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Composed Time Series

Source: vignettes/TimeSeries.Rmd
TimeSeries.Rmd

The function make_ts_function returns a time series function:

?make_ts_function

In studies, we will often need functions that output the value of some exogenous variable over time. These often hold information – the trace – from some other analysis, or trace function.

This package — ramp.func — has utilities to construct and modify multiplicative composed time series functions. Each function is the product of four components:

\[ x(t) = \bar x \times F_S(t) \times F_T(t) \times F_K(t) \]

  • \(\bar x\) is the mean value over an interval \((t_0, t_1)\):

  • \(F_S(t)>0\) is a seasonality function

  • \(F_T(t)>0\) is a trend function

  • \(F_K(t)>0\) is a shock function

When calling make_ts_function, the user can normalize with or without the shock function, and the value of the time interval is set by passing a vector of values with a non-zero range. A normalizing constant is set such that:

\[\int_{t_0}^{t_1} F_S(t)\; F_T(t); F_K(t)\; dt = \bar x\left( t_1-t_0 \right)\]

Example

Components

This creates a spline function:

set.seed(3482)
N <- 7
tt <- c(-365, seq(0, 10*365, length.out=N), 7300)
yy <- (c(1, rlnorm(N-1, 0, 1), 2, 1))^.4
tp <- makepar_F_spline(tt, yy, 2)
Ft <- make_function(tp)

This creates a seasonal pattern function:

sp <- makepar_F_sin(bottom = 0.1, pw=2)
Fs <- make_function(sp)

This creates a shock:

kp <- makepar_F_sharkbite(D=1000, L=300)
Fk <- make_function(kp)

We can plot the components on the same axes:

tm <- seq(0, 3650, by=5)
plot(tm, Ft(tm), type = "l", ylim = range(0, Ft(tm)), xlab = "Times", ylab = expression(list(F[S], F[T], F[K])))
lines(tm, Fs(tm))
lines(tm, Fk(tm))

Normalizing Constant

We let F1 be a function that gets normalized with the shocks in place (black); F2 the same function but normalized without the shocks (dark red), and F3 the function without the shocks (grey).

F1 <- make_ts_function(0.3, times=tm, 
                       season_par <- sp, trend_par <- tp, 
                       shock_par <- kp, norm_with_shocks = FALSE)
integrate(F1, 0, 3650)$value/3650
## [1] 0.1962408
F2 <- make_ts_function(0.3, times=tm, 
                        season_par <- sp, trend_par <- tp, 
                        shock_par <- kp, norm_with_shocks=TRUE)
integrate(F2, 0, 3650)$value/3650
## [1] 0.3
F3 <- make_ts_function(0.3, times=tm, 
                       season_par <- sp, trend_par <- tp, 
                       norm_with_shocks = TRUE)
integrate(F3, 0, 3650)$value/3650
## [1] 0.3

Note that F1 and F2 have the same shape, but the mean of F1 is lower because it ignored the shock to set the normalizing constant, which nullified one of the seasonal peaks.

plot(tm, F3(tm), type = "l", col = "grey", ylab = expression(list(F1, F2, F3)), xlab = "Time")
lines(tm, F1(tm), lwd=2)
lines(tm, F2(tm), col = "darkred")