# My Favorite Topic to Teach

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In this post I’m going to discuss a topic that I’m currently covering in UW – La Crosse’s MTH 353 Differential Equations course: Laplace transforms. While (like many math topics) I didn’t appreciate transforms as much as I should have when I learned them for the first time, they have become my favorite topic to teach in the undergraduate curriculum.

First, Pierre-Simon Laplace was an absolute crusher in the mathematical sciences. In addition to the transform method that bears his name, he’s responsible for a lot of the theoretical underpinnings of Bayesian statistics (one of Richard’s favorite topics), tidal flow, spherical harmonics, potential theory and Laplace’s equation, among many other things.

The Laplace transform, in its simplest application, transforms linear, generally inhomogeneous, constant-coefficient ordinary differential equations of time t into an algebraic equation of a (complex) frequency variable s.

What I love the most about Laplace transforms as a topic in the mathematics curriculum is that is requires students to apply techniques from earlier in their training. For example:

– Completing the square (elementary algebra)

– Horizontal translation of functions (elementary algebra)

– Improper integration (second-semester calculus)

– Partial fraction decomposition (second-semester calculus)

– Linear transformations (linear algebra/functional analysis)

– Elementary theory of linear ODEs (elementary differential equations)

Another nice thing about the Laplace transform is that it can handle discontinuous inhomogeneous (forcing) data like Heaviside step functions and Dirac delta functions, as well as forcing terms that aren’t their own derivatives like polynomials, sines, cosines and exponential functions. When viewing the solution to differential equations in this setting, in the space of functions of the variable s, one can clearly see how initial data and forcing data is propagated in time back in the original solution space.

If you’re interested in Laplace transforms, I’ve created some videos for my MTH 353 course, and they can be found here on my YouTube page!

# Multiple inheritance in Python

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The major reasons I switched my population modeling from R to Python is that Python is a nicer language (obligatory xkcd reference here).

Today, I was trying to figure out how multiple inheritance works. Multiple inheritance allows a Python class to inherit the attributes (e.g., functions) from other classes. In my specific context, I am trying to build a complex spatial population model. I want to be able to merge spatial and demographic attributes to create new population classes. However, I was getting tripped up about how to code it in Python.

