tpm_p {LaMa} | R Documentation |
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
)
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 |
byrow |
logical indicating if each transition probability matrix should be filled by row Defaults to |
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 |
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.
Value
array of transition probability matrices of dimension c(N,N,length(tod))
See Also
Other transition probability matrix functions:
generator()
,
tpm()
,
tpm_cont()
,
tpm_emb()
,
tpm_emb_g()
,
tpm_g()
Examples
# 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