Write down the four 4 nonlinear regression models covered in

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 Y_n = f(x_n, ) + Zn where f is the expectation function and x_n 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 (K^N – N^m)/mK^m which, on integration, gives N(t )=K N₀ /[ N₀+ (K^N – N₀^m) 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

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=2.18

= -0.815 => b=0.153

                             N= 0.153H^2.18

%// 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 Y_n = f(x_n, ) + Zn where f is the expectation function and x_n 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 (K^N – N^m)/mK^m which, on integration, gives N(t )=K N₀ /[ N₀+ (K^N – N₀^m) 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

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= 0.153H^2.18

%// Define points

X = [10 20 30 40 50];

Y = [ 25 70 380 550 610];

Hours (H)	10	20		40	50
No fo bacteria (N)	25	70	380	550	610