NOT FOR QUOTATION
WITHOUT THE PERMISSION OF THE AUTHORS
TOW- AIWANCED COMPUTER-ASSISTED MODELUUG
Y
S a w a r a g i H . h k a w a M. R y o b u Y. NakamoriApril
1 9 8 6CP-86-17
C o l l a b o r a t i v e P a p e r s r e p o r t work which has not been performed solely at t h e International Institute for Applied Systems Analysis and which h a s received only limited review. Views or opinions expressed herein d o not necessarily r e p r e s e n t those of t h e Institute, i t s National Member Organizations, or o t h e r organizations supporting t h e work.
INTERNATIONAL INSTITUTE FOR APPLIED SYSTEMS ANALYSIS 2361 Laxenburg, Austria
Preface
A mathematically e l a b o r a t e d modeling method alone cannot develop useful models of large-scale systems t h a t involve human activities. What i s needed as in- put t o t h e model-building p r o c e s s , besides measurement d a t a , i s t h e knowledge of e x p e r t s in r e l e v a n t fields. The problem is, then, what t y p e s of knowledge should o r c a n b e included in t h e modeling p r o c e s s and, more important, how d o w e manage them. The i n t e r a c t i v e method of d a t a handling (IMDH) presented in t h i s p a p e r develops l i n e a r models of complex systems through r e c u r s i v e interaction with t h e computer, systematically introducing t h e e x p e r t ' s knowledge about t h e s t r u c t u r e of t h e underlying system. It should b e emphasized t h a t t h e more one r e p e a t s dialo- gues with t h e computer, t h e more effectively knowledge c a n b e used t o develop and r e f i n e t h e model.
Contents
1. Introduction
2. Modeling Information 3. Modeling Procedures 4. Structural Analysis 5 . Interactive Modeling 6 . Conclusion
References
TOWARD ADVANCED COMPUTER-ASSISTED MODELING
X
~ a w a r a ~ i ~ ,H
~ukawa', M. ~yobu', andX
~ a k a n o r i ~1.
INTRODUCTIONOne of t h e difficulties in identifying multivariable, large-scale systems i s t h e determination of s t r u c t u r a l p a r a m e t e r s , i.e., t h e assumption of t h e forms of equa- tions. In e v e r y mathematical modeling context (e.g., Sage, 1977; Beck, 1979;
Mehra, 1980), t h e importance of t h e s t r u c t u r e formulation i s s t r e s s e d b e f o r e determination of t h e system p a r a m e t e r s . But g r e a t difficulties are encountered in extending t h e existing methodology t o ill-structured problems.
In uncertain environments t h a t involve experimentation and physical laws, two t y p e s of a p p r o a c h e s can b e used t o identify t h e optimum s t r u c t u r e . One a p p r o a c h i s t o s e l e c t a desirable s t r u c t u r e from a set of candidate s t r u c t u r e s using c e r t a i n c r i t e r i a , such as Bayesian comparison (Kashyap, 1977) o r p a t t e r n recognition (Vansteenkiste et al., 1979). The second a p p r o a c h i s t o compound a complex s t r u c - t u r e from a combination of simple s t r u c t u r e s , s t a r t i n g from a l i n e a r s t r u c t u r e (Young, 1977) o r a nonlinear basic function of coupled variables (Ivakhnenko.
1. Department of Computer Science. Kyoto Sangyo University, and The Japan Institute of S y s t e m Research, Kyoto, Japan
2. Department of Applied Mathematics and Physics, Kyoto University, Kyoto, Japan 3. The Japan Institute of Systems Research, Kyoto, Japan
4. IIASA, on leave from Department of Applied Mathematics, Konan University, Kobe, Japan
1968).
A
doubt, however, remains a b o u t t h e applicability of t h e s e a p p r o a c h e s to system-determined (Kalman, 1980) problems wherein t h e qualitative a s p e c t s tend to dominate.Data fitting of t h e r e g r e s s i o n t y p e t h a t i s often used in econometric modeling r e q u i r e s t r i a l - a n d - e r r o r methods in selecting a set of e x p l a n a t o r y variables. The stepwise or all-subset techniques implemented in a computer r e d u c e t h e burden on human e f f o r t t o some e x t e n t . But t h e i n t e r p r e t a t i o n of t h e r e s u l t s i s still a l a r g e t a s k b e c a u s e of difficulties in checking t h e validity of t h e hypothesis testing and in giving meaning to r e g r e s s i o n coefficients. Rethinking of t h e r e s u l t a n t equations i s not feasible when t h e number of equations i s l a r g e and t h e c a u s e e f f e c t relation- s h i p s between v a r i a b l e s are not known e x a c t l y in advance. Moreover, e x p e r i e n c e h a s taught u s t h a t s t a t i s t i c a l reliability does not e n s u r e applicability. To avoid un- n e c e s s a r y complication and o p e r a t i o n a l insignificance (Altman, 1980), s t r u c t u r a l considerations are c r u c i a l even in d a t a fitting of t h e r e g r e s s i o n type.
A s f a r as l i n e a r modeling i s concerned, t h e r e is t h e idea t h a t identification should depend on t h e d a t a a n d only on t h e d a t a (Kalman, 1980, 1983). But t h e ma- jority of p r a c t i c a l opinion emphasizes t h a t i t is difficult to build a model t h a t d o e s not r e f l e c t t h e outlook and bias of t h e modeler (e.g., Sage, 1977). The tendency f o r p r a c t i t i o n e r s to h a v e doubts a b o u t t h e mathematics and s t a t i s t i c s i s undeni- able. A l a r g e r a n g e of complexity i s methodologically undeveloped in t h e s e n s e t h a t n e i t h e r analytical n o r s t a t i s t i c a l methods are a d e q u a t e f o r dealing with t h e systems t h a t o c c u r in t h i s r a n g e (Klir, 1985). Thus, model building in uncertain en- vironments calls f o r c r a f t skills (Majone, 1984), where t h e word c r a f t i s used h e r e to d e s c r i b e t h e mixture of s c i e n c e a n d art t h a t i s essential f o r successful applica-
tion.
