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Title: Dynamic model abstraction
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Title: Dynamic model abstraction
Series Title: Department of Computer and Information Science and Engineering Technical Reports
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Language: English
Creator: Lee, Kangsun
Fishwick, Paul A.
Affiliation: University of Florida
University of Florida
Publisher: Department of Computer and Information Science and Engineering, University of Florida
Place of Publication: Gainesville, Fla.
Copyright Date: 1996
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DYNAMIC MODEL ABSTRACTION


Kangsun Lee
Paul A. 1 i-- !:

Dept. of Computer and Information Science and Engineering
University of Florida
Bldg. ('Si. Room 301
Gainesville, FL 32611


ABSTRACT

While complex behavior can be generated through
simple -1. I' as in chaotic and nonlinear -- 1 ii -
complex -I. in- are found where a -, i! study
contains multiple i,1! -1 i1 objects and interactions.
Through the use of I ........ ? ., we are able to simplify
and organize the complex I -1 i Every level within
the hierarchy !1i be refined into another level. Sys-
tem abstraction involves simplification through struc-
tural -- -I i t representation as well as through be-
havioral approximations of executed model structure.
There has been little work on creating a unified tax-
I.11. !.1 for model abstraction. We present such a tax-
i! i!.. and define two i!i ..i. sub-fields of model ab-
straction, while illustrating both sub-fields through
detailed examples. The introduction of this taxon-
I'!!! provides -- -I. 11i and simulation researchers with
a way in which to view and manage complex -1 11-

1 INTRODUCTION

Real world dynamic -1 I involve a large number
of variables and interconnections. Abstraction is a
technique of suppressing details and dealing instead
with the generalized, idealized model of a -- -,. i,
The need of abstract models and traversing levels
of abstractions are essential as complex models are
used in practice. Computational ItI !. i and rep-
resentational .... -ii'~ are main reasons of using ab-
stract models in simulation (1 i- i 1: 1987; 1 i-!I- I !:
1989; Zeigler 1972) and well as in programming lan-
guages (Berzins et al. 1986; Booch 1991).
Although i! !ii diverse areas iiil,1. abstraction
methods, no agreed-upon taxonomy has been devel-
oped to categorize and structure them with under-
lying characterization of a general approach. Our
goal is to clarify how abstraction methods relate to
each other under a uniform WI .,i ~n We define sys-
tem abstraction to be one of two l- I" -behavioral or
structural. In most cases, one should explore both


of these methods when constructing -- -1. it,- For in-
stance, when a -I. i! is first being designed, one
should construct it hierarchically, with simple sys-
tem I I" at first, graduating to more complex model
SI" later. ", inii, II I abstraction corresponds to
this iterative procedure (1 i-!h- I !: and Lee 1996; 1 i-! -
wick 1996a; 1 i- !: 1996c). After creating the hier-
archy, we iii want to isolate abstraction levels, so a
level can be executed apart from the rest of hierarchy
with no detailed internal structure. This is where the
behavioral approaches are ii!.1..- I1 In depth dis-
cussions of each abstraction technique follow in the
subsequent sections.
Our contribution is the formulation of a taxonomy
capturing two I I," of abstraction, which have gen-
erally been overviewed in separate disciplines. i it -
tural abstraction is found mostly in information on
design, whereas behavioral abstraction is strewn across
i 1 1, I, r 111- of computer science and simulation. Through
a unification in I ~1i i 1! -.1- -, we demonstrate that struc-
tural and behavioral methods are complementary as-
pects of -- 1 i1 abstraction. S, i i. 1, i I abstraction is
common in programming language development within
computer science as well as in simulation. Behavioral
abstraction is common in statistical i, 1- -i- and au-
tomatic control where -1 I abstractions are used in
lieu of more complicated model-based transfer func-
tions. Along with our discuss of the I i...i..,r we
present examples of each approach to complete the
discussion.
The paper is organized as follows : we present
the new -I. i, abstraction taxonomy with specific
methods of each category in section 2. Then we illus-
trate the abstraction I- ." using two scenarios and
show how abstraction methods perform in both lin-
ear and nonlinear -I -. i abstraction, in sections 3
and 4. We close with a summary of the taxonomy
and its advantages, with future goals to be achieved.










