lefko3 (version 6.2.0)

usher3: Two-parameter Usher function

Description

Function usher3() creates a vector of values produced by the two- parameter Usher function as applied with a user-specified time lag. The Usher function is given as \(\phi_{t+1} = \phi_t / (1 + e^{\alpha n_t + \beta})\). Here, if no separate_N vector is provided, then \(n_t = \phi_t\).

Usage

usher3(
  start_value,
  alpha,
  beta,
  time_steps = 100L,
  time_lag = 1L,
  pre0_subs = FALSE,
  pre0_value = 0,
  substoch = 0L,
  separate_N = NULL
)

Value

A numeric vector of values showing values projected under the two- parameter Usher function.

Arguments

start_value

A positive number to start the return vector in time 0.

alpha

The alpha parameter in the two-parameter Usher function.

beta

The beta parameter in the two-parameter Usher function.

time_steps

The number of time steps to run the projection. Must be a positive integer.

time_lag

A positive integer denoting the number of time steps back for the value of phi in the two-parameter Usher function.

pre0_subs

A logical value indicating whether to use a number other than that given in start_value for values of phi lagged from times prior to time 0.

pre0_value

A positive number to use for phi lagged from times prior to time 0. Only used if pre0_subs = TRUE.

substoch

An integer value indicating the kind of substochasticity to use. Values include: 0, no substochasticity enforced (the default); 1, all numbers must be non-negative; and 2, all numbers should be forced to the interval [0, 1].

separate_N

An optional numeric vector with values of N in each time, if phi is to be treated as different from N in the two-parameter model.

Examples

trial_run1 <- usher3(1, alpha = -0.5, beta = 0.005)
plot(trial_run1)

trial_run2 <- usher3(1, alpha = 0.5, beta = 0.005)
plot(trial_run2)

trial_run3 <- usher3(1, alpha = -5, beta = 0.005)
plot(trial_run3)

trial_run4 <- usher3(1, alpha = 5, beta = 0.005)
plot(trial_run4)

trial_run5 <- usher3(1, alpha = -25, beta = 0.005)
plot(trial_run5)

trial_run6 <- usher3(1, alpha = 25, beta = 0.005)
plot(trial_run6)

used_Ns <- c(10, 15, 12, 14, 14, 150, 15, 1, 5, 7, 9, 14, 13, 16, 17, 19,
  25, 26)
trial_run7 <- usher3(1, alpha = -0.5, beta = 0.005, separate_N = used_Ns)
plot(trial_run7)