The m o s t fascinating way to r e f l e c t t h e p r a c t i c a l knowledge and e x p e r i e n c e of analysts and e x p e r t s on model building i s computer-assisted analysis, which c a n develop t h e i r ideas and e x e r c i s e t h e i r judgment and intuition. Concepts f o r ad-
vanced computer-assisted modeling of d i f f e r e n t viewpoints are flourishing (e.g., Klir, 1979; Oren, 1979; Zeigler, 1984) a n d t h e a c t u a l design a n d implementation of i n t e r a c t i v e modeling systems h a s become quite a c t i v e (Gelovani and Yurchenko, 1983; Fedorov et al., 1984), stimulated by t h e r a p i d development of computers.
The advance of computer g r a p h i c s h a s facilitated t h e f u r t h e r development of computer-assisted modeling.
The i n t e r a c t i v e method of d a t a handling (IMDH) p r e s e n t e d in t h i s p a p e r w a s b o r n in such a n atmosphere, through a challenge to duplicate e x p e r t s ' mental models in t h e form of mathematical equations. But t h e p r a c t i c a l problems possess t h e i r own c h a r a c t e r i s t i c f e a t u r e s and await d i f f e r e n t developments. W e are f a r from t h e utopia where any kind of model c a n b e immediately developed with comput- er assistance. W e begin in t h i s p a p e r with t h e l i n e a r modeling of a system in which t h e qualitative a s p e c t i s dominant, b u t f o r which extensive knowledge and cumulat- ed e x p e r i e n c e are available. T h e r e are t h r e e c a t e g o r i e s of models, depending on t h e use: d e s c r i p t i v e models, predictive or forecasting models, and planning models.
E f f o r t s to develop t h e methodology of modeling in o r d e r to i n c r e a s e decision- making capability h a v e been made by s e v e r a l a u t h o r s (e.g., Elzas, 1983; Zeigler, 1984). Although o u r ultimate goal i s in t h i s direction, t h e p r e s e n t version of IMDH i s aimed at developing predictive or forecasting models.
In building a p r e d i c t i v e or forecasting model w e must s e p a r a t e c a u s e from ef- f e c t . The g r a p h t h e o r e t i c a p p r o a c h h a s been of g r e a t benefit in introducing as- symmetric causal dependence, in which t h e information as to which v a r i a b l e s ap- p e a r in which equations i s r e p l a c e d by a d i r e c t e d g r a p h with v a r i a b l e s as nodes (e.g., Lady, 1981). Although g r a p h t h e o r e t i c techniques seem to h a v e played a full p a r t only in t h e s t r u c t u r i n g of societal systems ( H a r a r y et at., 1965; R o b e r t s , 1976; Warfield, 1976, 1982; Linstone et at.. 1979; Lendaris, 1980), wide applications are a l s o r e p o r t e d in s e v e r a l fields, e.g., model simplification (Lady, 1981; War- field, 1981), l i n e a r systems t h e o r y (Tao and Hsia, 1982; Reinschke, 1984), and economic modeling (Royer, 1980).
IMDH i s a new t y p e of l i n e a r modeling p r o c e d u r e with computer assistance. I t r e q u i r e s t h a t a l l t h e responsibility f o r judgments as to t h e s t r u c t u r e of t h e model, t h e goodness of f i t , t h e o r d e r of t h e system, and t h e p r e d i c t i v e power should b e at- t r i b u t e d t o t h e analysts a n d t h e e x p e r t s , instead of using s t a t i s t i c a l o r information t h e o r e t i c a l c r i t e r i a . IMDH h a s t w o extremely different f e a t u r e s from t h e t r a d i - tional l i n e a r modeling methods.
F i r s t , i t uses a self-organization method, instead of t h e stepwise or all-subset p r o c e d u r e s , in selecting explanatory variables, which makes t h e modeling time considerably s h o r t e r a n d t o l e r a t e s t h e s c a r c i t y of d a t a points. The self- organization method used h e r e i s a modified version of t h e group method of d a t a handling (Ivakhnenko, 1968, 1970, 1971; lvakhnenko et al., 1979), t h a t i s based on h e u r i s t i c p r i n c i p l e s of self-organization and r e l i e s on bioengineering concepts.
Second, instead of hypothesis testing o r information t h e o r e t i c c r i t e r i a (Akaike, 1976; Rissanen, 1976) w e use t h e g r a p h t h e o r e t i c a l techniques t h a t a r e , t o some e x t e n t , similar t o some a s p e c t s of i n t e r p r e t i v e s t r u c t u r a l modeling (War- field, 1974). The digraph gives insight into t h e cause e f f e c t relationships p r e s e n t in t h e l i n e a r model. I t facilitates t h e interaction between analysts and t h e comput- er and t h e n makes rethinking of t h e model equations quite easy.
IMDH effectively reflects t h e e x p e r t s ' knowledge on t h e model and a s s i s t s analysts and e x p e r t s t o modify t h e model efficiently. Through t h e modeling pro- cess IMDH enlightens analysts about t h e underlying complex system, because t h e p r o c e s s of model building itself is a learning experience. IMDH a c c e p t s r e a c t i o n s of t h e analysts flexibly, and finally finds a n e l a b o r a t e model useful f o r t h e purpose in hand.
