9853

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.

SNAPSHOTS

  • [Snapshot]
  • [Snapshot]
  • [Snapshot]
    • Share:

Embed Interactive Demonstration New!

Just copy and paste this snippet of JavaScript code into your website or blog to put the live Demonstration on your site. More details »

Files require Wolfram CDF Player or Mathematica.









 
RELATED RESOURCES
Mathematica »
The #1 tool for creating Demonstrations
and anything technical.
Wolfram|Alpha »
Explore anything with the first
computational knowledge engine.
MathWorld »
The web's most extensive
mathematics resource.
Course Assistant Apps »
An app for every course—
right in the palm of your hand.
Wolfram Blog »
Read our views on math,
science, and technology.
Computable Document Format »
The format that makes Demonstrations
(and any information) easy to share and
interact with.
STEM Initiative »
Programs & resources for
educators, schools & students.
Computerbasedmath.org »
Join the initiative for modernizing
math education.
Step-by-step Solutions »
Walk through homework problems one step at a time, with hints to help along the way.
Wolfram Problem Generator »
Unlimited random practice problems and answers with built-in Step-by-step solutions. Practice online or make a printable study sheet.
Wolfram Language »
Knowledge-based programming for everyone.
Powered by Wolfram Mathematica © 2014 Wolfram Demonstrations Project & Contributors  |  Terms of Use  |  Privacy Policy  |  RSS Give us your feedback
Note: To run this Demonstration you need Mathematica 7+ or the free Mathematica Player 7EX
Download or upgrade to Mathematica Player 7EX
I already have Mathematica Player or Mathematica 7+