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5
Figure 2a . The linear regression of closing stock
prices from 2 companies given from Example 1 with the dependent variable is the LQ45 Index
using least square.
Figure 2b . The linear regression of closing stock
prices from 2 companies given from Example 1 with the dependent variable is also the LQ45 Index using
QR decompotition .
III. PARAMETERS DETERMINATION ON NONLINEAR LEAST SQUARE
Modellings based on the given data mostly deal with parameters determination. Some examples are using data taken from small industries and public offices in Salatiga and its
surroundings. Modelling of Total Investment and Its Efficiency in the District of Sidomukti Parhusip, 2009a is one example on economics
. The issue of ‘sapi glonggongan’ adding much drinking water to cows in the year 2009 inspired us to
measure an optimal weight of a cow Parhusip and Ayunani, 2009. To achieve optimal parameters, one needs to find ‘nice’ data for optimization purposes. Hessian matrix is
used to select data such that one may proceed an optimization procedure which fit to the used theories Parhusip H. A. 2009b.
3.1 Stevioside is a promising sweetener Parhusip and Martono, 2011
Leaves of Stevia rebaudiana Bertoni have been extracted in Chemisty Department of Science and Mathematics Faculty, SWCU in January
– March 2011 Using a quadratic function, percentage of stevioside is modelled Parhusip and Martono, 2011 as a function
of mass and time. Standard procedure of minimization problem is employed to find parameters in the objective function. We assume that the percentage of stevioside
follows
2 2
, m
t m
t S
:= S model 1
where t and m denote time and mass respectively and the parameters
,
.
are determined due to the given data. Standard least square leads to minimize the residual
function, i.e
2 1
mod ,
,
, ,
n i
el i
data i
S S
R
=
2 1
2 2
,
n i
i i
data i
m t
S
. 2 The critical conditions require
R
. Solving this nonlinear system, one yields
T
, ,
= 0.4201. 0.8696.- 0.0688
T
. The illustration of this function is depicted on Figure 1. To get the maximum percentage stevioside from the given minimum mass
will be the interest of this research. This leads to a minimax problem in optimization,i.e
max min
x S
S x
.
INDOMS,14 April 2012
6
As one kind of sweeteners, stevioside is promising. One
good news of using stevioside is that it is not disturbing a
fertility and reproduction of a user which was investigated
for mice Yodyingyuad and Bunyawong, 1991.
Figure1.
0688 .
8696 .
4201 .
,
2 2
m
t m
t S
. Since positive impact of using stevioside, study of stevioside is becoming attractive
and one needs to produce stevioside into easily and savely consumed. 3.2 Logistic model for crown diameter of Kailan
Crown diameter of Kailan has been modeled by logistic model Parhusip,2010. By introducing variable t as the time variable and Dt as the dependent variable of the
logistic model which is also known as Velhust model . We have
, D
D K
A Ae
K K
t D
kt
1a where D0 represents the initial diameter, and the parameters K and k must be
determined based on the given data. The standard procedure of least square leads to minimize the residual function. i.e
2 1
data ,
2 1
model ,
data ,
,
n i
t k
i n
i i
i
i
Ae K
K D
D D
k K
R
. 1b
At the first glance, it is not obvious how the system will appear from this function. Notice that the minimization requires to solve
R
.This is a nonlinear homogeneous system. Matrix-vector notations will shorten the equations. Let us denote X denotes a coloum
vector in
n
where each of its components defined by each component in the summation Eq.1b. Thus
,
k K
R X
X
T
such that 2
X X
K R
K
; 2
X X
k R
k
. 2
The
X
K
denotes gradient vector with respect to K. Similarly,
X
k
is the gradient vector with respect to k. Solving Eq.2, we get the value of each parameter, i.e K=0.9844
and k=3.6688. Substituting these parameters in the logistic model Eq.1, we obtain .
1.0119 9844
. 9844
.
6688 .
3 t
e t
D
Which is depicted on Figure 1.
Figure 1
. Logistic model of Kailan’s growth with K=0.9844 dan k=3.6688 Parhusip, 2010
Any algorithm with an iterative procedure requires an initial guess of solution Eq.2. There are several well-known algorithms such as Newton-method, Broyden
method, trust-region and using evolutionary algorithm Grosan and Abraham, 2008.
INDOMS,14 April 2012
7
3.3 Penalty Method with a noncoercive objective function