2. YODELING INFOWdATION
The f i r s t c r a f t r e q u i r e d i s t h e selection of descriptive variables. Let us write
as t h e set of v a r i a b l e s chosen by analysts o r e x p e r t s . The s e t X c a n include non- l i n e a r r e e x p r e s s i o n s o r time-delayed v a r i a b l e s of initial variables. Following t h e traditional usage, w e use t h e term l i n e a r model t o d e s c r i b e a set of equations whose s t r u c t u r a l p a r a m e t e r s are embedded linearly. Reexpression and time- shifting enable us t o analyze nonlinear relationships and multiple a u t o r e g r e s s i v e p r o c e s s e s , respectively.
A rigid assumption i s imposed h e r e t h a t t h e corresponding d a t a i s complete in t h e s e n s e t h a t t h e y are s c r e e n e d in advance t o avoid multicollinearity o r t h e influ- e n c e of outliers. This does not imply t h a t all t h e d a t a should b e measured abso- lutely c o r r e c t l y . Soft observation is allowed t o compensate f o r lacking o r e x t r a o r - dinary data. H e r e a f t e r , w e u s e t h e t e r m observation instead of measurement.
meaning t h a t observation includes d a t a estimated o r modified by t h e e x p e r t s . Let us write t h e observation sequence f o r t h e v a r i a b l e xi as
and t h e whole observation t a b l e as
O t h e r modeling information involved i s qualitative, i.e., t h e mental images of analysts o r e x p e r t s , among which t h e pairwise c a u s e e f f e c t relationships are f e d t o t h e computer in a matrix form. Let u s write
C =
( c i j ) , i , j=
1, 2 ,. . . ,
m,
as a n incidence matrix t h a t c h a r a c t e r i z e s t h e pairwise c a u s e effect relationships. In principle, t h e elements ofC
are defined byI
0 if zi never q f f e c t s z j , o r i=
j cij=
2 if zi c e r t a i n l y q f f e c t s z j1 otherwise
A basic assumption of o u r argument i s t h a t much of t h e s t r u c t u r e of t h e underlying system i s ambiguous. Because both t h e complexity a n d ambiguity of a n o b j e c t depend on t h e i n t e r e s t s and capabilities of t h e individual. filling in t h e in- cidence matrix is a l s o a c r a f t . But in-depth considerations are not r e q u i r e d ini- tially, r a t h e r , t h e way t o introduce such relationships should b e negative. H e r e , negative means t h a t t h e modeler should e n t e r into t h e computer a p a r t of his knowledge only, putting t h e 0 s a n d 2 s in t h e r i g h t places. The remaining ambigui- t i e s are r e s o l v e d a f t e r some i t e r a t i v e modeling sessions.
S t a r t i n g with t h i s a p r i o r i information, w e find a set of l i n e a r equations:
where Xi
=
X-
{zi1,
i=
1, 2 ,. . . ,
m , with t h e hope t h a t i t could d e s c r i b e t h e underlying complex system and b e c a p a b l e of predicting t h e behavior of t h e sys- tem. We s a y t h a t z j i s a n e x p l a n a t o r y v a r i a b l e f o r zi ifcryj +
0 , and t h a t zi i s a n explained v a r i a b l e if a y+
0 f o r at l e a s t one f ( + 0 ) .The modeling sessions are divided into two main stages. The f i r s t s t a g e i s de- voted t o finding a trade-off s t r u c t u r e between t h e e x p e r t s ' mental models a n d t h e computer models. The self-organization method i s used t o obtain l i n e a r equations a n d g r a p h t h e o r e t i c techniques are used f o r interaction. The r e q u i r e d human in- p u t i s knowledge of t h e s t r u c t u r a l image of t h e system. This s t a g e includes p a r t of t h e model verification, because t h e modeler should judge whether t h e model
behaves, in g e n e r a l as h e intends.
The second s t a g e is c o n c e r n e d with judgments about t h e validity of t h e model in t e r m s of i t s e x p l a n a t o r y and predictive powers. P r e p a r e d materials are residu- al plots and predictions. To check t h e p r e d i c t i v e power, some of t h e original d a t a are left unused during t h e m o d e l building. But d a t a concerning t h e r e s u l t s of poli- c i e s not implemented are generally not available, so s c e n a r i o analyses are p r e p a r e d . H e r e , both cumulative e x p e r i e n c e and d e e p insight into t h e system are r e q u i r e d .
Even p r o p e r l y t e s t e d models c a n t u r n out to b e inapplicable if sudden jumps o c c u r in some variables. The validity of a model of t h e black-box t y p e i s usually a s s u r e d only when t h e e x p l a n a t o r y v a r i a b l e s change within t h e d a t a r a n g e used in t h e modeling, having nearly equal c o r r e l a t i o n s with e a c h o t h e r . Since any mathematical model i s fatally tentative, t h e modeling sessions in
IMDH
are endless in principle. A l l of t h e modeling knowledges:1
X , D , C , computer m o d e l s , mental i m a g e s jwill b e refined in modeling sessions tomorrow and so b e different f r o m t h o s e of to- day.
3. MODELING PROCEDURES
The f i r s t t a s k of t h e computer i s to select t h e explanatory v a r i a b l e s and esti- mate t h e c o e f f i c i e n t s in e a c h equation using t h e information f X , D , C j
.
Let us define t w o s u b s e t s of&
as follows:The elements of X: are always chosen as e x p l a n a t o r y variables a n d t h o s e for are candidates of e x p l a n a t o r y v a r i a b l e s in x i . Let u s call t h e set of core vari- a b l e s and t h a t of optional v a r i a b l e s , as is usual in s t a t i s t i c a l terminology. The modeler c a n divide t h e observation set D i n t o t w o sets Db and D, ; t h e f o r m e r i s used for model building a n d t h e latter for checking t h e predictive power. The divi- sion c a n b e done a r b i t r a r i l y as long as t h e number of d a t a points in Db i s enough to
determine t h e p a r a m e t e r s in t h e model.
