1.9 -1.85 -1.8 -1.75 -1.7

-32

-30

-28

-26

-24

-22

-20

-18

-16

0 0.5 1 1.5 2 2.5 3

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.02 0.04 0.06 0.08 0.1

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

5y x siny x


-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

-30

-20

-10

0

10

20

30

0 0.5 1 1.5 2 2.5 3

0

0.2

0.4

0.6

0.8

1

5y x siny x

Parameters: 100;n ;4M ,1432

;0s ,99.0 noise - 10%

Algorithm “Forecast” Initial data: 0~ ( ) |niy y , ( )i iy y t , it a ih , 0a t , nb t ,

b ah

n .

zy - continuation y to point b h by value z :

, [ , ]( )

,ii

z ii

t a byy t

t b hz

For 0, ,s n ( ) 0 1( 1)( )

s k ks s sk n k

z s

C yy b

h

, sb b h sh

sw - non-negative weights (forecasting parameters)

( )0Re( ) ( ( ), ) |s s n

z z sy y b w - real distribution at the end at forecast ( )y b h z

0Id( ) ( , ) |s nsy id w - ideal gravitational distribution at the end

( )limgr ( )s

ss

t bid y t

( )2 20( ) ( ( ) ) minsn s s

s zsz w y b id

1*

0 0 2n ns s ss ss s

C w wz

h h

, where 1 1( 1)s k k

ssk n ks s

C yC id

h

Notion: *

[ , ]( | )a bz Forecast y

Algorithm “Forecast”

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

-2 -1.5 -1 -0.5 0 0.5 1

0

1

2

3

4

5

-2 -1.5 -1 -0.5 0 0.5 1

0

1

2

3

4

5

23 3( 1)y x x siny x

-2 -1.5 -1 -0.5 0 0.5 1-1

0

1

2

3

4

5

6

Agreement “Equilibrium” and “Forecast” 23 3 (10% )( 1)y x x noise

70.5 70.52 70.54 70.56 70.58 70.6 70.62 70.64 70.661308

1310

1312

1314

1316

1318

1320

1322

Real time series

La Fournaise volcano (Reunion Island)

DRAS: application to electric signals associated with the volcanic activity of La Fournaise volcano (Reunion Island) II

Time = 58.542

DRAS and FLARS

Algorithms DRAS (Difference Recognition Algorithm for Signals ) and FLARS (Fuzzy Logic Algorithm for Recognition of Signals) are a results of “smooth” modeling (in fuzzy mathematics sense) of interpreter’s logic, which searches for anomalies at the record.

DRAS and FLARS is really based on fuzzy mathematic principals.

Interpreter’s Logic

Local level:

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 correspond to more anomalous points (centres of fragments).

Global level:

The anomalies on the record are the uplifts on its rectification.

Interpreter’s Logic. Illustration.

Record

Local level - rectification of the record

Global level - searching the uplifts on rectification

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”

DRAS: block-scheme of the algorithmBackground measures

Record rectification

Record fragmentation

Potential anomaly on the record

Anomaly on the record

,

,y yL R

y

( )ky y kh

Record

DRAS: global observation

+

- parameter of global observation

δkhkh - model of global observation record at the point kh

δkhkh +h-h│k - k│+h

DRAS: recognition of potential anomaly on the record

( ) : ( ) , ,( )( )

( ) : ,

( ) : ( ) , ,( )( )

( ) : ,

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

One-sided 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: recognition of potential anomaly on the record. Illustration.

DRAS: recognition of anomaly on the record.

( )( ) ( )( ) ( )( )y y yD k L k R k

Genuine anomalis on the record y = A = {alternating-sign decreasing segments for (DαΦy)(k)}

DRAS: application to electric signals associated with the volcanic activity of La Fournaise volcano (Reunion

Island) I

Station – DON, direction - EW

FLARS: block-scheme of the algorithm

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. α[0,1] – vertical level of how extreme are the measure values

No anomaly on the record y NA = { kh Y : μ(k)<α}

Anomaly on the record y A = { kh Y : μ(k)α}

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

France)











Crystal. Monolith.



Forecast


DRAS. FLARS

Extremums on

time series.

