A Markov Chain describes a sequence of states where the probability of transitioning from states depends only the current state. Markov chains are useful in a variety of computer science, mathematics, and probability contexts, also featuring prominently in Bayesian computation as Markov Chain Monte Carlo. Here, we’re going to look at a relatively simple breed of Markov chain and build up some intuition using simulations and animations (two of my favorite things).
Binomial probability is the relatively simple case of estimating the proportion of successes in a series of yes/no trials. The perennial example is estimating the proportion of heads in a series of coin flips where each trial is independent and has possibility of heads or tails. Because of its relative simplicity, the binomial case is a great place to start when learning about Bayesian analysis. In this post, I will provide a gentle introduction to Bayesian analysis using binomial probability as an example.
This is a different kind of post, but one that I think is kind of fun. I currently live in Ottawa, which for those who don’t know, is the capital city of Canada. For a capital city, it’s fairly small, but it’s increasingly urbanizing (we just got lightrail transit). Segregated bicycle lanes and paths are becoming more common too and many of these paths have trackers on them that count how many bicycles cross a particular street or path each day.
This past summer, I watched a brilliant lecture series by Richard McElreath on Bayesian statistics. It honestly changed my whole outlook on statistics, so I couldn’t recommend it more (plus, McElreath is an engaging instructor). One of the most compelling cases for using Bayesian statistics is with a collection of statistical tools called linear mixed models or multilevel/hierarchical models. It’s common that data are grouped or clustered in some way.
If you’re anything like me, you probably set a lot of goals. Whether it’s to finish a paper by the end of the summer or to spend more time with friends and family, goals are what help motivate us to do something. Goals are also intimately tied to our feelings. You may have had the experience of falling behind in your goals, which made you upset, but ultimately motivated you to step up your efforts.
Emotion dynamics is the study of how emotions change over time. Sometimes our feelings are quite stable, but other times capricious. Measuring and predicting these patterns for different people is somewhat of a Holy Grail for emotion researchers. In particular, some researchers are aspiring to discover mathematical laws that capture the complexity of our inner emotional experiences - much like physicists divining the laws that govern objects in the natural environment.
Sporting events host witness to a wide range of human emotion. The emotional ups and downs are especially clear among invested fans. Fans experience the joy and excitement of a triumphant comeback, or the anxiety and disappointment of a loss. It is particularly interesting to see how emotions differ from two opposing fan groups watching the same match.
I decided to perform some simulations on how a crowd of fans would react during a basketball game.
Latent Profile Analysis (LPA) tries to identify clusters of individuals (i.e., latent profiles) based on responses to a series of continuous variables (i.e., indicators). LPA assumes that there are unobserved latent profiles that generate patterns of responses on indicator items.
Here, I will go through a quick example of LPA to identify groups of people based on their interests/hobbies. The data comes from the Young People Survey, available freely on Kaggle.
In this month’s post, I set out to create a visual network of emotions. Emotion Dynamics tells us that different emotions are highly interconnected, such that one emotion morphs into another and so on. I’ll be using a large dataset from an original study published in PLOS ONE by Trampe, Quoidbach, and Taquet (2015). Thanks to Google Dataset Search, I was able to locate this data. The data is collected from 11,000 participants who completed daily questionnaires on the emotions they felt at a given moment.
I was recently introduced to Google Dataset Search, an extension that searches for open access datasets. There I stumbled upon this dataset on childrens’ and adult’s ratings of facial expressions. The data comes from a published article by Vesker et al. (2018). Briefly, this study involved having adults and 9-year-old children rate a series of 48 faces on two dimensions of emotion, valence (positive vs. negative) and arousal (activated vs.