Create a Basis of Sine Waves with Given Phases

Basis of sine wave functions $s_k(t)$ with the phases $\phi_k$ given in phi corresponding to the periods $365.25 / k$.

Usage

sinBasis(dt, df = 7, phi, cst = TRUE, keepTrig = FALSE)

Arguments

dt: A vector with class "Date" or "POSIXct" representing the days at which the sine waves basis functions are to be evaluated.
df: Odd number as in tsDesign.
phi: Vector of $K$ phases where $K$ is equal to (df - 1) / 2.
cst: Logical. If TRUE a constant column with value 1.0 is added as a first column.
keepTrig: Logical. If TRUE, the cos-sin functions are joined as columns of the result. So the returned matrix then does not correspond to a basis. Used mainly for tests.

Value

A numeric matrix containing as columns the $K$ basis functions $s_k(t)$, and with its rows corresponding to the time $t$ given by Date. The names of the $K$

sine wave function have the prefix "sinjPhi" so are

"sinjPhi1", "sinPhij2", ... This is aimed to recall that the phases $\phi_k$ were given. If cst

is TRUE and keepTrig a constant column with value 1.0 is added, and if keepTrig is TRUE the trigonometric functions (with phase zero) are added as well.

Details

For a time $t$ the value of the $k$-th function is given by

$$s_k(t) := \sin\{2 \pi k [j_t - \phi_k] / 365.25)$$

for $k=1$ to K, where $j_t$ is the Julian day corresponding to $t$. This function is useful to provide a basis of functions with yearly seasonality that are suitable for meteorological variables. For instance the phase for the first harmonic of the daily temperature in France is such that the annual maximal temperature "normally" happen at the end of July.

Note

This functions uses the tsDesign function with type = "trigo".

Caution

This function is likely to be removed. It is safer to use tsDesign with the argument phi.

Examples

## Use only full years to maintain near orthogonality
df <- subset(Rennes, Year <= 2020 & Year >= 1960)
K <- 3
## Design of trigonometric functions
desTrig <- tsDesign(dt = df$Date, df = 2 * K + 1, type = "trigo")
fit <- lm(formula = TX ~ Cst + cosj1 + sinj1 + cosj2 + sinj2 + cosj3 + sinj3 - 1,
          data = data.frame(df, desTrig$X))
betaHat <- coef(fit)

## find the phases
phiHat <- phases(betaHat)
gamma <- attr(phiHat, "amplitude")

## Design of sine waves with prescribed phases
desSin <- tsDesign(dt = df$Date, df = 2 * K + 1,
                   type = "sinwave", phi = phiHat)
fit2 <- lm(formula = TX ~ Cst + sinjPhi1 + sinjPhi2 + sinjPhi3 -1,
           data = data.frame(df, desSin$X))
## should be zero
max(abs(predict(fit) - predict(fit2)))
#> [1] 1.24345e-14

## the amplitude and the coef
rbind(coef(fit2), c(NA, gamma))
#>           Cst sinjPhi1  sinjPhi2   sinjPhi3
#> [1,] 16.20282 7.937385 0.8769553 0.04824877
#> [2,]       NA 7.937385 0.8769553 0.04824877
j <- 3
indTrig <- paste0(c("cosj", "sinj"), j)
indSin <- paste0("sinjPhi", j)
plot(desTrig$X[1:366 , indTrig] %*% betaHat[indTrig], type = "l",
     col = "SpringGreen3", xlab = "", ylab = paste("harmonic", j))
lines(desSin$X[1:366, indSin] * gamma[j], col = "red", lwd = 2, lty = "dashed")