Finding the difference in growth rate between graphs

I have two set of graphs as follows:

x=[6,12,24,48,72,96];
control_vals=[126,363,672,1014,1358,1526];
test_vals=[83,43,112,412,714,826];

plot(x,control_vals,'*-','Color','k')
hold on
plot(x,test_vals,'o-','Color','r');title('G1')
ylim([0 2000]);hold off

control_vals2=[101,151,338,892,1142,1341];
test_vals2=[73,176,425,769,1021,1288];

figure;plot(x,control_vals2,'*-','Color','k')
hold on
plot(x,test_vals2,'o-','Color','m');title('G2')
ylim([0 2000]);hold off

I need to find out if the rate of growth is same between G1 control vs G2 control and G1 test vs G2 test.

I have tried out using curve fitting to approximate the above curve and using derivatives to find the answer but I'm getting some errors.

% For G1 Test (Red Curve in G1)
% Eq: y = - 0.003056992404*x^{3} + 0.4690737455*x^{2} - 9.431674809*x +
% 110.8789328 - Obtained using Interactive Fitting in MATLAB
p=[- 0.003056992404 0.4690737455 -9.431674809 110.8789328];
q = polyder(p)

q = 1×3

-0.0092 0.9381 -9.4317

tspan=[0 100];
y0=0;
[t,y]=ode45(@(t,y) (-0.0092*t^2+0.9381*t-9.4317),tspan,y0);
figure;plot(t,y);title('G1 Test- Differentiation')

% For G2 Test (Magenta Curve in G2)
% Eq: y = 0.000966*x^{3} - 0.2096*x^{2} + 25.49*x - 83.89 - Obtained using Interactive Fitting in MATLAB
p1=[0.000966 -0.2096 25.49 -83.89]

p1 = 1×4

0.0010 -0.2096 25.4900 -83.8900

q1=polyder(p1)

q1 = 1×3

0.0029 -0.4192 25.4900

[t,y]=ode45(@(t,y) (0.0029*t^2-0.4192*t+25.49),tspan,y0);
figure;plot(t,y);title('G2 Test- Differentiation')

Can someone guide me on the right method to find the difference in growth rate between these curves?

Appreciate your help in advance!

[Jon]

When you say "I have tried out using curve fitting to approximate the above curve and using derivatives to find the answer but I'm getting some errors." Please be specific and cut and paste the entire error message so other can see it.

In general, how are you defining "growth rate". Often the growth rate refers to the time constant in an exponential growth model. If this is how you are defining growth rate then you have to fit an exponential model to the data not a polynomial. If by growth rate you simple mean the instantaneous rate of change of the value, then fitting a polynomial and differentiating it seems like a reasonable approach. Be careful to use as low an order polynomial as you can othewise you will have spurious "ups and downs" in your fitted values which will lead to positive and negative derivative values.

[Aravind S]

Dear @Jon thank you for your reply and sorry for the delayed response.

By growth rate I meant to find at which how fast/slowly does G1 control vs G2 control and G1 test vs G2 test increases/decreases.

And when I try to find the derivate of the aferomentioned curves, I get some errors either due to ODE solver or some other issue. I've attached what I've tried now in code below.

Please let me know where I'm going wrong.

x=[6,12,24,48,72,96];
control_vals=[126,363,672,1014,1358,1526];
test_vals=[83,43,112,412,714,826];

plot(x,control_vals,'*-','Color','k')
hold on
plot(x,test_vals,'o-','Color','r');title('G1')
ylim([0 2000]);hold off

control_vals2=[101,151,338,892,1142,1341];
test_vals2=[73,176,425,769,1021,1288];

figure;plot(x,control_vals2,'*-','Color','k')
hold on
plot(x,test_vals2,'o-','Color','m');title('G2')
ylim([0 2000]);hold off

% For G1 Test (Red Curve in G1)
% Eq: y = - 0.003056992404*x^{3} + 0.4690737455*x^{2} - 9.431674809*x +
% 110.8789328 - Obtained using Interactive Fitting in MATLAB
p=[- 0.003056992404 0.4690737455 -9.431674809 110.8789328];
q = polyder(p)

q = 1×3

-0.0092 0.9381 -9.4317

tspan=[0 100];
y0=0;
[t,y]=ode45(@(t,y) (-0.0092*t^2+0.9381*t-9.4317),tspan,y0);
figure;plot(t,y);title('G1 Test- Differentiation')

