tpm_p: Build all transition probability matrices of a periodically inhomogeneous HMM

Description

Given a periodically varying variable such as time of day or day of year and the associated cycle length, this function calculates the transition probability matrices by applying the inverse multinomial logistic link (also known as softmax) to linear predictors of the form $$ \eta^{(t)}_{ij} = \beta_0^{(ij)} + \sum_{k=1}^K \bigl( \beta_{1k}^{(ij)} \sin(\frac{2 \pi k t}{L}) + \beta_{2k}^{(ij)} \cos(\frac{2 \pi k t}{L}) \bigr) $$ for the off-diagonal elements ($i \neq j$) of the transition probability matrix. This is relevant for modeling e.g. diurnal variation and the flexibility can be increased by adding smaller frequencies (i.e. increasing $K$).

Usage

tpm_p(
  tod = 1:24,
  L = 24,
  beta,
  degree = 1,
  Z = NULL,
  byrow = FALSE,
  ad = NULL,
  report = TRUE
)

Value

array of transition probability matrices of dimension c(N,N,length(tod))

Arguments

tod

equidistant sequence of a cyclic variable

For time of day and e.g. half-hourly data, this could be 1, ..., L and L = 48, or 0.5, 1, 1.5, ..., 24 and L = 24.

L

length of one full cycle, on the scale of tod

beta

matrix of coefficients for the off-diagonal elements of the transition probability matrix

Needs to be of dimension c(N *(N-1), 2*degree+1), where the first column contains the intercepts.

degree

degree of the trigonometric link function

For each additional degree, one sine and one cosine frequency are added.

Z

pre-calculated design matrix (excluding intercept column)

Defaults to NULL if trigonometric link should be calculated. From an efficiency perspective, Z should be pre-calculated within the likelihood function, as the basis expansion should not be redundantly calculated. This can be done by using trigBasisExp.

byrow

logical indicating if each transition probability matrix should be filled by row

Defaults to FALSE, but should be set to TRUE if one wants to work with a matrix of beta parameters returned by popular HMM packages like moveHMM, momentuHMM, or hmmTMB.

ad

optional logical, indicating whether automatic differentiation with RTMB should be used. By default, the function determines this itself.

report

logical, indicating whether the coefficient matrix beta should be reported from the fitted model. Defaults to TRUE, but only works if ad = TRUE.

Details

Note that using this function inside the negative log-likelihood function is convenient, but it performs the basis expansion into sine and cosine terms each time it is called. As these do not change during the optimisation, using tpm_g with a pre-calculated (by trigBasisExp) design matrix would be more efficient.

Examples

Run this code

# hourly data 
tod = seq(1, 24, by = 1)
L = 24
beta = matrix(c(-1, 2, -1, -2, 1, -1), nrow = 2, byrow = TRUE)
Gamma = tpm_p(tod, L, beta, degree = 1)

# half-hourly data
## integer tod sequence
tod = seq(1, 48, by = 1)
L = 48
beta = matrix(c(-1, 2, -1, -2, 1, -1), nrow = 2, byrow = TRUE)
Gamma1 = tpm_p(tod, L, beta, degree = 1)

## equivalent specification
tod = seq(0.5, 24, by = 0.5)
L = 24
beta = matrix(c(-1, 2, -1, -2, 1, -1), nrow = 2, byrow = TRUE)
Gamma2 = tpm_p(tod, L, beta, degree = 1)

all(Gamma1 == Gamma2) # same result