Multimodeling


Structural


Behavioral


Model Static

Homogeneous Heterogeneous


Dynamic


I ,.i i- 1: Proposed taxonomy for abstraction.


2 ABSTRACTION TAXONOMY

Fig. 2 illustrates our abstraction taxonomy. A -1. i 1
consists of data and model components. Data refers
to values obtained either by observation or arbitrary
assignment of values to model components. Model
components, which serve as fundamental building blocks
for models, take on the data values. Sample model
components include state and event ( -1 -!i 1 : 1: I'i).
We sub-define structural abstraction of a -1. 11, into
data abstraction and model abstraction.

Data Abstraction : abstraction of input, out-
put, time, parameter -l. 11 values or time-
dependent I i li. -

Model Abstraction : abstraction of dynamical
models.

Examples of data abstraction are symbolic value,
statistic mean and variance, interval, ratio and fuzzy
numbers. Data abstraction represents a way of com-
pressing time-dependent information. In construct-
ing a model, we further refine model abstraction to
homogeneous and heterogeneous abstraction (. li!!,
and 1 -!- ii !: 1992; 1 -l;- 1I !: 1991). For homogeneous-
structural abstraction, dynamical -1. I can be ab-
stracted with only one model I ". depending on the
level of information that one expects to receive from
:,i 1-i- Specific model I I," are required at til -
ent abstraction levels. For example, one would not
choose to model low-level ,1!. -i. 1 behavior with a
Petri net since a Petri net is an appropriate model
I ." for a particular sort of condition within a -1 i i,
where there is contention for resources by discretely-
defined moving entities. Examples of homogeneous-
structural abstraction are conceptual, declarative, func-
tional, constraint and spatial modeling. Detailed dis-
cussion on each model I I'- is shown in (1I i- !:
II' I'). Models must be multi-1 . 1 so that 1!iti. i-
ent abstraction levels of the model respond to tIl. i-
ent needs of the ;,ii 1 -I
In heterogeneous-structural abstraction, tIl!I I. I
abstraction levels of a -1. -11 are provided by allow-
ing either homogeneous or heterogeneous model I -


together under one structure. To incorporate differ-
ent levels together, we have constructed a multimod-
eling i ii. -i, ,, ,1. ,._ (1 i-I!- I. !: 1991; 1 !-1!;- 1. !: and Zei-
gler 1992; 1i-!!;- I. !: 1993; 1 i- !;- 1. 1: et al. 1994), which
provides a way of structuring a heterogeneous and ho-
mogeneous set of model I- i" together so that each
I I"* performs its part, and the behavior is preserved
as levels are mapped (1 i-!ih;- !: 1988; Zeigler 1972;
Zeigler 1990). Heterogeneous-structural abstraction
is equivalent to multimodeling in the sense that we
abstract a -1I 11, structurally using homomorphic re-
lationships of one level to another, providing multiple
level abstractions. While the multimodel approach is
sound for well-structured models defined in terms of
state space functions and set-theoretic components,
selecting -- -1. iI components in each level are depen-
dent on the next-lowest level. This implies that we are
unable to run each level ',.,.. 1 ... I, ,. il,, It is possible,
to obtain output for :i!! abstraction level but, nev-
ertheless, the -I. i! model must be executed at the
lowest levels of the hierarchy. A new definition and
i i !. ,1..1..- are needed to better handle abstraction
of -I. in- and components.
Behavioral abstraction is where a -I. i is ab-
stracted by its behavior. We replace a -1. i, compo-
nent with something more generic that approximates,
to some degree of accuracy, the behavior of the -1. i-
component at its refined levels. Therefore, discarding
the refined levels that define a -1 1i1 component will
still result in a complete behavioral description of a
-* -. i (I1 -! I. !: and Lee 1996). By incorporating
behavioral abstraction approaches into multimodel-
ing allows each level to be understood independently
of the others. This is why we put multimodeling on
the top of our I ,. ,,, ,,., .
We have two approaches of -1' f- i_ - 11i be-
havior:
,i 1I1, approach : one takes a -1. i and cap-
tures only the steady state output value instead
of a complete output trajectory. The input value
is defined to be the integral of time value over
the simulation trajectory.

