By the way by changing the graph representation we can give meaning even to non square matrices as described in this article https://www.math3ma.com/blog/matrices-probability-graphs
Let's say that in most cases you have a graph and you consider the corresponding matrix. Doing the inverse is not as useful in practice except in some cases as explained in the article.
If you had a lot of people and a lot of stocks, a low-rank representation of the matrix (probably not PCA per se with that particular matrix, but something closely related) could convey a lot of information about, e.g., submarkets and how they're valuated together. Or not, depending on how those prices covary over time.
For instance, you can normalize along the columns, and build a "recommender system" using matrix factorization.
With that, when a new person comes with a portfolio, the system will output a probability for this new person to acquire the other assets he doesn't have.
It's (the very basic) idea of how Netflix recommends movies.
Really, if your conclusions change depending on whether you measure in inches or centimeters, there’s something wrong with the analysis!
> When I try to get this point across about techniques like the PCA, I like to show that the measurement units strongly affect the inference.
In such a case the problem is not with PCA but with application. PCA is just a rotation of the original coordinate system that projects the data on new axes which are aligned with the directions of highest variability. It is not the job of PCA to parse out the origin of that variability (is it because of different units, or different effects).
> Really, if your conclusions change depending on whether you measure in inches or centimeters, there’s something wrong with the analysis!
To get a statistical distance one should: subtract the mean if the measurements differ in origin; divide by standard deviation if the measurements differ in scale; rotate (or equivalently compute Mahalanobis distance) if the measurements are dependant (co-vary). The PCA itself is closely related to Mahalanobis distance: Euclidian distance on PCA-transformed data should be equivalent to Mahalanobis distance on the original data. So, saying that something is wrong with PCA because it doesn't take units of measurement into account is close to saying that something is wrong with dividing by standard deviation because it doesn't subtract the mean.
1. Technically, the table you shared is better thought of as a two-dimensional tensor, rather than a "graph-like matrix" -- which as you point out must be a linear map from a (vector) space to itself.
2. While not technically "Principal Component Analysis", one could do "Singular Value Decomposition" for an arbitrarily shaped 2-tensor. Further, there are other decomposition schemes that make sense for more generic tensors.
3. (Rotations / linear combinations in such spaces) Given a table of stock holdings, it can be sensible to talk about linear combinations / rotations etc. Eg: The "singular vectors" in this space could give you a decomposition in terms of companies held simultaneously by people (eg: SAAS, energy sector, semiconductors, entertainment, etc). Likewise, singular vectors on the other side would tell you the typical holding patterns among people (and clustering people by those, eg. retired pensioner invested for steady income stream, young professional investing for long-term capital growth, etc). As it turns out, this kind of approximate (low-rank) factorization is at the heart of recommender systems.
For example, the transform from (Alive, Bob, Carol, Dan) to (Male, Female) is linear -- it's another matrix that you can compose with the individual-ownership one you have here.
Or, call your individual-ownership matrix A, and say that P is the covariance of daily changes to prices of the four stocks listed. Then A P A' is the covariance of daily changes to the peoples' wealths. The framing as linear algebra hasn't been useless.
I kinda get what you're saying though. Like, why would powers of this matrix be useful? It only makes sense if there's some implicit transform between prices and people, or vice versa, that happens to be an identity matrix.
You can make up a story. Say the people can borrow on margin some fraction of their wealth. Then say that they use that borrowing to buy stock, and that that borrowing affects prices. Composing all these transforms, you could get from price to price, and then ask what the dynamics are as the function is iterated.
But, ok, "I'm just going to do an SVD of the matrix and put it in a slide" isn't going to tell anybody much.
Maybe there's a use for a rank-one approximation to this system? Like, "this is pretty close to a situation where there's a single ETF with those stocks in these proportions, and where the people own the following numbers of shares in the ETF"? Maybe if you have millions of people and millions of stocks and wanted to simulate this "stock market" at 100Hz on a TI-83?
I dunno. You can make up stories.