Generates trigonometric variables at different frequencies.
Arguments
- frequency
Frequency of the series, number of periods per year (12, 4, 3, 2...)
- start, length
First date (array with the first year and the first period, for instance
c(1980, 1)) and number of periods of the output variables. Can also be provided with thesargument- s
time series used to get the dates for the trading days variables. If supplied the parameters
frequency,startandlengthare ignored.- seasonal_frequency
the seasonal frequencies. By default the fundamental seasonal frequency and all the harmonics are used.
Details
Denote by \(P\) the value of frequency (= the period) and
\(f_1\), ..., \(f_n\) the frequencies provides by seasonal_frequency
(if seasonal_frequency = NULL then \(n=\lfloor P/2\rfloor\) and \(f_i\)=i).
trigonometric_variables returns a matrix of size \(length\times(2n)\).
For each date \(t\) associated to the period \(m\) (\(m\in[1,P]\)),
the columns \(2i\) and \(2i-1\) are equal to:
$$
\cos \left(
\frac{2 \pi}{P} \times m \times f_i
\right)
\text{ and }
\sin \left(
\frac{2 \pi}{P} \times m \times f_i
\right)
$$
Take for example the case when the first date (date) is a January, frequency = 12
(monthly time series), length = 12 and seasonal_frequency = NULL.
The first frequency, \(\lambda_1 = 2\pi /12\) represents the fundamental seasonal frequency and the
other frequencies (\(\lambda_2 = 2\pi /12 \times 2\), ..., \(\lambda_6 = 2\pi /12 \times 6\))
are the five harmonics. The output matrix will be equal to:
$$
\begin{pmatrix}
\cos(\lambda_1) & \sin (\lambda_1) & \cdots &
\cos(\lambda_6) & \sin (\lambda_6) \\
\cos(\lambda_1\times 2) & \sin (\lambda_1\times 2) & \cdots &
\cos(\lambda_6\times 2) & \sin (\lambda_6\times 2)\\
\vdots & \vdots & \cdots & \vdots & \vdots \\
\cos(\lambda_1\times 12) & \sin (\lambda_1\times 12) & \cdots &
\cos(\lambda_6\times 12) & \sin (\lambda_6\times 12)
\end{pmatrix}
$$
Examples
trigonometric_variables(
frequency = 12,
length = 60,
start = c(2020, 1),
seasonal_frequency = 12
)
#> Series 1 Series 2
#> Jan 2020 1 -2.038010e-13
#> Feb 2020 1 2.011429e-15
#> Mar 2020 1 -2.469235e-13
#> Apr 2020 1 -4.111099e-14
#> May 2020 1 -2.900459e-13
#> Jun 2020 1 -8.423342e-14
#> Jul 2020 1 -3.331683e-13
#> Aug 2020 1 -1.273558e-13
#> Sep 2020 1 -3.762907e-13
#> Oct 2020 1 -1.704783e-13
#> Nov 2020 1 3.533420e-14
#> Dec 2020 1 -2.136007e-13
#> Jan 2021 1 -7.788221e-15
#> Feb 2021 1 -2.567231e-13
#> Mar 2021 1 -5.091064e-14
#> Apr 2021 1 -2.998455e-13
#> May 2021 1 -9.403307e-14
#> Jun 2021 1 -3.429680e-13
#> Jul 2021 1 -1.371555e-13
#> Aug 2021 1 6.865697e-14
#> Sep 2021 1 -1.802779e-13
#> Oct 2021 1 2.553455e-14
#> Nov 2021 1 -2.234003e-13
#> Dec 2021 1 -1.758787e-14
#> Jan 2022 1 -2.665228e-13
#> Feb 2022 1 -6.071029e-14
#> Mar 2022 1 -3.096452e-13
#> Apr 2022 1 -1.038327e-13
#> May 2022 1 -3.527676e-13
#> Jun 2022 1 -1.469551e-13
#> Jul 2022 1 5.885732e-14
#> Aug 2022 1 -1.900776e-13
#> Sep 2022 1 1.573490e-14
#> Oct 2022 1 -2.332000e-13
#> Nov 2022 1 -2.738752e-14
#> Dec 2022 1 -2.763224e-13
#> Jan 2023 1 -7.050994e-14
#> Feb 2023 1 -3.194448e-13
#> Mar 2023 1 -1.136324e-13
#> Apr 2023 1 -3.625673e-13
#> May 2023 1 -1.567548e-13
#> Jun 2023 1 4.905767e-14
#> Jul 2023 1 -1.998772e-13
#> Aug 2023 1 5.935251e-15
#> Sep 2023 1 -2.429996e-13
#> Oct 2023 1 -3.718717e-14
#> Nov 2023 1 -2.861221e-13
#> Dec 2023 1 -8.030959e-14
#> Jan 2024 1 -3.292445e-13
#> Feb 2024 1 -1.234320e-13
#> Mar 2024 1 -3.723669e-13
#> Apr 2024 1 -1.665544e-13
#> May 2024 1 -4.154893e-13
#> Jun 2024 1 2.450705e-13
#> Jul 2024 1 -3.864399e-15
#> Aug 2024 1 -2.527993e-13
#> Sep 2024 1 -5.017342e-13
#> Oct 2024 1 1.588256e-13
#> Nov 2024 1 -9.010925e-14
#> Dec 2024 1 -3.390441e-13