Solved – How to decompose a time series with multiple seasonal components

decompositionforecastingmultiple-seasonalitiesrtime series

I have a time series that contains double seasonal components and I would like to decompose the series into the following time series components (trend, seasonal component 1, seasonal component 2 and irregular component). As far as I know, the STL procedure for decomposing a series in R only allows one seasonal component, so I have tried decomposing the series twice. First, by setting the frequency to be the first seasonal component using the following code:

ser = ts(data, freq=48)
dec_1 = stl(ser, s.window="per")

Then, I decomposed the irregular component of the decomposed series (dec_1) by setting the frequency to be the second seasonal component, such that:

ser2 = ts(dec_1$time.series[,3], freq=336)
dec_2 = stl(ser2, s.window="per")

I'm not very confident with this approach. And I would like to know if there are any other ways to decompose a series that has multiple seasonalities. Also,I have noticed that the tbats() function in the R forecast package allows one to fit a model to a series with multiple seasonalities however, it doesn't say how to decompose a series with it.

Best Answer

R's forecast package bats() and tbats() functions can fit BATS and TBATS models to the data. The functions return lists with a class attribute either "bats" or "tbats". One of the elements on this list is a time series of state vectors, $x(t)$, for each time, $t$.

See http://robjhyndman.com/papers/complex-seasonality/ for the formula's and Hyndman et al (2008) for a better description of ETS models. BATS and TBATS are an extension of ETS.

For example:

fit <- bats(myTimeseries)
fit$x

In this case, each row of x will be on fourier-like harmonic.

There are also plot.tbats() and plot.bats() functions to automatically decompose and view the components.

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Solved – Interpreting time series decomposition using TBATS from R forecast package

In the user comments on this page, somebody asks about the interpretation of the level and slope, and also how to get the trend and residuals that the decompose() function provides. Hyndman remarks that there isn't a straight translation as decompose() and tbats() use different models. But if your TBATS model doesn't have a Box-Cox transformation, then the TBATS level is roughly the same as the decompose() trend. If on the other hand the model does apply the Box-Cox transformation, then you have to undo the transformation before interpreting the level as (roughly) the trend. At least that's how I interpret his response.

As for residuals and slope, they're not the same.

You can think of a basic decomposition has having a trend component, a seasonal component, and a residual component.

You can break the trend down further into a level and a slope. The level is essentially a baseline for the trend, and the slope is the the change per unit time.

The reason for breaking the trend down into a level and a slope is that some models support damped growth. Maybe you observe current growth, but you expect growth to diminish gradually over time, and you want your forecasts to reflect that expectation. The model supports this by allowing you to damp growth by applying a damping factor to the slope, making it converge toward zero, which means that the trend converges toward its level component.

There's not a straightforward answer to the question of how the level and slope combine to yield the trend. It depends on the type of model you are using. As a general statement, additive trend models combine them in an additive fashion and multiplicative trend models combine them in a multiplicative way. The damped variants of models combine the level with a damped slope. Hyndman's Forecasting with Exponential Smoothing book (hope it's ok to include the Amazon link—I have no affiliation whatsoever with the author) provides the exact equations on a per-model basis in Table 2.1.

Solved – time series decomposition/dtrending using splines

Could you use constrained B-splines from the R library cobs?

co <- cobs(x, y, lambda=-1)

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Solved – Interpreting time series decomposition using TBATS from R forecast package

Solved – time series decomposition/dtrending using splines

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