Exploring Exponential Models

Students explore the covid dataset, focusing on the growth in positive test cases for the state of Massachusetts. They try to fit different kinds of models to it, ultimately discovering the need for models beyond linear and quadratic functions. This section makes heavy use of interactive slider activities we’ve built in Desmos to support open-ended experimentation.

Starting in 2019, COVID-19 spread across the globe. Most of us heard terms like "flatten the curve" and "infection rate" in videos and on the news.

Even in 2020, very few people could have predicted the impact Covid would have on the world. But Data Scientists who were looking at the data knew differently. Let’s take a look at some of that data!

Let’s start out by looking at just one state. We’ll use Massachusetts for this investigation, but you can do your own investigation about any state you like after finishing this one!

The definition MA-table = filter​(​covid-table, is-MA​) filters our dataset, keeping only the rows for which state = "MA".

Linear models capture straight-line relationships, where one quantity varies proportionally based on another. In linear models, we expect the response variable to grow by equal amounts over equal intervals in the explanatory variable.

If we make the line go from the start to the peak of the curve (top line), almost all of the points bulge out below our line of best fit. If we make the line hit the bottom of the curve, all the points fall above it (bottom line). Splitting the difference (orange line) is better than both of those options, and we might even get a halfway decent S! But ultimately, straight-line, linear models just don’t behave like this curve, and we’ll never get the best-possible fit with them. The number of positive cases is growing too fast to be fit with a linear model that grows at a constant rate!

Maybe linear isn’t the way to go, here!

Quadratic models capture parabolic relationships, where one quantity varies based on the square of another. In quadratic models, we expect the response variable to grow by differing amounts over equal intervals in the explanatory variable.

Quadratic models change their rate of growth over time, which definitely makes them a better fit for this data than linear ones. It’s very likely we could find a quadratic model with a lower S-value than our linear model! But this data starts out almost flat and then suddenly takes off like a rocket - quadratic models just don’t have that kind of explosive growth, so our model will never be as good as it could be.

Having identified that the Covid scatter plot appears neither linear nor quadratic, students learn about characteristics of exponential functions in tabular, graphical, and function notation form.

Let’s review what we know about the behavior of the models we’ve seen so far:

Remember that linear functions grow by fixed intervals, so the rate of change is constant. In the table shown here, each time the x-value increases by 1, we see that the y-value increases by 2. This is true for any set of equal-sized intervals: a line needs to slope up or down at a constant rate in order to be a straight line!
If the "growth" is constant, the relationship is linear.

Quadratic functions grow by intervals that increase by fixed amounts! In the table to the right, the blue arrows show a differently-sized jump between identical intervals, meaning the function is definitely not linear! However, if we take a look at the difference between those differences(shown in red), we’re back to constant growth!
If the "growth of the growth" is constant, the relationship is quadratic.

Exponential functions grow even faster than quadratics.

When we calculate the growth between the y-values, we can immediately tell it’s not linear. When we try to calculate the "growth of the growth", we see that it’s not quadratic either.

If we calculate the "growth of the growth of the growth" (shown in green)…​ we still haven’t found a constant…​ but we should notice that each of these "growths" just repeats the original pattern of y-values! We could calculate the third difference, fourth difference, and so on - and never reach a stopping point.

Exponential functions grow so rapidly that looking for "what is added to y?" isn’t helpful at all.

In order to talk about the growth of an exponential function we need to identify the growth factor by asking ourselves "what is y being multiplied by?"

In this case, we can see that the y-values are doubling each time, so the growth factor is 2!

Exponential Functions can be written in the form: f(x) = ab^{(x - h)} + k

Most textbooks only present exponential functions with a horizontal shift of zero. When h=0, we can safely remove it and use this simplified form of the equation: f(x) = ab^x + k.

Let’s explore what each coefficient of f(x) = ab^x + k means!

Students extend their sampling techniques to exponential relationships. Students continue experimenting in Desmos, but eventually switch back to Pyret to formalize their understanding.

