Package 'NTS' reference manual

Title:	Nonlinear Time Series Analysis
Description:	Simulation, estimation, prediction procedure, and model identification methods for nonlinear time series analysis, including threshold autoregressive models, Markov-switching models, convolutional functional autoregressive models, nonlinearity tests, Kalman filters and various sequential Monte Carlo methods. More examples and details about this package can be found in the book "Nonlinear Time Series Analysis" by Ruey S. Tsay and Rong Chen, John Wiley & Sons, 2018 (ISBN: 978-1-119-26407-1).
Authors:	Ruey Tsay [aut], Rong Chen [aut], Xialu Liu [aut, cre]
Maintainer:	Xialu Liu <[email protected]>
License:	GPL (>= 2)
Version:	1.1.3
Built:	2024-11-18 04:53:54 UTC
Source:	https://github.com/cran/NTS

Estimation of Autoregressive Conditional Mean Models

Description

Estimation of autoregressive conditional mean models with exogenous variables.

Usage

ACMx(y, order = c(1, 1), X = NULL, cond.dist = "po", ini = NULL)
ACMx(y, order = c(1, 1), X = NULL, cond.dist = "po", ini = NULL)

Arguments

`y`	time series of counts.
`order`	the order of ACM model.
`X`	matrix of exogenous variables.
`cond.dist`	conditional distributions. "po" for Poisson, "nb" for negative binomial, "dp" for double Poisson.
`ini`	initial parameter estimates designed for use in "nb" and "dp".

Value

ACMx returns a list with components:

`data`	time series.
`X`	matrix of exogenous variables.
`estimates`	estimated values.
`residuals`	residuals.
`sresi`	standardized residuals.

Examples

x=rnorm(1000)*0.1
y=matrix(0,1000,1)
y[1]=2
lambda=matrix(0,1000,1)
for (i in 2:1000){
	lambda[i]=2+0.2*y[i-1]/exp(x[i-1])+0.5*lambda[i-1]
	y[i]=rpois(1,exp(x[i])*lambda[i])
}
ACMx(y,order=c(1,1),x,"po")
x=rnorm(1000)*0.1
y=matrix(0,1000,1)
y[1]=2
lambda=matrix(0,1000,1)
for (i in 2:1000){
	lambda[i]=2+0.2*y[i-1]/exp(x[i-1])+0.5*lambda[i-1]
	y[i]=rpois(1,exp(x[i])*lambda[i])
}
ACMx(y,order=c(1,1),x,"po")

Backtest for Univariate TAR Models

Description

Perform back-test of a univariate SETAR model.

Usage

backTAR(model, orig, h = 1, iter = 3000)
backTAR(model, orig, h = 1, iter = 3000)

Arguments

`model`	SETAR model.
`orig`	forecast origin.
`h`	forecast horizon.
`iter`	number of iterations.

Value

backTAR returns a list of components:

`model`	SETAR model.
`error`	prediction errors.
`State`	predicted states.

Backtest

Description

Backtest for an ARIMA time series model.

Usage

backtest(m1, rt, orig, h, xre = NULL, fixed = NULL, include.mean = TRUE)
backtest(m1, rt, orig, h, xre = NULL, fixed = NULL, include.mean = TRUE)

Arguments

`m1`	an ARIMA time series model object.
`rt`	the time series.
`orig`	forecast origin.
`h`	forecast horizon.
`xre`	the independent variables.
`fixed`	parameter constraint.
`include.mean`	a logical value for constant term of the model. Default is TRUE.

Value

The function returns a list with following components:

`orig`	the starting forecast origin.
`err`	observed value minus fitted value.
`rmse`	RMSE of out-of-sample forecasts.
`mabso`	mean absolute error of out-of-sample forecasts.
`bias`	bias of out-of-sample forecasts.

Examples

data=arima.sim(n=100,list(ar=c(0.5,0.3)))
model=arima(data,order=c(2,0,0))
backtest(model,data,orig=70,h=1)
data=arima.sim(n=100,list(ar=c(0.5,0.3)))
model=arima(data,order=c(2,0,0))
backtest(model,data,orig=70,h=1)

Kalman Filter for Tracking in Clutter

Description

This function implements Kalman filter to track a moving target under clutter environment with known indicators.

Usage

clutterKF(nobs, ssw, ssv, yy, ii)
clutterKF(nobs, ssw, ssv, yy, ii)

Arguments

`nobs`	the number of observations.
`ssw`	the standard deviation in the state equation.
`ssv`	the standard deviation for the observation noise.
`yy`	the data.
`ii`	the indicators.

