Type: Package
Title: Fitting Markov-Modulated Linear Regression Models
Version: 0.2.0
Maintainer: Nadezda Spiridovska <Spiridovska.N@tsi.lv>
Description: A set of tools for fitting Markov-modulated linear regression, where responses Y(t) are time-additive, and model operates in the external environment, which is described as a continuous time Markov chain with finite state space. Model is proposed by Alexander Andronov (2012) <doi:10.48550/arXiv.1901.09600> and algorithm of parameters estimation is based on eigenvalues and eigenvectors decomposition. Markov-switching regression models have the same idea of varying the regression parameters randomly in accordance with external environment. The difference is that for Markov-modulated linear regression model the external environment is described as a continuous-time homogeneous irreducible Markov chain with known parameters while switching models consider Markov chain as unobserved and estimation procedure involves estimation of transition matrix. These models have significant differences in terms of the analytical approach. Also, package provides a set of data simulation tools for Markov-modulated linear regression (for academical/research purposes). Research project No. 1.1.1.2/VIAA/1/16/075.
License: GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]
Encoding: UTF-8
LazyData: true
Imports: stats, pracma
RoxygenNote: 6.1.1
NeedsCompilation: no
Packaged: 2020-01-08 09:11:00 UTC; Nadezda Spiridovska
Author: Nadezda Spiridovska [aut, cre], Diana Santalova [ctb]
Repository: CRAN
Date/Publication: 2020-01-09 12:30:11 UTC

Calculating the average sojourn time in each state

Description

Calculating expectation of sojourn times in states for the observed time and for given initial state, using eigenvalues and eigenvectors.

Usage

Aver_soj_time(ii, tau_observed, Q)

Arguments

ii

number (scalar)

tau_observed

number (scalar), observed time

Q

Matrix (m x m), m - number of states

Details

Calculating expectation of sojourn times in states for the observed time (tau_observed) and if initial state is given (ii). Matrix Q is so-called Generator matrix: Q=\lambda-\Lambda, where \lambda is matrix with known transition rates from state $s_i$ to state $s_j$, and \Lambda is diagonal matrix with a vector (\Lambda_{1},...,\Lambda_{m} on the main diagonal, where m is a number of states of external environment. Eigenvalues and eigenvectors are used in calculations.

Value

Vector of average sojourn times in each state. Vector components in total should give observation time (tau_observed).

Examples

lambda <- matrix(c(0, 0.33, 0.45, 0), nrow = 2, ncol = 2, byrow = TRUE)
m <- nrow(lambda)
ld <- as.matrix(rowSums(lambda))
Lambda <- diag(as.vector(ld))
Generator <- t(lambda) - Lambda
Aver_soj_time(1,10,Generator)

Estimantion of unknown Markov-modulated linear regression model parameters using GLSM

Description

This function is used to fit Markov-modulated linear regression models with two states of external environment. This function estimates Markov-modulated linear regression model parameters, using GLSM. Function uses the algorithm based on eigenvalues and eigenvectors decompositions.

Usage

B_est(tGiven, initState, X, Y, lambda, W = TRUE)

Arguments

tGiven

Vector of the observed times (n x 1), n – number of observations

initState

Vector of the initial states (n x 1), n – number of observations

X

Matrix of predictors (n x k), n - number of observations, k - number of columns (k - 1 - number of regressors).

Y

Vector of the responses Y, n – number of observations

lambda

Matrix with the known transition rates \lambda_{i,j}, (m x m), m – number of states

W

an optional logical variable indicating should vector of weights be used in the fitting process or not. If TRUE, matrix with weights is used (that is, inverse values to tGiven – observed times).

Details

Function calculates the following expression: ![](vecB.png "Fig.2"), where vector of average sojourn times in each state $t_i$ is calculated using function Aver_soj_time, $t_i$ is an element of tGiven, $x_i$ is a vector of matrix X.

Value

Vector of estimated parameters \beta

Examples


lambda <- matrix(c(0, 0.33, 0.45, 0), nrow = 2, ncol = 2, byrow = TRUE)
Xtest <- cbind(rep_len(1,10),c(2,5,7,3,1,1,2,2,3,6), c(5,9,1,2,3,2,3,5,2,2))
tGiven <- matrix (c(6,4.8,1,2.6,6.4,1.7,2.9,4.4,1.5,3.4), nrow = 10, ncol = 1)
Y <- matrix(c (5.7, 9.9, 7.8, 14.5, 8.2, 14.5, 32.2, 3.8, 16.5, 7.7),nrow = 10, ncol = 1)
initState <- matrix (c(2,1,1,2,2,2,1,1,2,1),nrow = 10, ncol = 1)
B_est(tGiven,initState,Xtest,Y,lambda,W = 1)

