Crustal velocity field modelling with neural network and

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19 Οκτ 2013 (πριν από 3 χρόνια και 10 μήνες)

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A

comparison

of

the

ability

of

artificial

neural

network

and

polynomial

fitting

was

carried

out

in

order

to

model

the

horizontal

deformation

field
.

It

is

performed

by

means

of

the

horizontal

components

of

the

GPS

solutions

in

the

Cascadia

Subduction

Zone
.

One

set

of

the

data

is

used

to

calculate

the

unknown

parameters

of

the

model

and

the

other

is

used

only

for

testing

the

accuracy

of

the

model
.

The

problem

of

overfitting

(i
.
e
.
,

the

substantial

oscillation

of

the

model

between

the

training

points)

can

be

avoided

by

restricting

the

flexibility

of

the

neural

model
.

This

can

be

done

using

an

independent

data

set,

namely

the

validation

data,

which

has

not

been

used

to

determine

the

parameters

of

the

model
.


The

proposed

method

is

the

so
-
called

“stopped

search

method”,

which

can

be

used

for

obtaining

a

smooth

and

precise

fitting

model
.

However,

when

fitting

high

order

polynomial,

it

is

hard

to

overcome

the

negative

effect

of

the

overfitting

problem
.

The

computations

are

performed

with

Mathematica

software,

and

the

results

are

given

in

a

symbolic

form

which

can

be

used

in

the

analysis

of

crustal

deformation,

e
.
g
.

strain

analysis
.


Crustal velocity field modelling with neural network

and
polynomials

Piroska Zaletnyik
1
, Khos
r
o Moghtased
-
Azar
2



1

Department of Geodesy and Surveying, Budapest University of Technology and Economics


Hungary, piri@agt.bme.hu

2

Institut of Geodesy, University of Stuttgart


Germany, m
oghtased@gis.uni
-
stuttgart.de

The

adaptation

of

neural

networks

to

the

modeling

of

the

deformation

field

offers

geodesists

a

suitable

tool

for

describing

structural

deformation
.

Overfitting

problem

can

occur

in

higher

order

polynomials,

but

neural

network

overcomes

the

problem

thanks

to

stopped

search

method
.

The

greatest

advantage

of

this

method

is

that

the

solution

can

be

given

as

an

analytical

function,

which

could

be

use

to

compute

derivation

of

the

velocity

vectors

for

strain

analysis
.

The

first

author

wish
es

to

thank

to

the

Hungarian

E
ö
tv
ö
s

Fellowship

for

support
ing

her

visit

at

the

Department

of

Geodesy

and

Geoinformatics

of

the

University

of

Stuttgart

(Germany),

where

this

work

has

been

accomplished
.

Abstract

Results

Fig 1.GPS determined horizontal velocity field by Pacific Northwest Geodetic
Array (PANGA), which is plotted relative to North American Plate.

Introduction

Polynomial fitting


When

there

are

more

points

than

the

number

of

parameters,

there

is

a

possibility

for

adjustment

calculation,

i
.
e
.

for

polynomial

fitting
.

In

this

case

62

points

were

used

for

the

adjustment
.

The

calculations

were

carried

out

with

Mathematica

software
.


Figure
4

:
Northing velocity
model by polynomials


Figure
5

:
Northing velocity
model by neural networks

Conclusion

Acknowledgement

Neural network with stopped
search method

Testing set residuals (mm/year)

min

max

mean

Std.

Northing velocities

-
11.1

36.3

1.4

9.0

Easting velocities

-
9.3

17.3

1,8

6.1

Teaching set residuals (mm/year)

min

max

mean

Std.

Northing velocities

-
4.1

3.9

0.0

1.2

Easting velocities

-
7.9

8.2

0.0

2.8


Table1

:
The statistics of the differences between polynomial model and
real velocities in the 62 teaching points.


Testing set residuals (mm/year)

min

max

mean

Std.

Northing velocities

-
6.2

8.3

-
0.5

2.8

Easting velocities

-
4.6

8.9

0.2

3.8

Teaching set residuals (mm/year)

min

max

mean

Std.

Northing velocities

-
4.9

5.5

0.0

1.6

Easting velocities

-
6.8

9.5

0.2

3.1

The

GPS

measurements

to

determine

crustal

strain

rates

were

initiated

in

the

Cascadia

region

(US

Pacific

Northwest

and

south
-
western

British

Columbia,

Canada)

more

than

a

decade

ago,

with

the

first

campaign

measurements

in

1986

and

the

establishment

of

permanent

stations

in

1991
.

Nowadays,

continuous

GPS

data

from

the

Pacific

Northwest

Geodetic

Array

process
ed

by

the

geodesy

laboratory

serves

as

the

data

analysis

facility

for

the

Pacific

Northwest

Geodetic

Array

(PANGA)
.

This

organization

has

deployed

an

extensive

network

of

continuous

GPS

sites

that

measure

crustal

deformation

along

the

CSZ
.

Fig
.

1

illustrates

the

horizontal

velocity

field

along

the

Cascadia

margin

assuming

the

North

American

plate

to

be

stable
.



Table 2. The statistics of the differences between
3
D polynomial model
and real velocities in the 20 testing points.