Value

The function returns a list with the following components:

`xhat`	the fitted location.
`shat`	the fitted speed.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Examples

nobs <- 100; pd <- 0.95; ssw <- 0.1; ssv <- 0.5;
xx0 <- 0; ss0 <- 0.1; nyy <- 50;
yrange <- c(-80,80); xdim <- 2; ydim <- nyy;
simu <- simuTargetClutter(nobs,pd,ssw,ssv,xx0,ss0,nyy,yrange)
outKF <- clutterKF(nobs,ssw,ssv,simu$yy,simu$ii)
nobs <- 100; pd <- 0.95; ssw <- 0.1; ssv <- 0.5;
xx0 <- 0; ss0 <- 0.1; nyy <- 50;
yrange <- c(-80,80); xdim <- 2; ydim <- nyy;
simu <- simuTargetClutter(nobs,pd,ssw,ssv,xx0,ss0,nyy,yrange)
outKF <- clutterKF(nobs,ssw,ssv,simu$yy,simu$ii)

Check linear models with cross validation

Description

The function checks linear models with cross-validation (out-of-sample prediction).

Usage

cvlm(y, x, subsize, iter = 100)
cvlm(y, x, subsize, iter = 100)

Arguments

`y`	dependent variable.
`x`	design matrix (should include constant if it is needed).
`subsize`	sample size of subsampling.
`iter`	number of iterations.

Value

The function returns a list with following components.

`rmse`	root mean squares of forecast errors for all iterations.
`mae`	mean absolute forecast errors for all iterations.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Estimation of a CFAR Process

Description

Estimation of a CFAR process.

Usage

est_cfar(f, p = 3, df_b = 10, grid = 1000)
est_cfar(f, p = 3, df_b = 10, grid = 1000)

Arguments

`f`	the functional time series.
`p`	the CFAR order.
`df_b`	the degrees of freedom for natural cubic splines. Default is 10.
`grid`	the number of gird points used to construct the functional time series and noise process. Default is 1000.

Value

The function returns a list with components:

`phi_coef`	the estimated spline coefficients for convolutional function values, a (2*grid+1)-by-p matrix.
`phi_func`	the estimated convolutional function(s), a (df_b+1)-by-p matrix.
`rho`	estimated rho for O-U process (noise process).
`sigma`	estimated sigma for O-U process (noise process).

References

Liu, X., Xiao, H., and Chen, R. (2016) Convolutional autoregressive models for functional time series. Journal of Econometrics, 194, 263-282.

Estimation of a CFAR Process with Heteroscedasticity and Irregualar Observation Locations

Description

Estimation of a CFAR process with heteroscedasticity and irregualar observation locations.

Usage

est_cfarh(
  f,
  weight,
  p = 2,
  grid = 1000,
  df_b = 5,
  num_obs = NULL,
  x_pos = NULL
)
est_cfarh(
  f,
  weight,
  p = 2,
  grid = 1000,
  df_b = 5,
  num_obs = NULL,
  x_pos = NULL
)

Arguments

`f`	the functional time series.
`weight`	the covariance functions of noise process.
`p`	the CFAR order.
`grid`	the number of gird points used to construct the functional time series and noise process. Default is 1000.
`df_b`	the degrees of freedom for natural cubic splines. Default is 10.
`num_obs`	the numbers of observations. It is a t-by-1 vector, where t is the length of time.
`x_pos`	the observation location matrix. If the locations are regular, it is a t-by-(n+1) matrix with all entries 1/n.

Value

The function returns a list with components:

`phi_coef`	the estimated spline coefficients for convolutional function(s).
`phi_func`	the estimated convolutional function(s).
`rho`	estimated rho for O-U process (noise process).
`sigma`	estimated sigma for O-U process (noise process).

References

Liu, X., Xiao, H., and Chen, R. (2016) Convolutional autoregressive models for functional time series. Journal of Econometrics, 194, 263-282.

F Test for a CFAR Process

Description

F test for a CFAR process to specify the CFAR order.

Usage

F_test_cfar(f, p.max = 6, df_b = 10, grid = 1000)
F_test_cfar(f, p.max = 6, df_b = 10, grid = 1000)

Arguments

`f`	the functional time series.
`p.max`	the maximum CFAR order. Default is 6.
`df_b`	the degrees of freedom for natural cubic splines. Default is 10.
`grid`	the number of gird points used to construct the functional time series and noise process. Default is 1000.

Value

The function outputs F test statistics and their p-values.

References

Liu, X., Xiao, H., and Chen, R. (2016) Convolutional autoregressive models for functional time series. Journal of Econometrics, 194, 263-282.

F Test for a CFAR Process with Heteroscedasticity and Irregular Observation Locations

Description

F test for a CFAR process with heteroscedasticity and irregular observation locations to specify the CFAR order.

