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4

I would look to run a pre-optimization routine over the whole universe of 200+ ETFs. I would use this pre-optimization to reduce the universe to a cardinality that provides optimal diversification effects. You can do that by first looking at pair-wise correlations and then also run optimizations to reduce portfolio variance by utilizing the covariance ...

3

What you are looking for is an unsupervised learning algorithm algorithm: i.e an algorithm that will by itself determine the 3 most rational groups from your dataset. This method will allow you to choose the boundaries of the groups based on the dataset you provide and not by choosing some given fixed values. The algorithm I suggest you to use is the ...

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When I select assets for a portfolio given an universe, I tend to pick ones that span the beta spectrum, given your selected benchmark. I find that if your portfolio of assets have varying volatility or correlation, you can achieve better diversification. I didn't come up with the idea but it comes from a rotational system's framework from the link below: ...

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First, I assume that your price data are all from the same asset but spread over a certain time range. If you are looking for the distribution of the price of this asset on the real axis, you have plenty of methods (several fields in mathematics and statistics deal with this topic). As a first step you could make a histogram of your data. There you can see ...

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You may have a look at a list of clustering algos available in sklearn here, but I think all of them are of $O(n^2$) complexity. As well, have a look at the TSNE clustering algo, which is supposed to be $O(log(n)*n)$, but this may not be the fact depending on a particular implementation. A particular case in point is again Python sklearn implementation of ...

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I've used the graphical lasso for exactly this kind of thing in the past. You can control the degree of shrinkage, which determines the how tight the clusters become.

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One simple approach is Construct the cumulative probability function (CDF), which will be a step-function. Smooth the CDF; for example, by using splines or a kernel smoothing function. Calculate the slope of the smoothed CDF, giving a curvy linear PDF. In R, this could be done using the ecdf function and one of the kernel smoothers. Again, as vanguard2k ...

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