Write down the four (4) nonlinear regression models covered in class, i.e. both the name of the model and their functional form showing their respective regression coefficients. A rapidly growing bacteria has been discovered with a growth rate as shown in the table. Analyze this date using a power law model. First linearize the data in the table based on the power law model, call this new linearized data x and y. Next, determine the coefficients a_0 and a_1 of a linear regression y = a_0 + a_1x for the linearized power law data x and y, and compute the correlation coefficient for the liner regression. Note that you are not being asked to do a linear regression of the data in the table but rather a linear regression of the linearized version of that data. Once you compute a_0 and a_1 use these to determine the coefficients of the power law model. Write a Matlab script that uses the polyfit function to determine the regression coefficients of a second order polynomial fit to data in the table. In the same script, write down the code for using the polyval command to determine the value of the regression curve at the observed times. Solution Nonlinear regression is a statistical technique that helps describe nonlinear relationships in experimental data. A nonlinear regression model can be written Yn = f(xn, ) + Zn where f is the expectation function and xn is a vector of associated regressor variables or independent variables for the nth case. A nonlinear model is one in which at least one of the parameters appears nonlinearly. More formally, in a nonlinear model, at least one derivative with respect to a parameter should involve that parameter. Examples of a nonlinear model are: Y(t) = exp(at+bt2) Y(t) = at + exp(-bt). Important non linear growth models 1.Logistic Model This model is represented by the differential equation dN/dt = rN (1-N/K). Integrating, we get N(t) = K / [1+(K/No-1) exp(-rt)] The graph of N(t) versus t is elongated S-shaped and the curve is symmetrical about its point of inflexion. 2.Malthus Model If N(t) denotes the population size or biomass at time t and r is the intrinsic growth rate, then the rate of growth of population size is given by dN/dt = rN. Integrating, we get N(t) = No exp (rt), where No denotes the population size at t=0. Thus this law entails an exponential increase for r>0. Furthermore, N(t) à as t à , which cannot happen in reality. 3. Monomolecular Model This model describes the progress of a growth situation in which it is believed that the rate of growth at any time is proportional to the resources yet to be achieved, i.e. dN/dt = r(K-N), where K is the carrying size of the system. Integrating (3.3), we get N(t) = K-(K-No) exp (-rt). 4. Richards Model. This model is given by dN/dt = rN (KN – Nm)/mKm which, on integration, gives N(t )=K N0 /[ N0+ (KN – N0m) exp (-rt)]1/m. Unlike the earlier models, this model has four parameters Hours (H) 10 20 40 50 No fo bacteria (N) 25 70 380 550 610 b) N = b H m Log10.

1. Write down the four (4) nonlinear regression models covered in class, i.e. both the name of the
model and their functional form showing their respective regression coefficients. A rapidly
growing bacteria has been discovered with a growth rate as shown in the table. Analyze this date
using a power law model. First linearize the data in the table based on the power law model, call
this new linearized data x and y. Next, determine the coefficients a_0 and a_1 of a linear
regression y = a_0 + a_1x for the linearized power law data x and y, and compute the correlation
coefficient for the liner regression. Note that you are not being asked to do a linear regression of
the data in the table but rather a linear regression of the linearized version of that data. Once you
compute a_0 and a_1 use these to determine the coefficients of the power law model. Write a
Matlab script that uses the polyfit function to determine the regression coefficients of a second
order polynomial fit to data in the table. In the same script, write down the code for using the
polyval command to determine the value of the regression curve at the observed times.
Solution
Nonlinear regression is a statistical technique that helps describe nonlinear relationships in
experimental data. A nonlinear regression model can be written Yn = f(xn, ) + Zn where f is the
expectation function and xn is a vector of associated regressor variables or independent variables
for the nth case.
A nonlinear model is one in which at least one of the parameters appears nonlinearly. More
formally, in a nonlinear model, at least one derivative with respect to a parameter should involve
that parameter. Examples of a nonlinear model are:
Y(t) = exp(at+bt2)
Y(t) = at + exp(-bt).
Important non linear growth models
1.Logistic Model
This model is represented by the differential equation
dN/dt = rN (1-N/K).
Integrating, we get
N(t) = K / [1+(K/No-1) exp(-rt)]
The graph of N(t) versus t is elongated S-shaped and the curve is symmetrical about its point of
inflexion.
2.Malthus Model

