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Volodymyr STEPASHKO
Prof., Dr. Sci., Head of Department for Information Technologies of Inductive Modeling
Kyiv, March 2013
Inductive Modelingfrom historical perspective
International Research and Training Centre
for Information Technologies and Systemsof the National Academy of Sciences of
Ukraine
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Terminology evolution:
heuristic self-organization of models (1970s) inductive method of model building (1980s) inductive learning algorithms for modeling
(1992) inductive modeling (1998)
MGUA: Method of Group Using of Arguments
(direct translation of original name of the method)
GMDH: Group Method of Data Handling(standard international name starting from the very first
translation in USA)
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T h e o r y o f I n d u c t i v e M o d e l i n g
T h e o r y o f N o i s e - I m m u n i t y M o d e l i n g
G r o u p M e t h o d o f D a t a H a n d l i n g ( G M D H ) P o l y n o m i a l N e u r a l N e t w o r k ( P N N )
D a t a - B a s e d M o d e l i n g o f C o m p l e x S y s t e m s
S t r u c t u r a l I d e n t i f i c a t i o n o f C o m p l e x P l a n t s
G M D H - B a s e d I n f o r m a t i o n T e c h n o l o g y A S T R I D
E c o l o g i c a l P r o c e s s e s
E c o n o m i c a l S y s t e m s
T e c h n o l o g i c a l P l a n t s
Structure of Knowledge in the
Inductive Modeling Area
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II nn ff oo rr mm aa tt ii oo nn TT ee cc hh nn oo ll oo gg yy
AA SS TT RR II DD
GG MM DD HH
DD aa tt aa BB aa ss ee
PP rr aa cc tt ii cc aa ll EE xx pp ee rr ii ee nn cc ee
MM oo dd ee ll
PP rr ee dd ii cc tt ii oo nn
GMDH-Based Information Technology A S T R I D
for modeling complex processes from data
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Attempt to define IM: what is it?IM generally is a process of inductive transition from data to models under uncertainty conditions: limited size of data set: small sample of noisy data limited volume of a priori information: - unknown character and level of noise - inexact composition of relevant arguments (inputs) - unknown structure of relationships in an object
IM is the MGUA/GMDH based technique for model
self-organization IM is a technology for noise-immunity modeling MODELTHEORY DATA
Two opposite (but supplemental) approaches to model building
Deduction
Induction
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IM destination: what is this for?
There is a wide experience of IM using for the following problems to be solved:
Forecasting of complex processes Structural and parametric identification Data compression (via optimal approximation) Classification and pattern recognition (supervised
learning) Data clustering (unsupervised learning) Machine learning Data Mining Knowledge Discovery
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IM explanation: main tasksGeneral problem definition
Given: data set of n observations after m input
x1,x2,…xm and one output y
variables
Find: model y=f(x1,x2,…xm ,θ) with minimum variance of prediction
error, θ is unknown vector of model
parameters
GMDH Task: f *= arg minΦ C (f )
C (f ) is a model quality criterion Φ is a set of models, f Φ
Illustration: choose an optimal subset of monomials
out of the member set of Kolmogorov-Gabor polynomial:
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Two interdependent tasks
Φ – set of model structuresС – criterion of a model qualityStructure of a model:
)(minarg ff QmRf
Q – quality criterion for parameters estimation
),( ff Xfy
Estimation of parameters:
)(minarg* fCff
Discrete optimization task
Continuous optimization task
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DD AA TT AA (( ss aa mm pp ll ee ,, aa pp rr ii oo rr yy ii nn ff oo rr mm aa tt ii oo nn ))
CC hh oo ii cc ee oo ff aa mm oo dd ee ll cc ll aa ss ss SS tt rr uu cc tt uu rr ee gg ee nn ee rr aa tt ii oo nn
PP aa rr aa mm ee tt ee rr ee ss tt ii mm aa tt ii oo nn
CC rr ii tt ee rr ii oo nn mm ii nn ii mm ii zz aa tt ii oo nn AA dd ee qq uu aa cc yy aa nn aa ll yy ss ii ss FF ii nn ii ss hh ii nn gg tt hh ee pp rr oo cc ee ss ss
Main stages of the modeling process
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Main components of a method of inductive modeling
Method of inductive
modeling
Generator of model structures
Method of parameter estimation
Criterion of model selection
Class of models
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Basic Principles of GMDH as an Inductive Method
1. Generation of variants of the gradually complicated structures of models
2. Successive selection of the best variants using the freedom of choice of decisions
3. External criteria (based on the sample division) for the best model selection
Training subset А
Generation a set of models being complicated f Ф
Model quality evaluation -calculation of criterion С (f )
C min DATA SET
Checking subset В
f *
Data-driven inductive modelling process
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External Selection Criteria
Given data set: W = (X y), X [n x m], y [n x 1]Division into two subsets A and B :
nnny
yy
X
XXyXW BA
B
AW
B
AWWW
;;;
,,,,)( 1 WBAGyXXX GTGG
TGG
Parameter estimation by LSM for a model y =X :
Regularity criterion: 2
BWAW XXCB
Unbiasedness criterion:
2
ABBB XyAR
W=AB
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GMDH algorithms
Sorting-out Iterative
1. Combinatorial
(Exhaustive Search) COMBI
2. Multistage(Directed
Search) MULTI
3. Multilayered
Iterative MIA
4. Relaxation
Iterative RIA
General classification of GMDH algorithms
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Main types of model structure generators
1. COMBI GMDH = COMBInatorial algorithm
(sorting-out type)
All possible combinations are considered:
yν = Xν θν , ν =1,2,3,…,2m
Structural binary vector: d =(d1,d2,…,dm), dj
={0;1}
ExampleLevel 1: yi = αi xi i = 1,2,…,m
Level 2: yk = αkixi +αkj xj , i,j =1,2,…,m , k=1…Сm2
Level 3: Сm3 model structures of 3 arguments etc.
