Fit Gaussian mixture model with weighted observations

Hi everyone, looking at the help of fitgmdist, I cannot see that there is the possibility to weight observations. Is there a reason? Many functions of the Statistics and Machine Learning toolbox support weights. Does anyone have an idea how to include weights, or can anyone point me to an alternative?

[Adam Danz]

Would adding n duplicates of an observation be equivalent to applying a weight of n to that observation?

[Wolfgang Schwanghart]

This could be an option...

[Adam Danz]

If you end up giving that a try, keep in mind that the weights must be converted to integers and depending on how that's carried out, it could vastly increase the number of data points. Feel free to pull me in if you decide to go down this route and get stuck.

In a sense, by duplicating the values of the data being fit, you are strengthening their representation in the fit and that's kind of like weighting.

Answer 1

[Kaashyap Pappu]

The function fitgmdist fits a distribution to a given data set. This data set generally has points belonging to the same class therefore the ‘weight’ parameter is not needed, since you are essentially just fitting a distribution model to given data.

Functions such as fitcknn, fitcsvm have weights because those are classification models. Weights become essential when data from multiple classes is present for training, but there is a class imbalance, that is data points for each class are not in equal proportion. To account for this imbalance, weights are used and are essential input arguments.

Hope this helps!

Answer 2

[Jeff Miller]

It's not exactly clear (to me either) what it means to weight the different observations in this context, but maybe you have something like this in mind:

You have observations X(1:n) with weights W(1:n). Let sumW = sum(W).

Make a new dataset Y with (say) 10000 observations consisting of

round(W(1)/sumW*10000) copies of X(1)

round(W(2)/sumW*10000) copies of X(2)

etc--that is, round(W(i)/sumW*10000) copies of X(i)

Now use fitgmdist with Y. Every Y value will be weighted equally, but the different X's will have weights approximately proportional to their original W values--because their numbers will be in those proportions.

I hope that is clear.

[Wolfgang Schwanghart]

Thanks for your answer. Let me give some context: The data that I am working with are points with attributes that are located on a linear network. Imagine that this could be a road network and the points are accidents. Now I am interested where these accidents occur, for example as a function of distance from cross roads, and thus I calculate this distance along the network. A non-parametric estimate of the dependence of accidents on the distance can be obtained by kernel density estimator, but this estimation needs to account for the different frequency distribution of distances in the network. A nice solution to such a nonparametric estimate is shown here and relies on relative risk.

Now let's assume that I see two humps in the risk normalized kernel densities and I'd like to fit a Gaussian mixture model with two components to the observed distances. I think that I should also take account for the relative risk that reflects that some distances occur more often than others. My idea was to solve this with weighting the observations with their inverse relative risks. But maybe I am wrong ...

[Jeff Miller]

What about generating a lot of pseudo-observations from the risk normalized kernel densities and then fitting the gmm to those?

[Wolfgang Schwanghart]

Yes, this was my initial thought, and Adam Danz (see above) also came up with the idea. However, after giving the whole approach some thought, I think that the weighting scheme may not lead to the desired results. Rather, I think, we should normalize the probability density function as we obtain it from pdf(gmm,distance) with, let's say, a kernel density estimate of the distance values. I guess this will turn out increasingly difficult if we have models with many variables.

Answer 3

[Omkar Mulekar]

There seems to be an answer in this paper:

https://arxiv.org/pdf/1509.01509.pdf

They talk about a couple of methods for EM using weighted data. See if it's useful for you!