Dynamic approach: one needs to associate time-
dependent input and output f i ..11. -
System identification (L t!! and Soderstrom 1983;
Johansson 1993) is to abstract a -- -1. 11 by mathe-
matical models. Modeling the -I. i consists of se-
lecting a general, parameterized mathematical rep-
resentation and then tuning the parameters, so that
behavior predicted by the model coincides with mea-
surements from the real -I -. 11i Parameter estima-
tion procedure provides a search through parameter
space, effectively, to achieve a close-to optimal map-
ping between the actual values of the -1. i-i and the










Table 1: Sample abstraction categories and associ-
ated techniques.

Base Abstraction Type Abstraction Technique
Data Abstraction Symbolic Value
Mean, Variance
Interval, Ratio
rF,' Number
Si Ii I L I Abstraction Conceptual Modeling
Declaration Modeling
Functional Modeling
Constraint Modeling
Spatial Modeling
Multimodeling
Behavioral Abstraction Regression
System Identification
Neural Network
Wavelet
Genetic Algorithm



approximate abstract -- -I. i Commonly used pa-
rameter models are ARX, ARMAX, OE(Output Er-
ror) and BJ(Box-Jenkins) (Tan et al. 1995; Johans-
son 1993). Brief explanations of these models are
shown in sections 3.2.2.
Neural networks have been established as a gen-
eral approximation tool for fitting models from in-
put/output data (Cynbenko 1989; Tang et al. 1991;
Tang and1 i-!h- 1 !: 1993). From the -- -I. i! identifi-
cation perspective, a neural network is just another
model structure (I.iL!i_ and Soderstrom 1983; Bar-
ron 1989). The inputs are linearly combined at the
nodes of the hidden 1 -. (s) and then -,i.i., i. I to
a threshold-like non-lii,. ,il and then the proce-
dure is repeated until the output nodes are reached.
Backpropagation, recurrent and temporal neural net-
works have been shown to be applicable to model-
ing an identification (i i-lh- !I: and Lee 1996; Mills
et al. I''i ). On the other hand, recently intro-
duced wavelet decomposition achieves the same qual-
ity of approximation with a network of reduced size
by replacing the neurons by i- l." ,i-", i.e. com-
puting units obtained by cascading an ;!!l-. trans-
form and multidimensional wavelets (Ahang and Ben-
veniste 1992).
Table 1 summarizes the based categories along
with some sample abstraction techniques discussed so
far. Having defined the model abstraction taxonomy,
we now proceed to illustrate the different abstraction
techniques using following two examples.


B 1 F

Water

~- ~ CopperPot
Heatig Element
Knob






I ii.. 2: Boiling water -I it


3 EXAMPLE I: Boiling Water Model

Consider a pot of water in Fig. 2. Here we show a
picture of the boiling pot along with an input and
output trajectory. The input reflects the state of the
knob, which serves to specify external events for the
-* -. i, The output defines the temperature of the
water over time. Newton's law of cooling states that
Rqh = AT = TI T2 where Ti is the temperature of
the source (heating element), and T2 is the tempera-
ture of the water. qh is heat flow. Since T2 is our state
variable we let T = T2 for convenience. By combining
Newton's law with the capacitance law, and using the
law of capacitors in series, we arrive at:


k C1 = + C2
k =
RCIC2
k(T1 T)


3.1 Structural Abstraction

The structural approach to -- -I. i abstraction for
the boiling water is defined in a recent text (I i-!i-
wick I''I,) where the boiling water is included as a
-il,-.- -1. i within a -, ii of two flasks and a hu-
man operator who mixes the flasks once the liquids
are boiling. In the structural abstraction approach to
-1, 11 we first need to define our levels of abstrac-
tion and then choose which models I, to use at
each level.
We show part of the multimodel in I i,- 3 and
4. The first model is a compressed version of all the
hierarchy. Fig. 4 shows Newton's law of cooling in a
functional block form.


















