exponential model givenbest fitto the data. figure. exponential model of nonlinear regression for y...
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![Page 1: Exponential Model Givenbest fitto the data. Figure. Exponential model of nonlinear regression for y vs. x data 1](https://reader034.vdocuments.us/reader034/viewer/2022051214/56649f005503460f94c1655d/html5/thumbnails/1.jpg)
Exponential Model),( , ... ),,(),,( 2211 nn yxyx yxGive
nbest fit
bxaey to the data.
Figure. Exponential model of nonlinear regression for y vs. x data
bxaey
),(nn
yx
),(11
yx
),(22
yx
),(ii
yxibx
i aey
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Finding Constants of Exponential Model
n
i
bx
ir iaeyS
1
2
The sum of the square of the residuals is defined as
Differentiate with respect to a and b
021
ii bxn
i
bxi
r eaeya
S
021
ii bxi
n
i
bxi
r eaxaeyb
S
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Finding Constants of Exponential Model
Rewriting the equations, we obtain
01
2
1
n
i
bxn
i
bxi
ii eaey
01
2
1
n
i
bxi
n
i
bxii
ii exaexy
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Finding constants of Exponential Model
Substituting back into the previous equation
01
2
1
2
1
1
n
i
bxin
i
bx
bxn
ii
bxi
n
ii
i
i
i
i exe
eyexy
The constant b can be found through numerical methods such as bisection method.
n
i
bx
n
i
bxi
i
i
e
eya
1
2
1
Solving the first equation for a yields
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Example 1-Exponential Model
t(hrs) 0 1 3 5 7 9
1.000 0.891 0.708 0.562 0.447 0.355
Many patients get concerned when a test involves injection of a radioactive material. For example for scanning a gallbladder, a few drops of Technetium-99m isotope is used. Half of the Technetium-99m would be gone in about 6 hours. It, however, takes about 24 hours for the radiation levels to reach what we are exposed to in day-to-day activities. Below is given the relative intensity of radiation as a function of time.
Table. Relative intensity of radiation as a function of time.
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Example 1-Exponential Model cont.
Find: a) The value of the regression
constants A an
db) The half-life of Technetium-99m
c) Radiation intensity after 24 hours
The relative intensity is related to time by the equation tAe
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Plot of data
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Constants of the Model
The value of λ is found by solving the nonlinear equation
01
2
1
2
1
1
n
i
tin
i
t
n
i
ti
ti
n
ii
i
i
i
i ete
eetf
n
i
t
n
i
ti
i
i
e
eA
1
2
1
tAe
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Setting up the Equation in MATLAB
01
2
1
2
1
1
n
i
tin
i
t
n
i
ti
ti
n
ii
i
i
i
i ete
eetf
t (hrs) 0 1 3 5 7 9
γ 1.000
0.891
0.708
0.562
0.447
0.355
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Setting up the Equation in MATLAB
01
2
1
2
1
1
n
i
tin
i
t
n
i
ti
ti
n
ii
i
i
i
i ete
eetf
t=[0 1 3 5 7 9]gamma=[1 0.891 0.708 0.562 0.447 0.355]syms lamdasum1=sum(gamma.*t.*exp(lamda*t));sum2=sum(gamma.*exp(lamda*t));sum3=sum(exp(2*lamda*t));sum4=sum(t.*exp(2*lamda*t));f=sum1-sum2/sum3*sum4;
1151.0
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Calculating the Other Constant
The value of A can now be calculated
6
1
2
6
1
i
t
i
ti
i
i
e
eA
9998.0
The exponential regression model then is te 1151.0 9998.0
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Plot of data and regression curve
te 1151.0 9998.0
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Relative Intensity After 24 hrs
The relative intensity of radiation after 24 hours 241151.09998.0 e
2103160.6 This result implies that only
%317.61009998.0
10316.6 2
radioactive intensity is left after 24 hours.http://
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