How Far is xy From yx on Average for Quaternions?
Given two quaternions x and y, the product xy might equal the product yx, but in general the two results are different. How different are xy and yx on average? That is, if you selected quaternions x and y at random, how big would you expect the difference xy – yx to be? Since this difference would increase proportionately if you increased the length of x or y, we can just... Read more
Low-Rank Matrix Perturbations
Here are a couple of linear algebra identities that can be very useful, but aren’t that widely known, somewhere between common knowledge and arcane. Neither result assumes any matrix has low rank, but their most common application, at least in my experience, is in the context of something of... Read more
Linear Regression and Planet Spacing
Linear Regression and Planet Spacing A while back I wrote about how planets are evenly spaced on a log scale. I made a bunch of plots, based on our solar system and the extrasolar systems with the most planets, and said noted that they’re all roughly straight lines. Here’s the... Read more
Statistical Software Matters
This is a picture of all the genetic associations found in genome-wide association studies, sorted by chromosome. You can find more detail at the NHGRI GWAS catalog     There are two chromosomes with many fewer associations. One is the Y chromosome. There isn’t much there because there isn’t much... Read more
Partition numbers and Ramanujan’s approximation
The partition function p(n) counts the number of ways n unlabeled things can be partitioned into non-empty sets. (Contrast with Bell numbers that count partitions of labeled things.) There’s no simple expression for p(n), but Ramanujan discovered a fairly simple asymptotic approximation: How accurate is this approximation? Here’s a little Matheamtica code to see. p := PartitionsP... Read more
Talking About Clinical Significance
In statistical work in the age of big data we often get hung up on differences that are statistically significant (reliable enough to show up again and again in repeated measurements), but clinically insignificant (visible in aggregation, but too small to make any real difference to individuals). An example would be: a diet... Read more
Stirling Numbers, Including Negative Arguments
Stirling numbers are something like binomial coefficients. They come in two varieties, imaginatively called the first kind and second kind. Unfortunately it is the second kind that are simpler to describe and that come up more often in applications, so we’ll start there. Stirling numbers of the second kind... Read more
Fixed Points of Logistic Function
Here’s an interesting problem that came out of a logistic regression application. The input variable was between 0 and 1, and someone asked when and where the logistic transformation f(x) = 1/(1 + exp(a + bx)) has a fixed point, i.e. f(x) = x. So given logistic regression parameters a and b, when does the logistic curve... Read more
Relative Error in the Central Limit Theorem
If you average a large number independent versions of the same random variable, the central limit theorem says the average will be approximately normal. That is the absolute error in approximating the density of the average by the density of a normal random variable will be small. (Terms and conditions apply.... Read more
Quantifying Uncertainty with Bayesian Statistics
Whenever we’re working with data, there is necessarily uncertainty in our results. Firstly, we can’t collect all the possible data, so instead we randomly sample from a population. Accordingly, there is a natural variance and uncertainty in any data we collect. There is also uncertainty from missing data, systematic... Read more