First, t h e coefficients in equations of t h e form
are estimated by t h e method of l e a s t s q u a r e s f o r t h e v a r i a b l e s zi f o r which t h e c o r e sets are nonempty. Then, t h e residuals are calculated f o r t h e s e variables;
l e t us write t h e residual v a r i a b l e s as zi again, noticing t h a t t h e definite influences have a l r e a d y been accounted f o r . Finally, t h e self-organization method i s used t o s e l e c t additional explanatory variables f o r t h e v a r i a b l e s zi f o r which t h e optional sets %are nonempty. The final form of t h e equations i s written as
f o r t h e v a r i a b l e s x i , with t h e unions Xf U being nonempty.
The self-organization method implemented in t h e computer is a modified v e r - sion of t h e group method of d a t a handling proposed by Ivakhnenko (1968) and c a n b e r e g a r d e d as a specific algorithm of computer a r t i f i c i a l intelligence. The main
idea w a s inspired by t h e p r o c e s s of crossing and selecting plants t o obtain t h e b e s t possible hybrid a f t e r raising s e v e r a l generations of t h e plants. W e have adopted this idea in l i n e a r modeling and now explain t h e self-organization method used h e r e .
Suppose t h a t ! x i , x 2 ,
. . .
, x,,1
i s a s e t of candidates of explanatory v a r i a b l e s f o r t h e v a r i a b l e y.
The problem i s t o s e l e c t a n optimal subset of explanatory vari- a b l e s by which y could b e explained satisfactorily in terms of a l i n e a r equation.The p r o c e s s consists of s e v e r a l l a y e r s and in e a c h l a y e r new variables are intro- duced as hybrids of a p a i r of variables from t h e previous layer. Denote by x t and D: t h e candidate of explanatory v a r i a b l e and t h e d a t a s e t f o r model building in t h e kth l a y e r , respectively. The observation set
D:
i s divided f u r t h e r into t h e train- ing set D t l and t h e testing set0t2
; t h e former i s used f o r model development and t h e l a t t e r f o r selection of t h e p a r t i a l descriptions, i.e., b e t t e r hybrids. The algo- rithm c a n b e summarized as follows.A l g o r i t h m of t h e S e l f - O r g a n i z a t i o n Method.
S t e p 1. S e t k
=
1.If p
>
1 t h e n go t o s t e p 2.Otherwise estimate t h e coefficients of t h e equation:
by t h e method of l e a s t s q u a r e s with t h e d a t a
~t~ .
Go t o s t e p 6.
S t e p 2. Estimate t h e coefficients of l i n e a r equations in t h e form:
using t h e training d a t a s e t D t l and applying t h e method of least s q u a r e s , where i changes from 1 t o
,
C z , while s moves from 1 t o p-
1 and t from s+ I
to p. Note t h a t i and t h e p a i r ( s , t ) have one-to-one c o r r e s p o n - dence.S t e p 3 . Denoting by f f t h e estimated l i n e a r functions, let
k k k
yi = f i ( x s , x t ) i
= I ,
2 , .. .
, ,CzCalculate t h e mean s q u a r e e r r o r s between y and t h e yis, applying t h e testing d a t a set D t z
.
S t e p 4. Let
if p is even if p i s odd
S e l e c t q functions among all of t h e
5:s
s o t h a t t h e s e l e c t e d ones provide smaller mean s q u a r e e r r o r s t h a n t h e o t h e r s .If q
=
1 go t o s t e p 6.S t e p 5. Let p
=
q.
Denote again t h e s e l e c t e d functions byDefine t h e hybrid v a r i a b l e s f o r t h e n e x t l a y e r : xf
+' =
f ~ ( x , " , x ~ ) i= I,
2 ,. . .
, pand use t h e s e equations t o g e n e r a t e new d a t a sets
D : : '
andDt2" .
Let k
=
k +l. R e t u r n t o s t e p 2.S t e p 6. Find a function among those obtained in a l l t h e l a y e r s t h a t h a s t h e minimum mean s q u a r e e r r o r ; t h i s i s t h e final approximation. E x p r e s s t h i s final ap- proximation using t h e original variables by successive substitution.
Obviously, if t h e number of candidates p is less t h a n t h r e e , t h e y are chosen unconditionally. In o t h e r words, if t h e number of elements in t h e optional set Xf is l e s s t h a n t h r e e , t h e s e elements are t r e a t e d as if t h e y belong t o t h e c o r e set
.
From t h e p r a c t i c a l viewpoint, t h e smaller t h e number of explanatory vari- a b l e s is, t h e b e t t e r . In r e g r e s s i o n o r time s e r i e s analysis, t h e problem of determi- nation of t h e o r d e r of t h e equation i s stimulating and intensive r e s e a r c h . From o u r e x p e r i e n c e , t h e self-organization method described h e r e chooses a moderate number of explanatory v a r i a b l e s t h a t a r e , for some r e a s o n , difficult t o explain in t e r m s of mathematical terminologies.
4. STEZUCTURAL ANALYSIS
Even t h e e x p e r t s c a n hardly tell whether t h e obtained l i n e a r model i s ap- p r o p r i a t e o r not because t h e coefficients of a l i n e a r model d o not necessarily have p r a c t i c a l meaning. T h e r e f o r e , w e e x t r a c t t h e s t r u c t u r e of t h e l i n e a r model in t h e form of d i g r a p h s a n d show t h e s e to t h e e x p e r t s to a s s i s t t h e i r judgments.