Total number of models: pm = s Сms = 2m
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Main generator types
2. MULTI GMDH = MULTIstage (combinatorial- selective) algorithm (sorting-out type)
It selects only the most significant models and retrieves the exhaustive search result
In any stage s , the argument subset of a best model of previous stage is supplemented by anyone argument absent in this model by turn:
ysk = (X i
s-1 | xj )θs ,
s,j =1,2,…,m ; i=1,2,…,Fs-1 ; k = 1,2,…,(m – s)Fs-1
Fs is the freedom of choice in a stage s , usually Fs
≤ m
The whole number of generated models ~ m 3
(polynomial rise) vs. 2m (exponential rise) in COMBI
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3. MIA GMDH = Multilayered Iterative Algorithm (basic, classical, iterative type) Layer 1: yh = φh (xi ,xj ) , h = 1,2,…Cm
2
Layer 2: fk = φk (yi ,yj ) , k = 1,2,…CF2
Layer 3: gk = φk (fi , fj ) , k = 1,2,…CF2 etc.
Typical variants of the partial description φ : (a) yk
r+1=α0 +α1kyir +α2kyj
r , yi0 =xi , yj
0 =xj
(b) ykr+1=α1kyi
r +α2kyjr +α3kyi
ryjr
(c) ykr+1=α1kyi
r +α2kyjr +α3kyi
ryjr
+α4k(yir)2+α5k(yj
r)2
r = 1,2,…..; i,j =1,2…F ; k = 1,2,…CF2
F is the freedom of choice, usually F =m
Main generator types
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Classical Multilayered Iterative Algorithm
MIA GMDH
1st layer 2nd layer
)),(,( min arg*
*f
fXfyCRf
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4. RIA GMDH = Relaxational Iterative Algorithm
(non-classical, iterative type)
Any iteration (layer): fk = φk (yi ,xj ) , k = 1,2,…CF2
It considers pairs (yi ,xj) instead of (yi ,yj) in MIA
Typical variants of the partial description φ : (a) yk
r+1= α0+α1kyir+α2kxj , yi
0 =xi
(b) ykr+1= α1kyi
r +α2kxj +α3kyirxj
(c) ykr+1= α1kyi
r +α2kxj +α3kyirxj
+α4k(yir)2+α5k(xj)2
r= 1,2,…; i=1,2,…,F ; j=1,2,…,m ; k=1,2,…,mF
F is the freedom of choice, usually F =m
Main generator types
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Main concept: Self-organizing evolution of the model of
optimal complexity under uncertainty conditions
Main result: Complexity of the optimum forecasting model depends on the level of uncertainty in the data: the higher it is, the simpler (more robust) there must be the optimum model
Main conclusion: GMDH is the method for construction of
models with minimum variance of forecasting error
Basic Theoretical Results
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Reduction of optimal complexity s o when σ 2
growsHere true model contains: 3 relevant + 2 redundant arguments
Illustration to GMDH theory
Coordinates:complexity scriterion C (s)
Parameter:noise variance σ 2
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MIA GMDH as Polynomial Neural Network (PNN)
Illustration for inductive (forward) process of model construction
IM compared to ANN
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Optimal structure of the tuned GMDH net
Trained GMDH network (after backward tuning)Argument x2 appears to be redundant
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IM from CI perspectiveGMDH-based algorithms have typical features of Computational Intelligence tools: evolutionary-type computations network-like structures data-driven learning nature-inspired procedures – BICA !
