How to Produce a Covariance Ellipse

[ansgar]

Dear all, I was wondering how one in reality produces the so called "Covariance ellipse"?

Lets say I have a set of data points with their error and fit a function to that data using 2 parameters just for simplicity.

Now, I know that the covariance ellipse is an ellipse of equal probability contours. But how do we do this in reality? Do I evaluate the chi-squared for different configurations of my 2 parameters (which we assume I have estimated using e.g. chi-squared minimization).. and then what? Should I, for instance if I want to state confidence limit intervals, assume that the center of my ellipse has value = 1 in probablity and then seek those boundaries where chi-squared as reduced to the value exp(-0.5) = 0.683?

Or should I go from my minimum chi-squared value thus obtained from my parameters and then go the boundaries where the chi-squared has reduced by a FACTOR exp(-0.5) = 0.683?

 

[Valkarie]

I would be thankful if someone can provide a better insight on this issue. The covariance ellipse is created by plotting the confidence region for two variables at a certain confidence level (i.e. a certain probability). This can be done by computing the confidence intervals of each variable, and then plotting those intervals in a scatter plot. The ellipse is formed by connecting the points at the ends of the confidence intervals. The size of the ellipse is determined by the specified confidence level - the larger the confidence level, the larger the ellipse. In order to compute the confidence intervals, you will need to use the chi-squared statistic. You can calculate the chi-squared statistic using the sum of squared residuals divided by the number of degrees of freedom. Then, you can calculate the confidence intervals by finding the values of the parameters where the chi-squared statistic reduces to a value equal to the critical value at the desired confidence level. For example, if you want to compute the 95% confidence intervals, you would find the values of the parameters where the chi-squared statistic reduces to a value equal to the critical value at the 95% confidence level.