“Follow your own path”. Useful advice when navigating the ups and downs of life. It is also a useful primer for functional data! Sometimes when researchers collect data, they are interested in a “path” or trajectory of some measurable quantity over time. Imagine that you were able to know your heartbeat at any given moment during the day and visualize your heartbeat as a graph (trajectory), with time on the x-axis and beats per minute on the y-axis. It slows down when you rest, and it speeds up when you play LaserCatsTM.
Your pulse is a continuous process, which means that if it jumps from 70 bpm to 150 bpm it must visit every value in between the two. Furthermore, it seems reasonable to assume that your heart rate in the current moment depends on your heart rate from a few minutes ago. In the statistics world, we would say that your heart rate now is correlated with your heart rate in the past. Just as in life, every individual’s heart rate must follow its own path, and no two trajectories are identical. However, it may be reasonable to believe that there are underlying characteristics of your pulse that are similar to other humans. They tend to be elevated in the day and lower at night. The correlation between your heart rate 5 minutes ago and right now may be the product of a rhythm that is common to all humans. If we make a few assumptions, we can fit a model that describes the average value of this process over time and how the process is correlated with itself in the past and future.
If you had a high quality model, you could even make predictions about a future heart rate given enough information about the process up until the current time. Imagine that your goal was to maximize your heart rate at the end of an hour of working out, and you could choose whether to spend your next hour doing crossfit or jogging. Either one of these options could create the target rate, but the decision about which one performs better might be dependent on prior information, like the path of your heart rate through the day. The choice about whether to do crossfit or jogging could be tailored to an individual based on how their heart beat progression has looked up until the moment the decision needs to be made.
My research examines the underlying principles involved in modeling a continuous process and using it to predict a future outcome. I am currently working to apply it to a practical problem — depression in humans. Mental health workers use existing methodology to assign numerical scores to indicate severity of depression. Like heart rate, this is a value that may be constantly changing over a day, a week, or a year. We can tailor a treatment strategy using past information about the trajectory of someone’s depression score to find a medication that works best for that particular individual. I hope that the results of this research could be useful in improving the quality of life of people suffering from depression and that the underlying statistical tools could have broader applications in other fields.
Robert is a PhD Candidate whose research interests include computational statistics and machine learning. His current research focuses on functional Q-learning. We thought this posting was a great excuse to get to know a little more about him, so we we asked him a few questions!
Q: What do you find most interesting/compelling about your research?
A: I enjoy the challenge inherent in solving a problem that doesn’t have a standard solution. A lot of elementary statistics involves applying tools that are already well understood, but research allows you to develop something rigorous that could become a standard practice in the future.
Q: What do you see are the biggest or most pressing challenges in your research area?
A: Sequential decision making needs a lot of data, nonparametric problems tend to need a lot of data, so multi-stage, sequential problems that are estimated using nonparametric methods might need a lot^2 of data to be good.
Q: The poem Antigonish begins:
As I was going up the stair
I met a man who wasn’t there!
He wasn