parametric_dsmm: Parametric Drifting semi-Markov model specification
In dsmmR: Estimation and Simulation of Drifting Semi-Markov Models
parametric_dsmm R Documentation
Parametric Drifting semi-Markov model specification
Description
Creates a parametric model specification for a drifting
semi-Markov model. Returns an object of class
(dsmm_parametric, dsmm).
Usage
parametric_dsmm(
model_size,
states,
initial_dist,
degree,
f_is_drifting,
p_is_drifting,
p_dist,
f_dist,
f_dist_pars
)
Arguments
model_size
Positive integer that represents the size of
the drifting semi-Markov model n. It is equal to the length of a
theoretical embedded Markov chain
(J_{t})_{t\in \{0,\dots,n\}}, without the last state.
states
Character vector that represents the state space E.
It has length equal to s = |E|.
initial_dist
Numerical vector of s probabilities, that
represents the initial distribution for each state in the state space
E.
degree
Positive integer that represents the polynomial degree d
for the drifting semi-Markov model.
f_is_drifting
Logical. Specifies if f is drifting or not.
p_is_drifting
Logical. Specifies if p is drifting or not.
p_dist
Numerical array, that represents the probabilities of the
transition matrix p of the embedded Markov chain
(J_{t})_{t\in \{0,\dots,n\}} (it is defined
the same way in the nonparametric_dsmm function).
It can be defined in two ways:
-
If p is not drifting, it has dimensions of
s \times s.
-
If p is drifting, it has dimensions of
s \times s \times (d+1)
(see more in Details, Defined Arguments.)
f_dist
Character array, that represents the discrete sojourn time
distribution f of our choice.
NA is allowed for state transitions
that we do not wish to have a sojourn time distribution
(e.g. all state transition to the same state should have NA
as their value). The list of possible values is:
["unif", "geom", "pois", "dweibull", "nbinom", NA].
It can be defined in two ways:
-
If f is not drifting, it has dimensions of
s \times s.
-
If f is drifting, it has dimensions of
s \times s \times (d+1)
(see more in Details, Defined Arguments.)
f_dist_pars
Numerical array, that represents the parameters of the
sojourn time distributions given in f_dist. NA is allowed,
in the case that the distribution of our choice does not require
a parameter. It can be defined in two ways:
If f is not drifting, it has dimensions of
s \times s \times 2, specifying two possible
parameters required for the discrete distributions.
If f is drifting, it has dimensions of
s \times s \times 2 \times (d+1),
specifying two possible parameters required for the discrete
distributions, but for every single one of the i = 0, \dots, d
sojourn time distributions f_{\frac{i}{d}} that are required.
(see more in Details, Defined Arguments.)
Details
Defined Arguments
For the parametric case, we explicitly define:
The transition matrix of the embedded Markov chain
(J_{t})_{t\in \{0,\dots,n\}}, given in
the attribute p_dist:
If p is not drifting, it contains the values:
p(u,v), \forall u, v \in E,
given in an array with dimensions of s \times s,
where the first dimension corresponds to the previous state u and
the second dimension corresponds to the current state v.
If p is drifting, for i \in \{ 0,\dots,d \},
it contains the values:
p_{\frac{i}{d}}(u,v), \forall u, v \in E,
given in an array with dimensions of s \times s \times (d + 1),
where the first and second dimensions are defined as in the non-drifting
case, and the third dimension corresponds to the d+1 different
matrices p_{\frac{i}{d}}.
The conditional sojourn time distribution, given in the
attribute f_dist:
If f is not drifting, it contains the discrete
distribution names (as characters or NA), given in an
array with dimensions of s \times s,
where the first dimension corresponds to the previous state u,
the second dimension corresponds to the current state v.
If f is drifting, it contains the discrete
distribution names (as characters or NA) given in an
array with dimensions of s \times s \times (d + 1),
where the first and second dimensions are defined as in the
non-drifting case, and the third dimension corresponds to the
d+1 different arrays f_{\frac{i}{d}}.
The conditional sojourn time distribution parameters,
given in the attribute f_dist_pars:
If f is not drifting, it contains the
numerical values (or NA) of the corresponding
distributions defined in f_dist, given in
an array with dimensions of s \times s,
where the first dimension corresponds to the previous state u,
the second dimension corresponds to the current state v.
If f is drifting, it contains the
numerical values (or NA) of the corresponding
distributions defined in f_dist, given in an array
with dimensions of s \times s \times (d + 1),
where the first and second dimensions are defined as in the
non-drifting case, and the third dimension corresponds to the
d+1 different arrays f_{\frac{i}{d}}.
Sojourn time distributions
In this package, the available distributions for the modeling of the
conditional sojourn times, of the drifting semi-Markov model, used through
the argument f_dist, are the following:
Uniform (n):
f(x) = 1/n, for x = 1, 2, \dots, n, where n is a
positive integer.
This can be specified through the following:
-
f_dist = "unif"
-
f_dist_pars = (n, NA)
(n as defined here).
Geometric (p):
f(x) = p (1-p)^{x-1}, for
x = 1, 2, \dots, where p \in (0, 1)
is the probability of success.
This can be specified through the following:
-
f_dist = "geom"
-
f_dist_pars = (p, NA)
(p as defined here).
Poisson (\lambda):
f(x) = \frac{\lambda^{x-1} exp(-\lambda)}{(x-1)!}, for
x = 1, 2, \dots, where \lambda > 0.
