Chapter 11: Recurrent Neuro Fuzzy Systemsfuzzy.cs.ovgu.de/wiki/uploads/Lehre.NN2009/nn-07.pdfChapter...

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UZZYProf. Dr. Rudolf Kruse 1

Chapter 11:Recurrent Neuro Fuzzy Systems

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Neuro-fuzzy models are usually used if

Vague knowledge can be included into the solution(i.e. we know something about our data or a possible solution)

The solution should be interpretable in form of rules(i.e. we want to learn something about our data/problem)

From an applicational point of view the solution should be easy to implement, to use and to understand.

Interpretation is more important than performance

Motivation

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Conventional (feedforward) neuro-fuzzy models less appropriate for control / analysis of higher order dynamic systems or time series data:

obtained systems usually very complex (i.e. number of inputs and rules very high) obtained systems are usually hard to interpretperformance decreasesinitialization by use of linguistic rules usually impossible

Motivation

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Recurrent Neural Networks

Backward connections are used to model time delayed feedback(‚information of the past‘)

Mathematical complexity increases: RCNN‘s are universal approximators and can be used to model systems of higher order ordinary differential equations [Funashi/Nakamura, 1993]

Different architectures have been proposed, e.g.:

Hopfield networks [Hopfield, 1982]

Partially recurrent networks (e.g. recurrent multilayer perceptron[Puskorius/Feldkamp, 1994])

Fully recurrent networks

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Recurrent Neural Networks (Examples)

∑ Δ−+Δ−==i

jijijj ttextwttoftnetfto ))()(())(()(

∑ ++−=i

jijijj extwofo

dtdo

)(

(difference equation)

(differential equation)

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Recurrent Neural Networks

Basic learning methods:

Backpropagation through time [Rumelhart et al., 1986]

Real time recurrent learning [Williams and Zipser, 1989]

o1

o2

w12 w21

w11

w22

o1(0) w11(1)

o2(0)

o1(1)

o2(1)w22(1)

w12(1)

w21(1)

w11(t)

o1(t)

o2(t)w22(t)

w12(t)

w21(t)

o2(t-1)

o1(t-1)

0 1 t-1 t time

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Fuzzy System (hierarchical)

u

yx

R1 R2 R3

large

z

R4

Input variables: x, y Output variables: z Inner variables: u R1: If x is large then u is small R2: If x is small and y is large then u is zero R3: If y is small then u is large R4: If u is large then z is small R5: If u is small and y is large then z is zero R6: If y is small then z is large

R5 R6

small

large small

large zero small

large small

large small

large zero small

Subsystem z

Subsystem u

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Fuzzy System (hierarchical and recurrent)

u

yx

R1 R2 R3

large

z

R4

Input variables: x, y Output variables: z Inner variables: u R1: If x(t) is large and z(t-1) is large then u(t) is small R2: If x(t) is small and y(t) is large then u(t) is zero R3: If y(t) is small then u(t) is large R4: If z(t-1) is large then z(t) is small R5: If u(t) is small and y(t) is large then z(t) is zero R6: If y(t) is small then z(t) is large

R5 R6

small

large small

large zero small

large small

large small

large zero small

Subsystem z

Subsystem u

z-1

small

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Fuzzy System (recurrent)

A recurrent fuzzy system of the formRr : If y (t-1) is μr then y (t) is νr

can be used to approximate an initial value problem of the form),(yfy =& cty =)( 0

)()()(1 iiiiii ySyghyyftyy =⋅+=⋅Δ+=+

),(yfty=

ΔΔ cty =)( 0

(difference quotient)

),(1 iii yftyy ⋅Δ+=+ cy =0 (iterative solution)

)()( iii yghyyS ⋅+= (S defined by fuzzy system)

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Recurrent Fuzzy Systems (First order)

A first order recurrent fuzzy system consist of rules of the form:

If x(t-1) is A and y (t-1) is B’ then y (t) is B

RCFSrulebase

x(t-1)y(t)

z-1

y(t-1)

System state diagram

If y(t-1) is negative and x(t-1) is negative then y(t) is negativeIf y(t-1) is negative and x(t-1) is zero then y(t) is negativeIf y(t-1) is negative and x(t-1) is positive then y(t) is zeroIf y(t-1) is zero and x(t-1) is negative then y(t) is negativeIf y(t-1) is zero and x(t-1) is zero then y(t) is zeroIf y(t-1) is zero and x(t-1) is positive then y(t) is positiveIf y(t-1) is positive and x(t-1) is negative then y(t) is zeroIf y(t-1) is positive and x(t-1) is zero then y(t) is positiveIf y(t-1) is positive and x(t-1) is positive then y(t) is positive