Let X b e t h e set of v a r i a b l e s again and R b e a relation on X X X defined such t h a t ( x i , x j ) i s in R if and only if xi i s a n explanatory v a r i a b l e f o r x j in t h e l i n e a r model. W e introduce a digraph
where t h e elements of X are identified as v e r t i c e s and those of R as d i r e c t e d lines.
The v e r t i c e s are r e p r e s e n t e d by points and t h e r e i s a d i r e c t e d line, called a n arc, heading from xi t o x j if and only if ( x i , x j ) i s in R.
If t h e r e i s a path from x i t o z j , we say z j is r e a c h a b l e from xi and write
t, z j
where t h e path i s a sequence:
0 ( x i , z k Z k ' ' ' ' ' (xkt ' Z j ) ' Z j
If xi k z and z j k xi , we write
The digraph
GR
i s transitive, i.e.,if x i k z j and z j t, xk then xi k x k Hence t h e equivalence law holds with r e s p e c t t o
s,
i.e.,(i) xi g x j
(ii) x i x j -+ x j N
=
xi(iii) x i E x j , x j E z k -+ xi E x k
Let X' be t h e quotient s e t of X with r e s p e c t t o 2, i.e.,
We c a n now define t h e condensation digraph
GC
ofGR ,
identifyingX'
as t h e v e r t e x s e t . We draw a n a r c from x ' p t o x ' q if and only if p f q and, f o r some ver- tices, xi E x ' p and x j E z ' ~ , t h e r e i s a n a r c from xi t o x j inGR .
Finally, w e ob- tain a skeleton digraphGS,
which is a minimum-arc subdigraph ofGc
from which removal of any a r c would d e s t r o y t h e reachability p r e s e n t inGc .
W e show t h e s e d i g r a p h s t o t h e e x p e r t s in a session of IMDH and seek modification of t h e s t r u c t u r e of t h e model.This p r o c e s s of digraph modeling i s c a r r i e d out in t h e computer by a s e r i e s of matrix operation s t e p s . Many descriptions in t h e l i t e r a t u r e f o r obtaining skeleton digraphs a r e v e r y complicated. We show h e r e simple and efficient algorithms, in- cluding transitive closure, p a r t division, h i e r a r c h i c a l ordering, matrix condensa- tion, and skeletonizing. Let us use t h e same notation R f o r t h e corresponding ma- t r i x t o t h e relation R , defining t h a t R
=
( r i j ) , i , j=
1,2 , . . .
, m , and1 if ( x i s j ) is i n t h e r e l a t i o n R , o r i
=
j 0 otherwise.An interesting fact used in t h e matrix condensation i s t h a t if R i s a r e a c h a b i l i t y matrix, t h e n t h e following are equivalent:
(i) xi E x ,
(ii) t h e i t h row ( r e s p e c t i v e column) and t h e j t h row ( r e s p e c t i v e column) a r e identical.
The list of p r e p a r e d a r r a y s and t h e i r initial values a r e : R
=
( r i j ) , i , j=
1 , 2 ,. . .
, m : t h e given incidence matrix S=
(sij) : t h e skeleton matrix with undefined size n x nQ
=
(qij), qij=
rij, i , j=
1 ,. . .
, m : a dummy matrix v=
( v i ) , vi=
i , i=
1 , 2... .
, m : t h e index set a=
( a i ) , ai=
0, i=
1 , 2.. . .
, m : t h e p a r t indicator b=
(bi), bi=
0, i=
1 ,2
,. . .
, m : t h e level indicator c=
(ci), ci=
0, i=
1 ,2 , . . .
, m : t h e group indicator q=
(qi), qi=
0, i=
1 ,2, . . .
, m : a dummy v e c t o rThe final values of a r r a y s are: R becomes t h e t r a n s i t i v e c l o s u r e of t h e origi- nal o n e and i t s rows and columns are a r r a n g e d in t h e h i e r a r c h i c a l o r d e r . Rear- r a n g e d v a r i a b l e s are s t o r e d in t h e index set v , and a r r a y s a , b , and c s t o r e t h e p a r t s , levels, and g r o u p s to which t h e corresponding v a r i a b l e s belong, r e s p e c t i v e - ly. The algorithms to develop a d i g r a p h model are summarized as follows.
ALgorithm fir T r a n s i t i v e Closure.
S t e p 1. S e t i
=
0, s=
0.S t e p 2. Let i
=
i+
1. S e t j=
0.S t e p 3. Let j
=
j+
1. S e t t=
0, k=
0.S t e p 4. Let k
=
k+
1.If rU: x q t j
=
1, t h e n let t=
1 , k=
m . If k<
m , t h e n r e p e a t s t e p 4.otherwise if t
=
1 and rij=
0, then l e t rij=
1, s=
1.If j
<
m , t h e n r e t u r n to s t e p 3,otherwise if i
<
m , then r e t u r n t o s t e p 2, otherwise if s=
1, then r e t u r n t o s t e p 1, otherwise stop.Algorithm for Part Division.
S t e p 1. Let qij
=
maxf r i j t r j i1,
i , j=
1 , 2 , .. .
, m . S t e p 2. Take t h e transitive closure of Q=
( q i j ) . S t e p 3. S e t p a r t=
1.S t e p 4. Let i
=
i+
1.If i
>
m , then go t o s t e p 6,otherwise if ai
+
0, then r e p e a t s t e p 4 , otherwise let ai=
p a r t , and set j=
i.