This can be specified through the following:
-
f_dist = "pois"
-
f_dist_pars = (\lambda, NA)
Negative binomial (\alpha, p):
f(x)=\frac{\Gamma(x+\alpha-1)}{\Gamma(\alpha)(x-1)!}
p^{\alpha}(1-p)^{x-1}, for x = 1, 2,\dots, where
\Gamma is the Gamma function,
\alpha \in (0, +\infty) is the parameter describing the
target for number of successful trials, or the dispersion parameter
(the shape parameter of the gamma mixing distribution).
p is the probability of success, 0 < p < 1.
-
f_dist = "nbinom"
-
f_dist_pars = (\alpha, p)
(p as defined here)
Discrete Weibull of type 1 (q, \beta):
f(x)=q^{(x-1)^{\beta}}-q^{x^{\beta}}, for x=1,2,\dots, with
q \in (0, 1) is the first parameter (probability)
and \beta \in (0, +\infty) is the second parameter.
This can be specified through the following:
-
f_dist = "dweibull"
-
f_dist_pars = (q, \beta)
(q as defined here)
From these discrete distributions, by using "dweibull", "nbinom"
we require two parameters. It's for this reason that the attribute
f_dist_pars is an array of dimensions
s \times s \times 2 if f
is not drifting or s \times s \times 2 \times (d+1)
if f is drifting.
Value
Returns an object of the S3 class dsmm_parametric, dsmm.
It has the following attributes:
-
dist : List. Contains 3 arrays, passing down from the arguments:
-
p_drift or p_notdrift, corresponding to whether the
defined p transition matrix is drifting or not.
-
f_drift_parametric or f_notdrift_parametric,
corresponding to whether the
defined f sojourn time distribution is drifting or not.
-
f_drift_parameters or f_notdrift_parameters,
which are the defined f sojourn time distribution parameters,
depending on whether f is drifting or not.
-
initial_dist : Numerical vector. Passing down from the arguments.
It contains the initial distribution of the drifting semi-Markov model.
-
states : Character vector. Passing down from the arguments.
It contains the state space E.
-
s : Positive integer. It contains the number of states in the
state space, s = |E|, which is given in the attribute states.
-
degree : Positive integer. Passing down from the arguments.
It contains the polynomial degree d considered for the drifting of
the model.
-
model_size : Positive integer. Passing down from the arguments.
It contains the size of the drifting semi-Markov model n, which
represents the length of the embedded Markov chain
(J_{t})_{t\in \{0,\dots,n\}}, without the last state.
-
f_is_drifting : Logical. Passing down from the arguments.
Specifies if f is drifting or not.
-
p_is_drifting : Logical. Passing down from the arguments.
Specifies if p is drifting or not.
-
Model : Character. Possible values:
-
"Model_1" : Both p and f are drifting.
-
"Model_2" : p is drifting and f
is not drifting.
-
"Model_3" : f is drifting and p
is not drifting.
-
A_i : Numerical matrix. Represents the polynomials
A_i(t) with degree d that are used for solving
the system MJ = P. Used for the methods defined for the
object. Not printed when viewing the object.
References
V. S. Barbu, N. Limnios. (2008). semi-Markov Chains and Hidden semi-Markov
Models Toward Applications - Their Use in Reliability and DNA Analysis.
New York: Lecture Notes in Statistics, vol. 191, Springer.
Vergne, N. (2008). Drifting Markov models with Polynomial Drift and
Applications to DNA Sequences. Statistical Applications in Genetics
Molecular Biology 7 (1).
Barbu V. S., Vergne, N. (2019). Reliability and survival analysis for
drifting Markov models: modeling and estimation.
Methodology and Computing in Applied Probability, 21(4), 1407-1429.
T. Nakagawa and S. Osaki. (1975). The discrete Weibull distribution.
IEEE Transactions on Reliability, R-24, 300-301.
See Also
Methods applied to this object: simulate.dsmm, get_kernel.
For the non-parametric drifting semi-Markov model specification:
nonparametric_dsmm.
For the theoretical background of drifting semi-Markov models: dsmmR.
Examples
# We can also define states in a flexible way, including spaces.
states <- c("Dollar $", " /1'2'3/ ", " Z E T A ", "O_M_E_G_A")
s <- length(states)
d <- 1
# ===========================================================================
# Defining parametric drifting semi-Markov models.
# ===========================================================================
# ---------------------------------------------------------------------------
# Defining the drifting distributions for Model 1.
# ---------------------------------------------------------------------------
# `p_dist` has dimensions of: (s, s, d + 1).
# Sums over v must be 1 for all u and i = 0, ..., d.
# First matrix.
p_dist_1 <- matrix(c(0, 0.1, 0.4, 0.5,
0.5, 0, 0.3, 0.2,
0.3, 0.4, 0, 0.3,
0.8, 0.1, 0.1, 0),
ncol = s, byrow = TRUE)
# Second matrix.
p_dist_2 <- matrix(c(0, 0.3, 0.6, 0.1,
0.3, 0, 0.4, 0.3,
0.5, 0.3, 0, 0.2,
0.2, 0.3, 0.5, 0),
ncol = s, byrow = TRUE)
# get `p_dist` as an array of p_dist_1 and p_dist_2.
p_dist_model_1 <- array(c(p_dist_1, p_dist_2), dim = c(s, s, d + 1))
