Posts

I’m going over Chapter 5 in Casella and Berger’s (CB) “Statistical Inference”, specifically Section 5.5: Convergence Concepts, and wanted to document the topic of convergence in probability with some plots demonstrating the concept.

From CB, we have the definition of convergence in probability: a sequence of random variables $X_{1}, X_{2}, … X_{n}$ converges in probability to a random variable $X$, if for every $\epsilon > 0$,

In this post, I’m going to briefly cover the relationship between the Poisson distribution and the Multinomial distribution.

Let’s say that we have a set of independent, Poisson-distributed random variables $Y_{1}, Y_{2}… Y_{k}$ with rate parameters $\lambda_{1}, \lambda_{2}, …\lambda_{k}$. We can model the sum of these random variables as a new random variable $N = \sum_{i=1}^{k} Y_{i}$.

While most of my day-to-day research entails writing Python code, I also make heavy use of pre-written software. Most software comes pre-compiled, but whenever possible, I like to get access to the source code. I’m going to refer to some modifications I made to pre-existing packages – you can find those in my repository here.

I’ve been toying around with openCV for generating MRI images with synethetic motion injected into them. I’d never used this library before, so I tested a couple examples. Below I detail a few tools that I found interesting, and that can quickly be used to generate image transformations.

I recently attended Neurohackademy 2018, hosted by the University of Washington’s eScience Institute, and organized by Dr. Ariel Rokem and Dr. Tal Yarkoni.

This was a 2-week long event, beginning with a series of daily lectures, and ending with a fast-paced, high-intensity scramble to put together a beta (but working) version of some project coupling neuroimaging with software development. The lectures varied in topic, from how to test academic code and how to organize open source projects, to machine learning and algorithms for low-dimensional representations of neural recordings, to neuroethics (full lecture list here).

I just learned about TravisCI (actually, about continuous integration (CI) in general) after attending Neurohackademy 2018. We learned about CI from the perspective of ensuring that your code builds properly when you update files in your packages, incorporate new methods, refactor your code, etc. Pretty neat.

In putting together this blog, I wanted to be able to talk about various mathematical topics that I found interesting, which inevitably lead to using LaTex in my posts.

I’m currently using Atom as my editor (having converted from Sublime), and needed to install a bunch of packages first. First and foremost, I wanted to be able to render my markdown posts before hosting them on the blog, and consequentially needed a way to render LaTex. For this, I installed a few Atom packages:

In my previous post on dynamic mode decomposition, I discussed the foundations of DMD as a means for linearizing a dynamical system¹²³. In this post, I want to look at a way in which we can use rank-updates to incorporate new information into the spectral decomposition of our linear operator, $A$, in the event that we are generating online measurements from our dynamical system⁴ – see the citation below if you want a more-detailed overview of this topic along with open source code for testing this method.

In the next two posts, I want to talk briefly about an algorithm called Dynamic Mode Decomposition (DMD). DMD is a spatiotemporal modal decomposition technique that can be used to identify spatial patterns in a signal (modes), along with the time course of these spatial patterns (dynamics). As such, the algorithm assumes that the input data has both a spatial and a temporal component. We are interested in modeling how the system evolves over time.

In this post, I’ll be covering the basics of Multivariate Normal Distributions, with special emphasis on deriving the conditional and marginal distributions.

Given a random variable under the usual Gauss-Markov assumptions, with $y_{i} \sim N(\mu, \sigma^{2})$ with $e \sim N(0,\sigma^{2})$, and $N$ independent samples $y_{1}…y_{n}$, we can define vector $\mathbf{y} = [y_{1}, y_{2},…y_{n}] \sim N_{n}(\mathbf{\mu},\sigma^{2}I)$ with $\mathbf{e} \sim N_{n}(\mathbf{0},\sigma^{2}I)$. We can see from the covariance structure of the errors that all off-diagonal elements are 0, indicating that our samples are independent with equal variances.