Engineering, Technology & Applied Science Research Vol. 8, No. 4, 2018, 3130-3134 3130  
  

www.etasr.com Ahmad et al.: Developing A New Dimension of an Applied Exponential Model: Application… 
 

Developing A New Dimension of an Applied 
Exponential Model: Application in Biological 

Sciences 
 

W. M. A. W. Ahmad 
School of Dental Sciences 

Universiti Sains Malaysia, Malaysia 

R. A. A. Rohim 
Universiti Sains Malaysia 

Malaysia 

Y. Norhayati 
Universiti Sains Malaysia 

Malaysia 

N. A. Aleng  
School of Informatics and Applied Mathematics 

Universiti Malaysia Terengganu 
Malaysia 

Z. Ali 
School of Mathematical Sciences 

Universiti Sains Malaysia 
Malaysia 

 

 

Abstract—Modeling of exponential growth or decay is a nonlinear 
regression technique. In the real world, the exponential growth is 
often used to model population growth while the exponential 
decay is often used to a model declining population or a 
decreasing size. In this study, we try to improve the performance 
of exponential growth by adding bootstrap and fuzzy techniques. 
This gives us the option to perform analysis even when there is 
not enough data. The aim of the current work is to develop a new 
dimension of an applied exponential analysis with improved 
results. The suggested method was tested and applied to 
biological data. The gathered data was compared by measuring 
the average width of the predicted interval using least squares 
method and fuzzy method. The result shows that the average 
width of the predicted interval using least squares method was 
0.522 while using fuzzy method was 0.082. This indicated the 
superiority of the fuzzy regression methodology. Besides that, this 
paper provides the algorithm for the prediction of cell growth 
and inferences. 

Keywords-bootstrap; exponential growth; fuzzy regression; 
exponent decay; nonlinear regression  

I. INTRODUCTION  

The exponential function is one of the most important and 
widely occurring functions in physics and biology [1]. The 
exponential distribution has many applications in biology. 
Examples of the exponential application include the bacteria 
growth time and the decay time of bacterial pathogens with 
constant failure rates. Exponent distribution of an event is the 
probability of the event occurring in the next small time 
interval which does not vary through time, and time between 
events has a Poisson distribution. The exponential growth or 
decay follows a function N which changes with time in such a 
way that the change ΔN in N during a short time interval Δt is 
proportional to N and to Δt, as ΔN=kΔt, rearranging this 
equation, we obtain: 

N
k N

t





     (1) 

Therefore, the constant of proportionality k can be seen by 
rearranging equation (1). Then, we obtain k=(ΔN/N)/Δt. The 
constant k is then the fractional change (ΔN/N) in N per unit 
time Δt. The dimension of k is (time) 1 [2]. 

II. STATISTICAL THEORY AND METHODOLOGY 

The linear regression model is a powerful method for 
modeling and forecasting, especially in conventional regression 
analysis. For modeling purposes, data should be crisp and 
should follow a normal assumption, this will lead to better 
significant results [3]. In the current study, an exponential 
model was applied after transforming the data to a linear model 
with mathematical programming by assuming that the 
dependent variables were crisp while the independent variable 
was a symmetric fuzzy number. This paper provides an 
algorithm for the exponential growth model using cell cultured 
dataset. The basic analysis is to transform the nonlinear 
equation into a linear form. After transforming into linear 
regression, we will obtain an equation in the form of 
Y=β0+β1x1+ε. We transformed the nonlinear equation in order 
to get better results and better significant inferences. The 
random error term is added to make the model probabilistic 
rather than deterministic. The value of the coefficient βι 
determines the contribution of the independent variables and β0 
is the y-intercept [3, 4]. To be more accurate, a fuzzy regression 
can be rearranged as  

110 xZZY      (2) 

where the explanatory variables xi 
are assumed to be precise. 

However, according to (2), response variable Y is not crisp but 
is instead fuzzy in nature. For the fuzzy approach, Zi 

are 



Engineering, Technology & Applied Science Research Vol. 8, No. 4, 2018, 3130-3134 3131  
  

www.etasr.com Ahmad et al.: Developing A New Dimension of an Applied Exponential Model: Application… 
 

assumed symmetric fuzzy numbers which can be presented by 
the interval.

