MATLAB: Curve-fitting using the Curve Fitting Toolbox, gives wrong confidence intervals

Question

We have a set of data that looks something like this:

What we are trying to do is by using the curve fitting toolbox, apply a polynomial curve and plot the curve together with the confidence interval. Sounds simple right?

The MATLAB "fit" function returns a cfit object. The object has the following output:

fitresult =
      Linear model Poly2:
    fitresult(x) = p1*x^2 + p2*x + p3
    Coefficients (with 95% confidence bounds):
      p1 =    -0.01183  (-0.01667, -0.006986)
      p2 =       1.337  (1.088, 1.585)
      p3 =      -45.94  (-48.62, -43.25)

I'm seeing a problem there, the p3 value interval is a bit small, and by plotting it you can see how weird it looks:

It looks "backwards" and I'm not sure why. Does anyone know why?

Answer 1

Without your data, we cannot verify what you have done, or verify the coefficients, or verify the uncertainties produced around those estimates. Without it, I cannot show you the difference between various kinds of confidence limit curves. So hopefully, I can explain what you are seeing.

So in that vacuum, I would point out that you are thinking about how those limits would look on a simple first order model, and confusing things with a quadratic model. The limits shown on the plot are NOT 95% limits around the data, or anything like that. They are limits on the predictions of the model at each point. Why is this important? Again, it is a QUADRATIC model.

Think about it in terms of the quadratic term. At the upper end of your data where distance is in the neighborhood of 40-50, what happens with the quadratic term?

p1 =    -0.01183  (-0.01667, -0.006986)

So p1 varies CONSIDERABLY. We don't know the value of p1 at all well. So depending on the value of p1, those distances at 50 get SQUARED. They contribute hugely to the predicted value, because p1 itself has a relatively large variance. How does p1 enter into the model? It is the term:

p1*x^2

Again, the lines that you see drawn are lines around the PREDICTED value of the curve, NOT lines around the data.

Down near zero, what happens? we don't really care what the value of p1 is down there. Who gives a hoot, because x is near ZERO. Square zero, and who cares what p1 is?

Think of it like this, at x = 50, x^2 is 2500. So the cntribution to the prediction for the quadratic term is

2500*[-0.01667, -0.006986]
ans =
   -41.675000000000004 -17.465000000000000

At x=50, the uncertainty is huge around the prediction due to the quadratic term. At x=0, the quadratic term is irrelevant.

So your problem is in not understanding what those confidence lines mean, and how they are computed. They are NOT confidence limits around the data in any form. And they are not quite the same as what you would expect for a simple first order linear model, so I think you may be letting your preconceived notions sway your thinking.