Statistics @ Home

Sampling from the Singular Normal

stats.at.home@gmail.com (Justin and Rachel Silverman) — Sat, 27 Oct 2018 00:00:00 +0000

Following up the previous post on sampling from the multivariate normal, I decided to describe in more detail the situation where the covariance matrix or precision matrix is singular (e.g., it is not positive definite). A normal distribution with such a singular covariance/precision matrix is referred to as a singular normal distribution. Here is 100 samples from a two dimensional example:

Notice that a singular normal essentially has less dimensions (in this case 1 dimension) than the dimension of the random variable (in this case 2 dimensions).

Sampling from Multivariate Normal (precision and covariance parameterizations)

stats.at.home@gmail.com (Justin and Rachel Silverman) — Fri, 19 Oct 2018 00:00:00 +0000

Two things are motivating this quick post. First, I have seen a lot of R code that is slower than it should be due to unoptimized sampling from a multivariate normal. Second, yesterday I spend a frustrating few hours tracking down a bug that ultimately was due to a slight subtlety in sampling from the multivariate normal parameterized by a precision matrix (the inverse of a covariance matrix).

Key Idea: It is easy to draw univariate standard (e.g., zero mean and unit variance) normal random variables. In fact most programming languages provide efficient vectorized (e.g., parallelized) algorithms for doing this. In contrast, it is challenging to draw multivariate random variables directly. Motivated by this fact, the approach I discuss below transform samples from standard normal random variables into samples from the desired multivariate normal random variable¹.

The Bridge Gods

stats.at.home@gmail.com (Justin and Rachel Silverman) — Thu, 18 Jan 2018 00:00:00 +0000

The Problem

My parents like to play Bridge. It’s a great card game and totally worth the time it takes to learn all the rules. And there are many. My parents play on opposing teams since my mom has the opinion that one should never partner up with his/her actual partner, for the sake of the marriage. Which is saying something about being Bridge teammates since by being on opposite teams, when you lose, your partner wins.

Bayesian Decision Theory Made Ridiculously Simple

stats.at.home@gmail.com (Justin and Rachel Silverman) — Thu, 12 Oct 2017 00:00:00 +0000

Bayesian Decision Theory is a wonderfully useful tool that provides a formalism for decision making under uncertainty. It is used in a diverse range of applications including but definitely not limited to finance for guiding investment strategies or in engineering for designing control systems. In what follows I hope to distill a few of the key ideas in Bayesian decision theory. In particular I will give examples that rely on simulation rather than analytical closed form solutions to global optimization problems. My hope is that such a simulation based approach will provide a gentler introduction while allowing readers to solve more difficult problems right from the start.

Plotting a Sequential Binary Partition on a Tree in R

stats.at.home@gmail.com (Justin and Rachel Silverman) — Wed, 20 Sep 2017 00:00:00 +0000

For users of PhILR (Paper, R Package), and also for users of the ILR transform that wan to make use of the awesome plotting functions in R. I wanted to share a function for plotting a sequential binary partition on a tree using the ggtree package. I recently wrote this for a manuscript but figured it might be of more general use to others as well.

In its simplest form a sequential binary partition can be represented as a binary tree.

Visualizing the Multinomial in the Simplex

stats.at.home@gmail.com (Justin and Rachel Silverman) — Thu, 14 Sep 2017 00:00:00 +0000

Lately I have been working on figures for a manuscript. In this process I created a few visualizations that I thought might help others visualize the Multinomial distribution. I will focus on describing how counting processes introduce uncertainty into estimates of relative abundances and I will end with a discussion of how understanding the Multinomial has impacted my view of analyses of sequence count data (e.g., data from 16s studies of the microbiome, RNA-seq, and more). Here I have chosen to focus on the Multinomial distribution, however, much of what I discuss also relates to the Multivariate Hypergeometric Distribution as well.

Does Gauss Love Me More in the Kitchen?

stats.at.home@gmail.com (Justin and Rachel Silverman) — Sun, 27 Aug 2017 00:00:00 +0000

The Idea

First things first, Gauss is our dog.

Since I am able to work from home, my dog Gauss and I spend a lot of time together. As a result, I like to think I know why he does what he does. But of course I will never really know - though, it’s nice to think that I do. Both of us being a creatures of habit, we have fallen into a nice routine during the day - one where he sleeps the day away and comes to get me around 4pm for some outdoor training/playing. I have noticed that whenever I do anything interesting or out of the norm, he is right there, waiting to see if he can benefit from the activity. Most remarkably, it feels like whenever we are in the kitchen, he sits down right in the middle of everything waiting for scraps and food that drops on the floor.

