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ETS model - non linearity. Weekly Challenge #170

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Asteroid

Alexeryx version 2019.2.5.62427 - installed in the morning

ETS (Summer1 + Summer2) = 702.87

ETS (Summer1) + ETS (Summer2) = 356.67 + 339.68 = 696.35 <> ETS (Summer1 + Summer2)

On my mind the forecast of a sum it should be equal the sum of a forecasts:

- trend is additive

- seasonalyty should have same periods

And if I have a huge ammount off addends I am interested in additive forecast.

Nebula
Nebula

Hi @klyuka 

 

I believe that your conjecture "forecast of a sum it should be equal the sum of a forecasts:" can only be true if the individual forecasts can be shown to be completely independent.  In "Optimal combination forecasts for hierarchical time seriesby Rob J.Hyndman, Roman A.Ahmed, et al., reproduced in full here, the authors discuss this topic in detail.   The key point they make here is the following

 

The various components of the hierarchy can interact in varying and complex ways. A change in one series at one level, can have a consequential impact on other series at the same level, as well as series at higher and lower levels

 

leading to 

 

Of course, it is also possible to forecast all series at all levels independently, but this has the undesirable consequence of the higher level forecasts not being equal to the sum of the lower level forecasts. Consequently, if this method is used, some adjustment is then carried out to ensure the forecasts add up appropriately. These adjustments are usually done in an ad hoc manner.

 

They then proceed to discuss a method whereby they use a regression model to reconcile and combine the forecasts across all the levels. 

 

Dan 

 

 

Asteroid

Thank you for your idea, Dan!

 

I know nothing about TS. Correlation between processes makes is a case.

 

This work if processes have correlation with lags. For example - Markov chain:

* Process 1 - people come on first part of the lecture:

  0->1 with 20% probability, 1->0 with 100% probability. E(next | 0) = 2/10,  E(next | 1) = 0  

* Process 2 - people deside the next day after visited part one to visit part two with 50% probability.

  E(next | 1) = 0, E(next | 0) = 1/10 once we do not know Process 1 realization but E(next | process1 = 1) = 1/2

 

Once we see total 1 today (without knowing history) - 2/3 it is process 1, 1/3 - it is project 2.

E(next total | total = 1) = 1/3*2/10 + 2/3 * 1/2 = 4/10

 

However if two processes are correlated by depending on the third process

w1 = F(w3, independent 1), w2 = G(w3, independed 2)

itprobably that the sum is additive.

But if F and  G are nonlinear and have noises than knowing other process will give more information on W3 (that is not observed).

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