Finding Best Fit for Log Scaled Graphs

Hi, I am attempting to find the best fit for a log scaled graph but have no idea how to do it. My code is as below,

x = [0, 0.00027036151034654, 0.00115093883458806, 0.00223432233808507, 0.00539981900723518, 0.00809536518833309, 0.0111270389729589, 0.014777933095254, 0.0189175724121603, 0.0235573556054278, 0.0287025567374752, 0.0343600870656987, 0.0582919612991634, 0.0682154843903124, 0.0790320917901928, 0.0914334718128318, 0.104973874288574, 0.119722291281348, 0.135699592328357, 0.240771355163788, 0.278078798463323, 0.314454995935481, 0.561768676836072]
y = [0.30371074055921, 0.290537098717707, 0.324374702814966, 0.337252335934006, 0.38897792790656, 0.403745877906912, 0.415277428739248, 0.427209812186528, 0.438112837361668, 0.448214463712894, 0.457622869575205, 0.466422006319539, 0.51823177314633, 0.526699880477541, 0.534668807004341, 0.543031975970824, 0.55093440322382, 0.558408580520752, 0.565435121097348, 0.622450007115967, 0.631025115825886, 0.637361264649512, 0.685778962718182]

set(gca,'YScale','log')
set(gca,'XScale','log')
plot(x,y)

I tried using polyfit but can't seem to find the right approach to get a best fit still. I still want to use the log-scaled graph, so I think simply logging the x and y values doesn't work too.

Any help will be much appreciated!!!

Answer 1

[Star Strider]

It is best to fit the power function using a nonlinear approach:

x = [0, 0.00027036151034654, 0.00115093883458806, 0.00223432233808507, 0.00539981900723518, 0.00809536518833309, 0.0111270389729589, 0.014777933095254, 0.0189175724121603, 0.0235573556054278, 0.0287025567374752, 0.0343600870656987, 0.0582919612991634, 0.0682154843903124, 0.0790320917901928, 0.0914334718128318, 0.104973874288574, 0.119722291281348, 0.135699592328357, 0.240771355163788, 0.278078798463323, 0.314454995935481, 0.561768676836072]
y = [0.30371074055921, 0.290537098717707, 0.324374702814966, 0.337252335934006, 0.38897792790656, 0.403745877906912, 0.415277428739248, 0.427209812186528, 0.438112837361668, 0.448214463712894, 0.457622869575205, 0.466422006319539, 0.51823177314633, 0.526699880477541, 0.534668807004341, 0.543031975970824, 0.55093440322382, 0.558408580520752, 0.565435121097348, 0.622450007115967, 0.631025115825886, 0.637361264649512, 0.685778962718182]

objfcn = @(b,x) b(1).*x.^b(2);

[B,resnorm] = fminsearch(@(b) norm(y - objfcn(b,x)), rand(2,1))

figure
plot(x, y, 'p')
hold on
plot(x, objfcn(B,x), '-r')
hold off
grid
hold off
text(0.25, 0.45, sprintf(' y = %.3f\\cdotx^{%.3f}', B))

figure
loglog(x, y, 'p')
hold on
plot(x, objfcn(B,x), '-r')
hold off
grid
hold off
text(5E-4, 0.45, sprintf(' y = %.3f\\cdotx^{%.3f}', B))

producing (for the loglog plot):

.