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Problem 1 (Handwritten) – Nonlinear Regression.
(13 points)
A University of Florida researcher performs an experiment to determine the growth rate of
bacteria
k
(bacteria per day) as a function of oxygen consumption
c
(mg/L). Tabulated data from
the experiment and a corresponding plot of the data (open circles) are provided below, along
with a regression fit of the data performed by the researcher (solid line).
c
0.5
0.8
1.5
2.5
4
k
1.1
2.3
5.2
7.7
8.9
0
1
2
3
4
5
0
2
4
6
8
10
c (mg/L)
k (bacteria per day)
To generate this regression fit, the researcher modeled the data using the nonlinear relationship
2
1
2
2
ac
k
=
+
(1)
and linearized this equation using transformation of variables. After performing the regression fit
for the linearized model, the researcher calculated the values of the regression coefficients
1
a
and
2
a
for the original nonlinear model and used these values in Eq. (1) to generate the solid line
shown in the figure above.
a)
Use transformation of variables to
derive
the linearized version of Eq. (1) that was used by
the researcher for regression analysis. Assume the linearized model is given by
yp
xq
=
+
where
p
and
q
are regression coefficients. Provide expressions in the spaces below for
y
as a
function of
k
,
x
as a function of
c
, and how
p
and
q
as functions of
1
a
and
2
a
.
y = _____________________________
x = _____________________________
p = _____________________________
q = _____________________________
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b) Formulate a linear system of equations
Ax
b
=
that could be solved for the regression
coefficients
p
and
q
in the linearized model (Note: DO NOT solve these equations). Keep
only two significant figures for all values used in your linear system of equations. Record
you’re
A
matrix and
b
vector in the spaces provided below:
____
____
____
____
____
____
____
____
____
____
____
____
____
____
____
p
q
⎡⎤
⎡
⎤
⎢⎥
⎢
⎥
⎢
⎥
=
⎢
⎥
⎢
⎥
⎣⎦
⎢
⎥
⎢
⎥
⎣
⎦
c) Once the researcher solved for the regression coefficients
p
and
q
in the linearized model,
show how he calculated regression coefficients
1
a
and
2
a
in the original nonlinear model.
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This note was uploaded on 09/05/2011 for the course EGM 3344 taught by Professor Raphaelhaftka during the Spring '09 term at University of Florida.
 Spring '09
 RAPHAELHAFTKA
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