1 i,iL. 3: Six state automaton controller for the boil-
ing water multimodel.


I i,''i.- 5: 1.! 'I'-' time vs. Temperature.


I ,,.. i 4: Decomposition of Heating state.


3.2 Behavioral Abstraction
3.2.1 Static Approach
In the static approach, we're interested only in the fi-
nal (i.e., steady state) temperature of the water. Our
two inputs are: (1) total amount of elapsed time for
the input trajectory and (2) integral value of the in-
put trajectory integrated over time. The output is
the temperature of the water at the elapsed time. A
graph of elapsed time versus temperature is shown
in Fig. 5. This information is obtained directly from
the i1i. 1.1 il_ simulation of the boiling water sys-
tem. We chose a subset of all possible input time
,, i i. ., r in such a way that some nonlinearity was
introduced into the graph in Fig. 5. This was done to
challenge the behavioral parameter estimation meth-
ods in creating a good fit. This explains why Fig. 5
contains a small area of discontinuity in the region
between steady state temperature values of 20 and
40.

Linear Regression

In general, a polynomial fit to data in vectors x and
y is a function p of the form:

p(x) = cizX" C + ... + cd (3)

The degree is n and the number of ...i. !. !!, is
d = n + 1. The regression .... ti ,. cl,c2,...,c,
are determined by solving a -- -1. 1, of simultaneous


linear equations: Ac = y (Law and Kelton 1991).
Fig. 6 shows the result. The approximation is poor
in the graph's central region because linear regres-
sion is done by polynomial fit, and so it generates a
.... I . ... ,,1, increasing function.

3.2.2 Dynamic Approach
In the dynamic approach, we're interested in time
dependent behavior. In this case, we are concerned
not only in the steady state temperature but also the
way in which the temperature changes over time. For
this approach, we chose a -I. I with just one input
and one output, both time- ,- i,,_ i .I .. i. The
input is the input "!:!i. .I off/knob i o! trajectory and
the output is the temperature trajectory.

Linear System Identification

The Box-Jenkins method is a frequently used -- -1 11
identification method in time series ; 1 i 1 -i- (Tang,
de Almeida, and i-.!- !. !: 1991; Tang and i-1!;- I. !:
1993; The MathWorks 1991). Its structure is given
by


B(q)
y(t) = F(q)u(t


nk)) + e(t)
D (q)


with


y(t)
B(q)
F(q)
C(q)
D(q)


bl + b2q-1 +... + babq-b
+flq-i +... + f,fq -
1 + lq-1 + ... + Cncaq-
S-dlq- 1+... + ddqa-"d


The numbers nb, nc, nd and nf are the orders of
the respective polynomials and q is the shift opera-
tor. The number nk is the number of .1 1 - from
input to output. Fig. 7 shows the approximation re-
sult. Successful identification of y(t) depends on how
well we guess the values of nb, nc, nd, nf and nk.






















1 !i. 6: Linear regression.


I i,. i. 7: Box-Jenkins method.


Heuristics and "- -.I." 1, rules," if available, aid us in
choosing parameters. For example, too large a value
for a parameter results in computational i [!! I!IL, -
to generate y(t), while too small a value results in a
rough estimation. We often had to tune parameters
by hand in order to get a good approximation.

4 EXAMPLE II: Hematopoiesis Model

Though the abstraction methods discussed so far were
good at linear -- -I. i abstraction, non-linear -I. ii,
abstraction involves more complex behavior mapping.
In this section, we show how these abstraction meth-


I ,i- -,- 8: Abstraction error in Box-Jenkins method.


