I’ve been working on patching some holes in my understanding of RNA sequencing. I worked with gene expression data in the last days of the microarray, so I (used to) have some understanding of that technology, but RNA-Seq is a little more mysterious to me.
In a way I’m quite happy to take the more abstract “box of numbers” point of view and focus on the type of thing you can do with a box of numbers but I wanted to have an idea of what RNA-Seq data looks like, on average, for, say, a certain type of bacterium. And you don’t just download RNA-Seq data and load it into R. There are all kinds of mysterious steps. The box of numbers philosophy will only take you so far.
But I am not neglecting the box of numbers. I’ve been looking at Greenacre et al. (2023), a kind of spicy preprint on transformation approaches to compositional data. Remember that compositions sum to some constant: we grabbed five marbles and three of them were blue so we know how many reds there are. There are two types of marble but only one degree of freedom: all of the data lie on the line described by \(\text{reds} + \text{blues} = 5\) and the only information we can gain is relative. There are three blue marbles for every two red ones. How many in total? Nobody knows.
A widely-used approach is to lean into this and consider logarithmic ratios (logratios) like \(\log \frac{\text{blues}}{\text{reds}}\), where this particular one is called an additive logratio. This puts the data in an unconstrained space, and considering ratios gets us out of some problems with coherence: If I analyze compositions of the form \((x, y, z)\) and consider the ratio \(x/y\) I will reach the same conclusions as you do if you only consider the subcomposition \((x, y)\), no matter how each of us normalizes the vector.
But the additive logratio transformation is considered crude because it’s not an isometry: it doesn’t preserve distances. To not preserve distances is to distort the data, so you’d ideally like to have an isometry. An example of a linear isometry is rotation by \(\theta\) degrees. A triangle is mathematically the same triangle if you turn it on its head. It isn’t if you stretch it.
The centered logratio transformation preserves distances. Here instead of dividing by a certain element of the vector you divide by the geometric mean of the vector. But the resulting coordinates depend on the entire vector so we still have to consider ratios of the sort \(\frac{x/gm(x,y)}{y/gm(x,y)} = \frac{x}{y}\) to get the subcompositional coherence we had earlier. So various other, more complicated transformations were conceived, and since they are more complicated their interpretation is more subtle.
Getting to the point (if any): Greenacre et al. (2023) is a spicy paper because they argue that in high dimensions there are many transformations that are near-isometries. This means we could just use the additive log ratio for its simplicity and wouldn’t be sacrificing much in terms of geometry.
But even better: since dividing by zero or taking the logarithm of zero are undefined, all logratio transformations have a zeros problem. The mathematicians are happy to just define this problem away, saying that this whole theory of logratios, etc., is for strictly positive data. But that doesn’t help the data analyst. Zeros are pretty common in real data! Various hacky solutions exist (like adding a small arbitrary constant); Greenacre et al. (2023) opens the door for something like Box–Cox, which tolerates zeros and has the logarithm as a limiting case.
The weather in Denmark is 5 degrees C, partially cloudy. Like a good summer day back home, really. Conditions are now such that I can take an alternate bike route through the forests, etc. It’s pretty good, pretty scenic at times, but I did end up with water up past my ankles at one point.
openalexR is an R package to interface with openalex.org.
Putting information that you’ve found useful on a simple web page with no ads, no scripts, no ulterior motives—this is a valuable online service.
Mathoverflow Q: is there a fast way to check if [millions of matrices have] any small eigenvalues? Yes: do it on the GPU.
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