I know Gauss loves me, but I wonder if I am more valuable to him in certain rooms? Does he “love” me more in the kitchen?

A New(?) Regression Clustering Algorithm

stats.at.home@gmail.com (Justin and Rachel Silverman) — Sun, 13 Aug 2017 00:00:00 +0000

Motivation

I am a fan of the Stack Exchange forums. In particular, I like Cross Validated and Stack Overflow. An interesting question regarding clustering was posted recently. Essentially someone had the following dataset.

plot(Retirees)

Essentially the poster wanted a way of clustering the observations into the “lines” that are fairly easy to observe in the data. I am going to ignore the fact that these lines are actually the result of artifact (e.g., conversion of discrete values to percentages and then plotting the percentages vs. a variable used to calculate the percentages) and just pretend they are real as I think its still an interesting problem. I am actually going to simulate some non-artifactual data and use this as well.

Building the ILR from the ALR Transform

stats.at.home@gmail.com (Justin and Rachel Silverman) — Thu, 10 Aug 2017 00:00:00 +0000

Following up on a recent post on limitations of the ALR and Softmax transforms, I wanted to briefly show how we can derive an Isometric Log-Ratio transform from the Additive Log-Ratio (ALR) transform.

The ILR transform is just an orthonormal basis in the simplex with respect to the Aitchison metric (which follows naturally from using log-ratios - I will probably have a post explaining this more in the future). We are going to use the Gram-Schmidt orthonomalization process to build an orthonormal basis given a set of vectors which are not orthonormal (the coordinates defined by our ALR transform). This is actually the same approach used when the ILR transform was first described. In particular we will use the QR decomposition which is essentially the same as the Gram-Schmidt orthonomalization.

We can do better than the ALR or Softmax Transform

stats.at.home@gmail.com (Justin and Rachel Silverman) — Wed, 09 Aug 2017 00:00:00 +0000

In multiple places in the Compositional Data Analysis literature (for example here and here) people refer to the Additive Log-Ratio transform (ALR) as “not preserving metric concepts”. But what exactly does this mean and how can we visualize this problem?

Here I am going to briefly describe how this problem can be seen with the ALR transform and then show how the Isometric Log-Ratio (ILR) transform does not have this problem.

Note 1: The ALR transform is just a softmax transform that decreases the dimension of the transformed space by 1. The softmax transform has 1 too many dimensions compared to the dimensions of information in the simplex.

Follow-up on Error Analysis

stats.at.home@gmail.com (Justin and Rachel Silverman) — Wed, 02 Aug 2017 00:00:00 +0000

I wanted to write a quick post responding to a question that we received about our last post (Error Analysis Made Ridiculously Simple).

Can you give an example of how to generate an estimate of the error? It’s easy enough when measuring a table, as long as your meter stick is accurate: measure 1,000 times and make an inference. But in a setting where you don’t actually know the true outcome – let’s say you are trying to model household income – I’m not sure how to generate a reasonable guess of the size of the error. I suppose sometimes surveys provide error bands, but I don’t really trust them. The implication would be that they have some estimate of “true” income, which seems implausible. Even the administrative data they might be matching the surveys to is measured with error (e.g. people lie on tax returns, etc.).

Error Analysis Made Ridiculously Simple

stats.at.home@gmail.com (Justin and Rachel Silverman) — Fri, 21 Jul 2017 00:00:00 +0000

Introduction

All measurements have uncertainty. This is not a subjective opinion but an objective fact that should never be ignored. In light of this, I have always been curious about how infrequently uncertainty is actually taken into account in science.

Stochastic Loading of Microfluidic Droplets

stats.at.home@gmail.com (Justin and Rachel Silverman) — Sat, 08 Jul 2017 00:00:00 +0000

Droplet-based microfluidics are emerging as a useful technology in various fields of biomedicine. Both droplet digital PCR and droplet based culture methods require that droplets are created with either a single DNA molecule or a single cell per droplet. Obviously it is difficult to individually place DNA molecules or cells into droplets, instead people turn to stochastic models to estimate the distribution of cells per droplet, tuning the experimental parameters to achieve an acceptable distribution.

