Least-Squares Estimation of an Ellipse

Given the ellipse , a random sample of uniformly distributed abscissae in the interval is generated. From these a set of ordinates is obtained by adding to each corresponding point on the ellipse a normally distributed error with mean 0 and standard deviation 0.1. The set of points thus obtained is used to estimate the parameters of the ellipse with the least-squares method.

In the graph, the dots represent the sample points generated according to the procedure. The blue curve is the original ellipse, while the red one is the estimated ellipse for a sample of size . Even for a relatively small sample size the result of the estimation appears generally to be rather good. However, using the slider to increase n shows that the estimation does not always improve with increasing sample size, contrary to intuition. The mean of the squared differences between the estimated ellipse and the true (original) one at each of the observed abscissae does not seem to decrease as n grows, as can be seen in the inset.

The reason for this odd behavior is that sample points abscissae far apart from the center have a disproportionate influence on the least-squares calculations, since the differences to be measured with respect to the curve increase enormously due to the effect of the slope of the ellipse near the extremes of the horizontal axis.

Press the button at the bottom for new samples.