 
Zi 

can be expressed as a fuzzy set given by 

 1w1c1 a,aZ  where ica  is the center and iwa is radius or 
associated vagueness. This reflects the confidence in the 

regression coefficients around ica  in terms of symmetric 
triangular membership function. So, the relationship is also 
considered to be fuzzy. This  w1c1i a,aZ  can be written 
as  R1L11 a,aZ  with w1c1L1 aaa  and w1c1R1 aaa  . 
In fuzzy regression methodology, parameters are estimated by 
minimizing total vagueness in the model. Using 

 1w1c1 a,aZ , we can write the fuzzy regression as  

 0w0c a,ay xa,a 1w1c     (3) 

 

Data Collection 

Main Algorithm : 

 Algorithm of an  
Exponential with 
Combining Fuzzy 

Method  

End Process

\*Using secondary data*\

Interpret The Output 

Performing Exponential 
Growth or Decay  to 

Estimate the Parameter

Adding
 Bootstrapping to the 

Main Algorithm 

\*Methodology building for 
bootstrap and this methodology 
will be add to the main algorithm*\

\* Main methodology *\

\*Performing an analysis*\

\*Obtained an output*\

\*Finish*\

 
Fig. 1.  Flowchart of the integrated exponential calculation. 

Thus this can be written as 1j1c0cjc xaay  then it can be 

written straightly as jw 0w 1w 1jy a a x  . As jwy  
represents 

radius and so cannot be negative, therefore on the right-hand 
side of the equation

1j1w0wjw xaay  , absolute values of 

ijx are taken. Suppose there are m data points, each comprising 

of   row1na  vector. The parameters Zi are estimated by 
minimizing the quantity, which is the total vagueness of the 
model-data set combination, subject to the constraint that each 
data point must fall within the estimated value of response 
variable. This can be visualized as the following linear 

programming problem, minimized  
m

0w 1w 1j
j 1

a a x


 and subject 

to:     0c 1c 1j 0w 1w 1j ja a x a a x Y     

    0c 1c 1j 0w 1w 1j ja a x a a x Y     

and 0aiw  . A simple procedure is commonly used to solve 
the linear programming problem [5]. The present data is a 
sample of the results obtained from [6] which characterize the 
proliferative capacity of mesenchymal stem cells. The data are 
composed of two variables which are the days of culture (X) 
and population doubling level (Y) – see Table I.  

TABLE I. X AND Y DESCRIPTION 

Variable Description 
X A total number of time (days) between each cell passage. 

Y 
Population doubling level is the proportion of cells count at 
80% confluency over the original number of cells seeded. 

 

A. Exponential Growth Transforming to Linear Form 
Exponential growth formula and exponential decay formula 

are given by bXY Ae . The procedure to transform the growth 
and decay formula into a linear form follows. The equation for 
the linear form for exponential growth is given by (4): 

 

ln ln( ) ln( ) ln( ) ln(A)
bX bX

Y Ae A e bx      (4) 

B. Calculations of an Exponential Cell Growth using SAS 
Algorithm 

 First step: Data for exponential cell growth should enter in 
SAS algorithm as follows. 

Data Cell_Growth;  
input x y lny;  
datalines;  
0.00 38.00 3.64  
5.00 39.31 3.67  
8.00 39.74 3.68 
10.00 40.98 3.71 
13.00 43.10 3.76 
17.00 45.78 3.82 
20.00 59.15 3.89 
22.00 49.90 3.91 
24.00 53.98 3.99 
28.00 57.46 4.05 
31.00 61.03 4.11 
34.00 63.80 4.16 
37.00 65.52 4.18 
40.00 68.54 4.23 
44.00 72.62 4.29 
47.00 75.42 4.32 
50.00 79.38 4.37 
53.00 83.31 4.42 
;  
run;  

 Second step: Adding bootstrapping to the calculation. The 
following algorithm calculates the data using a bootstrap 
method and prints out the data.  

%MACRO bootstrap(data=_last_, booted=booted, boots=2,  
seed=1234);  
DATA &booted;  
pickobs = INT(RANUNI(&seed)*n)+1;  
SET &data POINT = pickobs NOBS = n;  
REPLICATE=int(i/n)+1;  



  

 

 

C. 

sum

T

P

par
exp
at 
obt

mo

Engineerin

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i+1;  
IF i > n*&bo
RUN;  
%MEND bo
ods rtf file='a
%bootstrap(d
run;  
proc print da
run;  

Third step: 
equation. T
parameters a

Title “Expon
ods graphics 
proc nlin dat
parameters A
model y = A
ods output E
run;  

Fourth step
according to

Proc reg data
model lny=x
run;  
ods rtf  
close;  

Parameter E

The SAS ou
mmary of the 

TABLE II. 

