How to Plot Confidence Contours in Matlab for a Non-Gaussian Distribution?

[aymer]

Hello..

I am dealing with a cosmological parameter estimation problem. I have a sum of squares function (chi-squared) of two parameters and I have minimized it using fminsearch, to find the best fit. Now, I want to plot 1-sigma, 2-sigma confidence contours for this. My parameter probability distribution may not be Gaussian. So , I don't want to do it in the usual way of adding 2.3 to chi-squared minimum and drawing the contour for 1-sigma. How can I draw a contour that encloses 68% of the area of the surface?

thanx

 

[Chalnoth]

Hello..

I am dealing with a cosmological parameter estimation problem. I have a sum of squares function (chi-squared) of two parameters and I have minimized it using fminsearch, to find the best fit. Now, I want to plot 1-sigma, 2-sigma confidence contours for this. My parameter probability distribution may not be Gaussian. So , I don't want to do it in the usual way of adding 2.3 to chi-squared minimum and drawing the contour for 1-sigma. How can I draw a contour that encloses 68% of the area of the surface?

thanx

Well, if you want the entire contour that may not be Gaussian, you're probably going to want to use a Monte-Carlo Markov Chain, unless you already have an analytical result for the two-parameter probability distribution. Most people use cosmomc for MCMC's with cosmological parameter estimation. Cosmomc also includes a program, getdist, which can be used to produce a variety of different contour plots for chain outputs. It uses Matlab for the actual plotting.

 

[aymer]

I already have an mcmc chain of parameter values and corresponding chi-square values. I have not used cosmomc though. Can you give me an idea how are the confidence regions estimated from the chain?

 

[bapowell]

I already have an mcmc chain of parameter values and corresponding chi-square values. I have not used cosmomc though. Can you give me an idea how are the confidence regions estimated from the chain?

To obtain the posterior distribution, you chop the parameter's prior range into bins. Then, you count the number of chain steps that fall into each bin. The number in each bin is proportional to the marginal probability that the parameter value falls in that bin. Confidence intervals are then set by the parameter values within which a specified % of chain points lie. To obtain smooth 2D error contours, functions like getdist perform an interpolation on the 2D grid of binned chain points.

 

[alphy]

Hello there,

Thank you for reaching out with your question. Confidence contours can be a useful tool in parameter estimation, especially when dealing with non-Gaussian distributions. In Matlab, there are a few different ways to create confidence contours for your data.

One approach is to use the contour function, which allows you to specify the contour levels based on a given percentage of the total area. For example, for a 68% confidence contour, you can use the following code:

contour(x,y,Z,[chi2min chi2min+2.3],'ShowText','on')

where x and y are the parameter values, Z is the chi-squared function, and chi2min is the minimum value of chi-squared found by fminsearch. This will create a contour that encloses 68% of the total area under the chi-squared surface.

Another approach is to use the confidenceplot function from the Statistics and Machine Learning Toolbox. This function allows you to specify the confidence level and will automatically calculate the appropriate contour levels for you. For example, for a 68% confidence contour, you can use the following code:

confidenceplot(x,y,Z,'Confidence',0.68)

Both of these methods will give you a contour that encloses 68% of the total area, regardless of the shape of your parameter probability distribution.

I hope this helps and best of luck with your cosmological parameter estimation problem!