Rule base of a simple limited integrator

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Recurrent Fuzzy Systems (Higher order example)

Using rules of the formIf x(t-1) is Ai and s(t-1) is Ci then s(t) is Ci’If y(t-1) is Bi and s(t-1) is Ci then y(t) is Bi’

we obtain the following system: x(t-1)

s(t)z-1

s(t-1)

y(t)

z-1

y(t-1)

s(t-1)

So, we are able to define the following functions:)),(),(()( 11 −−= ii txtsfts ))(),(()( 11 −−= ii tstyfty

These functions may be used to compute a system of two first order differential equations.

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A Hierarchical Hybrid Model

∏

∑

∏ ∏

…

ν1(1)

… …

…

∑

…

x1 xn

y1 yk

∑

xj

yj

νl(j)

…

νk(n)

z-1

)1(1μ

)1(1nμ

…

… )(1

jμ )( jn j

μ … )(1

nμ )(nnn

μ

∑…

yn

∏…

based on recurrent hierarchical fuzzy system

heuristics and GA applied for rule base learning

optimization using gradient descent considering time delayed feed back

interpretability enforced by:

coupled fuzzy sets

constraints

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A Hierarchical Hybrid Model (optimization)

∑ ∑∑==

==T

t kk

T

t

tEtEE0

2

0

))((21)(

⎪⎩

⎪⎨⎧ −=

otherwise.0 ,given is at time node

for output theif)()()( tkotytotE

kkk

k

∑ ∂∂

−=∂

∂

k

kk p

tytEptE )()()(

∑

∑

∈

∈

∂∂

∂∂

+∂∂

=

∂∂

∂∂

+∂

∂∂∂

=∂

∂

)(:

)(:

)(

)(

)()()(

)()(

)()()()(

)()()(

rConr

r

r

k

p

k

rConr

r

r

k

p

kk

krl

krl

pta

taty

pfty

pta

taty

ppf

pfty

pty

ν

ν

Learning method minimizes error:

Gradient for parameter p:

∑

∑

∈

∈

∂∂

∂∂

+∂∂

=

∂∂

∂∂

+∂

∂∂∂

=∂

∂

)(:

)(:

)(

)(

)()()(

)()(

)()()()(

)()()(

rAnti

i

i

r

p

r

rAnti

i

i

r

p

rr

i

rj

i

rj

ptx

txta

pfta

ptx

txta

ppf

pfta

pta

μ

μ

with

with

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Finally we obtain:

∑ ∑∈ ∈ ⎥

⎥

⎦

⎤

⎢⎢

⎣

⎡

⎟⎟⎟

⎠

⎞

⎜⎜⎜

⎝

⎛

∂∂

∂∂

+∂∂

⋅∂∂

+∂∂

=∂

∂

)(: )(:)( )(

)()()(

)()(

)()(

)()()(

rConr rAnti

i

i

r

p

r

r

k

p

kk

krl

i

rjptx

txta

pfta

taty

pfty

pty

ν μ

where

⎪⎪⎪⎪⎪

⎩

⎪⎪⎪⎪⎪

⎨

⎧

∂∂∂

∂∂

−∂

−∂

∂∂

−∂∂

∂∂

∂

=∂

∂

input. externalan )(

if0

feedback, local a )(

if)1(

),1(feedback delayed timea )(

if)1(

),(feedback alhierarchic a )(

if)(

)(

ptx

ptx

ptx

typtx

pty

typtx

pty

ptx

i

ij

jij

jij

i

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Updating parameter:

Time complexity: O( (#variables)4 · (#fuzzy sets)3 · (#rules) )Memory complexity: O( (#variables)2 · (#fuzzy sets) )

Online learning: )1()()1()()1( −Δ⋅+Δ⋅−+=+ tptptptp ββ , with

∑ ∂∂

=∂

∂−=Δ

k

kkpp p

tytEptEtp )()()()( ηη

Batch learning: ∑∑=

−

=

+ Δ⋅+Δ⋅−+=T

t

nT

t

nnn tptppp1

)1(

1

)()()1( )()()1()0()0( ββ

∑ ∂∂

=∂

∂−=Δ

k

kkpp

n

ptytE

ptEtp )()()()()( ηη

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Rule Base Learning

Learning a hierarchically structured rule base of localsubsystems

Here: Domains are partitioned

Use of rule templates to define subsystems, e.g.:

Two learning methods:

Heuristics: Iterative creation of rules

Genetic Algorithm

IF (x[t-1] LIKE ‘*S1') AND (v[t-1] LIKE '*S1') THEN (x[t] LIKE '*S0') IF (y[t] LIKE '*') AND (v[t-1] LIKE '*S2') THEN (v[t] LIKE '*S0').