I found an example on StackOverflow, but I knew I would need to recode it myself to remember. Plus, it’s a physics example, and I like biology story problems better. Here’s the example  I created:

``` class apples: def myName(self): print "apples"```

class oranges:
def myName(self):
print “oranges”

class fruit1(apples, oranges):
pass

class fruit2( oranges, apples):
pass

f1 = fruit1()
f2 = fruit2()
f1.myName()
f2.myName()

Note that if you run the code, each fruit object produces different a different name. This demonstrates the order of multiple inheritance in Python as well as the concept. Also, note how the function myName() is overloaded (which simply demonstrates inheritance).

# Clustering Applied to Quarterback Play in the NFL

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In this blog post I want to talk a bit about unsupervised learning. As some of you that know me may know, I am relatively new to data science and machine learning, having my formal educational training in applied mathematics/mathematical biology. My interest in machine learning came not through mathematical biology or ecology, but through studying football.

Using ProFootballFocus data (I am a data scientist for PFF) we can study the quality of quarterback play through the process of grading players on every play of every game of every season.  To do so, it’s the most efficient to “cluster” quarterback seasons into buckets of similar seasons.  The best way to do this (do date) is through k-means clustering.

While there are many references on k-means clustering in the literature and on the web, I’ll briefly summarize the idea in this blog.  K-means clustering is an unsupervised learning algorithm that aims to partition a data set of n observations into k clusters where each observation belongs to one and only one cluster with the nearest mean.  Visually, one can think of a cluster as a collection of objects in m-dimensional space that are “close” to each other.  Below is an example of clustering quarterbacks from the 2016 season by their proportions of positively-graded and negatively-graded throws.  Different clusters are visualized with different colors:

As a part of our in-depth study of quarterback play at PFF, we clustered quarterbacks on the composition of their play-by-play grades in various settings (when under pressure, when kept clean, with using play action).  This gave us a tier-based system in which to evaluate the position throughout the PFF era (2006-present).  In 2016 the only quarterback that was in our top cluster on all throws, throws when from a clean pocket, throws when under pressure, and throws on third and long was New England Patriots’ star Tom Brady.

Stay tuned for more of an in-depth look at the quarterback position by visiting profootballfocus.com both in-season and during the offseason.

# Random versus fixed effects

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Wrapping my head around random versus fixed effects took me a while in graduate school. In part, this is because multiple definitions exist. Within ecology, the general definition I see is that a fixed effect is estimated by itself whereas a random effect comes from a higher distribution. Two examples drilled this home for me and helped it click.

First, the question: “Do we care about the specific group or only that the groups might be having an impact?” helped me see the difference between fixed and random effects. For example, if we were interested in air quality data as a function of temperature across cities, city could be either a fixed or random effect. If city was a fixed effect, then we would be interested the air quality at that specific city (e.g., the air quality in New York, Los Angles, and Chicago). Conversely, if city as a random effect, then we would not care about a specific city, only that a city might impact the results due to city specific conditions.

Second, an example in one of Marc Kerry’s book on WinBugs drilled home the point. Although he used WinBugs, the R package lme4 can be used to demonstrate this. Additionally, although his example was something about snakes, a generic regression will work. (I mostly remember the figure and had to recreate it from memory. It was about ~5 or 6 years ago and I have not been able to find the example in his book to recreate it, hence I coded this from memory). Here’s the code

library(ggplot2)
library(lme4)

population = rep(c(“a”, “b”, “c”), each = 3)
intercept = rep( c(1, 5, 6), each = 3)
slope = 4
sd = 2.0

dat = data.frame(
population = population,
interceptKnown = intercept,
slopeKnown = slope,
sdKnown = sd,
predictor = rep(1:3, times = 3))

dat\$response = with(dat,
rnorm(n = nrow(dat), mean = interceptKnown, sd = sdKnown) +
predictor * slopeKnown
)

## Run models
lmOut <- lm(response ~ predictor + population, data = dat)
lmerOut <- lmer( response ~ predictor + (1 | population), data = dat)

## Create prediction dataFrame
dat\$lm <- predict(lmOut, newData = dat)
dat\$lmer <- predict(lmerOut, newData = dat)

ggplot(dat, aes(x = predictor, y = response, color = population)) +
geom_point(size = 2) +
scale_color_manual(values = c(“red”, “blue”, “black”)) +
theme_minimal() +
geom_line(aes(x = predictor, y = lm)) +
geom_line(aes(x = predictor, y = lmer), linetype = 2)

Which produces this figure:

Example of a fixed-effect intercept (solid line) compared to a random-effect (dashed line) regression analysis.

Play around with the code if you want to explore this more. At first, I could not figure out how to make the dashed lines be farther apart from the solid lines. Change the simulated standard deviation to see what happens. Hint, my initial guess of decreasing did not help.

# Test Driven Development

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Recently at work, I’ve been building a complex, spatially-explicit population model. The model is complex enough that I started using Python to program it because R did not easily allow me to program the model. Initially while developing the model, I used informal “testing” to make it produced the correct results. For example, I would write a test script to plot simple results and make sure they outputs looked okay. However, this approach was not satisfactory and it was suggested to me that I use Test Drive Development (TDD).

With TDD, I write a small unit test and then program a function or few of lines of code to answer to test. The test is written in a second script file. After writing the new model code, I run the script test file and make sure the test passes. As part of Python’s “Batteries included” philosophy, base Python even comes with a module for unit testing built in.

This seems simple enough, but I now love TDD! With TDD, I know my code does what I think it is doing (something that is not always easy with complicated models or code). Also, I know (can change my code, but not it’s behavior! For example, if I try to improve a function, I now simply re-run the test script to make sure I didn’t change or break anything.

Although seemingly overkill for simple ecological models, TDD improves the quality and reproducibility of our models. Also, using TDD makes me a better and more confident programmer. My only regret is that I did not start using TDD earlier! For anyone wanting to learn TDD, I found this to be a helpful introduction as well as the Python documentation on unit testing.