S t e p 5. Let j
=
j+
1.If j
<
m and a j+
0, then r e p e a t s t e p 5, otherwise if qij=
1, then a j=
p a r t . If j<
m , then r e p e a t s t e p 5,otherwise if i
<
m , thenpart=
p a r t+
1 and r e t u r n t o s t e p 4.S t e p 6. Let p a r t
=
maxtq
{ .If p a r t
=
1, then stop, otherwise set s=
m . S t e p 7. Let s=
s-
1. S e tt =
0.S t e p 8. Let
t = t +
1.If at
>
at thenswap a t and at , swap vt and vt , swap r t j a n d r t + l , j , j
=
1 , 2 . .. . ,
m , swap r j t andrjtt j=
1 , 2 ,..., m . Ift <
s , then r e p e a t s t e p 8,otherwise if s
>
1, then r e t u r n t o s t e p 7, otherwise stop.Algorithm for Level Division.
S t e p 1. S e t Level
=
0 , part=
0,t =
0.S t e p 2. Let
part = p a r t +
1. S e t s= t +
1, c=
0 , d=
0.S t e p 3. Let
t = t +
1.If
at = p a r t ,
then l e t c=
c+
1, and ift <
m , then r e p e a t s t e p 3.If
a t # p a r t ,
t h e n l e tt = t
-1.S e t h
= t .
S t e p 4. Let
LeveL = LeveL +
1. S e t i=
s-
1.S t e p 5. Let i
=
i+
1.If
i > t
, then go t o s t e p 9,otherwise if bi # 0, then r e p e a t s t e p 5, otherwise s e t
r =
0 ,a =
0 , j=
s-
1.Step 6. Let j
=
j+
1.If j
> t
, then go t o s t e p 7,otherwise if b j # 0, t h e n r e p e a t s t e p 6,
otherwise l e t
r = r + rij
anda = a + rij
Xrji .
If j
< t
, then r e p e a t s t e p 6.S t e p 7. If
r = a ,
t h e n l e t d=
d+
1, qd=
i.If i
< t
, then r e t u r n t o s t e p 5, otherwise set L=
0.S t e p 8. L e t L
=
L+
1.If bQl
=
0 , t h e n l e t bQ1= LeveL.
If L
<
d , then r e p e a t s t e p 8,otherwise if d
<
c , then r e t u r n t o s t e p 4.S t e p 9. Let h
=
h-
1. S e t k=
s-
1.S t e p 10. L e t k
=
k+
1.If bk
>
b k + l , then swap bk and b k + l , swapak
andak
, swapvk
andvk
, swaprkj
andr k
+ l , j , j=
1 , 2 ,. . . .
m ,swap
r j k and^^,^+^,
j=
1 , 2,.. .
, m . If k<
h , t h e n r e p e a t s t e p 10,otherwise if
h >
s, then r e t u r n t o s t e p 9, otherwise ift <
m , then r e t u r n t o s t e p 2, otherwise stop.Algorithm for G r o u p Division.
Step 1. Let g r o u p
=
1, level=
0, t=
0.Step 2. Let level
=
level+
1. S e t s=
t+
1.Step 3. L e t t
=
t+
1.If t
<
m and bt=
l e v e l , then r e p e a t s t e p 3.If bt
+
l e v e l , then l e t t=
t-
1.S e t h
=
t . Step 4. S e t i=
s-
1.Step 5 . L e t i
=
i+
1.If i
>
t , then go t o s t e p 8,otherwise if ci
+
0 , then r e p e a t s t e p 5 , otherwise s e t ci=
g r o u p , j=
i.
Step 6. Let j
=
j+
1.If j
>
t then r e t u r n t o s t e p 5 ,otherwise if c j
+
0 , then r e p e a t s t e p 6, otherwise s e t q=
0, c=
0 .Step 7 . Let q
=
q+
1.If r i p
=
r j q , then c=
c+
1.If q
<
m , then r e p e a t s t e p 7,otherwise if c
=
m , then c j=
g r o u p . If j<
t , then r e t u r n t o s t e p 6,otherwise if i
<
t , then g r o u p=
group+
1 and r e t u r n t o s t e p 5 . Step 8. Let h=
h-
1. S e t k=
s-
1.Step 9. Let k
=
k+
1.If ck
>
ck thenswap ck and ck , swap bk and bk , swap ak and ak , swap v k and vk , swap r k j and r k j , j
=
1 , 2 ,. . .
, m ,swap r j k a n d r j , k + l , j
=
1 , 2 , .. .
, m . If k<
h , then r e p e a t s t e p 9,otherwise if h
>
s , then r e t u r n t o s t e p 8, otherwise ift <
m , then r e t u r n t o s t e p 2, otherwise stop.ALgorithm fir C o n d e n s a t i o n a n d S k e l e t o n i z i n g . S t e p 1 . S e t q l
=
1, i=
1.S t e p 2. L e t i
=
i+
1.If ci
=
ci t h e n l e t qi=
0, o t h e r w i s e l e t qi=
1.If i
<
m , t h e n r e p e a t s t e p 2.S t e p 3. L e t n
=
c , . S e t i=
0, k=
0 . S t e p 4. Let i=
i+
1.If i
>
m , t h e n g o to s t e p 6,o t h e r w i s e if qi
=
0, t h e n r e p e a t s t e p 4,o t h e r w i s e l e t k
=
k+
1 a n d set h=
0, j=
0.S t e p 5. L e t j
=
j+
1.If j
>
m , t h e n r e t u r n t o s t e p 4 ,o t h e r w i s e if q j
=
0, t h e n r e p e a t s t e p 5, o t h e r w i s e let h=
h+
1 .If k # h , t h e n l e t s k h
=
r i j.