# `f_dist` has dimensions of: (s, s, d + 1).
# First matrix.
f_dist_1 <- matrix(c(NA, "unif", "dweibull", "nbinom",
"geom", NA, "pois", "dweibull",
"dweibull", "pois", NA, "geom",
"pois", NA, "geom", NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_2 <- matrix(c(NA, "pois", "geom", "nbinom",
"geom", NA, "pois", "dweibull",
"unif", "geom", NA, "geom",
"pois", "pois", "geom", NA),
nrow = s, ncol = s, byrow = TRUE)
# get `f_dist` as an array of `f_dist_1` and `f_dist_2`
f_dist_model_1 <- array(c(f_dist_1, f_dist_2), dim = c(s, s, d + 1))
# `f_dist_pars` has dimensions of: (s, s, 2, d + 1).
# First array of coefficients, corresponding to `f_dist_1`.
# First matrix.
f_dist_1_pars_1 <- matrix(c(NA, 5, 0.4, 4,
0.7, NA, 5, 0.6,
0.2, 3, NA, 0.6,
4, NA, 0.4, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_1_pars_2 <- matrix(c(NA, NA, 0.2, 0.6,
NA, NA, NA, 0.8,
0.6, NA, NA, NA,
NA, NA, NA, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second array of coefficients, corresponding to `f_dist_2`.
# First matrix.
f_dist_2_pars_1 <- matrix(c(NA, 6, 0.4, 3,
0.7, NA, 2, 0.5,
3, 0.6, NA, 0.7,
6, 0.2, 0.7, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_2_pars_2 <- matrix(c(NA, NA, NA, 0.6,
NA, NA, NA, 0.8,
NA, NA, NA, NA,
NA, NA, NA, NA),
nrow = s, ncol = s, byrow = TRUE)
# Get `f_dist_pars`.
f_dist_pars_model_1 <- array(c(f_dist_1_pars_1, f_dist_1_pars_2,
f_dist_2_pars_1, f_dist_2_pars_2),
dim = c(s, s, 2, d + 1))
# ---------------------------------------------------------------------------
# Parametric object for Model 1.
# ---------------------------------------------------------------------------
obj_par_model_1 <- parametric_dsmm(
model_size = 10000,
states = states,
initial_dist = c(0.8, 0.1, 0.1, 0),
degree = d,
p_dist = p_dist_model_1,
f_dist = f_dist_model_1,
f_dist_pars = f_dist_pars_model_1,
p_is_drifting = TRUE,
f_is_drifting = TRUE
)
# p drifting array.
p_drift <- obj_par_model_1$dist$p_drift
p_drift
# f distribution.
f_dist_drift <- obj_par_model_1$dist$f_drift_parametric
f_dist_drift
# parameters for the f distribution.
f_dist_pars_drift <- obj_par_model_1$dist$f_drift_parameters
f_dist_pars_drift
# ---------------------------------------------------------------------------
# Defining Model 2 - p is drifting, f is not drifting.
# ---------------------------------------------------------------------------
# `p_dist` has the same dimensions as in Model 1: (s, s, d + 1).
p_dist_model_2 <- array(c(p_dist_1, p_dist_2), dim = c(s, s, d + 1))
# `f_dist` has dimensions of: (s, s).
f_dist_model_2 <- matrix(c( NA, "pois", NA, "nbinom",
"geom", NA, "geom", "dweibull",
"unif", "geom", NA, "geom",
"nbinom", "unif", "dweibull", NA),
nrow = s, ncol = s, byrow = TRUE)
# `f_dist_pars` has dimensions of: (s, s, 2),
# corresponding to `f_dist_model_2`.
# First matrix.
f_dist_pars_1_model_2 <- matrix(c(NA, 0.2, NA, 3,
0.2, NA, 0.2, 0.5,
3, 0.4, NA, 0.7,
2, 3, 0.7, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_pars_2_model_2 <- matrix(c(NA, NA, NA, 0.6,
NA, NA, NA, 0.8,
NA, NA, NA, NA,
0.2, NA, 0.3, NA),
nrow = s, ncol = s, byrow = TRUE)
# Get `f_dist_pars`.
f_dist_pars_model_2 <- array(c(f_dist_pars_1_model_2,
f_dist_pars_2_model_2),
dim = c(s, s, 2))
# ---------------------------------------------------------------------------
# Parametric object for Model 2.
# ---------------------------------------------------------------------------
obj_par_model_2 <- parametric_dsmm(
model_size = 10000,
states = states,
initial_dist = c(0.8, 0.1, 0.1, 0),
degree = d,
p_dist = p_dist_model_2,
f_dist = f_dist_model_2,
f_dist_pars = f_dist_pars_model_2,
p_is_drifting = TRUE,
f_is_drifting = FALSE
)