Usage

F_test_cfarh(
  f,
  weight,
  p.max = 3,
  grid = 1000,
  df_b = 10,
  num_obs = NULL,
  x_pos = NULL
)
F_test_cfarh(
  f,
  weight,
  p.max = 3,
  grid = 1000,
  df_b = 10,
  num_obs = NULL,
  x_pos = NULL
)

Arguments

`f`	the functional time series.
`weight`	the covariance functions for noise process.
`p.max`	the maximum CFAR order. Default is 3.
`grid`	the number of gird points used to construct the functional time series and noise process. Default is 1000.
`df_b`	the degrees of freedom for natural cubic splines. Default is 10.
`num_obs`	the numbers of observations. It is a t-by-1 vector, where t is the length of time.
`x_pos`	the observation location matrix. If the locations are regular, it is a t-by-(n+1) matrix with all entries 1/n.

Value

The function outputs F test statistics and their p-values.

References

Liu, X., Xiao, H., and Chen, R. (2016) Convolutional autoregressive models for functional time series. Journal of Econometrics, 194, 263-282.

F Test for Nonlinearity

Description

Compute the F-test statistic for nonlinearity

Usage

F.test(x, order, thres = 0)
F.test(x, order, thres = 0)

Arguments

`x`	time series.
`order`	AR order.
`thres`	threshold value.

Value

The function outputs the test statistic and its p-value, and return a list with components:

`test.stat`	test statistic.
`p.value`	p-value.
`order`	AR order.

Examples

y=rnorm(100)
F.test(y,2,0)
y=rnorm(100)
F.test(y,2,0)

Generate a CFAR Process

Description

Generate a convolutional functional autoregressive process.

Usage

g_cfar(
  tmax = 1001,
  rho = 5,
  phi_list = NULL,
  grid = 1000,
  sigma = 1,
  ini = 100
)
g_cfar(
  tmax = 1001,
  rho = 5,
  phi_list = NULL,
  grid = 1000,
  sigma = 1,
  ini = 100
)

Arguments

`tmax`	length of time.
`rho`	parameter for O-U process (noise process).
`phi_list`	the convolutional function(s). Default is the density function of normal distribution with mean 0 and standard deviation 0.1.
`grid`	the number of grid points used to construct the functional time series. Default is 1000.
`sigma`	the standard deviation of O-U process. Default is 1.
`ini`	the burn-in period.

Value

The function returns a list with components:

`cfar`	a tmax-by-(grid+1) matrix following a CFAR(p) process.
`epsilon`	the innovation at time tmax.

References

Liu, X., Xiao, H., and Chen, R. (2016) Convolutional autoregressive models for functional time series. Journal of Econometrics, 194, 263-282.

Generate a CFAR(1) Process

Description

Generate a convolutional functional autoregressive process with order 1.

Usage

g_cfar1(
  tmax = 1001,
  rho = 5,
  phi_func = NULL,
  grid = 1000,
  sigma = 1,
  ini = 100
)
g_cfar1(
  tmax = 1001,
  rho = 5,
  phi_func = NULL,
  grid = 1000,
  sigma = 1,
  ini = 100
)

Arguments

`tmax`	length of time.
`rho`	parameter for O-U process (noise process).
`phi_func`	convolutional function. Default is density function of normal distribution with mean 0 and standard deviation 0.1.
`grid`	the number of grid points used to construct the functional time series. Default is 1000.
`sigma`	the standard deviation of O-U process. Default is 1.
`ini`	the burn-in period.

Value

The function returns a list with components:

`cfar1`	a tmax-by-(grid+1) matrix following a CFAR(1) process.
`epsilon`	the innovation at time tmax.

References

Liu, X., Xiao, H., and Chen, R. (2016) Convolutional autoregressive models for functional time series. Journal of Econometrics, 194, 263-282.

Examples

phi_func= function(x)
{
 	return(dnorm(x,mean=0,sd=0.1))
}
y=g_cfar1(100,5,phi_func,grid=1000,sigma=1,ini=100)
phi_func= function(x)
{
 	return(dnorm(x,mean=0,sd=0.1))
}
y=g_cfar1(100,5,phi_func,grid=1000,sigma=1,ini=100)

Generate a CFAR(2) Process

Description

Generate a convolutional functional autoregressive process with order 2.

Usage

g_cfar2(
  tmax = 1001,
  rho = 5,
  phi_func1 = NULL,
  phi_func2 = NULL,
  grid = 1000,
  sigma = 1,
  ini = 100
)
g_cfar2(
  tmax = 1001,
  rho = 5,
  phi_func1 = NULL,
  phi_func2 = NULL,
  grid = 1000,
  sigma = 1,
  ini = 100
)

Arguments

`tmax`	length of time.
`rho`	parameter for O-U process (noise process).
`phi_func1`	the first convolutional function. Default is 0.5x^2+0.5x+0.13.
`phi_func2`	the second convolutional function. Default is 0.7x^4-0.1x^3-0.15*x.
`grid`	the number of grid points used to construct the functional time series. Default is 1000.
`sigma`	the standard deviation of O-U process. Default is 1.
`ini`	the burn-in period.

