Fourier Series Model with Two Harmonics (Sine-Cosine Form)
Source:R/fourier_series_double.R
fourier_series_double.RdThis function implements a generalized Fourier series model with two harmonics in the sine-cosine form. The model is often used to capture seasonal or periodic dynamics in time series data, such as temperature, precipitation, or disease cases.
Arguments
- t
A numeric vector representing time points (e.g., day of the year or week of the year).
- beta0
A numeric value representing the intercept term (fixed at 0 by default).
- a_1
A numeric value representing the amplitude of the first cosine term (first harmonic).
- b_1
A numeric value representing the amplitude of the first sine term (first harmonic).
- a_2
A numeric value representing the amplitude of the second cosine term (second harmonic).
- b_2
A numeric value representing the amplitude of the second sine term (second harmonic).
- p
A numeric value representing the period of the seasonal cycle (e.g., 52 for weekly data over a year).
Details
The model is defined as follows:
$$\beta_t = \beta_0 + a_1 \cos\left(\frac{2 \pi t}{p}\right) + b_1 \sin\left(\frac{2 \pi t}{p}\right) + a_2 \cos\left(\frac{4 \pi t}{p}\right) + b_2 \sin\left(\frac{4 \pi t}{p}\right)$$
The model includes an intercept term beta0 (set to 0 by default) and two harmonics with coefficients a_1, b_1, a_2, and b_2. The period p controls the periodicity of the series.
This function is based on the sine-cosine form of the Fourier series. For more details, see the Wikipedia page on Fourier series.
This function is commonly used for modeling seasonal dynamics in environmental or epidemiological data, where periodic patterns are observed.
Examples
if (FALSE) { # \dontrun{
# Example usage with weekly data (p = 52 weeks in a year)
time_points <- 1:52
beta0 <- 0
a_1 <- 1.5
b_1 <- -0.5
a_2 <- 0.8
b_2 <- 0.3
p <- 52
predictions <- fourier_series_double(time_points, beta0, a_1, b_1, a_2, b_2, p)
print(predictions)
} # }