get_kernel: Obtain the Drifting semi-Markov kernel

Description

This is a generic method that computes and returns the drifting semi-Markov kernel.

Usage

get_kernel(obj, t, u, v, l, klim = 100)

Value

An array with dimensions of $s \times s \times k_{max} \times (n + 1)$, giving the value of the drifting semi-Markov kernel $q_{\frac{t}{n}}(u,v,l)$ for the corresponding $(u,v,l,t)$. If any of $u,v,l$ or $t$ are specified, we obtain the element of the array for their given value.

Arguments

obj

An object that inherits from the S3 classes dsmm, dsmm_fit_parametric, or dsmm_fit_nonparametric, dsmm_nonparametric or dsmm_parametric.

t

Optional, but recommended. Positive integer specifying the instance $t$ of the visited states.

u

Optional. Can be either of the two options below:

Character specifying the previous state $u$, e.g. u = "a".
Positive integer, specifying a state in the state space $E$. For example, if $E = \{a, c, g, t\}$ and u = 1, it corresponds to the state $a$, if u = 2, it corresponds to the state $c$.

v

Optional. Can be either of the two options below:

Character specifying the next state $v$, e.g. v = "c".
Positive integer, specifying a state in the state space $E$. For example, if $E = \{a, c, g, t\}$ and v = 3, it corresponds to the state $c$, if v = 4, it corresponds to the state $t$.

l

Optional. Positive integer specifying the sojourn time $l$ that is spent in the previous state $u$.

klim

Optional. Positive integer. Used only when obj inherits from the S3 classes dsmm_parametric or dsmm_fit_parametric. Specifies the time horizon used to approximate the $d + 1$ sojourn time distributions if $f$ is drifting, or just $1$ sojourn time distribution if $f$ is not drifting. Default value is 100.

A larger value will result in a considerably larger kernel, which has dimensions of $s \times s \times klim \times (n + 1)$, which will increase the memory requirements and will slow down considerably the simulate.dsmm() method. However, this will lead to better estimations through fit_dsmm(). (dsmm_parametric, fit_dsmm, simulate.dsmm)

Details

The drifting semi-Markov kernel is given as the probability that, given at the instance $t$ the previous state is $u$, the next state state $v$ will be reached with a sojourn time of $l$: $$q_{\frac{t}{n}}(u,v,l) = P(J_{t}=v,X_{t}=l|J_{t-1}=u),$$ where $n$ is the model size, defined as the length of the embedded Markov chain $(J_{t})_{t\in \{0,\dots,n\}}$ minus the last state, $J_t$ is the visited state at the instant $t$ and $X_{t} = S_{t}-S_{t-1}$ is the sojourn time of the state $J_{t-1}$. Specifically, it is given as the sum of a linear combination: $$q_{\frac{t}{n}}(u,v,l)= \sum_{i = 0}^{d}A_{i}(t)\ q_{\frac{i}{d}}(u,v,l),$$ where $A_i, i = 0, \dots, d$ are $d + 1$ polynomials with degree $d$ that satisfy certain conditions (see dsmmR) and $q_{\frac{i}{d}}(u,v,l), i = 0, \dots, d$ are $d + 1$ semi-Markov kernels. Three possible model specifications are described below. We will use the exponentials $(1), (2), (3)$ to distinguish between the drifting semi-Markov kernel $q_\frac{t}{n}$ and the semi-Markov kernels $q_\frac{i}{d}$ used in Model 1, Model 2 and Model 3.

Model 1

In this case, both $p$ and $f$ are "drifting" between $d + 1$ fixed points of the model, hence the "drifting" in drifting semi-Markov models. Therefore, the semi-Markov kernels $q_{\frac{i}{d}}^{\ (1)}$ are equal to:

$$q_{\frac{i}{d}}^{\ (1)}(u,v,l) = {p_{\frac{i}{d}}(u,v)}{f_{\frac{i}{d}}(u,v,l)},$$

where for $i = 0, \dots, d$ we have $d + 1$ Markov Transition matrices $p_{\frac{i}{d}}(u,v)$, and $d + 1$ sojourn time distributions $f_{\frac{i}{d}}(u,v,l)$, where $d$ is the polynomial degree.