# p drifting array.
p_drift <- obj_par_model_2$dist$p_drift
p_drift
# f distribution.
f_dist_notdrift <- obj_par_model_2$dist$f_notdrift_parametric
f_dist_notdrift
# parameters for the f distribution.
f_dist_pars_notdrift <- obj_par_model_2$dist$f_notdrift_parameters
f_dist_pars_notdrift
# ---------------------------------------------------------------------------
# Defining Model 3 - f is drifting, p is not drifting.
# ---------------------------------------------------------------------------
# `p_dist` has dimensions of: (s, s).
p_dist_model_3 <- matrix(c(0, 0.1, 0.3, 0.6,
0.4, 0, 0.1, 0.5,
0.4, 0.3, 0, 0.3,
0.9, 0.01, 0.09, 0),
ncol = s, byrow = TRUE)
# `f_dist` has the same dimensions as in Model 1: (s, s, d + 1).
f_dist_model_3 <- array(c(f_dist_1, f_dist_2), dim = c(s, s, d + 1))
# `f_dist_pars` has the same dimensions as in Model 1: (s, s, 2, d + 1).
f_dist_pars_model_3 <- array(c(f_dist_1_pars_1, f_dist_1_pars_2,
f_dist_2_pars_1, f_dist_2_pars_2),
dim = c(s, s, 2, d + 1))
# ---------------------------------------------------------------------------
# Parametric object for Model 3.
# ---------------------------------------------------------------------------
obj_par_model_3 <- parametric_dsmm(
model_size = 10000,
states = states,
initial_dist = c(0.3, 0.2, 0.2, 0.3),
degree = d,
p_dist = p_dist_model_3,
f_dist = f_dist_model_3,
f_dist_pars = f_dist_pars_model_3,
p_is_drifting = FALSE,
f_is_drifting = TRUE
)
# p drifting array.
p_notdrift <- obj_par_model_3$dist$p_notdrift
p_notdrift
# f distribution.
f_dist_drift <- obj_par_model_3$dist$f_drift_parametric
f_dist_drift
# parameters for the f distribution.
f_dist_pars_drift <- obj_par_model_3$dist$f_drift_parameters
f_dist_pars_drift
# ===========================================================================
# Parametric estimation using methods corresponding to an object
# which inherits from the class `dsmm_parametric`.
# ===========================================================================
### Comments
### 1. Using a larger `klim` and a larger `model_size` will increase the
### accuracy of the model, with the need of larger memory requirements
### and computational cost.
### 2. For the parametric estimation it is recommended to use a common set
### of distributions while only the parameters are drifting. This results
### in higher accuracy.
# ---------------------------------------------------------------------------
# Defining the distributions for Model 1 - both p and f are drifting.
# ---------------------------------------------------------------------------
# `p_dist` has dimensions of: (s, s, d + 1).
# First matrix.
p_dist_1 <- matrix(c(0, 0.2, 0.4, 0.4,
0.5, 0, 0.3, 0.2,
0.3, 0.4, 0, 0.3,
0.5, 0.3, 0.2, 0),
ncol = s, byrow = TRUE)
# Second matrix.
p_dist_2 <- matrix(c(0, 0.3, 0.5, 0.2,
0.3, 0, 0.4, 0.3,
0.5, 0.3, 0, 0.2,
0.2, 0.4, 0.4, 0),
ncol = s, byrow = TRUE)
# get `p_dist` as an array of p_dist_1 and p_dist_2.
p_dist_model_1 <- array(c(p_dist_1, p_dist_2), dim = c(s, s, d + 1))
# `f_dist` has dimensions of: (s, s, d + 1).
# We will use the same sojourn time distributions.
f_dist_1 <- matrix(c( NA, "unif", "dweibull", "nbinom",
"geom", NA, "pois", "dweibull",
"dweibull", "pois", NA, "geom",
"pois", 'nbinom', "geom", NA),
nrow = s, ncol = s, byrow = TRUE)
# get `f_dist`
f_dist_model_1 <- array(f_dist_1, dim = c(s, s, d + 1))
# `f_dist_pars` has dimensions of: (s, s, 2, d + 1).
# First array of coefficients, corresponding to `f_dist_1`.
# First matrix.
f_dist_1_pars_1 <- matrix(c(NA, 7, 0.4, 4,
0.7, NA, 5, 0.6,
0.2, 3, NA, 0.6,
4, 4, 0.4, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_1_pars_2 <- matrix(c(NA, NA, 0.2, 0.6,
NA, NA, NA, 0.8,
0.6, NA, NA, NA,
NA, 0.3, NA, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second array of coefficients, corresponding to `f_dist_2`.
# First matrix.
f_dist_2_pars_1 <- matrix(c(NA, 6, 0.5, 3,
0.5, NA, 4, 0.5,
0.4, 5, NA, 0.7,
6, 5, 0.7, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_2_pars_2 <- matrix(c(NA, NA, 0.4, 0.5,
NA, NA, NA, 0.6,
0.5, NA, NA, NA,
NA, 0.4, NA, NA),
nrow = s, ncol = s, byrow = TRUE)
# Get `f_dist_pars`.
f_dist_pars_model_1 <- array(c(f_dist_1_pars_1, f_dist_1_pars_2,
f_dist_2_pars_1, f_dist_2_pars_2),
dim = c(s, s, 2, d + 1))
# ---------------------------------------------------------------------------
# Defining the parametric object for Model 1.
# ---------------------------------------------------------------------------
obj_par_model_1 <- parametric_dsmm(
model_size = 4000,
states = states,
initial_dist = c(0.8, 0.1, 0.1, 0),
degree = d,
p_dist = p_dist_model_1,
f_dist = f_dist_model_1,
f_dist_pars = f_dist_pars_model_1,
p_is_drifting = TRUE,
f_is_drifting = TRUE
)
cat("The object has class of (",
paste0(class(obj_par_model_1),
collapse = ', '), ").")