Parameter Es

A 35
b 0

The obtained

Y e97.35

Exponential 
rameter estim
ponential grow
the certain po
tain 

ln (35.97y ln

ln 3.5827y  

So the param

3.5827

 

lny

Std Error




The upper li

odel are compu

ng, Technology

r.com 

oots THEN ST

otstrap;  
abc.rtf' style=j
data= Cell_Gr

ata=booted;  

estimating th
The algorithm
according to th

nential Equatio
/ imagename 
a=booted plot

A=1 b=-1;  
*exp(b*x);  
stSummary=su

p: Estimating 
o the transform

a=booted;  
x;  

Estimation for 

utput for the pa
output follows

PARAMETER E

timate 
Appr

Std Er
5.9723 0.256
.0160 0.0002

d equation is g

X016.0
  

growth formu
mation, we ob
wth equation w
oint ˆ 35.97Y 

0.016735 ) (Xe ln

78 0.0160 x
 

meter estimatio

 
78 0.0160

0.2566 0.0

x


imits of predic
uted using (7)

y & Applied Sci

TOP;  

ournal;  
rowth, boots=2

he parameters
m below is 
he data. 

on”;  
= “Exponentia
s=fit;  

ummExp;  

the linear f
m bootstrap dat

Exponential C

arameter estim
s. 

STIMATE OF AN E

ox 
rror 

Approxim

66 35.4509 
219 0.0155 

 

iven by (5): 

ula is given by
btain (6). Usin
we can estimat

0.0167 Xe . Takin

(35.9735) (ln e

on and standard

000219
 

ction interval 
: 

ience Research

Ahmad et al.: D

2);  

s of an expon
used to est

al Equation”; 

form of regre
ta. 

Cell Growth 

mation follows

XPONENTIAL EQU

mate 95% Confid
Limits 

36.4937
0.0164

 (5)

y 
bXY Ae .

ng the equati
te the growth o
ng a logarithm

0.016 )X   

d error are giv

 (6)

for the expon

h V

Developing A N
 

nential 
timate 

ession 

s. The 

UATION 

ence 

After 
ion of 
of cell 
m, we 

ven as 

nential 

ln

ln

y

y

and
mo

 

 

ln y

ln y

D.

(FL
be
sum
esti

Vol. 8, No. 4, 20

New Dimension


 
3.58278

3.83938

y

y

 



d the lower lim
del are compu


 
3.58278

3.32618

y

y

 



F

 Calculation o
Growth 
The algorithm

LS) for expone
visualized b

mmarizes the
imates. 

ods rtf file='ab

018, 3130-3134

n of an Applied 

 
  

0.2566 0

0.016219

 



mits of predic
uted using (8): 

 
 

0.2566 0

0.015781  

 



Fig. 2.  The pl

Fig. 3.  Fit d

of Fuzzy Leas

m below calc
ential growth.
by the follow
e optimizatio

abc.rtf' style=jo

4 

d Exponential M


0.0160 0.000

x



ction interval f

.0160 0.0002

x



 

lot of an exponent

 

diagnostic for lny

t Squares(FLS

culates the fu
 The full set 
wing program

on results fo

ournal;  

3132  

Model: Applicat

0219 x
 (7)

for the expone

219  x
 (8)

tial 

y. 

S) For Expone

uzzy least sq
of calculation
mming. Table
or the param

tion… 

ential 

 

ential 

quares 
ns can 
e III 
meter 



  

Engineerin

www.etasr

proc nlp;  
min Y;  
decvar a0c a0
bounds a0w>
lincon a0c+1
lincon a0c+5
lincon a0c+2
lincon a0c+5
lincon a0c+1
lincon a0c+5
lincon a0c+0
lincon a0c+5
lincon a0c+2
lincon a0c+8
lincon a0c+0
lincon a0c+2
lincon a0c+5
lincon a0c+5
lincon a0c+5
lincon a0c+4
lincon a0c+2
lincon a0c+2
lincon a0c+1
lincon a0c+2
lincon a0c+1
lincon a0c+2
lincon a0c+1
lincon a0c+2
lincon a0c+8
lincon a0c+3
lincon a0c+0
lincon a0c+2
lincon a0c+5
lincon a0c+1
lincon a0c+4
lincon a0c+8
lincon a0c+8
lincon a0c+2
lincon a0c+3
lincon a0c+4
lincon a0c+1
lincon a0c+5
lincon a0c+2
lincon a0c+5
lincon a0c+1
lincon a0c+5
lincon a0c+0
lincon a0c+5
lincon a0c+2
lincon a0c+8
lincon a0c+0
lincon a0c+2
lincon a0c+5
lincon a0c+5
lincon a0c+5
lincon a0c+4
lincon a0c+2
lincon a0c+2
lincon a0c+1
lincon a0c+2

