Fourier Series Model with Two Harmonics (Sine-Cosine Form)
+ +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).
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+- beta0 +
A numeric value representing the intercept term (fixed at 0 by default).
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+- a1 +
A numeric value representing the amplitude of the first cosine term (first harmonic).
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+- b1 +
A numeric value representing the amplitude of the first sine term (first harmonic).
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+- a2 +
A numeric value representing the amplitude of the second cosine term (second harmonic).
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+- b2 +
A numeric value representing the amplitude of the second sine term (second harmonic).
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+- p +
A numeric value representing the period of the seasonal cycle (e.g., 52 for weekly data over a year).
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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 a1, b1, a2, and b2. 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) {
+# Example usage with weekly data (p = 52 weeks in a year)
+time_points <- 1:52
+beta0 <- 0
+a1 <- 1.5
+b1 <- -0.5
+a2 <- 0.8
+b2 <- 0.3
+p <- 52
+
+predictions <- fourier_series_double(time_points, beta0, a1, b1, a2, b2, p)
+print(predictions)
+}
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+