Estimantion of the variance of the response Y

Description

This function is used for calculation of the variance of the respone Y (Var(Y))

Usage

VarY(bb, sigma, i, x, tau, la)

Arguments

bb

Matrix (k x m), k - number of columns (k - 1 - number of regressors), m - number of states, m = 2,3,.. .

sigma

Scalar, the standard deviation of the disturbance term

i

number (scalar), initial state

x

Row-vector of the matrix of predictors X (1 x k), k - number of columns.

tau

number (scalar), observed time

la

Matrix with the known transition rates \lambda_{i,j}, (m x m), m – number of states

Details

Function calculates the following expression: ![](varY.png "Fig.3"), where vector of average sojourn times in each state $t_i$ is calculated using function Aver_soj_time

Value

Estimantion of the variance of the response Y, scalar

Examples

Xtest <- cbind(rep_len(1,10),c(2,5,7,3,1,1,2,2,3,6), c(5,4,1,2,3,2,3,5,2,2))
tGiven <- matrix (c(0.9,1.18,1,1.6,1.4,1.7,1.9,1.45,1.5,2.14), nrow = 10, ncol = 1)
initState <- matrix (c(2,1,1,2,2,2,1,1,2,1),nrow = 10, ncol = 1)
lambda <- matrix(c(0, 0.33, 0.45, 0), nrow = 2, ncol = 2, byrow = TRUE)
beta <- matrix(c(1, 2, 3, 4, 6, 8), nrow = 3, ncol = 2, byrow = TRUE)
VarY(beta,1,2,Xtest[3,],10,lambda)

Preparing data for parameter estimation procedure

Description

Regressors matrix formation taking into account observation times and initial states. Kronecker product is used.

Usage

Xreg(tGiven, initState, X, lambda)

Arguments

tGiven

Vector of the observed times (n x 1), n – number of observations

initState

Vector of the initial states (n x 1), n – number of observations

X

Matrix of predictors (n x k), n - number of observations, k - number of columns (k - 1 - number of regressors).

lambda

Matrix with the known transition rates \lambda_{i,j}, (m x m), m – number of states

Details

Function calculates the following expression ![](matrix.png "Fig.1"), where vector of average sojourn times in each state is calculated using function Aver_soj_time.

Value

Matrix (n x 2k)

Examples

Xtest <- cbind(rep_len(1,10),c(2,5,7,3,1,1,2,2,3,6), c(5,9,1,2,3,2,3,5,2,2))
tGiven <- matrix (c(6,4.8,1,2.6,6.4,1.7,2.9,4.4,1.5,3.4), nrow = 10, ncol = 1)
initState <- matrix (c(2,1,1,2,2,2,1,1,2,1),nrow = 10, ncol = 1)
lambda <- matrix(c(0, 0.33, 0.45, 0), nrow = 2, ncol = 2, byrow = TRUE)
Xreg(tGiven, initState, Xtest, lambda)

Simulation of the vector of responses Y. Data preparation stage for simulation.

Description

Additional function to be used for simulation purposes (academical or research). Simulating the vector of responses Y according to the formula (see details).

Usage

Ysimulation(t, I, X, lambda, sigma = 1, beta)

Arguments

t

Vector of the observed time (n x 1), n – number of observations

I

Vector of the initial states (n x 1), n – number of observations

X

Matrix of predictors (n x k), n - number of observations, k - number of columns (k - 1 - number of regressors).

lambda

Matrix with the known transition rates \lambda_{i,j}, (m x m), m – number of states

sigma

Scalar, the standard deviation of the disturbance term

beta

Matrix (k x m), k - number of columns (k - 1 - number of regressors), m - number of states, m = 2,3,.. .

Details

The i-th response $Y_i$ is defined by the following formula: $Y_i(t)=x_i\beta + Z_i sqrtt, i=1,...,n.$ The vector with stationary probabilities is user-defined vector.

Value

Vector with new response values of vector Y (n x 1)

Examples

Xtest <- cbind(rep_len(1,10),c(2,5,7,3,1,1,2,2,3,6), c(5,4,1,2,3,2,3,5,2,2))
tGiven <- matrix (c(0.9,1.18,1,1.6,1.4,1.7,1.9,1.45,1.5,2.14), nrow = 10, ncol = 1)
initState <- matrix (c(2,1,1,2,2,2,1,1,2,1),nrow = 10, ncol = 1)
lambda <- matrix(c(0, 0.33, 0.45, 0), nrow = 2, ncol = 2, byrow = TRUE)
beta <- matrix(c(1, 2, 3, 4, 6, 8), nrow = 3, ncol = 2, byrow = TRUE)
Ysimulation(tGiven,initState,Xtest,lambda,1,beta)

Transformation of vector with initial states I for various observations. Data preparation stage for simulation.