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Rule Base Learning

Heuristics:Two learning parts:

Rule base learning (iterative creation of rules)Re-assigning consequents

Genetic Algorithm: IF (x[t-1] IS neg2 AND v[t-1] IS neg1) THEN x IS neg0IF (x[t-1] IS neg2 AND v[t-1] IS zero1) THEN x IS neg0IF (x[t-1] IS neg2 AND v[t-1] IS pos1) THEN x IS zero0IF (x[t-1] IS zero2 AND v[t-1] IS neg1) THEN x IS neg0IF (x[t-1] IS zero2 AND v[t-1] IS zero1) THEN x IS zero0IF (x[t-1] IS zero2 AND v[t-1] IS pos1) THEN x IS pos0IF (x[t-1] IS pos2 AND v[t-1] IS neg1) THEN x IS zero0IF (x[t-1] IS pos2 AND v[t-1] IS zero1) THEN x IS pos0IF (x[t-1] IS pos2 AND v[t-1] IS pos1) THEN x IS pos0

IF (x[t-1] IS neg1 AND v[t-1] IS neg2) THEN v IS zero0IF (x[t-1] IS neg1 AND v[t-1] IS zero2) THEN v IS pos0IF (x[t-1] IS neg1 AND v[t-1] IS pos2) THEN v IS pos0IF (x[t-1] IS zero1 AND v[t-1] IS neg2) THEN v IS neg0IF (x[t-1] IS zero1 AND v[t-1] IS zero2) THEN v IS zero0IF (x[t-1] IS zero1 AND v[t-1] IS pos2) THEN v IS pos0IF (x[t-1] IS pos1 AND v[t-1] IS neg2) THEN v IS neg0IF (x[t-1] IS pos1 AND v[t-1] IS zero2) THEN v IS neg0IF (x[t-1] IS pos1 AND v[t-1] IS pos2) THEN v IS zero0

Sub system

1

Ordered Rule Base

Sub system

2

112123133233123112

Chromosome

Assume full rule baseEach Chromosome encodes complete rule base (Pittsburgh approach)

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Software Implementation

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A Simple Physical System: Mass On a Spring

With the physical laws

xmamF &&⋅=⋅= (Newton’s second law)

xclcF ⋅−=Δ⋅−= (Hooke’s law)

we can describe the system by a second order differential equation

xmcx −=&& 0)0()0(,)0( 0 === xvxx &, with

or by a system of two first order differential equations

vx =& .xmcv −=&and

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Application Example (Spring-mass model)

Initial partitioning:

(FS’s: neg0, neg1, neg2, zero0, zero1, zero2,pos0, pos1, pos2)

Initial rules (not covered by simulation):Rule 1: IF (x@t-1 IS zero2 AND v@t-1 IS zero1) THEN x IS zero0Rule 2: IF (x@t-1 IS zero1 AND v@t-1 IS zero2) THEN v IS zero0

Training data: Simulation of the modell for a period of 20 sec.

Templates used for rule base learning:IF (x[t-1] LIKE '*2') AND (v[t-1] LIKE '*1') THEN (x[t-0] LIKE '*0') IF (x[t-1] LIKE '*1') AND (v[t-1] LIKE '*2') THEN (v[t-0] LIKE '*0')

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-6.5

-4.5

-2.5

-0.5

1.5

3.5

5.5

0 10 20 30 40 50 60 70 80 90 100 110 120 130 140 150 160 170 180 190

v train v (x given) v (v given) v (x and v given)

0

200

400

600

800

1000

1200

1400

1600

1800

2000

0 500 1000 1500 2000 2500 3000 3500 4000

cycle

x and v given

-8

-6

-4

-2

0

2

4

6

8

-1.5 -1 -0.5 0 0.5 1 1.5

x

v

training data(Runge-Kutta)

initial rule base

x given

v given

x and v given

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Rule 9: IF (x[t-1] IS neg2 AND v[t-1] IS neg1) THEN x IS neg0Rule 11: IF (x[t-1] IS neg2 AND v[t-1] IS zero1) THEN x IS neg0Rule 13: IF (x[t-1] IS neg2 AND v[t-1] IS pos1) THEN x IS zero0Rule 7: IF (x[t-1] IS zero2 AND v[t-1] IS neg1) THEN x IS neg0Rule 1: IF (x[t-1] IS zero2 AND v[t-1] IS zero1) THEN x IS zero0Rule 15: IF (x[t-1] IS zero2 AND v[t-1] IS pos1) THEN x IS pos0Rule 5: IF (x[t-1] IS pos2 AND v[t-1] IS neg1) THEN x IS zero0Rule 3: IF (x[t-1] IS pos2 AND v[t-1] IS zero1) THEN x IS pos0Rule 17: IF (x[t-1] IS pos2 AND v[t-1] IS pos1) THEN x IS pos0