Convextime series


Monotonousness on time series

0~ ( ( )) |nix x t , it a ih , b a

hn , 0t a , nt b , lim gr - construction of gravitational limit

Definition: x increasing gravitationally (decreasing) on [ , ]a b , if lim lim

i it t t tgr x gr x

( lim lim

i it t t tgr x gr x

) [ , ]it a b

Example: limgr - gravitational global :

( ) ( )lim : [ , ]

( )i

i i

j t jj i

t t t j

x t tgr x t a t

t

( )

i

jt j

i

t ht

t h

,( ) ( )

lim : [ ]( )

i

i i

j t jj i

t t t j

bx t t

gr x t tt

( )

i

jt j

i

b h tt

b h t

x -dynamics on [ , ]a b - consequent alteration of maximum opposite gravitational monotonousness. Differently from ordinary monotonousness, gravitational ones can intersect with each other

a d c b a d c b

Extremums on time series

[ , ]arg max ( ) ( ) : [ , ]tt d c

M x t t t a b

[ , ]arg min ( ) ( ) : [ , ]tt d c

m x t t t a b

| |( ) 1

max(| |,| |)tt t

tt a t b h

a d c b a d c b a d c b

Extremums on time series

Convexity on time seriesA function is convex on [ , ]a b , if ( )x t is monotonous on [ , ]a b

-point of excesses for ( )x t , if is extremum for ( )x t

Time serios: ( ) ( ) ( ( ))x Sm x Sm x Sm Sm x

Definition.

1) Time series ( )ix t is convex on [ , ] ha b , if ( ( ))Sm Sm x is gravitationally monotonous on [ , ]a b h

2) - точка point of excesses for ( )ix t , if is extremum for ( ( ))Sm Sm x

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Crystal. Monolith.



Forecast


DRAS. FLARS

Extremums on

time series.

Convextime series


Elementary measures

( )x t – time series on [ , ] ha b R , [ , ]t a b

( )

( )( )[ , ]

( )

( )

l i

l s

r i

r s

t

x ttt a b

t

t

minimal to the left

to which extent maximal to the leftfrom on

canbe considered minimal to the right

maximal to the right

b a – parameter of one-sided overview

Construction ( )l i t

( )lix t – upper derivation x from ( )x t to the left of t on [ , ]t t :

( ) ( ( ) ( )) : [ , ] ( ) ( )lix t x t x t t t t x t x t

Let us set:

( , ) ( ( ) ( )) : [ , ] ( ) ( )

( , ) ( ( ) ( )) : [ , ] ( ) ( )

li li li li li

li li li li li

x t x t x t t a b x t x t

x t x t x t t a b x t x t

and ( ) ( ( , ), ( , ))li li li

x t n x t x t ,

where n – fuzzy comparison. Normalization: if is any elementary measure, then value ( )t is close to 1 (-1), if a corresponding property in point t is weak (strong).

Geometrical measures

T - any fuzzy conjunction ( minT , T )

(1,1,1,1)

( 1,1,1,1)

( 1, 1, 1, 1)

(1,1,1,1)

( 1,1,1,1)

( 1, 1, 1, 1)

( ) ( ( ), ( ), ( ), ( ))

( ) ( ( ), ( ), ( ), ( ))

( ) ( ( ), ( ), ( ), ( ))

li l s ri rs

li l s ri rs

li l s ri rs

t T t t t t

t T t t t t

t T t t t t

Examples:

1) (1,1,1,1)( )t - measure “background”

2) (1,1, 1,1)( )t - measure “beginning of mountain”

3) ( 1,1,1,1)( )t - measure “end of mountain”

4) (1, 1, 1,1)( )t - measure “left slope”

5) ( 1,1,1, 1)( )t - measure “right slope”

6) ( 1,1, 1,1)( )t - measure “hollow”

7) (1, 1,1, 1)( )t - measure “peak”

Geometrical measure “background”

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Га

л

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Geometrical measure “Beginning of mountain”

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л

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Geometrical measure “left slope”

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л

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Geometrical measure “peak”

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Thank you

Discrete Mathematical Analysis: theory and geophysical applications.

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Transcript of Discrete Mathematical Analysis: theory and geophysical applications.