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Sin

A Generalized Trigonometric Seasonal Pattern

Source: vignettes/Sin.Rmd
Sin.Rmd

A constructor for seasonality functions drawn from a generalized family involving trigonometric functions is returned by make_function.sin with the associated makepar_F_sin that returns functions of the form: \[F_S(t) = c\left(1 + \epsilon + \sin\left(\frac{2 \pi (t-\tau)}P\right)\right)^q\]

  • \(c\) or norm is a normalizing constant

  • \(\tau\) or phase sets the timing of the peak

  • \(\epsilon \geq 0\) or bottom is a shape parameter: increasing the values of \(\epsilon\) reduces the variance

  • \(q \geq 0\) or pw is a shape parameter

  • \(P \geq 0\) or period sets the period of the function

p1 = makepar_F_sin()
S1 <- make_function(p1)

The default normalizing constant is equal to the period (by default, \(365\)) so that if \(S\) is multiplied by some other constant, \(m,\) the average daily value of the function over a year is \(1.\)

integrate(S1, 0, 365)$val
## [1] 365
tt <- seq(0, 3*365, by=5)
plot(tt, S1(tt), type ="l", xlab = "Time (in Days)", ylab = expression(S(t)))

p2 = makepar_F_sin(phase=120)
S2 <- make_function(p2)
plot(tt, S1(tt), type ="l", xlab = "Time (in Days)", ylab = expression(S(t)))
lines(tt, S2(tt), col = "blue")

The function can return a vector of \(N\) functions, each one configured as if \(N=1\)

p3 = makepar_F_sin(phase = c(0,120), N=2)
S3 <- make_function(p3)
s3 <- S3(tt) 
plot(tt, s3[1,], type ="l", xlab = "Time (in Days)", ylab = expression(S(t)))
lines(tt, S1(tt), col = "yellow", lty=2)
lines(tt, s3[2,], col = "blue")
lines(tt, S2(tt), col = "orange", lty=2)

p4 <- makepar_F_sin(bottom=.5)
p5 <- makepar_F_sin(bottom=2)
p6 <- makepar_F_sin(pw=3)
p7 <- makepar_F_sin(pw=6)
S4 <- make_function(p4)
S5 <- make_function(p5)
S6 <- make_function(p6)
S7 <- make_function(p7)

The shape parameters make it easy to configure a seasonality function with a range of features:

clrs = turbo(7)
plot(tt, S7(tt), type ="n", xlab = "Time (in Days)", ylab = expression(S(t)))
lines(tt, S1(tt), col = clrs[1])
lines(tt, S4(tt), col = clrs[2])
lines(tt, S5(tt), col = clrs[3])
lines(tt, S6(tt), col = clrs[5])
lines(tt, S7(tt), col = clrs[7])
text(1000, 3.5, expression(p==6), col=clrs[7])
text(1000, 3, expression(p==3), col=clrs[5])
text(1000, 2.5, expression(epsilon==0.5), col=clrs[2])
text(1000, 2, expression(epsilon==2), col=clrs[3])

Multiple Periods 

p1 <- makepar_F_sin()
p2 <- makepar_F_sin(period=1, bottom=1)
p12 <- makepar_F_product(p1, p2) 
S2 <- make_function(p12)

This plots the function over a stretch of values:

tt <- seq(120, 210, length.out=365*50)
plot(tt, S2(tt), type = "l", xlab = "Time (in Days)")

integrate(S2, 0, 365, rel.tol=1e-3)$value
## [1] 133225
p12a<- makepar_F_product(p1, p2, P=365, tol = 1e-3) 
S2a <- make_function(p12a)
integrate(S2a, 0, 365, rel.tol=1e-3)$value 
## [1] 365