Thus, the drifting semi-Markov kernel will be equal to:

$$q_{\frac{t}{n}}^{\ (1)}(u,v,l) = \sum_{i = 0}^{d} A_i(t)\ q_{\frac{i}{d}}^{\ (1)}(u,v,l) = \sum_{i = 0}^{d} A_i(t)\ p_{\frac{i}{d}}(u,v)f_{\frac{i}{d}}(u,v,l) $$

Model 2

In this case, $p$ is drifting and $f$ is not drifting. Therefore, the semi-Markov kernels $q_{\frac{i}{d}}^{\ (2)}$ are equal to: $$q_{\frac{i}{d}}^{\ (2)}(u,v,l)={p_{\frac{i}{d}}(u,v)}{f(u,v,l)}.$$

Thus, the drifting semi-Markov kernel will be equal to: $$q_{\frac{t}{n}}^{\ (2)}(u,v,l) = \sum_{i = 0}^{d} A_i(t)\ q_{\frac{i}{d}}^{\ (2)}(u,v,l) = \sum_{i = 0}^{d} A_i(t)\ p_{\frac{i}{d}}(u,v)f(u,v,l) $$

Model 3

In this case, $f$ is drifting and $p$ is not drifting.

Therefore, the semi-Markov kernels $q_{\frac{i}{d}}^{\ (3)}$ are now described as: $$q_{\frac{i}{d}}^{\ (3)}(u,v,l)={p(u,v)}{f_{\frac{i}{d}}(u,v,l)}.$$

Thus, the drifting semi-Markov kernel will be equal to: $$q_{\frac{t}{n}}^{\ (3)}(u,v,l) = \sum_{i = 0}^{d} A_i(t)\ q_{\frac{i}{d}}^{\ (3)}(u,v,l) = \sum_{i = 0}^{d} A_i(t)\ p(u,v)f_{\frac{i}{d}}(u,v,l) $$

Examples

Run this code

# Setup.
states <- c("Rouen", "Bucharest", "Samos", "Aigio", "Marseille")
emc <- create_sequence(states, probs = c(0.3, 0.1, 0.1, 0.3, 0.2))
obj_model_2 <- fit_dsmm(
    sequence = emc,
    states = states,
    degree = 3,
    f_is_drifting = FALSE,
    p_is_drifting = TRUE
)

# Get the kernel.
kernel_model_2 <- get_kernel(obj_model_2)
cat(paste0("If no further arguments are made, kernel has dimensions ",
           "for all u, v, l, t:\n",
           "(s, s, k_max, n + 1) = (",
           paste(dim(kernel_model_2), collapse = ", "), ")"))

# Specifying `t`.
kernel_model_2_t <- get_kernel(obj_model_2, t = 100)
# kernel_model_2_t[ , , , t = 100]
cat(paste0("If we specify t, the kernel has dimensions for ",
           "all the remaining u, v, l:\n(s, s, k_max) = (",
           paste(dim(kernel_model_2_t), collapse = ", "), ")"))

# Specifying `t` and `u`.
kernel_model_2_tu <- get_kernel(obj_model_2, t = 2, u = "Aigio")
# kernel_model_2_tu["Aigio", , , t = 2]
cat(paste0("If we specify t and u, the kernel has dimensions for ",
           "all the remaining v, l:\n(s, k_max) = (",
           paste(dim(kernel_model_2_tu), collapse = ", "), ")"))

# Specifying `t`, `u` and `v`.
kernel_model_2_tuv <- get_kernel(obj_model_2, t = 3,
                                 u = "Rouen", v = "Bucharest")
# kernel_model_2_tuv["Rouen", "Bucharest", , t = 3]
cat(paste0("If we specify t, u and v, the kernel has dimensions ",
           "for all l:\n(k_max) = (",
           paste(length(kernel_model_2_tuv), collapse = ", "), ")"))

# It is possible to ask for any valid combination of `u`, `v`, `l` and `t`.