# Fitting a parameterized 2D distribution

I frequently deal with datasets that rely on a binary classification based on a Pulse-Shape-Distribution (PSD) discrimination value. This is based on a 2d fit of the PSD value vs the amplitude of the data, a typical graph is show below

The fits you see are the means and +/- 1 sigma for the two bands, where the mean and sigma is determined by chunking the 2d distribution along the x axis and performing a two gaussian fit along each slice. This works in most cases but sometimes the fits can bounce around a lot and on the lower end are less accurate.

What I would like to do is to somehow parameterize the means of the gaussian to a functional form. For reasons based on the underlying physics of the data, it can be assumed that the mean of the lower distribution is constant and the mean of the upper is a monotonically decreasing curve (a cubic would probably be easiest to model).

The integral of the two distributions relative to each other is not important, nor is the distribution of values along the x-axis. This changes for each dataset and is not important. What is needed is just the fits you see above.

It seems that there are two options, both of which I am having a hard time thinking about how to implement: * Continue doing a 1d slicing fit, but constrain the values of the means in such a way that they follow a functional form * Fit a 2D distribution directly, but unsure of how to deal with the bin value and amplitude distribution not mattering.

Is there a specific name, class of algorithms, or recasting of the problem that can help me on the path to implement a sort of 1.5-dimensional fit?

• What would be the problem with first finding the means exactly as you do now, and then fitting your constant/cubic function to the resulting curve? – amoeba Feb 10 '14 at 23:46
• That would work to some degree, but in the above plot you can see the means dip downwards. This is not a real effect, but rather that the top gaussian gets overwhelmed by more statistics in the lower. By parameterizing the means with a form that doesn't dip downwards it would prevent that effect. – abnowack Feb 11 '14 at 0:46