Computes 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:
$$
\begin{pmatrix}
\cos(\lambda_1) & \sin (\lambda_1) & \cdots &
\cos(\lambda_6) & \sin (\lambda_6) \\
\cos(\lambda_1\times 2) & \sin (\lambda_1\times 2) & \cdots &
\cos(\lambda_6\times 2) & \sin (\lambda_6\times 2)\\
\vdots & \vdots & \cdots & \vdots & \vdots \\
\cos(\lambda_1\times 12) & \sin (\lambda_1\times 12) & \cdots &
\cos(\lambda_6\times 12) & \sin (\lambda_6\times 12)
\end{pmatrix}
$$