# ---------------------------------------------------------------------------
# Generating a sequence from the parametric object.
# ---------------------------------------------------------------------------
# A larger klim will lead to an increase in accuracy.
klim <- 20
sim_seq <- simulate(obj_par_model_1, klim = klim, seed = 1)
# ---------------------------------------------------------------------------
# Fitting the generated sequence under the same distributions.
# ---------------------------------------------------------------------------
fit_par_model1 <- fit_dsmm(sequence = sim_seq,
states = states,
degree = d,
f_is_drifting = TRUE,
p_is_drifting = TRUE,
estimation = 'parametric',
f_dist = f_dist_model_1)
cat("The object has class of (",
paste0(class(fit_par_model1),
collapse = ', '), ").")
cat("\nThe estimated parameters are:\n")
fit_par_model1$dist$f_drift_parameters
dsmmR documentation built on Sept. 14, 2024, 9:09 a.m.
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| parametric_dsmm | R Documentation |
Parametric Drifting semi-Markov model specification
Description
Creates a parametric model specification for a drifting
semi-Markov model. Returns an object of class
(dsmm_parametric, dsmm).
Usage
parametric_dsmm(
model_size,
states,
initial_dist,
degree,
f_is_drifting,
p_is_drifting,
p_dist,
f_dist,
f_dist_pars
)
Arguments
model_size |
Positive integer that represents the size of
the drifting semi-Markov model |
states |
Character vector that represents the state space |
initial_dist |
Numerical vector of |
degree |
Positive integer that represents the polynomial degree |
f_is_drifting |
Logical. Specifies if |
p_is_drifting |
Logical. Specifies if |
p_dist |
Numerical array, that represents the probabilities of the
transition matrix
|
f_dist |
Character array, that represents the discrete sojourn time
distribution
|
f_dist_pars |
Numerical array, that represents the parameters of the
sojourn time distributions given in
|
Details
Defined Arguments
For the parametric case, we explicitly define:
The transition matrix of the embedded Markov chain
(J_{t})_{t\in \{0,\dots,n\}}, given in the attributep_dist:If
pis not drifting, it contains the values:p(u,v), \forall u, v \in E,given in an array with dimensions of
s \times s, where the first dimension corresponds to the previous stateuand the second dimension corresponds to the current statev.If
pis drifting, fori \in \{ 0,\dots,d \}, it contains the values:p_{\frac{i}{d}}(u,v), \forall u, v \in E,given in an array with dimensions of
s \times s \times (d + 1), where the first and second dimensions are defined as in the non-drifting case, and the third dimension corresponds to thed+1different matricesp_{\frac{i}{d}}.
The conditional sojourn time distribution, given in the attribute
f_dist:If
fis not drifting, it contains the discrete distribution names (as characters orNA), given in an array with dimensions ofs \times s, where the first dimension corresponds to the previous stateu, the second dimension corresponds to the current statev.If
fis drifting, it contains the discrete distribution names (as characters orNA) given in an array with dimensions ofs \times s \times (d + 1), where the first and second dimensions are defined as in the non-drifting case, and the third dimension corresponds to thed+1different arraysf_{\frac{i}{d}}.
The conditional sojourn time distribution parameters, given in the attribute
f_dist_pars:If
fis not drifting, it contains the numerical values (orNA) of the corresponding distributions defined inf_dist, given in an array with dimensions ofs \times s, where the first dimension corresponds to the previous stateu, the second dimension corresponds to the current statev.If
fis drifting, it contains the numerical values (orNA) of the corresponding distributions defined inf_dist, given in an array with dimensions ofs \times s \times (d + 1), where the first and second dimensions are defined as in the non-drifting case, and the third dimension corresponds to thed+1different arraysf_{\frac{i}{d}}.
Sojourn time distributions
In this package, the available distributions for the modeling of the
conditional sojourn times, of the drifting semi-Markov model, used through
the argument f_dist, are the following:
Uniform
(n):f(x) = 1/n, forx = 1, 2, \dots, n, wherenis a positive integer. This can be specified through the following:-
f_dist = "unif" -
f_dist_pars= (n,NA) (nas defined here).
-
Geometric
(p):f(x) = p (1-p)^{x-1}, forx = 1, 2, \dots,wherep \in (0, 1)is the probability of success. This can be specified through the following:-
f_dist="geom" -
f_dist_pars= (p,NA) (pas defined here).
-
Poisson
(\lambda):f(x) = \frac{\lambda^{x-1} exp(-\lambda)}{(x-1)!}, forx = 1, 2, \dots,where\lambda > 0. This can be specified through the following:-
f_dist="pois" -
f_dist_pars= (\lambda,NA)
-
Negative binomial
(\alpha, p):f(x)=\frac{\Gamma(x+\alpha-1)}{\Gamma(\alpha)(x-1)!} p^{\alpha}(1-p)^{x-1}, forx = 1, 2,\dots,where\Gammais the Gamma function,\alpha \in (0, +\infty)is the parameter describing the target for number of successful trials, or the dispersion parameter (the shape parameter of the gamma mixing distribution).pis the probability of success,0 < p < 1.-
f_dist="nbinom" -
f_dist_pars= (\alpha, p) (pas defined here)
-
Discrete Weibull of type 1
(q, \beta):f(x)=q^{(x-1)^{\beta}}-q^{x^{\beta}}, forx=1,2,\dots,withq \in (0, 1)is the first parameter (probability) and\beta \in (0, +\infty)is the second parameter. This can be specified through the following:-
f_dist="dweibull" -
f_dist_pars= (q, \beta) (qas defined here)
-
From these discrete distributions, by using "dweibull", "nbinom"
we require two parameters. It's for this reason that the attribute
f_dist_pars is an array of dimensions
s \times s \times 2 if f
is not drifting or s \times s \times 2 \times (d+1)
if f is drifting.