Description

Additional function to be used for simulation purposes (academical or research). Transforming of vector with initial states I for various observations with respect to stationary distribution of the states for the random environment.

Usage

randomizeInitState(StatPr, X, p = 1)

Arguments

StatPr

Vector (m x 1), m - number of states, m = 2,3,.. .The vector with stationary probabilities, user-defined vector.

X

Matrix (n x k), n - number of observations, k - number of columns (k - 1 - number of regressors). The matrix is needed to get the number of observations.

p

Scalar (from 1 to +inf), random number for simulation. The default value is 1.

Details

The initial states (m - number of states, m = 2,3,...) for various observations are independent and are chosen with respect to stationary distribution of the states for the random environment. The vector with stationary probabilities is user-defined vector.

Value

Vector with new initial states, according to stationary distribution of the states for the random environment.

Examples

Xtest <- cbind(rep_len(1,10),c(2,5,7,3,1,1,2,2,3,6), c(5,9,1,2,3,2,3,5,2,2))
StatPr <- matrix (c(0.364,0.242,0.394), nrow = 3, ncol = 1)
randomizeInitState(StatPr,Xtest,1)

Transformation of the observed time vector tau. Data preparation stage for simulation

Description

Additional function to be used for simulation purposes (academical or research). Transforming of the observed time vector tau according to user preferences. Random disturbances are entered in the initial values of the vector tau. The expectation of new observed times coincides with initial values of vector tau.

Usage

randomizeTau(tau, p, k0 = 2, k1 = 1)

Arguments

tau

Vector (n x 1), n - number of observations

p

Scalar (from 1 to +inf), random number for simulation. The default value is 1

k0

Scalar (from 1 to +inf). Multiplicative parameter for transforming the initial value The default is k0 = 2.

k1

Scalar. The number of digits after the comma when rounded. The default is 1.

Details

Initial values of observation times are multiplied by a random value ($tau_i$ x k x rnd(0, 1)). All times are independent and time of ith observation has uniform distribution on (0, k$tau_i$).

Value

Vector with new observation times, according to user preferences.

Examples

tGiven <- matrix (c(6,4.8,1,2.6,6.4,1.7,2.9,4.4,1.5,3.4), nrow = 10, ncol = 1)
randomizeTau(tGiven,1,2,2)

Transformation of regressors' matrix X. Data preparation stage for simulation

Description

Additional function to be used for simulation purposes (academical or research). Transforming the matrix of regressors according to user preferences. Random disturbances (according to a chosen distribution) are entered in the initial values of the matrix X. The expectation of the resulting matrix coincides with the initial matrix X.

Usage

randomizeX(X, p = 1, k0 = 1, k1 = 0.5, k2 = 1, k3 = 1)

Arguments

X

Matrix (n x k), n - number of observations, k - number of columns (k - 1 - number of regressors). Note, that 1st column contains only ones (1) (intercept)

p

Scalar (from 1 to +inf), random number for simulation. The default value is 1

k0

Scalar. Number from 1 to 3 (distribution selection). k0 = 1 - uniform distribution (RV Z ~ U (K1, k2)). k0 = 2 - exponential distribution (RV Z ~ exp(lambda)). k0 = 3 - Gamma distribution (RV Z ~ gamma(shape, rate)). The default value is k0 = 1.

k1

Scalar. 1) If k0 = 1, then k1 is a left boundary of uniform distribution (RV Z ~ U(k1, k2)). 2) If k0 = 2, then k1 is a parameter lambda of exponential distribution (RV Z ~ exp(lambda)). 3) If k0 = 3, then k1 is a shape parameter of Gamma distribution (RV Z ~ gamma(shape, rate)). The default is k1 = 0.5.

k2

Scalar. 1) If k0 = 1, then k2 is a right boundary of uniform distribution (RV Z ~ U(k1, k2). 2) If k0 = 3, then k2 is a rate parameter of Gamma distribution (RV Z ~ gamma(shape, rate)). The default is k2 = 1.

k3

Scalar. The number of digits after the comma when rounded. The default value is 1.

Details

Random perturbations are added to the initial values of matrix X elements ($X_i,j$ + rnd), which are distributed according to a chosen distribution (possible alternatives: uniform, exponential and gamma distribution).

Value

New transformed matrix of regressors (n x k), according to user preferences.

Examples

Xtest <- cbind(rep_len(1,10),c(2,5,7,3,1,1,2,2,3,6), c(5,9,1,2,3,2,3,5,2,2))
randomizeX(Xtest,1,1,1,2,2)