2. If N(t) denotes the population size or biomass at time t and r is the intrinsic growth rate, then the
rate of growth of population size is given by
dN/dt = rN.
Integrating, we get
N(t) = No exp (rt),
where No denotes the population size at t=0. Thus this law entails an exponential increase for
r>0. Furthermore, N(t) à as t à , which cannot happen in reality.
3. Monomolecular Model
This model describes the progress of a growth situation in which it is believed that the rate of
growth at any time is proportional to the resources yet to be achieved, i.e.
dN/dt = r(K-N),
where K is the carrying size of the system. Integrating (3.3), we get
N(t) = K-(K-No) exp (-rt).
4. Richards Model.
This model is given by dN/dt = rN (KN – Nm)/mKm which, on integration, gives N(t )=K N0 /[
N0+ (KN – N0m) exp (-rt)]1/m.
Unlike the earlier models, this model has four parameters
Hours (H)
10
20
40
50
No fo bacteria (N)
25
70
380
550
610
b)

3. N = b H m
Log10 N = logb + mLog10H (below data are tabulated in excel)
xi
Log10N
1
1.301
1.4771
1.6021
1.699
7.079181246
xi
yi
Log10H
1.3979
1.8451
2.5798
2.7404
2.7853
11.34851417
yi
xi.yi
1.3979
2.4005
3.8107
4.3902
4.7322
16.73153836
xiyi
(xi)^2
1
1.6927
2.1819
2.5666
2.8865
10.32766154
(xi)^2

4. m=2.18
= -0.815 => b=0.153
N= 0.153H2.18
%// Define points
X = [10 20 30 40 50];
Y = [ 25 70 380 550 610];
Nonlinear regression is a statistical technique that helps describe nonlinear relationships in
experimental data. A nonlinear regression model can be written Yn = f(xn, ) + Zn where f is the
expectation function and xn is a vector of associated regressor variables or independent variables
for the nth case.
A nonlinear model is one in which at least one of the parameters appears nonlinearly. More
formally, in a nonlinear model, at least one derivative with respect to a parameter should involve
that parameter. Examples of a nonlinear model are:
Y(t) = exp(at+bt2)
Y(t) = at + exp(-bt).
Important non linear growth models
1.Logistic Model
This model is represented by the differential equation
dN/dt = rN (1-N/K).
Integrating, we get
N(t) = K / [1+(K/No-1) exp(-rt)]
The graph of N(t) versus t is elongated S-shaped and the curve is symmetrical about its point of
inflexion.
2.Malthus Model
If N(t) denotes the population size or biomass at time t and r is the intrinsic growth rate, then the
rate of growth of population size is given by
dN/dt = rN.
Integrating, we get
N(t) = No exp (rt),
where No denotes the population size at t=0. Thus this law entails an exponential increase for

5. r>0. Furthermore, N(t) à as t à , which cannot happen in reality.
3. Monomolecular Model
This model describes the progress of a growth situation in which it is believed that the rate of
growth at any time is proportional to the resources yet to be achieved, i.e.
dN/dt = r(K-N),
where K is the carrying size of the system. Integrating (3.3), we get
N(t) = K-(K-No) exp (-rt).
4. Richards Model.
This model is given by dN/dt = rN (KN – Nm)/mKm which, on integration, gives N(t )=K N0 /[
N0+ (KN – N0m) exp (-rt)]1/m.
Unlike the earlier models, this model has four parameters
Hours (H)
10
20
40
50
No fo bacteria (N)
25
70
380
550
610
b)
N = b H m
Log10 N = logb + mLog10H (below data are tabulated in excel)
xi
Log10N
1
1.301

6. 1.4771
1.6021
1.699
7.079181246=
xi
yi
Log10H
1.3979
1.8451
2.5798
2.7404
2.7853
11.34851417=
yi
xi.yi
1.3979
2.4005
3.8107
4.3902
4.7322
16.73153836
xiyi
(xi)^2
1
1.6927
2.1819
2.5666
2.8865
10.32766154
(xi)^2
m=(nxiyi-xiyi)/(n(xi)^2-(xi)^2)
m=2.18
= -0.815 => b=0.153
N= 0.153H2.18
%// Define points
X = [10 20 30 40 50];

7. Y = [ 25 70 380 550 610];
Hours (H)
10
20
40
50
No fo bacteria (N)
25
70
380
550
610