I i,..i- 9: Cell concentration vs. time for 1. 1 T =
20 .1 -


ods perform under non-linear conditions.
Our model deals with the regulation of hematopoiesis,
the formation of blood cell elements in the body.
Hematopoiesis is the process of blood creation in the
body. White and red blood cells are produced in bone
marrow. From the marrow they enter the blood cir-
culatory -- -. i i As the i.-.- _- I level decreases in the
body, there is a feedback back to the bone marrow-
which produces more cells.
Mackey and (;! - (. I I: and Glass 1977) pro-
vide a 1. 1 ,- model for hematopoieses of the following
form:


dP(t) AO^P(t T)
dt 0 '' +P'(t T)


gP(t)


where, A : the flux of cells into the blood stream,
P(t) : the concentration of cells(the population species)
in the circulating blood (cells/l .,.., ), g : day-1, cell
loss rate per .1 and T : maturation l. 1 ,
We use A as an input. Depending on the matura-
tion *1. 1 T, we can generate different solutions. In
a lower maturation l. 1 the -- -. i i shows periodic
behaviors, but, as the 1. 1 moves upward, nonpe-
riod f i ., .1i. appear. Fig. 9 shows a nonperiodic
trajectory when the l. 1 ,- is 20.
Since we are interested in abstracting the time
dependent behavior of cell concentration in the cir-
culating blood, we will restrict out experiments to
dynamic-behavioral abstractions. Also, to see how
abstraction techniques perform under heavy nonperi-
odic and nonlinearity, we choose maturation l.1 20.
Now, the dynamic-behavioral abstraction of hematopoiesis
model is to approximate equation (5) with a discrete
model of the form


P(t) = f(P(t 1),..., P(t na))


where f is a nonlinear function to be estimated with
order na.


f























1 i,Li 10: Hematopoiesis model for .1 1 ,- = 20 1 -
with increased sampling period: abstraction target.


Small sampling period for the discretization makes
the order of the discrete model very high due to the
long dependence of P(t) on P(t 20), which results
in numerical [lt! i il l to compute the optimal func-
tion of f. Therefore, increasing sampling period is
needed as long as the discretization is not too rough.
Fig. 10 .1i 1,1 the time trajectory for the total con-
centration of blood cells when the sampling period is
increased by 100, which introduces more nonlinearity
and ii,-1 1,.ii- We choose Fig. 10 as the abstraction
target and use A for input.


ADALINE neural network


A I .\i. \ 1. was developed by \\ ih .l..- and Hoff (\\ i.h ..'
and 1i i I 1', -.). Their neural network model differs
from the perceptronq in that Al 1. 1.I\ neurons have
a linear transfer function. The ADA LI.M. network
also enables the \\ i. h, ..; -Hoff learning rule, known as
the Least Mean Square (I..l IS) rule, to ;,li -I weights
and biases according to the magnitude of errors.
The Al i. .1.\ neural network for the hematopoiesis
model performs abstraction as shown in Fig. 11. An
A I. \ .1\ 1. neural network takes initial weights and
biases, an input signal and a target signal, and then
ti 1 i the signal adaptively based on input .1. 1 and
learning rate parameters. In most cases, input de-
lay can be guessed by the modeled -1, i, itself. For
hematopoiesis model, we know an output at time t is
determined by 20 most recent inputs, which could be
inferred by the the [. I -lt!!1. i. i-ii I! equation (5). A
proper learning rate is determined by repetitive trials
until a good fit is achieved with fewer learning-stage
perturbations.

CONCLUSIONS

We have presented a new I i,...!..i, for model ab-
stractions in dynamic -- -1I iT,- The taxonomy of ab-


I i,,.i- 11: ADA 1. \1. network for hematopoiesis
model


I 1,,iL,- 12: Abstraction Error in ADA 1. \1. network


traction I- with multimodeling at the top level,
is constructed by model engineering perspective: when
a -I 11, is first being developed, one should use struc-
tural abstraction to organize the whole -- -, ii hier-
archically with simple -, 11, l I" and then grad-
uate to more complex model i" Below the struc-
tural abstraction, each component is black-box with
no detailed internal structure. Behavioral abstrac-
tion is used to represent those black-boxes by approx-
imating the behavior of the -I. il1 components. By
combining structural and behavioral abstraction to-
gether, each level of abstraction is independent from
the lower abstraction levels, so a level can be exe-
cuted apart from the rest of the hierarchy. These two
concepts: structural and behavioral abstraction are
blended together to form a comprehensive I i,., ,i, ,,r .
In addition to the taxonomy, we discussed several ab-
straction methods according to the categories they
belong to and showed how they perform in linear and
nonlinear -- -I. il abstractions. We felt it important
to provide both linear and nonlinear models since one
technique 1i fare well for one I *" of -1, 11, and
then poorly on the other.
Given that we have developed this a ,i...!..!- a
good question is "What to do with it?" We are de-
veloping a -- -I. i! called MOOSE (1 i-I- I: l''I; it,.I