The

o
verfitting

problem

means

that

the

error

of

the

teaching

set

is

decreasing

while

the

error

of

the

testing

set

is

growing,

in

other

words

the

network

excessively

fits

the

teaching

points

which

is

illustrated

by

Fig
.

3
.



Overfitting problem

Fig. 2. Overfitting problem


Comparing

the

results

of

Table

1

and

Table

2

we

recognize

a

significant

difference

between

the

deviations

of

the

teaching

and

the

testing

set
.

The

determined

model

by

polynomial,

work
s

well

only

in

the

teaching

points

but

between

them

it

does

not

work

as

well
.

The

testing

set,

which

was

not

used

during

the

determination

of

the

model,

is

also

needed

in

order

to

qualify

the

results
.

As

a

classical

approximation

model,

3
D

polynomial

fitting

technique

is

used

to

build

continuous

velocity

field

as

a

function

of

geodetic

coordinates
.

Displacement

vector

which

can

be

derived

from

GPS

observations

have

east,

north

and

up

components

in

topocentric

coordinates
.

F
or

modeling

the

horizontal

displacement

field

we

use

only

the

north

and

the

east

elements
.

Accuracy

of

modeling

is

determined

by

differences

between

true

values

and

values

estimated

by

3
D

polynomial

fitting
.

When

we

increase
e

the

degree

of

the

polynomial,

accuracy

is

increase
ing

up

to

the

6
th

degree,

but

above

that

started

to

decrease
,

because

of

the

deterioration

of

the

conditions

of

the

equations

(
ill

conditioned

equations)
.

The

6
th

order

polynomial

was

the

best

fitting

model
.

In

this

case

28

points

are

needed,

because

a

two
-
variable

6
th

order

polynomial

has

28

parameters
.


A

central

issue

in

choosing

the

most

suitable

model

for

a

given

problem

is

selecting

the

right

structural

complexity
.

Clearly,

a

model

that

contains

too

few

parameters

will

not

be

flexible

enough

to

approximate

important

features

in

the

data
.

If

the

model

contains

too

many

parameters,

it

will

approximate

not

only

the

data

but

also

the

noise

in

the

data
.


Overfitting

may

be

avoided

by

restricting

the

flexibility

of

the

neural

model

in

some

way
.

The

Neural

Networks

package

in

Mathematica

offers

a

few

ways

to

handle

the

overfitting

problem
.

All

solutions

rely

on

the

use

of

a

second,

independent

data

set,

the

so
-
called

validation

data,

which

has

not

been

used

to

train

the

model
.

One

way

to

handle

this

problem

is

the

stopped

search

method
.



Stopped

search

refers

to

obtaining

the

network

s

parameters

at

some

intermediate

iteration

during

the

training

process

and

not

at

the

final

iteration

as

it

is

normally

done
.

During

the

training

the

values

of

the

parameters

are

changing

to

reach

the

minimum

of

the

mean

square

error

(MSE)
.

Using

validation

data,

it

is

possible

to

identify

an

intermediate

iteration

where

the

parameter

values

yield

a

minimum

MSE
.

At

the

end

of

the

training

process

the

parameter

values

at

this

minimum

are

the

ones

used

in

the

delivered

network

model
.


In

order

to

avoid

the

overfitting

problem

by

means

of

stopped

search

method,

we

will

need

more

data
.

A

learning

set

and

a

validation

set
.

Hence,

we

have

to

divide

the

used

teaching

set

(
62

points)

into

two

sets,

the

first

will

be

the

learning

set

with

42

points

and

the

remaining

20

points

will

be

the

validation

set
.



In

fig
.

3

we

can

see

the

errors

of

the

learning

and

the

teaching

set

during

the

learning

procedure

of

the

neural

network

model

for

the

northing

velocities
.

The

errors

of

the

learning

set

(continuous

line)

decrease

during

the

whole

procedure,

but

the

errors

of

the

validation

set

(dashed

line)

are

decreasing

only

until

the

262
nd

iteration

step,

from

that

point

are

growing
.




The

maximum

number

of

iteration

was

500
,

but

the

best

parameter

set

is

the

one

calculated

at

the

262
nd

iteration

step
.

In

the

model

in

the

hidden

layer

7

neurons

(nodes)

were

used
.

The

s
election

of

number

of

neurons

is

basically

depends

on

the

number

of

known

points
.

In

fact,

by

having

more

known

data

we

can

increase

the

numbers

of

neurons
.

Let

us

check

the

statistics

of

the

residuals

for

the

whole

teaching

set

(
62

points)

in

Table

3
.

Fig.3. Errors of
learning
(continuous line)
and validation set
(dashed line) during
stopped search
method


Table 3. The statistics of the differences between neural network
model and real velocities in the 62 teaching points
.

Table 4. The statistics of the differences between neural network model
and real velocities in the 20 testing points
.

Let’s

see

the

differences

between

the

neural

network

model

and

the

real

velocities

in

the

20

testing

points

(Table

4
.
)

Using

neural

network

model

with

stopped

search

technique

we

can

obtain

a

smooth

and

good

fitting

model,

while

in

the

case

of

high

order

polynomial

model

there

are

substantial

oscillations

between

the

teaching

points
.

See

fig
.

4

and

5
.