Principle Component Analysis
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Principle Component Analysis
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Re: Principle Component Analysis
#2The way the blogpost details the process of using the spreadsheet is nice - giving one simple example and then scaling the number of stocks clarifies when this sort of thing would be useful.
However, it would have been nice to have a bit of explanation/background on how the PCA algorithm actually works.
Re: Principle Component Analysis
#3Nice to see PCA in an HN article, it's a very powerful tool.
For those struggling to get the example in this article, I find PCA easier to understand given visual examples, and in less dimensions (try http://en.wikipedia.org/wiki/File:GaussianScatterPCA.png)
Note how this dataset is two dimensional in nature, and PCA yields two vectors. The first gives the direction of the greatest variation, and the next gives the variation orthogonally to the first.
An awesome use of PCA is for facial detection, a method called 'Eigenfaces' http://en.wikipedia.org/wiki/Eigenface
Re: Principle Component Analysis
#4PCA goes far deeper than meets the eye. For instance, it's a well-known phenomenon that too much dimensionality can actually drive predictor performance to random, but PCA can mitigate that. It's a basically the bread and butter of practical unsupervised learning.
Re: Principle Component Analysis
#5The author is basically using a linear algebra tool for creating orthogonal basis vectors of a matrix of stock prices. (The PCA is like eigenvector decomposition, but it works on rectangular matrices too. In fact, unlike many operations, it's very fast on unbalanced rectangular matrices!) Since these vectors are, by definition, uncorrelated, they can be very useful in building CAPM-balanced stock portfolios.
Using the PCA is great in this situation, but people often run into traps when using these sorts of spectral-decomposition methods on real world data.
The most obvious is that they try to interpret what the vectors "represent". Sometimes this is reasonable -- if you did a similar experiment on the stock price of energy companies, the strongest vector probably really would be closely correlated with the price of oil. But aside from unusual situations like that, interpreting the "meaning" of spectral vectors is a fool's errand.
Re: Principle Component Analysis
#6Nice to see PCA in an HN article, it's a very powerful tool. For those struggling to get the example in this article, I find PCA easier to understand given visual examples, and in less dimensions (try http://en.wikipedia.org/wiki/File:GaussianScatterPCA.png ) Note how this dataset is two dimensional in nature, and PCA yields two vectors. The first gives the direction of the greatest variation, and the next gives the…
eigenfaces are very cool. i did a bunch of work with them a few years ago.
Re: Principle Component Analysis
#7PCA is a very useful tool in lots of places. But be warned that when you use it on stocks, you'll find correlations, make your investment, then discover that during a financial crisis all sorts of things that were not previously correlated, now are. Thus your analysis falls apart at exactly the moment you would least want it to do so.
Incidentally if you take answers to a wide variety of questions that are meant to test intelligence, how the component of your score on the first component on a PCA analysis should be fairly well correlated with IQ or your SAT score. The second component should be reasonably well correlated to the difference between your math and verbal scores on the SAT. And people have much less variability on the third component than on the first two.
Re: Principle Component Analysis
#8The author is basically using a linear algebra tool for creating orthogonal basis vectors of a matrix of stock prices. (The PCA is like eigenvector decomposition, but it works on rectangular matrices too. In fact, unlike many operations, it's very fast on unbalanced rectangular matrices!) Since these vectors are, by definition, uncorrelated, they can be very useful in building CAPM-balanced stock portfolios. Using th…
It's often true that you can figure out what the first handful (say, 3 to 6) PCA components mean, in a large problem.
The first is usually the mean of the quantities. It is typical in practice to compute PCA by using the SVD of the data itself; if you subtract the mean first, then of course it will not appear as the first PCA component. In matlab, this is literally a one-liner using the svd of the original data -- not even forming a covariance matrix.
No, the people who do this don't care if you know what the Karhunen-Loeve decomposition is, they just use the one-liner:
[U,D] = svd(X)
The post should have plotted the time history of the 6 stocks together with the time history of each PC, then some pattern might have suggested itself. The first PC could be as simple as "GOOG,AMZN,AAPL,AKAM going up, MSFT steady, and FB going down", given the stocks mentioned and their weighting.
The classic examples are (mentioned elsewhere on the thead) the eigenfaces example (on Wikipedia), where PCA was used for faces, and various features like eyes, foreheads, and mouths are emphasized, plus "second-order" features like edges around the eyes, noses, and mouths. If you try it yourself, what you find is that adding more of these second-order features to a face (literally, adding, as in:
new_image = old_image + alpha * second_order_feature
People have done the same thing with natural images, and out pop things like 2d wavelets (the Gabor filters, http://en.wikipedia.org/wiki/Gabor_filter). It's somewhat magical, because you went in with no information, and out pops this structure, which also characterizes (surprise!) the human visual cortex.
Other classic examples are in atmosphere/weather analysis, where ENSO ("el nino") will pop out of analysis of temperature and pressure fields in the Pacific ocean.
Re: Principle Component Analysis
#9PCA can also be used for compression
http://www.willamette.edu/~gorr/classes/cs449/Unsupervised/p...
Also worth noting is apache mahout supports PCA - you can perform this type of analysis on large matrices pretty easily these days
Re: Principle Component Analysis
#10This expository post lined up the 6 stocks and computed the SVD of the time history of all 6 together. This shows how the 6 stocks correlate.
You can do it another way. Run a sliding window across one single stock, line up all the resulting vectors, and then take the SVD of (err...apply PCA to) that. That is, if you started with a single-stock time history:
x1, x2, x3...
z1 = [x1 x2 x3]
z2 = [x2 x3 x4]
z3 = [x3 x4 x5]
This will extract seasonable variability (on all kinds of scales -- not just annual). One name for it is Singular Spectrum Analysis (http://en.wikipedia.org/wiki/Singular_spectrum_analysis)