Value
Returns an object of the S3 class dsmm_parametric, dsmm.
It has the following attributes:
-
dist: List. Contains 3 arrays, passing down from the arguments:-
p_driftorp_notdrift, corresponding to whether the definedptransition matrix is drifting or not. -
f_drift_parametricorf_notdrift_parametric, corresponding to whether the definedfsojourn time distribution is drifting or not. -
f_drift_parametersorf_notdrift_parameters, which are the definedfsojourn time distribution parameters, depending on whetherfis drifting or not.
-
-
initial_dist: Numerical vector. Passing down from the arguments. It contains the initial distribution of the drifting semi-Markov model. -
states: Character vector. Passing down from the arguments. It contains the state spaceE. -
s: Positive integer. It contains the number of states in the state space,s = |E|, which is given in the attributestates. -
degree: Positive integer. Passing down from the arguments. It contains the polynomial degreedconsidered for the drifting of the model. -
model_size: Positive integer. Passing down from the arguments. It contains the size of the drifting semi-Markov modeln, which represents the length of the embedded Markov chain(J_{t})_{t\in \{0,\dots,n\}}, without the last state. -
f_is_drifting: Logical. Passing down from the arguments. Specifies iffis drifting or not. -
p_is_drifting: Logical. Passing down from the arguments. Specifies ifpis drifting or not. -
Model: Character. Possible values:-
"Model_1": Bothpandfare drifting. -
"Model_2":pis drifting andfis not drifting. -
"Model_3":fis drifting andpis not drifting.
-
-
A_i: Numerical matrix. Represents the polynomialsA_i(t)with degreedthat are used for solving the systemMJ = P. Used for the methods defined for the object. Not printed when viewing the object.
References
V. S. Barbu, N. Limnios. (2008). semi-Markov Chains and Hidden semi-Markov Models Toward Applications - Their Use in Reliability and DNA Analysis. New York: Lecture Notes in Statistics, vol. 191, Springer.
Vergne, N. (2008). Drifting Markov models with Polynomial Drift and Applications to DNA Sequences. Statistical Applications in Genetics Molecular Biology 7 (1).
Barbu V. S., Vergne, N. (2019). Reliability and survival analysis for drifting Markov models: modeling and estimation. Methodology and Computing in Applied Probability, 21(4), 1407-1429.
T. Nakagawa and S. Osaki. (1975). The discrete Weibull distribution. IEEE Transactions on Reliability, R-24, 300-301.
See Also
Methods applied to this object: simulate.dsmm, get_kernel.
For the non-parametric drifting semi-Markov model specification: nonparametric_dsmm.
For the theoretical background of drifting semi-Markov models: dsmmR.
Examples
# We can also define states in a flexible way, including spaces.
states <- c("Dollar $", " /1'2'3/ ", " Z E T A ", "O_M_E_G_A")
s <- length(states)
d <- 1
# ===========================================================================
# Defining parametric drifting semi-Markov models.
# ===========================================================================
# ---------------------------------------------------------------------------
# Defining the drifting distributions for Model 1.
# ---------------------------------------------------------------------------
# `p_dist` has dimensions of: (s, s, d + 1).
# Sums over v must be 1 for all u and i = 0, ..., d.
# First matrix.
p_dist_1 <- matrix(c(0, 0.1, 0.4, 0.5,
0.5, 0, 0.3, 0.2,
0.3, 0.4, 0, 0.3,
0.8, 0.1, 0.1, 0),
ncol = s, byrow = TRUE)
# Second matrix.
p_dist_2 <- matrix(c(0, 0.3, 0.6, 0.1,
0.3, 0, 0.4, 0.3,
0.5, 0.3, 0, 0.2,
0.2, 0.3, 0.5, 0),
ncol = s, byrow = TRUE)
# get `p_dist` as an array of p_dist_1 and p_dist_2.
p_dist_model_1 <- array(c(p_dist_1, p_dist_2), dim = c(s, s, d + 1))
# `f_dist` has dimensions of: (s, s, d + 1).
# First matrix.
f_dist_1 <- matrix(c(NA, "unif", "dweibull", "nbinom",
"geom", NA, "pois", "dweibull",
"dweibull", "pois", NA, "geom",
"pois", NA, "geom", NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_2 <- matrix(c(NA, "pois", "geom", "nbinom",
"geom", NA, "pois", "dweibull",
"unif", "geom", NA, "geom",
"pois", "pois", "geom", NA),
nrow = s, ncol = s, byrow = TRUE)
# get `f_dist` as an array of `f_dist_1` and `f_dist_2`
f_dist_model_1 <- array(c(f_dist_1, f_dist_2), dim = c(s, s, d + 1))
# `f_dist_pars` has dimensions of: (s, s, 2, d + 1).
# First array of coefficients, corresponding to `f_dist_1`.
# First matrix.
f_dist_1_pars_1 <- matrix(c(NA, 5, 0.4, 4,
0.7, NA, 5, 0.6,
0.2, 3, NA, 0.6,
4, NA, 0.4, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_1_pars_2 <- matrix(c(NA, NA, 0.2, 0.6,
NA, NA, NA, 0.8,
0.6, NA, NA, NA,
NA, NA, NA, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second array of coefficients, corresponding to `f_dist_2`.
