It's Not Just You

It's not just you...

No, really, you're not the only one who's confused by Nick Rowe's metaphor here, that Rowe labels as Friedman's Thermostat.

It is the case that, for very a particular technical definition of correlation, the setting of your thermostat is not directly correlated with the amount of power used to maintain that set point.

However, it is also the case that, if you set your thermostat to 74 degrees Fahrenheit, you will save approximately 6 percent power (in summer, in winter reverse the two for the same effect) than if you had set your thermostat to 72 degrees Fahrenheit.

So, how can both of these statements be true at the same time? It has to do with a very particular definition of correlation, and a technical result from so defining the object in question.

To see what's going on, look at Temperature (T) in your house as a function of time (t), T(t) over some period, we'll use a 24 hour time for convenience. Now, let be the average temperature over that 24 hour period. It is the case then that we have an additional function T(t)-<T>, call it dT(t) for delta orchange or fluctuation.

What's interesting about dT(t), when you measure it, is that, if your a/c system is all in good shape, then dT(t) oscillates regularly around zero; alternatively, T(t) osciallates around the average value of <T>.

Put another way, your house spends as much time slightly above <T> as it does slightly below <T>. Thus, your a/c turns on, the temperature drops, the a/c cuts off, the temperature rises. Rinse and repeat.

So what's this mean? Well, if you then integrate over dT(t) over your 24 hour period, you just get zero. Or, if you integrate T(t), you get <T>, the average. The first measures the fluctuations, the second the average. If your system is in good repair, there should be no bias in the fluctuations, and thus you get zero.

So how then a zero correlation? Well, in one of its technical definitions, a correlation function can be referring strictly to the integral over dT(t), in which case multiplying by a constant (power use while the a/c is running) doesn't change the integral. You just get zero again.

Or, if you instead use a slightly different definition for correlation, the one that refers to the integral over T(t), you get a correlation of 1. Which doesn't tell you anything either, since you get 1 for 74 degrees on the thermostat, and 1 for 72 degrees on the thermostat as well.

How then do you tell the difference?

There are at least three ways. One of them is something that I've hidden: the normalization. Correlations are often defined similarly to probabilities, and so restricted between a range of -1 to 1 inclusive.

However, for thermostats and temperature, this normalization condition is inverted by multiplying by <T> and <P>, the average temperature and power, respectively. In which case, the product <T><P> will be slightly different for 72 degrees versus 74 degrees. And that's how your power company knows that you used more electricity cooling your house at 72 degrees than you would have cooling it at 74 degrees.

No really, that's pretty much exactly what you kilowatt-hours charge on your electric bill is. You could divide by various things and turn the total kilowatt-hours used per month into a correlation, but your total bill won't change. You still used a given number of kilowatt-hours total for that month, and you would have used less or more if you'd used a slightly different thermostat setting.

That's one way to measure a non-trivial correlation, what's another?

Use the absolute value for T(t)-<T>, rather than the scalar value. In which case, abs(dT(t)) is positive definite, which in turn sums to a non-zero number over any 24 hour period, and you then in turn get a real, and useful measure. If you've ever stumbled over the difference between velocity (a vector) and speed (a positive definite scalar), it's precisely the same thing.

The third method is just a variation of these two, and in fact counts the same thing (to within a scalar conversion factor if your system is in good shape): measure the number of times your system turns on in a given period. In summer, the system will come on more often at 72 than at 74, and you will again have a useful correlation measure between power usage and temperature setting.

Caveats? Well, for a physical system, one thing that's lurking here is that you can re-run the experiment. You can go in and set your thermostat at 72, count the number of cycles, and then go in tomorrow and redo the experiment at 74. If today and tomorrow have essentially the same weather, you can readily measure the difference.

In macroeconomics and history, to pick two examples, you don't often get the chance to re-run an experiment. The best you can hope for, to this outsider, is to find moments in time where today looks enough like yesterday that comparing the two gives you something close to a controlled comparison.

Note also that this gives you one way to tell, assuming the differences don't wash out in the variations due to the weather, that your system is in good running condition. This is one of the ways, in fact, that smart thermostat systems can help. Over time, your data set will be robust enough so that "assuming your system is in good shape" notes above will then give your system a potential means of self-diagnosing when something is going wrong.

Meaning: this summer went ok, next summer goes ok, third summer and all of a sudden your thermostat notices that it can no longer count on the relations it's been measuring. Call a repair team out before something goes really wrong.

I could go into measurement theory; indeed, this is why quantum mechanics (the folks) scratch their head when they hear about quantum mechanics (the popular speculative topic) from others. What's a measurement, indeed?

I won't. Because really what we're talking about is the difference between training and practice or craft. When I teach a class on something, I'll of necessity need to give you the highlights. These are what I hope you'll retain, and I'll hope as well that if you stick with the subject, in using it you'll learn the finer points.

Which is fine for those who go on to practice and learn the craft. It's when you get popular speculation based on the highlights only that you run into these mass confusion things. Terminology that everyday practitioners use verus how that terminology is understood by observers often turns into the most difficult hurdle.