A question on Fourier seasonal model

[hardmann]

Dear Tom:

A additive seasonal model can be easily understanded, however, fourier seasonal model is more complicated. How to understand separate two-states and variable lambda?
how to specify SW matrix. Could you give further detial and examples, even comparison between additive and fourier model.

Best regard.
Hardmann

[TomDoan]

The two types of seasonal models are covered in some detail as part of the State-Space e-course. The additive model is more appropriate when the seasonal is driven primarily by calendar events such as Christmas since it says that this December is similar to last December, but January isn't necessarily similar to December. The Fourier model is more appropriate when it's weather-driven since it's a sum of smooth curves.

Examples of the Fourier model are durkp167.rpf, harveyp095.rpf, westp257.rpf, westp318.rpf and westp387.rpf. The Harvey example, for instance, is energy consumption, which would be an obvious situation for the Fourier rather than the additive.

The values for the variances on the components will vary quite a bit from application to application.

[hardmann]

appreciate

[hardmann]

Dear Tom:
I had read E_course of state space and struture time series model(harvey,A.C), however, I am still confused.I do not fully understand description as follows:
1. (1-L)*(1+L)*(1-2cos(2*PI/S*L+L^2))*...The second degree terms represent the harmonics of the seasonal: the cycles that are an exact fraction of the seasonal. For monthly data, these are cycles of 12 months, 6 months, 4 months, 3 months and 12/5 months. (The two month cycle is covered by the two “real” terms). Any regular cycle with those lengths will, of necessity, also repeat at 12 months. what's meaning of 'The two month cycle is covered by the two “real” terms'.
2. The harmonic or trigonometric or Fourier seasonal model is to include a separate (independent) two-state model for each of the harmonics, with an extra one state model for the period 2 harmonic. why and how an extra one state work?
3. for generlly form
st = cos(lambda)*st-1 + sin(lambda)*sstar_t-1 + kt
sstar_t = -sin(lambda)*st-1 + cos(lambda)*sstar_t-1 + kstar_t
where lambda=2*PI*j/S,j=1,...,S-1

for quarterly data, could you give expression of transition equation inluding both st and s_star_t.

Best reagrd
hardmann

[TomDoan]

Dear Tom:
I had read E_course of state space and struture time series model(harvey,A.C), however, I am still confused.I do not fully understand description as follows:
1. (1-L)*(1+L)*(1-2cos(2*PI/S*L+L^2))*...The second degree terms represent the harmonics of the seasonal: the cycles that are an exact fraction of the seasonal. For monthly data, these are cycles of 12 months, 6 months, 4 months, 3 months and 12/5 months. (The two month cycle is covered by the two “real” terms). Any regular cycle with those lengths will, of necessity, also repeat at 12 months. what's meaning of 'The two month cycle is covered by the two “real” terms'.

The two real root factors combine to give you (1-L)(1+L)=1-L^2.

2. The harmonic or trigonometric or Fourier seasonal model is to include a separate (independent) two-state model for each of the harmonics, with an extra one state model for the period 2 harmonic. why and how an extra one state work?

There aren't multiple roots for the period 2 harmonic. It's either down-up-down or up-down-up. The other harmonics have a pair of complex roots which allows for different phases.

3. for generlly form
st = cos(lambda)*st-1 + sin(lambda)*sstar_t-1 + kt
sstar_t = -sin(lambda)*st-1 + cos(lambda)*sstar_t-1 + kstar_t
where lambda=2*PI*j/S,j=1,...,S-1

for quarterly data, could you give expression of transition equation inluding both st and s_star_t.

The one "complex root" is with lambda=pi/2, so cos(lambda) is 0 and sin(lambda)=1. If you work through successive substitutions, you will see that eventually that would give you s_t approximately s_{t-4}

If what's confusing you is how you get from

ss seasonal.gif (924 Bytes) Viewed 9472 times

to the representation with the s and s*, that follows from the transformation of the model by

ss transform.gif (2.62 KiB) Viewed 9472 times

If A and B are diagonalizable with the same set of eigenvalues, you can make this model transformation. (That's a sufficient, but not a necessary condition for being able to do the model transformation). You can verify that that's the case here.