5.6 Forecasting using transformations

[Amarjit_Chandhial]

Hi,

Quantiles (such as the median) are preserved under monotonic transformations. 5.6 Forecasting using transformations | Forecasting: Principles and Practice (3rd ed) (otexts.com) does not explain why "the back-transformed point forecast will not be the mean of the forecast distribution", and "will usually be the median of the forecast distribution (assuming that the distribution on the transformed space is symmetric)"?
Also, why a Taylor Series expansion produces the bias-adjusted back transformed means Forecasting with transformations • fable (tidyverts.org)?

I'd be grateful for an explanation.

Amarjit

[robjhyndman]

In general, for a random variable Y and a function f, E(f(Y)) is not equal to f(E(Y)). But if f is monotonically increasing, then Pr(Y <= y) = Pr(f(Y) <= f(y)).

To compute E(f(Y)), we can expand f using a Taylor series expansion, and then compute the expectation of the first few terms. See Rob J Hyndman - The forecast mean after back-transformation for further details.

[Amarjit_Chandhial]

Thanks.

E is the expected value or mean. Pr() denotes probability.

I'll accept that latter regarding Taylor Series expansion.

However, there is no mention of median in the former, I don't understand:

@robjhyndman

But if f is monotonically increasing, then Pr(Y <= y) = Pr(f(Y) <= f(y)).

An example would be useful.

[robjhyndman]

The median is m such that Pr(Y <= m) = 0.5. Therefore , Pr(f(Y) <- f(m)) = 0.5 and f(m) is the median of the transformed variable.

[Amarjit_Chandhial]

Sorry, I don't understand?