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FourierFeatures

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Multivariate

Fourier Features for time series seasonality.

Fourier Series terms can be used as explanatory variables for the cases of multiple seasonal periods and or complex / long seasonal periods [1], [2]. For every seasonal period, \(sp\) and fourier term \(k\) pair there are 2 fourier terms sin_sp_k and cos_sp_k:

  • sin_sp_k = \(sin(\frac{2 \pi k t}{sp})\)

  • cos_sp_k = \(cos(\frac{2 \pi k t}{sp})\)

Where \(t\) is the elapsed time since the beginning of the seasonal period and \(sp\) the total time of the seasonal period.

The transformed output is a series that contains all requested Fourier terms.

Warning: the output will contain only the Fourier terms under default settings, and discard the original columns of the input data, to avoid multiplication of the original data in a pipeline or FeatureUnion. To keep the original columns, set keep_original_columns=True.

Names of the columns are generated as follows: additional columns with the naming convention stated above (sin_sp_k and cos_sp_k). The numbers of Fourier terms \(K\) in the fourier_terms_list determines the number of Fourier terms that will be used for each seasonal period, i.e., Fourier terms \(k = 1\dots K\) (integers), cos and sine, will be generated for the seasonality \(sp\) at the same list index. For example, consider sp_list = [12, “Y”] and fourier_terms_list = [2, 1]. This says that we compute 2 (2 cos, 2 sine) Fourier terms for seasonality 12 periods, and 1 Fourier term (1 cos and 1 sine) for seasonality 1 year. The transformed series will then have columns with the following names: “cos_12_1”, “sin_12_1”, “cos_12_2”, “sin_12_2”, “cos_Y_1”, “sin_Y_1”

The implementation is based on the fourier function from the R forecast package [3]

Quickstart

python
from sktime.transformations.fourier import FourierFeatures

estimator = FourierFeatures(sp_list: list [float ], fourier_terms_list: list [int ], freq: str | None=None, keep_original_columns: bool | None=False)

Parameters(4)

sp_listList[float and/or str]

List of seasonal periods. Can be defined with the following options:

fourier_terms_listList[int]

List of number of fourier terms (\(K\)) per corresponding (\(sp\)); each \(K\) matches to one \(sp\) of the sp_list. For example, if sp_list = [7, “Y”] and fourier_terms_list = [3, 9], the seasonality of 7 timesteps will have 3 sin_sp_k and 3 cos_sp_k fourier terms and the yearly seasonality “Y” will have 9 sin_sp_k and 9 cos_sp_k fourier terms.

freqstr, optional, default = None

Only used when X has a pd.DatetimeIndex without a specified frequency. Specifies the frequency of the index of your data. The string should match a pandas offset alias:

https://pandas.pydata.org/pandas-docs/stable/user_guide/timeseries.html#offset-aliases

keep_original_columnsboolean, optional, default=False

Keep original columns in X passed to .transform()

Examples

>>> from sktime.transformations.fourier import FourierFeatures
>>> from sktime.datasets import load_airline
>>> y = load_airline ()
>>> transformer = FourierFeatures (sp_list = [12, "Y" ], fourier_terms_list = [4, 1 ])
>>> y_hat = transformer. fit_transform (y)

References

[1]

Hyndsight - Forecasting with long seasonal periods: https://robjhyndman.com/hyndsight/longseasonality/

[2]

Hyndman, R.J., & Athanasopoulos, G. (2021) Forecasting: principles and practice, 3rd edition, OTexts: Melbourne, Australia. OTexts.com/fpp3. Accessed on August 14th 2022.