If j
<
m , t h e n r e p e a t s t e p 5,o t h e r w i s e if i
<
m , t h e n r e t u r n t o s t e p 4.S t e p 6. S e t i
=
0.S t e p ? . L e t i
=
i+
1. S e t j = i . S t e p 8 . Let j=
j+
1. S e t k=
j . S t e p 9. Let k=
k+
1.If s j i x s k j
=
1, t h e n l e tski =
0.If k
<
n , t h e n r e p e a t s t e p 9,o t h e r w i s e if j
<
n-
1 , t h e n r e t u r n t o s t e p 8, o t h e r w i s e if i<
n-
2. t h e n r e t u r n t o s t e p 7 , o t h e r w i s e s t o p .The s k e l e t o n d i g r a p h c a n b e drawn as follows. F i r s t w e w r i t e elements of t h e g r o u p i n d i c a t o r c o n e b y o n e in a c i r c l e from t o p to bottom, e x c e p t f o r t h e same elements as a p p e a r e d b e f o r e . Then w e d r a w a n arc between t h e c i r c l e s if t h e c o r r e s p o n d i n g e n t r y of t h e s k e l e t o n m a t r i x i s 1. Finally, w e amend t h e f o r m a t of t h e h i e r a r c h y t o f a c i l i t a t e i n t e r p r e t a t i o n of t h e skeleton.
5. INTERACTIVE MODELING
H e r e w e summarize t h e whole p r o c e s s of IMDH. A s mentioned a l r e a d y , t h e modeling sessions consist of two main stages. The f i r s t s t a g e i s devoted t o finding a trade-off s t r u c t u r e between t h e computer models and t h e e x p e r t s ' mental models.
The dialogue continues until t h e c a u s e effect r e l a t i o n in t h e computer model be- comes s a t i s f a c t o r y . The second s t a g e i s r e l a t e d t o judgments of t h e explanatory and p r e d i c t i v e powers of t h e computer model obtained in t h e f i r s t stage. If t h e model is not s a t i s f a c t o r y , t h e n t h e modeling p r o c e s s i s r e p e a t e d from t h e begin- ning. The whole p r o c e s s i s schematized in Figure 1 and t h e dialogues are summar- ized as follows.
The F i r s t Stage D i a l o g u e .
S t e p 1. ( E z p e r t ) e d i t s t h e set of d e s c r i p t i v e system v a r i a b l e s and p r e p a r e s t h e observation t a b l e .
S t e p 2. ( E x p e r t ) introduces t h e c a u s e e f f e c t relationships between variables.
S t e p 3. ( C o m p u t e r ) finds a l i n e a r model, i.e., a set of l i n e a r equations using t h e self-organization method.
S t e p 4. ( C o m p u t e r ) displays t h e c a u s e e f f e c t relationships embedded in t h e l i n e a r model in terms of h i e r a r c h i c a l digraphs.
S t e p 5. ( E x p e r t ) amends t h e digraph by adding o r removing arcs in i t , if neces- s a r y . If t h e amendments c a u s e changes in t h e c a u s e e f f e c t relationships in t h e l i n e a r model, t h e n t h e modeling session r e t u r n s t o s t e p 2. otherwise i t p r o c e e d s t o t h e second s t a g e dialogue.
The Second S t a g e D i a l o g u e .
S t e p 6. ( C o m p u t e r ) provides residual plots and predictions, and a l s o a s s i s t s t h e s c e n a r i o analysis.
S t e p 7. ( E x p e r t ) looks f o r t h e equations t h a t have weak explanatory and predic- tive powers. If t h e r e are such equations, t h e modeling session r e t u r n s t o t h e beginning.
T h e r e are s e v e r a l points t h a t are fascinating in computer-assisted modeling and r e q u i r e sophisticated computer software f o r effective interaction. They in- clude:
(1) Data screening and transformation of variables.
(2) Introduction of t h e initial version of c a u s e e f f e c t relationships.
(3) Format of and substantial amendments t o digraphs.
(4) Reflection of amendments in t h e digraphs on t h e incidence matrix.
(5) Graphic displays of t h e residuals and predictions.
(6) I n t e r a c t i v e s c e n a r i o analysis.
W e are now developing t h e computer software f o r t h e method proposed in t h i s p a p e r . The detailed treatments of t h e s e points are described in a s e p a r a t e publi- cation (Nakamori, et at., 1985).
A s a n important application of IMDH, w e have been engaged in a regional economic-forecasting model f o r Kyoto, Japan. H e r e we p r e s e n t a brief summary of a r e s u l t obtained using IMDH. The selected variables a r e shown in Table 1. Be- sides t h e s e original variables, one- and two-year time-delayed v a r i a b l e s are t a k e n into consideration. After four-time repetitions of t h e p r o c e s s of IMDH, we and t h e e x p e r t s r e a c h e d a final agreement on t h e incidence matrix, as shown in Table 2, where t h e time-delayed variables are assumed t o have t h e same dependencies as t h e original ones. From t h i s matrix t h e forecasting model w a s obtained, as shown in Table 3 , and t h e corresponding digraph i s shown in Figure 2.
The d a t a used in t h e model is from 1960 and 1976 and t h e predictions of t h e obtained model a r e summarized in Table 4. This r e s u l t i s fairly s a t i s f a c t o r y from t h e viewpoint of t h e consumed time f o r modeling, which w a s about 27 hours, includ- ing calculations and discussions. Generally, i t i s v e r y difficult t o modify a large- scale model once obtained because of t h e c o s t and time. IMDH overcomes t h i s diffi- culty.
OBSERVATION TABLE RELATIONSHIPS
LINEAR MODELING
0
I RES I DUAL PLOTS. P R E D I CT I ONS , SCENAR 10 ANALYS I S I
1 yes
Figure 1 . Structure of the i n t e r a c t i v e method of data handling.
Table 1. Selected variables in modeling.
