Forecasting in Python with Prophet

Re:Forecasting in Python with Prophet by CapmCrackaWaka on Friday February 22, 2019 @06:28PM (#19230157)

The assumption that your target is normally distributed about the expected value. Let's say you take the log of your target, and run a regression using RMSE. You now assume that log(Y) is normally distributed. Now, for a given sample, your model outputs E[log(Y)|X]. However, exp(E[log(Y)|X]) is not the same as E[exp(log(Y)|X], which is what you really want. It can be shown mathematically that in order to get to E[exp(log(Y)|X], you need to multiply by an offset factor, which in this case is exp(var(E[log(Y)]-log(Y)/2), which is how you would normally transform between the mean of a normal distribution to a lognormal distribution.

Here is a paper which addresses the issue in the introduction. https://www.ncbi.nlm.nih.gov/pmc/articles/PMC4024993/

[ Parent | on HN ]

Re:Forecasting in Python with Prophet by wenc on Friday February 22, 2019 @08:32PM (#19230971)

Let me see if I understood you correctly. Let Y represent a random variable representing stationary time series data. Let g(.) be the forecast function.

1) Log-transformation: log(Y)

2) Forecast results of transformed data: E[g(log(Y))]

3) Inversion of forecast results: exp(E[g(log(Y))])

4) However, what we really want is the expectation of the transformed-forecasted-inversed results, which is E[exp(g(log(Y)))].

Jensen's inequality states that for a convex function φ,

φ(E[X]) <= E[φ(X)]

And equality is only attained if either X is constant or φ is affine. Since neither is (generally) the case here, therefore

exp(E[g(log(Y))]) < E[exp(g(log(Y)))]

An offset factor ε is needed to correct (3) to (4)

E[exp(g(log(Y)))] = exp(E[g(log(Y))]) * ε

Did I get that right? (Offset changed to multiplicative factor)

[ Parent | on HN ]

Re:Forecasting in Python with Prophet by CapmCrackaWaka on Friday February 22, 2019 @09:05PM (#19231126)

Yes, exactly. However in this case, the offset is multiplicative.

[ Parent | on HN ]

Re:Forecasting in Python with Prophet by wenc on Saturday February 23, 2019 @09:19AM (#19233396)

Understood. To add one more point, I'm noticing the reason the above works the way it does is because most forecast algorithms output an expected value instead of a random variable, hence the results are E[g(log(Y)] instead of just g(log(Y)).

It strikes me that if you package the entire thing as a random variable:

Z = exp(G(log(Y)))

and use a different kind of forecast function G : Y -> Y' where Y, Y' ~ Normal, then we don't need the multiplicative factor -- which can be difficult to calculate for an arbitrary transformation. We can just get the expected value of Z, ie. E[Z] = E[exp(G(log(Y)))]. This is not done in the article, but in theory it could be.

[ Parent | on HN ]