A Post on Tournament Designs

stats.at.home@gmail.com (Justin and Rachel Silverman) — Sat, 01 Jul 2017 00:00:00 +0000

The Problem

When hosting our annual Matzah Hunt event, we wanted to come up with a cool and unusual way of picking teams. We decided that we wanted each participant to complete each puzzle only once and to complete each puzzle with a different partner. Here are the precise details of the problem:

There are 12 participants, 6 puzzles and 6 (20-minute) rounds. Teams of 2 will attempt to solve each puzzle each round. We want to create teams of 2 such that every participant attempts each puzzle one time with a new partner each time.

Measure Theory Made Ridiculously Simple

stats.at.home@gmail.com (Justin and Rachel Silverman) — Mon, 26 Jun 2017 00:00:00 +0000

During my first few years of medical school I became a big fan of the [Subject] Made Rediculously Simple book series (as in Clinical Microbiology Made Rediculously Simple). I found that the authors did a great job of simplifying the subject matter, sometimes to the point of absurdity, while getting the core concepts across in a memorable way. For some time now I have wished that similar tools were available for mathematics. While I may not have the same artistry or comical flare that those authors have, here I attempt to take the Made Rediculously Simple flare to explain some of the core concepts of measure theory. This post is designed for those who have a background in basic calculus and recognize that measure theory plays some important (although perhaps cryptic) role in modern probability and statistics.

Eternal Sunshine of the Endless Flight

stats.at.home@gmail.com (Justin and Rachel Silverman) — Mon, 12 Jun 2017 00:00:00 +0000

I am sitting on a plane from Rome to Philadelphia, marveling at how quickly we can move around the globe. There is a 6 hour time-zone difference between Rome and Philadelphia. My flight took off at 11am in Rome and was set to arrive 9 hours and 45 minutes later at 3:45pm in Philadelphia. This got me thinking, what would your daily schedule look like if you continuously flew west?

Fitting Non-Linear Growth Curves in R

stats.at.home@gmail.com (Justin and Rachel Silverman) — Wed, 07 Jun 2017 00:00:00 +0000

A few months ago I offered to help a friend fit a bunch of microbial growth curves using R. When I was looking over possible solutions I was quite supprised by how little information was available online. Apart from the R package grofit (which after playing around with I decided seemed a little over-designed for my uses) I found very limited recources or code available. As a result of this I wanted to share a few functions I wrote to quickly fit non-linear growth models. I was specifically asked to help fit growth curves using the gompertz function and this is what I demonstrate below. I hope that this example gives some insight into how to fit non-linear models in R, beyond simply gompertz gorwth curves.

2017 Matzah Hunt Potions Puzzle

stats.at.home@gmail.com (Justin and Rachel Silverman) — Sun, 04 Jun 2017 00:00:00 +0000

For the past 3 years we have put together an event for our friends and family that we have named Matzah Hunt. We put on this event in place of/in addition to a Passover Seder and it is an afternoon and evening full of a series of math/logic puzzles which ultimately lead to a hunt for the Missing Matzah, hence the name, Matzah Hunt. Over the years, the puzzles have become more involved and creative, the decorations have become more elaborate, and luckily, the attendance has grown. It ends up being a wonderfully fun day that we look forward to each year.

Sampling Covariance Matricies with Fixed Total Variance

stats.at.home@gmail.com (Justin and Rachel Silverman) — Thu, 01 Jun 2017 00:00:00 +0000

Introduction

I have been thinking a lot about the concept of Total Variance recently. Total variance (which can be defined as the trace of a covariance matrix) is a measure of global dispersion that has been particularly useful for me when building multivariate models. However, for some reason, I have yet to see this concept discussed much outside of compositional data analysis (see pg. 35 of Lecture Notes on Compositional Data Analysis) or Principle Component Analysis.

About Us

stats.at.home@gmail.com (Justin and Rachel Silverman) — Mon, 01 Jan 0001 00:00:00 +0000

Hello! And welcome to Stats@Home. We are Justin and Rachel Silverman, PhD students in Statistics and Biostatistics. This blog came out of some creative and statistical musing, side projects, and dissertation tangents. Our goal in writing this blog is to explore some questions or puzzles that we find interesting with a statistical flare.

Some of our statistical interests include machine learning, clinical trials, compositional data, combinatorial analysis, probability theory, but we enjoy all kinds of statistical and probabilistic puzzles. Our recreational interests mostly involve the outdoors and our dog, Gauss, who may end up being the subject of some of our posts.