The population in Kyoto City Little age (age: 0-14)
Productive age (age: 15-64) Old age (age: 6 5 -
Birth
Daytime population of the primary industry Daytime population of the secondary industry Daytime population of the tertiary industry Usual population of the primary industry Usual population of the secondary industry Usual population of the tertiary industry
1 1 . The population within Kyoto zone (except Kyoto City) 11.LAOU Little age out of Kyoto City (age: 0-14)
12.PAOU Productive age out of Kyoto City (age: 15-64) 13.OAOU Old age out of Kyoto City (age: 6 5 -
14.BIOU Birth out of Kyoto City
15.DPOUl Daytime population of the primary industry out of Kyoto City
16.DPOU2 Daytime population of the secondary industry out of Kyoto City
17.DPOU3 Daytime population of the tertiary industry out of Kyoto City
18.UPOUl Usual population of the primary industry out of Kyoto City
19.UPOU2 Usual population of the secondary industry out of Kyoto City
20.UPOU3 Usual population of the tertiary industry out of Kyoto City
1 1 1 . The industries (Primary industry)
21.PI Primary industry (Secondary industry)
22. CON Construction industry 23 .TEX Textile industry
24.MAC Machine and metalworking industry 25.OTSE Other industry
26.MIN Mining industry (Tertiary industry)
27. WHO Wholesale trade 28.RET Retail trade 29. SER Service
30. PUB Public service 31 .OTER Others
32. COL 0 Commercial
33. INL I ndus try 34. HOUL Hous i ng 35. OTL Others
V. The others 36.CIN Civil income
37.GAP General accounts of Kyoto prefecture 38. SAP Special accounts of Kyoto prefecture 39. GAC General accounts
40. SAC Special accounts 41 .SIGH Sightseer
42.ROAR Road area
T a b l e 2.
1 1 . LAOU 12. PAOU 13.OAOU 14.BIOU 15.DPOUl
26.MIN 27.WHO 28. R E T 29. SER 30. PUB 3 1 . OTER
32 .COL 33.INL 34.HOUL 35. OTL 36.CIN 37.GAP 38. SAP 39.GAC 40. SAC 41 .SIGH 42. ROAR
T h e i n c i d e n c e m a t r i x just b e f o r e the final s e s s i o n .
T a b l e 3. A regional e c o n o m i c forecasting model using IMDH.
PAOU)
- I1.5125(UPOU2)-I-0.0092(DPOU2) DPOU2)-2-1.8641(DPOU2)-2
0.5823(UPOU3)-I-0.0549(DPOU3) 8(DPOU3)-1+0.1079(DPOU3)-2+0.
1.0062(DPOU1)-1+1.71O6~UPOU1) DPOU3)-2+0.2951(DPOU3)-1
3.7767(UPOU2)-I-1.9758(DPOU2) UPOU2)-2-0.0001(BIOU)-1
1.2461(UPOU3)-i-0.2813(OAOU)- 0.4257(GAP)-1+0.3160(PI)-i+O.
GAC)
- I.7281(HOUL)-2+12.0319(INL)-2- 3.8366(UPOU3)-1+0.2085(PAOU)- .6142(DPOU3)-1+0.2901(TEX)-1 2.9858(INL)-2+0.2786(1NL)-i+O CON)-1+0.9670(ROAR)-1
2.0803(CIN)-1-1.4501(CIN)-2+0
CON)
- :,L e v e 1 L e v e 1 L e v e
1L e v e 1 L e v e 1 L e v e l L e v e 1 L e v e l L e v e 1 L e v e 1
F i g u r e 2. T h e s k e l e t o n d i g r a p h c o r r e s p o n d i n g to t h e i n c i d e n c e m a t r i x .
-
23-
Table
4 .Economic forecasting by the obtained model.
ate of Growth(%) ( Pa;i7;ecords Forecasts
Year
1 9 7 61
1 9 7 7 1 9 7 8 1 9 7 9Pure production Primary industry Secondary industry
Manufacturing T E X M A C O T S E C O N
M I N
Tertiary industry W H O
R E T S E R
1 ~ o m ~ o n e n t Rat
io
( %Past records Forecasts Yea:
-9761
1977-1 pure product
ion Primary industry Secondary industry
Manufacturing T E X M A C O T S E C O N
1 Amount (lO6yen) 1 Past records Forecasts Year
1 9 7 5 1 9 7 61
1 9 7 7 1 9 7 81 0 0 . 0 0 . 4 3 4 . 6 2 8 . 6 8 . 8 8 . 6 1 1 . 2 6 . 0
M I N
Tertiary industry W H O
R E T S E R
Pure production Primary industry Secondary industry
I
Manufacturing T E X M A C O T S E C O N
M I N
Tertiary industry W H O
R E T S E R
0 . 0 6 5 . 0 2 0 . 3 6 . 9 3 7 . 8
6. CONCLUSION
IMDH starts with a belief in t h e p r e p a r e d observation and, a f t e r i t e r a t i v e modeling sessions, i t develops and r e f i n e s both t h e computer models and t h e human mental models. Computer models can b e obtained even when t h e amount of d a t a i s s c a r c e , owing t o t h e self-organization method, and easily modified with t h e assis- t a n c e of graph-theoretic techniques.
Because t h e modeling c a n b e done at low c o s t and in a s h o r t time and because t h i s method intends t o develop tentative models, a v a r i e t y of applications i s ex- pected. Actually, w e are now engaging in t h e development of regional economic forecasting models of Kyoto. Japan, as p r e s e n t e d briefly in t h e previous section.
Also, as a collaborative work with t h e IIASA Regional Water Policy P r o j e c t (Pro- ject Leader: S.A. Orlovski) and i t s successive p r o j e c t (Decision S u p p o r t Systems f o r Managing L a r g e International Rivers), w e are developing and elaborating a computer system t o obtain water r e s o u r c e s models usable in decision s u p p o r t sys- tems.
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