Ack! I don’t have time to do justice to this right now, but any readers need to know if you don’t already that the geniuses at Desmos seem to be making a matrix calculator: https://www.desmos.com/matrix.

Having read that, you might rightly say, I can’t get to everything in my curriculum as it is, why are you bringing up matrices? (You might also say, Tim, I thought you were a data guy, what does this have to do with data?)

Let me address that first question (and forget the second): I’m about to go do a week of inservice in a district that, for reasons known only to them, have put matrices in their learning goals for high-school math. Their goal seems to be to learn procedures for using matrices to solve systems of linear equations.

I look at that and think, surely there are more interesting things to do with matrices. And there are!

One is to use matrices as transformation operators, like doing reflections and rotations using 2×2 matrices in the plane. Just using 0 and ±1, you can do a lot of great stuff. You can even introduce symmetry groups. But not in this post.

Because you can also use them to model the weather, and that’s what we’ll do, using the Desmos tool to help with calculation.

A very simple model

Suppose we say, this is California, mostly it doesn’t rain. So we’ll make a model of the weather that goes like this: every (simulated) day, you roll a die. If you roll 6, it rains. Otherwise it’s sunny.

This will have some long-term behavior that kinda-sorts resembles real weather but also really doesn’t. We can ask, what aspects of the simulated weather this algorithm generates are realistic, and what are not?

For one thing, 1/6 is probably not the fraction of days with precipitation. (If you want to find out the real proportion, you could play with my recent portal to weather data. See? I am a data guy!) We could adjust the model so the probability was right. We won’t do that, though, because I want to stick with dice.

We could point out that “sunny or rainy” is too coarse. We might want temperature, wind, amount of precipitation. Agreed. But that’s not where I’m going either.

Instead, we might note that the weather tends to be streaky. That’s another way of saying that the weather one day is not independent of the weather on another day: if it’s sunny today, there is a big chance that it will be sunny tomorrow—bigger than if it’s raining today.

So we will make a model that takes today’s weather into account when predicting tomorrow’s weather.

A Markov model

Here’s the model we will study:

If today is sunny, roll a die: if it’s a 6, tomorrow will be rainy, otherwise it’s still sunny.

If today is rainy, roll a die: if it’s a 5 or 6, tomorrow will be rainy again, otherwise, it will be sunny.

[image error]Diagram for our Markov model for the weather

We can now give the class a task: everybody model a month. We can ask all sorts of what-do-you-notice questions. We can assess the streakiness of sunny or rainy days, and so forth. We can muse about how to change the model to make it more like Seattle, or whether we should have different models in different seasons, or what our model should really depend on.

But let’s focus on this question: in this model, on average, what fraction of the days are rainy?

It should be clear that it ought to be more than 1/6, because of the rule for rainy days.

Do the data bear this out? Sure: especially when we have a class where everybody has done 30 days. We aggregate the data and see (for a fictitious class of 20, 600 days run in CODAP) that we have 479 sunny days and 121 rainy days, for a ratio of almost exactly 4:1, or 20% rainy days. About one-fifth. Which is more than one-sixth, as expected.

You might already be objecting: “but you have to start with either rain or sun—that will bias your results.” Not as much as you might think! More on this later.

How could we figure this out theoretically?

There is an amazingly elegant way that I will let you figure out. My theory about this is that we will usually approach problems first in an inefficient way, and only later will we see the cool, elegant solution—often inspired by the result we get inelegantly.

One such inelegant way is to start making a tree diagram. I will spare you this. Instead, I’ll modify that tree diagram and follow Gerd Gigerenzer and his colleagues in using natural frequencies.

Imagine that we have six situations (six, because dice), each starting with a sunny day. Of those six, on average, five will be sunny the next day and one will be rainy. That is, the six equally-likely outcomes—assuming we start sunny—are SS, SS, SS, SS, SS, and SR.

To do the next level, we have to start with 36 sunny days and let them progress; we get six each of our results from the previous paragraph. That’s 30 SS and 6 SR.

Of the 30 days that began SS, 5/6 (or 25) will still be sunny: SSS
The other 5 will be SSR
Of the six that began SR, four will be sunny: SRS
And the other two are still rainy: SRR.

If we look just at the most recent day, the sun:rain ratio is 29:7, or just under 1/5.

Finally, using matrices

At this point, we could move on to the next level, the third new day, perhaps starting with 216 days; or we could do this again starting with a rainy day, or we could count up all the days instead of just the last—but I want to get to the matrices while I have the time.

We can use matrices to do the calculation. Suppose we represent the initial sunny state with a vector like this:

[image error]

That one in the top is for sunny and the zero is for rainy.

I can make a matrix that evolves that state to the next day. I think figuring out that matrix would be a great challenge for students, but I haven’t done that part with actual students yet, so I will just write it here as a matrix A:

[image error]
Think deeply about why this is the matrix that represents the model