# First matrix.
f_dist_2_pars_1 <- matrix(c(NA, 6, 0.4, 3,
0.7, NA, 2, 0.5,
3, 0.6, NA, 0.7,
6, 0.2, 0.7, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_2_pars_2 <- matrix(c(NA, NA, NA, 0.6,
NA, NA, NA, 0.8,
NA, NA, NA, NA,
NA, NA, NA, NA),
nrow = s, ncol = s, byrow = TRUE)
# Get `f_dist_pars`.
f_dist_pars_model_1 <- array(c(f_dist_1_pars_1, f_dist_1_pars_2,
f_dist_2_pars_1, f_dist_2_pars_2),
dim = c(s, s, 2, d + 1))
# ---------------------------------------------------------------------------
# Parametric object for Model 1.
# ---------------------------------------------------------------------------
obj_par_model_1 <- parametric_dsmm(
model_size = 10000,
states = states,
initial_dist = c(0.8, 0.1, 0.1, 0),
degree = d,
p_dist = p_dist_model_1,
f_dist = f_dist_model_1,
f_dist_pars = f_dist_pars_model_1,
p_is_drifting = TRUE,
f_is_drifting = TRUE
)
# p drifting array.
p_drift <- obj_par_model_1$dist$p_drift
p_drift
# f distribution.
f_dist_drift <- obj_par_model_1$dist$f_drift_parametric
f_dist_drift
# parameters for the f distribution.
f_dist_pars_drift <- obj_par_model_1$dist$f_drift_parameters
f_dist_pars_drift
# ---------------------------------------------------------------------------
# Defining Model 2 - p is drifting, f is not drifting.
# ---------------------------------------------------------------------------
# `p_dist` has the same dimensions as in Model 1: (s, s, d + 1).
p_dist_model_2 <- array(c(p_dist_1, p_dist_2), dim = c(s, s, d + 1))
# `f_dist` has dimensions of: (s, s).
f_dist_model_2 <- matrix(c( NA, "pois", NA, "nbinom",
"geom", NA, "geom", "dweibull",
"unif", "geom", NA, "geom",
"nbinom", "unif", "dweibull", NA),
nrow = s, ncol = s, byrow = TRUE)
# `f_dist_pars` has dimensions of: (s, s, 2),
# corresponding to `f_dist_model_2`.
# First matrix.
f_dist_pars_1_model_2 <- matrix(c(NA, 0.2, NA, 3,
0.2, NA, 0.2, 0.5,
3, 0.4, NA, 0.7,
2, 3, 0.7, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_pars_2_model_2 <- matrix(c(NA, NA, NA, 0.6,
NA, NA, NA, 0.8,
NA, NA, NA, NA,
0.2, NA, 0.3, NA),
nrow = s, ncol = s, byrow = TRUE)
# Get `f_dist_pars`.
f_dist_pars_model_2 <- array(c(f_dist_pars_1_model_2,
f_dist_pars_2_model_2),
dim = c(s, s, 2))
# ---------------------------------------------------------------------------
# Parametric object for Model 2.
# ---------------------------------------------------------------------------
obj_par_model_2 <- parametric_dsmm(
model_size = 10000,
states = states,
initial_dist = c(0.8, 0.1, 0.1, 0),
degree = d,
p_dist = p_dist_model_2,
f_dist = f_dist_model_2,
f_dist_pars = f_dist_pars_model_2,
p_is_drifting = TRUE,
f_is_drifting = FALSE
)
# p drifting array.
p_drift <- obj_par_model_2$dist$p_drift
p_drift
# f distribution.
f_dist_notdrift <- obj_par_model_2$dist$f_notdrift_parametric
f_dist_notdrift
# parameters for the f distribution.
f_dist_pars_notdrift <- obj_par_model_2$dist$f_notdrift_parameters
f_dist_pars_notdrift
# ---------------------------------------------------------------------------
# Defining Model 3 - f is drifting, p is not drifting.
# ---------------------------------------------------------------------------
# `p_dist` has dimensions of: (s, s).
p_dist_model_3 <- matrix(c(0, 0.1, 0.3, 0.6,
0.4, 0, 0.1, 0.5,
0.4, 0.3, 0, 0.3,
0.9, 0.01, 0.09, 0),
ncol = s, byrow = TRUE)
# `f_dist` has the same dimensions as in Model 1: (s, s, d + 1).
f_dist_model_3 <- array(c(f_dist_1, f_dist_2), dim = c(s, s, d + 1))
# `f_dist_pars` has the same dimensions as in Model 1: (s, s, 2, d + 1).
f_dist_pars_model_3 <- array(c(f_dist_1_pars_1, f_dist_1_pars_2,
f_dist_2_pars_1, f_dist_2_pars_2),
dim = c(s, s, 2, d + 1))
