Help using poly fit to determine equation constants of power function and plotting the linear form

• x axis (t) 172 241 392 483 608 812 959 1305
• y axis (h) .20 .60 2.0 4.0 8.0 20.0 40.0 80.0

Using the above data I am required to...

• Enter the data as vectors
• Plot the data and by changing the axes determine the function type
• Using polyfit, determine the equation constants
• Create a plot (using the linear form of the graph)that includes the highlighted data points along with your best fit equation

I entered the data as vectors...

t=[172,241,392,483,608,812,959,1305]

h=[0.200,0.6.00,2.00,4.00,20.00,40.00,80.00]

I determined the data represents a power function by plotting it on a log log series

I THINK I determined the constants using the following commands...

>> p=polyfit(log(t),log(h),1)

p =

    2.9861  -17.0002

If I am correct the 2.9861 represents m and -17.0002 represents b in the equation y=m*x^b

I really don't know how to graph the linear form correctly and I am unsure of how to do it to include the "highlighted data points"

I attempted to do it using the commands...

>> m=p(1)

m =

    2.9861

>> b=p(2)

b =

-17.0002

>> y=[m*t+b];

>> plot(y)

I am pretty sure the graph isn't correct and there aren't any highlighted points... Any input on what I did wrong and should do differently would be GREATLY appreciated. Thank you in advance.

Answer 1

[Jonathan LeSage]

You just need to transform the linear fit coefficients (the outputs from polyfit) from a logarithm scale back to the linear scale you started with. Since you started by transforming your data into a log-log scale, the linear fit will be on the log-log scale.

Your fit (from polyval): log(h) = m*log(t) + b

Returning to the linear scale (take the exponential of each side): h = t^m*exp(b)

Here is some code that illustrates these transformations in side-by-side plots:

% Original data
t = [172,241,392,483,608,812,959,1305];
h = [0.200,0.600,2.00,4.00,8.00,20.00,40.00,80.00];

% Transformation to logarithmic scale
log_t = log(t);
log_h = log(h);

% Linear fit of data in logarithmic scale
fit = polyfit(log_t,log_h,1);

% Generate linear fit curve in the logarithmic scale
xVec = linspace(log_t(1),log_t(end),100);
log_h_fit = polyval(fit,xVec);

% Plot logarithmic data and fit
subplot(1,2,1);
plot(xVec,log_h_fit,log_t,log_h,'o');
grid on;
legend('Fit','Data')
title('log-log Data Plot')

% Convert from logarithmic scale to linear scale
m = fit(1);
b = fit(2);
h_fit = t.^m*exp(b);

% Plot linear scale
subplot(1,2,2)
plot(t,t.^fit(1)*exp(fit(2)),t,h,'o');
grid on;
legend('Fit','Data');
title('Original scale plot')