standing for multimodeling object oriented simula-
tion environment, in which the I ., >,, ,,r is to be ap-
plied. MOOSE models are constructed using a graph-
ical user interface which begins with the user speci-
fying an object oriented class hierarchy. This proce-
dure takes advantage of structural abstraction. For
exploiting behavioral model abstraction, our current
plans are to provide two or three basic techniques and
allow the user to choose which they would like. More-
over, we are developing a semi-automated approach
to developing behavioral abstractions of multimodel
components which can benefit most from the compu-
tational gain afforded by not having to simulate at
the lowest level.

ACKNOWLEDGMENTS

We would like to acknowledge the following funding
sources which have contributed towards our study
of modeling and implementation of a multimodeling
simulation environment: (1) Rome Laboratory, Griff-
iss Air Force Base, New York under contract I ;1 ii.1 2'-
95-C-0267 and grant 1 ;111i'- -95-1-0031; (2) Depart-
ment of the Interior under grant 14-45-0009-1544-154
and the (3) National Science Foundation Engineering
Research Center (i. i:l) in Particle Science and Tech-
!i..1 .- at the University of Florida (with Industrial
Partners of the 1 1) under grant EEC-94-02989.

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AUTHOR BIOGRAPHIES


Paul A. Fishwick is an Associate Professor in the
Department of Computer and Information Science
and Engineering at the University of Florida. He re-
ceived the PhD in Computer and Information Sci-
ence from the University of P. !!-- I >;1 i in 1986.
He also has six years of industrial/government pro-
duction and research experience working at Newport
News I! il i i,, gand Dry Dock Co. (doing CAD/CAM
parts definition research) and at NASA I. ii_1. Re-
search Center (-1 11 i engineering data base models
for structural. H i;,,,. .i ;_i. His research interests are
in computer simulation modeling and ;, i! 1 -;- meth-
ods for complex -I. i,,- He is a senior member of
the IEEE and the Society for Computer Simulation.
He is also a member of the IEEE Society for Systems,
Man and Cybernetics, ACM and AAAI. Dr. I i I- I! I
founded the comp.simulation Internet news group
(Simulation Digest) in 1987. He has chaired work-
shops and conferences in the area of computer sim-


ulation, and will serve as General ('!h ,!! of the 2000
\\ iiii. Simulation Conference. He was chairman of
the IEEE Computer Society technical committee on
simulation (T( 'Si. !) for two years (1988-1990) and he
is on the editorial boards of several journals including
the AC.1 I........ ..... on Modeling and Computer
Simulation, IEEE 1,.. ..... .'.. on ,.* ....'. Man and
C., 1 l ..... ..I I ... of the Society for Com-
puter Simulation, International Journal of Computer
Simulation, and the Journal of S.,~i'. ... Engineering.
Dr. i I i, !: WWW home page is http://www. cis.
ufl. edu /f ishw ick and his E-mail address is fishw
ick@ cis.ufl.edu.


Kangsun Lee received the B.S and M.S degree in
Computer Science from Ewha Womans Uiii. -il
Korea in 1992 and 1994, respectively. !I.- is cur-
rently a Ph.D student in the Computer and Infor-
mation Sciences and Engineering department at the
University of Florida, Gainesville. Her research inter-
ests are in Modeling iii. I!. I. l. I Abstraction tech-
niques and simulation. Kangsun Lee's WWW home
page is http://www.cis.ufl.edu/~kslee and her
E-mail address is kslee@cis.ufl.edu




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