# ---------------------------------------------------------------------------
# Parametric object for Model 3.
# ---------------------------------------------------------------------------
obj_par_model_3 <- parametric_dsmm(
model_size = 10000,
states = states,
initial_dist = c(0.3, 0.2, 0.2, 0.3),
degree = d,
p_dist = p_dist_model_3,
f_dist = f_dist_model_3,
f_dist_pars = f_dist_pars_model_3,
p_is_drifting = FALSE,
f_is_drifting = TRUE
)
# p drifting array.
p_notdrift <- obj_par_model_3$dist$p_notdrift
p_notdrift
# f distribution.
f_dist_drift <- obj_par_model_3$dist$f_drift_parametric
f_dist_drift
# parameters for the f distribution.
f_dist_pars_drift <- obj_par_model_3$dist$f_drift_parameters
f_dist_pars_drift
# ===========================================================================
# Parametric estimation using methods corresponding to an object
# which inherits from the class `dsmm_parametric`.
# ===========================================================================
### Comments
### 1. Using a larger `klim` and a larger `model_size` will increase the
### accuracy of the model, with the need of larger memory requirements
### and computational cost.
### 2. For the parametric estimation it is recommended to use a common set
### of distributions while only the parameters are drifting. This results
### in higher accuracy.
# ---------------------------------------------------------------------------
# Defining the distributions for Model 1 - both p and f are drifting.
# ---------------------------------------------------------------------------
# `p_dist` has dimensions of: (s, s, d + 1).
# First matrix.
p_dist_1 <- matrix(c(0, 0.2, 0.4, 0.4,
0.5, 0, 0.3, 0.2,
0.3, 0.4, 0, 0.3,
0.5, 0.3, 0.2, 0),
ncol = s, byrow = TRUE)
# Second matrix.
p_dist_2 <- matrix(c(0, 0.3, 0.5, 0.2,
0.3, 0, 0.4, 0.3,
0.5, 0.3, 0, 0.2,
0.2, 0.4, 0.4, 0),
ncol = s, byrow = TRUE)
# get `p_dist` as an array of p_dist_1 and p_dist_2.
p_dist_model_1 <- array(c(p_dist_1, p_dist_2), dim = c(s, s, d + 1))
# `f_dist` has dimensions of: (s, s, d + 1).
# We will use the same sojourn time distributions.
f_dist_1 <- matrix(c( NA, "unif", "dweibull", "nbinom",
"geom", NA, "pois", "dweibull",
"dweibull", "pois", NA, "geom",
"pois", 'nbinom', "geom", NA),
nrow = s, ncol = s, byrow = TRUE)
# get `f_dist`
f_dist_model_1 <- array(f_dist_1, dim = c(s, s, d + 1))
# `f_dist_pars` has dimensions of: (s, s, 2, d + 1).
# First array of coefficients, corresponding to `f_dist_1`.
# First matrix.
f_dist_1_pars_1 <- matrix(c(NA, 7, 0.4, 4,
0.7, NA, 5, 0.6,
0.2, 3, NA, 0.6,
4, 4, 0.4, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_1_pars_2 <- matrix(c(NA, NA, 0.2, 0.6,
NA, NA, NA, 0.8,
0.6, NA, NA, NA,
NA, 0.3, NA, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second array of coefficients, corresponding to `f_dist_2`.
# First matrix.
f_dist_2_pars_1 <- matrix(c(NA, 6, 0.5, 3,
0.5, NA, 4, 0.5,
0.4, 5, NA, 0.7,
6, 5, 0.7, NA),
nrow = s, ncol = s, byrow = TRUE)
# Second matrix.
f_dist_2_pars_2 <- matrix(c(NA, NA, 0.4, 0.5,
NA, NA, NA, 0.6,
0.5, NA, NA, NA,
NA, 0.4, NA, NA),
nrow = s, ncol = s, byrow = TRUE)
# Get `f_dist_pars`.
f_dist_pars_model_1 <- array(c(f_dist_1_pars_1, f_dist_1_pars_2,
f_dist_2_pars_1, f_dist_2_pars_2),
dim = c(s, s, 2, d + 1))
# ---------------------------------------------------------------------------
# Defining the parametric object for Model 1.
# ---------------------------------------------------------------------------
obj_par_model_1 <- parametric_dsmm(
model_size = 4000,
states = states,
initial_dist = c(0.8, 0.1, 0.1, 0),
degree = d,
p_dist = p_dist_model_1,
f_dist = f_dist_model_1,
f_dist_pars = f_dist_pars_model_1,
p_is_drifting = TRUE,
f_is_drifting = TRUE
)
cat("The object has class of (",
paste0(class(obj_par_model_1),
collapse = ', '), ").")
# ---------------------------------------------------------------------------
# Generating a sequence from the parametric object.
# ---------------------------------------------------------------------------
# A larger klim will lead to an increase in accuracy.
klim <- 20
sim_seq <- simulate(obj_par_model_1, klim = klim, seed = 1)
# ---------------------------------------------------------------------------
# Fitting the generated sequence under the same distributions.
# ---------------------------------------------------------------------------
fit_par_model1 <- fit_dsmm(sequence = sim_seq,
states = states,
degree = d,
f_is_drifting = TRUE,
p_is_drifting = TRUE,
estimation = 'parametric',
f_dist = f_dist_model_1)
cat("The object has class of (",
paste0(class(fit_par_model1),
collapse = ', '), ").")
cat("\nThe estimated parameters are:\n")
fit_par_model1$dist$f_drift_parameters
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