2. Working with Time Series

Mauna Loa atmospheric carbon dioxide time series (5/5)

Cyclic components cont.

We will now use this frequency data to separate the noise from the other cyclic components.

Noise is defined as the random variations that occur from observation to observation in a dataset. This means that they must only occur in the very high frequency components of this frequency data.

The frequencies selected in the next graph will be removed from the frequency data by filtering them out, thereby removing the noise.

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Frequencies to be removed
The frequencies selected to be removed.

Once these frequencies have been removed, then the remaining frequency data is inverted by processing with the inverse Fourier Transform, so as to get data in the real (or time) domain.

Real data or data in the time domain can be processed using the Fourier Transform to get frequency data, and then this dataset can be inverted through the inverse Fourier Transform to get real or time domain data.

Doing this gives the result shown below, which represents the non-annual cyclic components in the data.

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Residuals dataset
The residuals dataset that remains after the high frequency components have been filtered out. You will note that this dataset is much smoother than the residuals used as the source for this filtering task.

 

Now, we can subtract this signal from the signal that we started this page with, to get the noise component.

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Remaining residuals
The residuals that remain after the residuals dataset above are removed from the dataset used as the source to this analysis. This dataset still contains some structure.

What do you think of this? Do you think that it still contains some cyclic components? It does seem to, even though it is very small. Maybe we should just have filtered out the highest frequency?

So, now you can see that a time series can consist of:

  1. a long term trend, which may or may not be cyclic itself if the time series is long enough
  2. an annual cycle that can actually change from year to year
  3. other cyclic components
  4. a random or noise component