Well, that is not entirely true. The future-prediction experts whose works I am carefully perusing note that there is also a family of future prediction procedures that expect the future to be pretty much exactly like the past and the present.

Other than that there are only two methods in play for predicting the future: linear and exponential.

NB: I am ignoring futures with seasonality. The futures that I am predicting do not need to worry about Christmas or whatever.

OK, well this paper that I’m reading actually has five types of non-seasonal futures. But it didn’t really come out any say that early enough in the paper for me. It starts out with things like “Automatic forecasts of large numbers of univariate time series are often needed in business” (yes, I believe that). From there it goes on to give a full taxonomy of the models, describe the related literature, and define a lot of notation. But it makes you read through all the equations in order to realize how primitive our understanding of future prediction really is.

It’s not until you read through all of the equations that you realize that there are only five (or two, or three) types of futures.

  1. There is the “San Diego weather” future. Whatever is happening today will happen tomorrow. This is actually not a great example because we do have seasonality! We have many, many seasons in San Diego. Not too long ago, we had First Summer. Then we had Ant Season. Now we are in Second Summer. Next is Fire Season. After that is Second Spring. That’s followed by Rain. Then we have First Spring. After that comes May Gray and June Gloom (this is one season whose name changes partway through). And then we cycle back to First Summer.

  2. The linear future. In the linear future, you do a lot of calculations in order to find Just The Right Slope. A whole bunch of parameters are optimized to figure out how much of the past goes into brewing your magical Goldilocks slope. Then you take the last of your real data points and you just keep adding this same magical slope to it for however many steps you want to predict you future. Seriously, that is the world-class, state-of-the-art in future prediction. Just set your future loose on a linear trajectory.

  3. Well, there is also the damped linear model. You have both a slope \(b_t\) and a constant parameter \(\phi \in (0, 1)\), and you predict the value \(h\) steps in the future by \(\hat{y}_{t+h\vert t} = \ell_t + \phi_h b_t\), where \(\phi_h = \phi + \phi^2 + \phi^3 + \cdots + \phi^h\). In this case \(\ell_t\) is the value that you predicted at step \(t\).

  4. Exponential future. Just like the linear future, except you forecast with \(\hat{y}_{t+h\vert t} = \ell_t b_t^h\).

  5. Damped exponential future. Like above. Instead of raising \(b_t\) to the \(h\) power, you raise it to the \(\phi_h\) power.

According to the people who make their academic livelihood predicting the future and assessing different methods of predicting the future, these five models (and the 10 variants that come up if your future cares about Christmas—five of them for a an affine Christmas and five of them for a multiplicative Christmas) are the world’s best future predicting methods.

This is a little bit disappointing and a little bit underwhelming. How can you impress people if you are predicting a future based on one (or two) simple-to-understand paramaters?

And that must be why people have developed very sophisticated future-predicting neural networks. No one knows how the algorithm is predicting the future, so it can not be belittled for being purely linear. (Instead, it is likely an overfit polynomial, but, whatever.)