Can't seem to get polyfit to work.

When using the following bit to fit a 3rd degree polynomial through my data points, I can't seem get a correct result. Here, x_outside and y_outside contain the data points. I want to find a line of points that contains the average y-values of the blue lines, so a line in the middle of the blue lines.

x_outside = x_points(k);
y_outside = y_points(k);
figure
plot(x_outside,y_outside,'.')

xSize=length(x_outside);
ySize=length(y_outside);

p  = polyfit(x_outside, y_outside, 3);

x1 = linspace(min(x_outside)-10,max(x_outside)+10);
y1 = polyval(p, x1);
hold on
plot(x1,y1)

I get the following result (the fit seems to be reflected around the line y=-(9000/1200)x+9000) ):

However, when use exactly the same code in order to fit a hyperbola I made to test the code, I do seem to get a good fit,

x_outside = linspace(-100,100,100);
y_outside = x_outside.^2;
figure
plot(x_outside,y_outside,'.')

xSize=length(x_outside);
ySize=length(y_outside);

p  = polyfit(x_outside, y_outside, 2);

x1 = linspace(min(x_outside)-10,max(x_outside)+10);
y1 = polyval(p, x1);
hold on
plot(x1,y1)

I get:

Do you guys have an idea what could be going wrong?

Answer 1

[John D'Errico]

I'd suggest that you are confused. Nothing you do will fit a cubic polynomial well to that data.

It seems to have two points where the function approaches a singularity. You are trying to fit a square peg into a round hole. Worse, it seems you have two curves in that data set.

Even if I split your data into two pieces, the curve you have is not a cubic polynomial. Polynomials simply do not have that fundamental shape, because no polynomial ever has a derivative singularity in it (thus a point of infinite slope.)

Thus, lets try a cubic fit to the first 154 data points in your data.

plot(x_outside(1:154),y_outside(1:154))

I simply used the basic fitting tools in plot itself, which is exactly what polyfit will give you. As you can see, a polynomial will not approximate a function well with a singularity at the ends.

Now, is this better than what you show? OF COURSE IT IS!!!!!! Because what you showed was the result of fitting a cubic polynomial to TWO SETS OF DATA.

polyfit(x_outside,y_outside,3)
ans =
      4.43419574654287e-06       -0.0083952855040473         -2.47023950662053          8133.82849470259

So of course that generates complete crap. But surely you need to expect that?