Main advantages of IM algorithms: automatic evolving of model structure and parameters self-organizing net structures (node and layer numbers) fast learning (locally optimized nodes)
Equally, IM may be attributed to Soft Computing means: inductive inference procedures precedence-based reasoning fuzzy GMDH realizations
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Some real-world applications of IM Modelling of economical processes: - prediction of tax revenues and inflation - system prediction of power indicators Modelling of ecological processes: - activity of microorganisms in soil and green
algae in water under influence of heavy metals - irrigation of trees by processed wastewaters Simulation in medicine: - self-monitoring of diabetes - prediction of drugs effectiveness Integral evaluation of the state of complex
multidimensional systems - economic safety - investment activity - ecological state of water reservoirs Technology of informational support of
operative management decisions
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Illustration example 1: Comparison of prediction quality of the real inflation process (USA, 1999) using regr. analysis (LSM) and GMDH
0
0,02
0,04
0,06
0,08
0,1
0,12
0,14
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
INF
LA
TIO
N
MONTHS
Data LSM GMDH
Note: learning interval = 15 months (training+checking)
prediction points (3 months) indicated by arrows
Y – inflation rateX1 – personal savings; X4 – personal
consumption;X2 – total unemployed number; X5 – personal
incomes;X3 – interest rates; X6 – GDP
LSM model structure (static model):
Y = a1X1 + a2X2 + a3X3 + a4X4 + a5X5 + a6X6
GMDH model structure (static model):
Y = a1X1 + a3X3 + a6X6
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Illustration example 2: Ukraine budget revenues
Y – budget revenuesX1 – cash income of populationX2 – cash disbursements and savings of populationX3 – consumer price index (%)X4 – price index in light industryX5 – GDP indexX6 – total retail turnoverX7 – total employmentX8 – light industry employmentX9 – wage (nominal) X10 – wage index (real, %) X11 – money supply X12 – US$ official course X13 – account payable betveen enterprises
X14 – expenditure of the consolidated budget
Dependence of prediction efficiency for tax revenues on the statistical sample volume is investigated.
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11121 2 3 4 5 6 7 8 91011121 2 3 4 5 6 7 8 91011121 2 3 4 5 6 7 8 91011121 2 3 4 5 6 7 8 9
Data NA=14
Training Prediction
1995 1996 1997 1998 1999
Fig.1. Prediction quality (14 training points)
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11121 2 3 4 5 6 7 8 91011121 2 3 4 5 6 7 8 91011121 2 3 4 5 6 7 8 91011121 2 3 4 5 6 7 8 9
Data NA=18
Training Prediction
1995 1996 1997 1998 1999
Fig.2. Prediction quality (18 training points)
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Data NA=29
Training Prediction
1995 1996 1997 1998 1999
Fig.3. Prediction quality (29 training points)
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Main developed IM tools
Information Technology ASTRID (IRTC, Kyiv)www.mgua.irtc.org.ua
KnowledgeMiner (Frank Lemke, Berlin)www.knowledgeminer.net
FAKE GAME (Pavel Kordik et al., Prague)http://ncg.felk.cvut.cz
GMDH_Shell (Oleksiy Koshulko, Kyiv)http://www.gmdhshell.com
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Main centers of IM research
IRTC ITS NANU, Kyiv, Ukraine NTUU “KPI”, Kyiv, Ukraine KNURE, Kharkiv, Ukraine KnowledgeMiner, Berlin, Germany CTU in Prague, Czech Sichuan University, Chengdu, China
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IM development prospects
The most promising directions:
1. Theoretical investigations Study of GMDH performance for cases of different noise
properties and model classes, new external criteria, etc.
2. Integration of IM, NN and CI best developments Algorithms with heterogenic and fuzzy neurons,
immune networks, GA and nature-inspired estimators etc.
3. Paralleling Constructing computational schemes oriented to multi-core and cluster computer architectures
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The most promising directions :
4. Preprocessing Constructing the optimal combination of preprocessing procedures
5. Ensembling Modelling and forecasting based on weighted averaging
of a group of the best models
6. Intellectual interface Knowledge-based interactive modes with strong support
and control of user’s activity, default modes, etc.
7. Case studies New applications to real-world tasks of different nature
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Main Web sources on GMDH
Basic home page: info, books, articles, reviews, software
http://www.gmdh.net Professional research group site: ITIM
Departmentdevelopments, staff, info, Proceedings of ICIM/IWIMs
http://www.mgua.irtc.org.uaResearch&training group at CTU in Prague: Developments, staff, ICIM/IWIM home pages
http://cig.felk.cvut.czBusiness site: info, software, applications
http://www.knowledgeminer.netOpen-source site: info, parallel algorithms
http://opengmdh.org
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THANK YOU !
Volodymyr STEPASHKO
Address: Prof. Volodymyr Stepashko, International Centre for ITS of NANU Akademik Glushkov Prospekt 40 Kyiv, MSP, 03680, Ukraine
Phone: +38 (044) 526-30-28 Fax: +38 (044) 526-15-70 E-mail: [email protected]: : www.mgua.irtc.org.ua