Rule 10: IF (x[t-1] IS neg1 AND v[t-1] IS neg2) THEN v IS zero0Rule 12: IF (x[t-1] IS neg1 AND v[t-1] IS zero2) THEN v IS pos0Rule 14: IF (x[t-1] IS neg1 AND v[t-1] IS pos2) THEN v IS pos0Rule 8: IF (x[t-1] IS zero1 AND v[t-1] IS neg2) THEN v IS neg0Rule 2: IF (x[t-1] IS zero1 AND v[t-1] IS zero2) THEN v IS zero0Rule 16: IF (x[t-1] IS zero1 AND v[t-1] IS pos2) THEN v IS pos0Rule 6: IF (x[t-1] IS pos1 AND v[t-1] IS neg2) THEN v IS neg0Rule 4: IF (x[t-1] IS pos1 AND v[t-1] IS zero2) THEN v IS neg0Rule 18: IF (x[t-1] IS pos1 AND v[t-1] IS pos2) THEN v IS zero0

)()()( 11 −− ⋅Δ+= iii tvttxtx )()()( 11 −− ⋅Δ−= iii txtmctvtv

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Recurrent Neuro-Fuzzy Systems (Cooperative approach)

Motivation/Problem: Modeling, identification and simulation of viscoelastic models in virtual reality applications

Existing approaches: Model creation usually expensive, cannot be optimized by measured data

Idea: Support developer with a generic framework for model creation and definition / optimization of its parameters:

Enable use of standard 3D models (triangular meshes)

Fuzzy system for interactive definition of model parametersand initialization of the learning process

Recurrent neural network architecture for simulation and model optimization if measured data is available

Cooperative neuro-fuzzy architecture

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RCNN describing viscoelastic models

ai vi pi

fj

sj

cj ak vk pk

+1

-1

)1()())(()( −+== tototttenfto pvcpppϖϖϖϖϖ

)1()())(()( −+== tototttenfto vacvvvϖϖϖϖϖ

))()(())(()(1

texttowwttenfto ffi

imaaa +== ∑ ϖϖϖϖ

)()())(()( tostocttenfto scccfffϖϖϖϖϖ +==

))()(())(()(21

totofttenfto ppssssϖϖϖϖϖϖ −==

))()(())(()(21

totofttenfto vvccccϖϖϖϖϖϖ −==

node node

viscosity

spring constantmass mass

xmamF &&⋅=⋅=ldlcF ⋅−Δ⋅=

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Parameter determination (Learning considerations)

Requirements:Learning by use of time series data (node positions)Changing external forces Missing data for (inner) nodesMissing data for „intermediate“ time steps

Problems:Large number of time steps between attractorsGreat time delay between weight modification and effect on positionsMissing data

Combination of BPTT and RTRL with teacher forcing

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Parameter determination (Initialization)

Initialization by „real“ physical parameters

Initialization by use of a fuzzy system:Fuzzy system describes the relations between vague knowledge of the object and the network parameters

Some fuzzy rules for the description of tissue

if hardness is soft and elasticity is bigthen spring constant is low

if shiftability is high then spring constant is very low and

viscosity is highif consistency is chapped

then fraction force is low andspring constant is very high

if consistency is slightly hardthen spring constant is slightly high and

viscosity is low

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Parameter determination (Initialization)

Screenshot of the tool 'Elastodynamic Shape Modeler'

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SUSILAP-G(SUrgical SImulator for LAParoscopy in Gynaecology)

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Conclusions

Recurrent fuzzy systems can be used to approximate dynamic systems

Proposed recurrent neuro-fuzzy system can be used to:

optimize recurrent and/or hierarchical fuzzy systems

learn rule base by use of rule templates

Constraints must be used more carefully than in feed-forward systems

Chapter 11: Recurrent Neuro Fuzzy Systemsfuzzy.cs.ovgu.de/wiki/uploads/Lehre.NN2009/nn-07.pdfChapter...

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