ng, Technology

r.com 

0w a1c a1w;  
>=0, a1w>=0; 
13*a1c-a0w-13
5*a1c-a0w-5*a
20*a1c-a0w-20
5*a1c-a0w-5*a
13*a1c-a0w-13
5*a1c-a0w-5*a
0*a1c-a0w-0*a
5*a1c-a0w-5*a
24*a1c-a0w-24
8*a1c-a0w-8*a
0*a1c-a0w-0*a
24*a1c-a0w-24
5*a1c-a0w-5*a
50*a1c-a0w-50
53*a1c-a0w-53
40*a1c-a0w-40
22*a1c-a0w-22
20*a1c-a0w-20
17*a1c-a0w-17
28*a1c-a0w-28
17*a1c-a0w-17
20*a1c-a0w-20
13*a1c-a0w-13
22*a1c-a0w-22
8*a1c-a0w-8*a
37*a1c-a0w-37
0*a1c-a0w-0*a
28*a1c-a0w-28
50*a1c-a0w-50
13*a1c-a0w-13
44*a1c-a0w-44
8*a1c-a0w-8*a
8*a1c-a0w-8*a
28*a1c-a0w-28
34*a1c-a0w-34
40*a1c-a0w-40
13*a1c+a0w+1
5*a1c+a0w+5*
20*a1c+a0w+2
5*a1c+a0w+5*
13*a1c+a0w+1
5*a1c+a0w+5*
0*a1c+a0w+0*
5*a1c+a0w+5*
24*a1c+a0w+2
8*a1c+a0w+8*
0*a1c+a0w+0*
24*a1c+a0w+2
5*a1c+a0w+5*
50*a1c+a0w+5
53*a1c+a0w+5
40*a1c+a0w+4
22*a1c+a0w+2
20*a1c+a0w+2
17*a1c+a0w+1
28*a1c+a0w+2

y & Applied Sci

 
3*a1w<=3.76;
a1w<=3.67;  
0*a1w<=3.89;
a1w<=3.67;  
3*a1w<=3.76;
a1w<=3.67;  
a1w<=3.64;  
a1w<=3.67;  
4*a1w<=3.99;
a1w<=3.68;  
a1w<=3.64;  
4*a1w<=3.99;
a1w<=3.67;  
0*a1w<=4.37;
3*a1w<=4.42;
0*a1w<=4.23;
2*a1w<=3.91;
0*a1w<=3.89;
7*a1w<=3.82;
8*a1w<=4.05;
7*a1w<=3.82;
0*a1w<=3.89;
3*a1w<=3.76;
2*a1w<=3.91;
a1w<=3.68;  
7*a1w<=4.18;
a1w<=3.64;  
8*a1w<=4.05;
0*a1w<=4.37;
3*a1w<=3.76;
4*a1w<=4.29;
a1w<=3.68;  
a1w<=3.68;  
8*a1w<=4.05;
4*a1w<=4.16;
0*a1w<=4.23;
13*a1w>=3.76
*a1w>=3.67;  
20*a1w>=3.89
*a1w>=3.67;  
13*a1w>=3.76
*a1w>=3.67;  
*a1w>=3.64;  
*a1w>=3.67;  
24*a1w>=3.99
*a1w>=3.68;  
*a1w>=3.64;  
24*a1w>=3.99
*a1w>=3.67;  
50*a1w>=4.37
53*a1w>=4.42
40*a1w>=4.23
22*a1w>=3.91
20*a1w>=3.89
17*a1w>=3.82
28*a1w>=4.05

ience Research

Ahmad et al.: D

;  

;  

;  

;  

;  

;  
;  
;  
;  
;  
;  
;  
;  
;  
;  
;  

;  

;  
;  
;  
;  

;  
;  
;  
6;  

9;  

6;  

9;  

9;  

7;  
2;  
3;  
1;  
9;  
2;  
5;  

h V

Developing A N
 

E.

