## 0.1 Introduction

This example illustrates how to fit a discrete-time regime-switching linear model in the dynr package.

## 0.2 Data

First, create a dynr data object using dynr.data. Here our observed variable is called “EMG” . In additon, we are to use a covariate, “self”, in our measurement model, thus it also needs to be specified in the data step.

require(dynr)

data(EMGsim)
dd <- dynr.data(EMGsim, id='id', time='time', observed='EMG', covariates='self')

## 0.3 Measurement Model

Next, we specify the measurement model using prep.measurement. Parameters are indicated by parameter names (e.g.mu_0 and beta_0 in the following code), and fixed values are indicated by “fixed”. In this case, our model has two regimes and the measurement models differ in these two regimes. The values.* arguments specify the starting values and fixed values. The params.* arguments specify the free parameter names or the reserved word “fixed” for fixed parameters. Hence values.int gives the values of the intercept and params.int gives the parameter names of the intercept. Parameters with the same name are constrained to be equal. Likewise values.exo specifies the values of the loadings associated with the exogenous covariate(s) and params.exo specifies the fixed values and parameter names in the covariate loadings. These arguments are lists lists of two matrices: the first matrix is for the measurement model of the first regime; the second matrix is for the second regime. The measurement models for the regimes are written out as follows:

Regime 1: $$EMG(t)=\mu_0 + lEMG(t) + beta_0\times self$$
Regime 2: $$EMG(t)=\mu_1 + lEMG(t) + beta_1\times self$$

$$lEMG$$ is the latent state EMG variable.

recMeas <- prep.measurement(
values.int=list(matrix(0, 1, 1), matrix(1, 1, 1)),
params.int=list(matrix('mu_0', 1, 1), matrix('mu_1', 1, 1)),
values.exo=list(matrix(0, 1, 1), matrix(1, 1, 1)),
params.exo=list(matrix('beta_0', 1, 1), matrix('beta_1', 1, 1)),
obs.names = c('EMG'),
state.names=c('lEMG'),
exo.names=c("self"))

## 0.4 Dynamic Model

In the next chuck, we specify our dynamic models by first specifying the covariance matrices of measurement errors and dynamic noises using prep.noise, and then specifying the dynamic functions using prep.matrixDynamics. The dynamic models are:

Regime 1: $$lEMG(t+1)=\phi_0\times lEMG(t) + w(t)$$
Regime 2: $$lEMG(t+1)=\phi_1\times lEMG(t) + w(t)$$
$$w(t)\sim N(0,dynNoise)$$

We assume the same dynamic noise process applies to both regimes, hence we only have one matrix for dynamic noise specification (*.latent). We also assume there is no measurement error in this model, shown by fixing values.observed to a 0 matrix in prep.noise.

recNoise <- prep.noise(
values.latent=matrix(1, 1, 1),
params.latent=matrix('dynNoise', 1, 1),
values.observed=matrix(0, 1, 1),
params.observed=matrix('fixed', 1, 1))

recDyn <- prep.matrixDynamics(
values.dyn=list(matrix(.1, 1, 1), matrix(.8, 1, 1)),
params.dyn=list(matrix('phi_0', 1, 1), matrix('phi_1', 1, 1)),
isContinuousTime=FALSE)

## 0.5 Regime Probabilities

In the next step, we specify transition probability matrix between the regimes using prep.regimes. This transition probabilities from time t to time t+1 are:

Regime1(t+1) Regime2(t+1)
Regime1(t) $\frac{exp(p11)}{exp(p11)+exp(0)}$ $\frac{exp(0)}{exp(p11)+exp(0)}$
Regime2(t) $\frac{exp(p21)}{exp(p21)+exp(0)}$ $\frac{exp(0)}{exp(p21)+exp(0)}$

(Here, $$p11$$ and $$p21$$ are model parameters. )

recReg <- prep.regimes(
values=matrix(0, 2, 2),
params=matrix(c('p11', 'p21', 'fixed', 'fixed'), 2, 2))

## 0.6 Initial Values

After that, we specify values at time t=0 using prep.initial. These values are used to initialize the recursive algorithm (extended Kalman filter) that dynr uses. The *.inistate arguments specify the initial (starting) states of the latent state variables. The *.inicov arguments specify the starting covariance matrix of the latent state variables. The *.regimep specifies the initial probabilities of the two regimes.

recIni <- prep.initial(
values.inistate=matrix(0, 1, 1),
params.inistate=matrix('fixed', 1, 1),
values.inicov=matrix(1, 1, 1),
params.inicov=matrix('fixed', 1, 1),
values.regimep=c(1, 0),
params.regimep=c('fixed', 'fixed'))

## 0.7 Dynr Model

Now we put together everything we’ve previously specified in dynr.model. This code connects the recipes we’ve written up with our data and writes a c file in our working directory. We can inspect c functions that go with each recipe in the c file.

rsmod <- dynr.model(dynamics=recDyn, measurement=recMeas, noise=recNoise, initial=recIni, regimes=recReg,
data=dd, outfile="RSLinearDiscrete.c")

## 0.8 Tex Options

We can check our model specifications in a neatly printed pdf file using the following code.

The printex command is used to write the model into a Latex file, with a name given by the outFile argument. Then, the tools::texi2pdf command generates a pdf file from the latex file we just created. The system command prints out the pdf file:

We can also print out the model in R, instead of generating a Latex file, using the command plotFormula.

printex(rsmod, ParameterAs=rsmod\$param.names, printInit=TRUE,printRS=TRUE,
outFile="RSLinearDiscrete.tex")
tools::texi2pdf("RSLinearDiscrete.tex")
system(paste(getOption("pdfviewer"), "RSLinearDiscrete.pdf"))

## 0.9 Optimization Step and Results

Finally, it is time to cook dynr (i.e. fit our model through parameter optimization)!

yum <- dynr.cook(rsmod)

And serve!

summary(yum)
##             names  parameters       s.e.    t-value    ci.lower
## phi_0       phi_0  0.31414604 0.05506106  5.7054119  0.20622834
## phi_1       phi_1  0.92725308 0.02727326 33.9986131  0.87379847
## beta_0     beta_0  0.01190633 0.03974690  0.2995536 -0.06599617
## beta_1     beta_1  0.45385563 0.17534348  2.5883805  0.11018873
## mu_0         mu_0  2.98574402 0.08881840 33.6162779  2.81166315
## mu_1         mu_1  4.17992677 0.47141965  8.8666791  3.25596124
## dynNoise dynNoise  0.23228631 0.01500866 15.4768173  0.20286987
## p11           p11  4.49335395 0.58626479  7.6643762  3.34429607
## p21           p21 -4.50750828 0.71461456 -6.3076077 -5.90812708
##             ci.upper
## phi_0     0.42206373
## phi_1     0.98070770
## beta_0    0.08980882
## beta_1    0.79752253
## mu_0      3.15982488
## mu_1      5.10389231
## dynNoise  0.26170275
## p11       5.64241183
## p21      -3.10688949
##
## -2 log-likelihood value at convergence = 748.24
## AIC = 766.24
## BIC = 804.18