The circle --- redux

In the previous example, we were given a circle, whose center we guessed and tried to validate this by computing the predicted radius, point-by-point. In the present situation, we suppose that the only data available to us is the arc of a circle (even though we can see all of it) and the analysis proceeds in the attempt to least squares fit a quadratic form, , to the data. Our principle observation is that this sort of least squares fit is extremely sensitive to errors in the data. It is so sensitive that meaningful results are impossible to achieve without further assumptions.

Narrowly distributed data

We sample and look at several points taken on one about one half of one quadrant of the circle. See below.

This data is pasted into Excel as you see below.

This is fit to the model

with the resulting curve given to be



This plot in no way resembles the circle It does not even properly even look like a rounded curve. The correct formula in Excel is an array formula given by linest(range of right side, range of left side, false, false). Here the first false is to suppress the use of the constant in the least squares model. The second false suppresses the display of statistics associated with the plot.

How was the formula constructed? Within Excel there is a linear estimator function. With it, one can make a linear line to fit any data. So we applied this function to the data in the chart below. That is we determined by least squares a fit of that data to the linear model .

The reason for this is that the correct formula for the circle is , and this means that the curve has the form Therefore the one "1" in the model above leads to disaster.

If we replace the model by a new model then we obtain . The graph of this equation shown below and this nearly approximates the circle.

Notice that the circle has the correct radius, nearly, but the center is above the -axis. Can you see that possibly, the origin was digitized slightly below the -axis. Could this account for the small error? If so, this is certainly a factor that should be always considered.

Another lesson we learn here is that is is important to select the correct model. If we were real clever, we could see that the correct equation should have the form . Now if this holds for each point we have determined, then

Subtracting gives , Solving for results in

Computing these values for the numbers above gives the following set of

We see that the value is about This is exactly correct. However, making this calculation relied on having even more knowledge about the circle. Moreover, if you recall our starting point, we presupposed only that we had a small cluster of data --- and did not know it was actually a circle.

The lesson to learn here is to use the right model for the right situation.