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I am very new to time series. I am wonder why would we want to detrend or removing some seasonality of some time series remove important information about the time series itself?

For example this is from my notes enter image description here

By removing noise, doesn't that create a white noise process. A white noise process itself is random so no prediction can be made?

Thank you!!

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The key point is that when removing trend, seasonality or noise, we don't just delete information. We take that information apart in order to analyze separately each part of the behaviour. For example, if we are interested in seasonal effects, we still have the seasonal variation we have removed, isolated from other components.

When we remove seasonality and trend, we can tell what is the effect of the trend, what is the effect of the seasons, and what is the effect that isn't accounted for by season nor trend and that should reveal another - hopefully interesting - phenomenon.

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One way to think about the seasonality problem is this: Is seasonality of interest or is it just a nuisance?

How you deal with seasonality depends on the answer to this question.

As an example, if you are interested in describing the pattern of seasonality in a time series of monthly temperatures, then seasonality is of interest and you will not want to remove it - rather, you will want to directly model it (e.g., using sine and cosine terms).

However, if you are interested in seeing the association between two time series (e.g., monthly temperatures and monthly relative humidity in the same area for a multi-year period), then you would want to remove the seasonality present in the two time series first and then correlate the residual series to get at the association of interest.

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