% For G2 Test (Magenta Curve in G2)
% Eq: y = 0.000966*x^{3} - 0.2096*x^{2} + 25.49*x - 83.89 - Obtained using Interactive Fitting in MATLAB
p1=[0.000966 -0.2096 25.49 -83.89]

p1 = 1×4

0.0010 -0.2096 25.4900 -83.8900

q1=polyder(p1)

q1 = 1×3

0.0029 -0.4192 25.4900

[t,y1]=ode45(@(t,y1) (0.0029*t^2-0.4192*t+25.49),tspan,y0);
figure;plot(t,y1);title('G2 Test- Differentiation')

y(numel(y1)) = 0;
plot((y-y1))

Thanks for your help in advance!

[Jon]

Can you please be more specific about the errors? When you say " when I try to find the derivate of the aferomentioned curves, I get some errors either due to ODE solver or some other issue", what exactly goes wrong. Please cut an paste any error messages

[Aravind S]

@Jon I don't specifically get any error message in MATLAB. By error I meant the graphs which I'm getting as output seems to be incorrect theoretically or I'm not sure how to interpret the results I'm getting.

If you please guide me on how to approach the problem furthur it would be of great help as I'm out of ideas.

Appreciate your help in advance!

Answer 1

[Jon]

I'm assuming you want to characterize the growth rates, by the instantaneous rate of change of your control and test values. By definition, the instantaneous rate of change is the first derivative of the function.

If this is what you want then one approach is to fit your data with a smooth curve, for example a polynomial, and then analytically differentiate the resulting polynomials to get the derivatives. The code below illustrates how you could do this.

I'm not understanding why you were using ode45, which is a differential equation solver, but maybe I am missing something about what you were trying to do.

% Parameters
numExp = 2; % number of experiments
polyOrder = 3; % order of polynomial used to fit data

% Assign test and control values for the two tests 
x=[6,12,24,48,72,96];
control_vals=[126,363,672,1014,1358,1526];
test_vals=[83,43,112,412,714,826];

control_vals2=[101,151,338,892,1142,1341];
test_vals2=[73,176,425,769,1021,1288];


% Combine data into matrices so we can just loop and not cut and paste code
% (cutting and pasting is bad because if you have an error in one place you
% have to remember to fix it in all the places you pasted to.)
Control = [control_vals;control_vals2]; % rows are experiments G1,G2
Test = [test_vals;test_vals2];

% Fit the data with polynomials and differntiate them
% preallocate arrays to hold polynomial coefficients
Pcntrl = zeros(numExp,polyOrder+1);
Ptest = zeros(numExp,polyOrder+1);
Dcntrl = zeros(numExp,polyOrder);
Dtest = zeros(numExp,polyOrder);
% loop through experiments
for k = 1:numExp
    % fit
    Pcntrl(k,:) = polyfit(x,Control(k,:),polyOrder);
    Ptest(k,:) =  polyfit(x,Test(k,:),polyOrder);
    % differentiate
    Dcntrl(k,:) = polyder(Pcntrl(k,:));
    Dtest(k,:) = polyder(Ptest(k,:));

end

% Visualize the curve fits to make sure they look OK
xfit = linspace(x(1),x(end),1000); % x values for fitted data
for k = 1:numExp
    figure
    plot(x,Control(k,:),'ok',xfit,polyval(Pcntrl(k,:),xfit),'-k',...
        x,Test(k,:),'*r',xfit,polyval(Ptest(k,:),xfit),'-r')
    legend('Control','Control Fit','Test','Test Fit','Location','northwest')
    title("G" + string(k))
end

% Compare test and control rates of change (derivatives) for both experiments
for k = 1:numExp
    figure
    plot(xfit,polyval(Dcntrl(k,:),xfit),'-k',...
        xfit,polyval(Dtest(k,:),xfit),'-r')
    legend('Control','Test')
    title("Rate of Change G" + string(k))
end

[Aravind S]

Thank you so much for your reply @Jon.

I have to make an interpretation of the graphs on how much change has occured between test and control. Does the Rate of change graph plotted above give me the information directly or do I have do something else (for example find the change over the curve and average it out)?

Appreciate your help in advance!

[Jon]

How you want to analyze the data really requires an understanding of the system you are studying and what is important for your specific situation. I suggest that you talk to your colleagues, mentors, advisors, somebody that is familiar with the type of study you are doing and has experience to advise you.

Once you can express what it is you want to calculate, people on MATLAB answers can help you with your problems transforming those mathematical ideas into code.