Symmetric Gaussian matrices The previous post looked at the distribution of eigenvalues for very general random matrices. In this post we will look at the eigenvalues of matrices with more structure. Fill an n by n matrix A with values drawn from a standard normal distribution and let Mbe the average of A and its transpose, i.e. M = ½(A + AT). The eigenvalues... Read more

This year’s World Cup in Russia was the most watched sporting event in history. GlobalWebIndex reports that up to 3.4 billion people – around half of the world’s population – watched some part of the tournament. As with past World Cups, a global prediction market emerged allowing spectators to... Read more

If all you know about a person is that he or she is around 5′ 7″, it’s a toss-up whether this person is male or female. If you know someone is over 6′ tall, they’re probably male. If you hear they are over 7″ tall, they’re almost certainly male.... Read more

In my two SciPy 2018 co-taught tutorials, I made the case that ECDFs provide richer information compared to histograms. My main points were: We can more easily identify central tendency measures, in particular, the median, compared to a histogram. We can much more easily identify other percentile values, compared... Read more

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

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

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 plot for our solar system,... Read more

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

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

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