a0w
upp
the 

and
mo

cap
hig
was

Vol. 8, No. 4, 20

New Dimension

lincon a0c+17
lincon a0c+20
lincon a0c+13
lincon a0c+22
lincon a0c+8*
lincon a0c+37
lincon a0c+0*
lincon a0c+28
lincon a0c+50
lincon a0c+13
lincon a0c+44
lincon a0c+8*
lincon a0c+8*
lincon a0c+28
lincon a0c+34
lincon a0c+40
Y= a0w*36+7
run;  
ods rtf close;

Parameter Es
Cell Growth 
Parameter

w=0.041176, a
per prediction
equation as fo




ln 3.598

     0.015

ln 3.6400

y

y







d the lower lim
del is compute



3.5

    0.0

3.55

l yn

lny






Table IV sh

pable of handli
hly correlated
s found to be

018, 3130-3134

n of an Applied 

7*a1c+a0w+1
0*a1c+a0w+2
3*a1c+a0w+1
2*a1c+a0w+2
*a1c+a0w+8*
7*a1c+a0w+3
*a1c+a0w+0*
8*a1c+a0w+2
0*a1c+a0w+5
3*a1c+a0w+1
4*a1c+a0w+4
*a1c+a0w+8*
*a1c+a0w+8*
8*a1c+a0w+2
4*a1c+a0w+3
0*a1c+a0w+4
727*a1w;  

Fig. 4. 

stimation for F

estimates
a1c=0.015294
limit for expo

ollows: 

824 0.04117

294 0.00000

00 0.015294x







mit of predict
ed using the eq

598824 0.041

015294 0.000

57648 0.0152






hows that fuz
ing situations 

d. From this T
0.522, while 

4 

d Exponential M

7*a1w>=3.82
0*a1w>=3.89
3*a1w>=3.76
2*a1w>=3.91
a1w>=3.68;  
7*a1w>=4.18
a1w>=3.64;  
8*a1w>=4.05
0*a1w>=4.37
3*a1w>=3.76

44*a1w>=4.29
a1w>=3.68;  
a1w>=3.68;  
8*a1w>=4.05
4*a1w>=4.16

40*a1w>=4.23

 

Plot lny vs x  

Fuzzy Least Sq

are given:
4 and a1w=0.0
onential model




76

00 x

x
  

tion interval f
quation: 




1176

0000

294

x

x
   

zzy regression
in which pred

Table, the ave
the one for F

3133  

Model: Applicat

2;  
9;  
6;  
;  

8;  

5;  
7;  
6;  
9;  

5;  
6;  
;  

quare Exponen

 a0c=3.598
000000. The f
l is computed u

 (9)

for the expone

(10) 

n methodolog
dictor variable
erage width fo
FR was only 0

tion… 

ntial 

8824, 
fuzzy 
using 

ential 

gy is 
es are 
or LS 
0.082, 



  
ind
me

P

T

M

pro
gro
SA
res
wit
dat
con
Ta
to 
fou
per
lin
LS

Sa
no
Sa

[1] 

 

Engineerin

www.etasr

dicating ther
ethodology.  

TAB

Parameter Esti

a0c 3.59
a0w 0.04
a1c 0.01
a1w 0.00

TABLE IV. 

Method of Least

 (LP) - UP
1, 2,..., 36

i i

i 
 

II

This paper 
ogramming m
owth cell mo

AS software. 
searcher with 
th good accur
ta developed 
nventional on

able IV. In [1]
classical crisp

und that mod
rformance of a

near regression
S method. 

Authors wou
ains Malaysia 
.1001/PPSG/8

ains Malaysia).

R. J. Tallarida
with Computer

ng, Technology

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reby the sup

Fig. 5.  R

BLE III. 

imate 
Grad

98824 
41176 
15294 
00000 7

AVERAGE W

t Squares (LS) 

 
36

1

36

0.522

i
i

Width




  

 LP: Lo

II. SUMMAR

gives an e
method of boo
odeling nonlin
The aim of 
an alternativ

racy predictio
a better acc

ne. The differe
, authors discu
p estimated to
dified linear 
an estimated m
n model had 

ACKNOW

uld like to exp
for providin

8012278, Scho
. 

REFE
a, R. B. Murray,
r Programs, Sprin

y & Applied Sci

periority of 

 

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dient Objective 
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RY AND DISCUS

explanation f
otstrap appro
near regressio
the algorithm
e programmin

on result. Fuzz
curacy model
ence on the re
ussed Zadeh's 
o the least squ

estimates of
model. They c
better perform

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