by []). Below we show how to execute the MatLab function
fminsearch
with a user defined function
sumsq
vonBert
, including an initial guess and the data sets.
1
[ p1 , J ,
f l a g
]
= fminsearch ( @sumsq
vonBert , [ 1 0 0 , 0 . 1 ] , [ ] , tdfish , l d f i s h )
MatLab returns the best fitting parameter values in the vector
p
1
(which could be used in another
iteration if one is uncertain of convergence), the least sum of square errors,
J
, and a variable
flag
, which is 1 if MatLab thinks
fminsearch
has converged and 0 if it failed to converge. For our
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particular problem, MatLab returns the results
p
1
= [92
.
401
,
0
.
14553],
J
= 1
,
107
.
3, and
flag
= 1,
so the best fitting parameters are
L
*
= 92
.
401
and
b
= 0
.
14553
and our best model for the length of Lake Trout as it ages is
L
(
t
) = 92
.
401
1

e

0
.
14553
t
.
A MatLab script file is developed to graph this best fitting model and the data. This best fitting
model is shown with the data in the graph below. The script,
vonBert
plot.m
, is available from the
Lecture notes (including initial clear and final text and output controls, which are omitted here for
clarity).
1
2
c l e a r
% Clear
previous
d e f i n i t i o n s
3
f i g u r e
(1)
% Assign
f i g u r e
number
4
c l f
% Clear
previous
f i g u r e s
5
hold
o f f
% Start
with
f r e s h
graph
6
7
mytitle
=
’ Length
of
Lake
Trout ’
;
%
T i t l e
8
xlab =
’Age
( Years ) ’
;
% X

l a b e l
9
ylab =
’ Length
(cm) ’
;
% Y

l a b e l
10
11
load
(
’ f i s h d a t ’
) ;
% Provide
vectors
of
Lake
Trout
data
12
13
tt
=
l i n s p a c e
(0 ,20 ,500) ;
% t
domain
of
function
14
Lt = 92.404
*
(1

exp
(

0.14553
*
tt ) ) ;
% Function
f o r
von
Bertalanffy
15
16
plot
( tt , Lt ,
’b

’
,
’ LineWidth ’
, 1 . 5 ) ;
% Plot
model
17
hold
on
% Plots
Multiple
graphs
18
plot
( tdfish , l d f i s h ,
’ bo ’
,
’ LineWidth ’
, 1 . 5 ) ;
% Plot
data
with
c i r c l e s
19
20
grid
% Adds
Gridlines
21
22
xlim ( [ 0
2 0 ] ) ;
% Defines
l i m i t s
of
graph
23
ylim ( [ 0
100]) ;
24
25
f o n t l a b s
=
’ Times New Roman ’
;
% Font
type
used
in
l a b e l s
26
x l a b e l
( xlab ,
’ FontSize ’
,14 ,
’FontName ’
, fontlabs ,
’ i n t e r p r e t e r ’
,
’ latex ’
) ;
27
% x

Label
s i z e
and
font
28
y l a b e l
( ylab ,
’ FontSize ’
,14 ,
’FontName ’
, fontlabs ,
’ i n t e r p r e t e r ’
,
’ latex ’
) ;
29
% y

Label
s i z e
and
font
30
t i t l e
( mytitle ,
’ FontSize ’
,16 ,
’FontName ’
,
’ Times New Roman ’
,
’ i n t e r p r e t e r ’
,
’ latex ’
) ;
31
%
T i t l e
s i z e / font
32
set
(
gca
,
’ FontSize ’
,12) ;
% Axis
t i c k
font
s i z e
33
34
print

depsc
vonBert .
eps
% Create
f i g u r e
as
EPS
f i l e
35
%print

djpeg
vonBert . jpg
% Create
f i g u r e
as
JPEG
f i l e
The graph of the von Bertalanffy model with the relevant data is below.
0
2
4
6
8
10
12
14
16
18
20
0
10
20
30
40
50
60
70
80
90
100
Age (Years)
Length (cm)
Length of Lake Trout
Relation Between Weight and Length
The next step in the modeling for this problem is to find a functional relationship between the
weight and the length of the Lake Trout. This is known as
Allometric
modeling or a
Power Law
relationship. Specifically, we examine a relationship of the form
W
=
kL
a
.
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 Fall '08
 staff