Now that you’re familiar with exponential functions, let’s use them to model this Covid data!

On Exponential Models: f(x) = ab^x + k you’ll see a note about the use of ~1 to tell Pyret to prioritize speed over precision. Unlike most calculators, Pyret usually prioritizes precision.

In a math classroom, this is the difference between ²/₃ rendering as 0.3 or being rounded to 0.666666667.

In data processing, choosing speed over precision can have ethical or technical consequences!

For example:

1) When calculating a path over an extremely long distance, being off by just a billionth of a degree could result in a Mars-bound rocket missing its destination.

2) For an extremely large population like China, rounding to 10 decimal places might result in discounting an entire group of people!

Chances are, your guesses got less accurate as the number of days increased!

Why was it so much guesstimate the farther out we got, when the number of days was always increasing by a fixed amount?

We are creatures of nature, so our brains are designed to be really good at working with things we see all the time. It’s normal to see groups of 2, 5, or even 10 or 100, and we have a pretty good intuition for comparing group sizes as long as they’re small.

But when numbers grow really, really, really fast…​we get lost! Our brains lose track of differences when two numbers get really enormous.

Mathematically, the number line is composed of equal intervals forever. But we don’t actually process it that way at all.

Exponential growth poses a problem for those of us with human brains, because the numbers get so big, so fast that it can be difficult to wrap our heads around it!

This may have played a role in the sluggish response of many countries, and the tragic loss of life and decrease in public health that followed.

Fortunately, there’s another mathematical tool that can help us get control of these wildly gigantic numbers. (Stay tuned!)

Students discuss an example of Simpson’s Paradox, which motivates splitting a dataset into grouped samples using filters. They then discover another motivation for filtering: scatter plots like our covid dataset show multiple correlations, instead of just one. Finally, they learn how to filter a dataset and apply that knowledge to filtering the Covid dataset into samples grouped by state.

A college is looking at housing data for a sample of students and comparing choices among those students who’ve decided what their major will be to choices made by students who are undecided about their major:

It looks like the two variables are significantly related: undecided majors are more likely to live on campus than decided ones!

But there’s a third variable hiding in the background: freshmen college students are far less likely to have picked a major than seniors, and they are much more likely to live on campus.

When we filter by this important third variable, it turns out that for both Freshmen and Non-Freshmen, there is no correlation between between deciding on a major and living on- or off-campus.

What looks like a correlation between having-a-major and living-on-campus is actually a correlation between age and living-on-campus.

A third variable lurking in the data can play tricks by obscuring relationships between two other variables - or by creating the appearance of a relationship where none exists! Normally we think that the more data we include in our sample the more clearly we’ll see any potential relationships. But in certain circumstances the correlations in our sub-groups cancel each other out when we put the groups together. This is called Simpson’s Paradox.

Sometimes filtering the data into subsets is the only way to see what’s really going on. That’s exactly what this starter file does, by filtering the data for Massachusetts only.

Datasets like the one used in our Covid Spread Starter File are very difficult to model all at once, because there will always be lots of points that are far from any single function. But it’s not that there’s no relationship between the x- and y-variables. Instead, we have several sub-groups each with their own very strong relationships, and another variable lurking in the background.

In fact, the scatter plot for all our New England states didn’t look much like a scatter plot all! It looks like someone took a marker and drew in five different curvy lines.

Diseases spread more rapidly in densely-populated areas, since it’s easier for the infection to jump from one person to another. Unfortunately, we can’t see the density data in our table, so that dimension is missing from our dataset! This is exactly what happened in our college example: we couldn’t see the age of the students, which skewed our interpretation of the scatter plot.

These patterns are so distinct from one another that we’re going to need to make more than one model.

The filter function consumes a Table and a helper function! The helper function is used on every Row of the Table, producing true or false. The filter function takes all the Rows for which the helper produced true, and combines them all into a new table.

It’s extremely common for students to think that filtering a table changes the original table. This is NOT how it works in Pyret! Instead, the filter function always produces a new table, containing only the Rows for which the supplied function evaluates to true.