Value

The function returns a list with components:

`cfar2`	a tmax-by-(grid+1) matrix following a CFAR(1) process.
`epsilon`	the innovation at time tmax.

References

Liu, X., Xiao, H., and Chen, R. (2016) Convolutional autoregressive models for functional time series. Journal of Econometrics, 194, 263-282.

Examples

phi_func1= function(x){
return(0.5*x^2+0.5*x+0.13)
}
phi_func2= function(x){
return(0.7*x^4-0.1*x^3-0.15*x)
}
y=g_cfar2(100,5,phi_func1,phi_func2,grid=1000,sigma=1,ini=100)
phi_func1= function(x){
return(0.5*x^2+0.5*x+0.13)
}
phi_func2= function(x){
return(0.7*x^4-0.1*x^3-0.15*x)
}
y=g_cfar2(100,5,phi_func1,phi_func2,grid=1000,sigma=1,ini=100)

Generate a CFAR(2) Process with Heteroscedasticity and Irregular Observation Locations

Description

Generate a convolutional functional autoregressive process of order 2 with heteroscedasticity, irregular observation locations.

Usage

g_cfar2h(
  tmax = 1001,
  grid = 1000,
  rho = 1,
  min_obs = 40,
  pois = 5,
  phi_func1 = NULL,
  phi_func2 = NULL,
  weight = NULL,
  ini = 100
)
g_cfar2h(
  tmax = 1001,
  grid = 1000,
  rho = 1,
  min_obs = 40,
  pois = 5,
  phi_func1 = NULL,
  phi_func2 = NULL,
  weight = NULL,
  ini = 100
)

Arguments

`tmax`	length of time.
`grid`	the number of grid points used to construct the functional time series.
`rho`	parameter for O-U process (noise process).
`min_obs`	the minimum number of observations at each time.
`pois`	the mean for Poisson distribution. The number of observations at each follows a Poisson distribution plus min_obs.
`phi_func1`	the first convolutional function. Default is 0.5x^2+0.5x+0.13.
`phi_func2`	the second convolutional function. Default is 0.7x^4-0.1x^3-0.15*x.
`weight`	the weight function to determine the standard deviation of O-U process (noise process). Default is 1.
`ini`	the burn-in period.

Value

The function returns a list with components:

`cfar2`	a tmax-by-(grid+1) matrix following a CFAR(1) process.
`epsilon`	the innovation at time tmax.

References

Liu, X., Xiao, H., and Chen, R. (2016) Convolutional autoregressive models for functional time series. Journal of Econometrics, 194, 263-282.

Examples

phi_func1= function(x){
return(0.5*x^2+0.5*x+0.13)
}
phi_func2= function(x){
return(0.7*x^4-0.1*x^3-0.15*x)
}
y=g_cfar2h(200,1000,1,40,5,phi_func1=phi_func1,phi_func2=phi_func2)
phi_func1= function(x){
return(0.5*x^2+0.5*x+0.13)
}
phi_func2= function(x){
return(0.7*x^4-0.1*x^3-0.15*x)
}
y=g_cfar2h(200,1000,1,40,5,phi_func1=phi_func1,phi_func2=phi_func2)

Create Dummy Variables for High-Frequency Intraday Seasonality

Description

Create dummy variables for high-frequency intraday seasonality.

Usage

hfDummy(int = 1, Fopen = 10, Tend = 10, days = 1, pooled = 1, skipmin = 0)
hfDummy(int = 1, Fopen = 10, Tend = 10, days = 1, pooled = 1, skipmin = 0)

Arguments

`int`	length of time interval in minutes.
`Fopen`	number of dummies/intervals from the market open.
`Tend`	number of dummies/intervals to the market close.
`days`	number of trading days in the data.
`pooled`	a logical value indicating whether the data are pooled.
`skipmin`	the number of minites omitted from the opening.

Examples

x=hfDummy(5,Fopen=4,Tend=4,days=2,skipmin=15)
x=hfDummy(5,Fopen=4,Tend=4,days=2,skipmin=15)

Full Information Propagation Step under Mixture Kalman Filter

Description

This function implements the full information propagation step under mixture Kalman filter with full information proposal distribution and Rao-Blackwellization, no delay.

Usage

MKF.Full.RB(
  MKFstep.Full.RB,
  nobs,
  yy,
  mm,
  par,
  II.init,
  mu.init,
  SS.init,
  xdim,
  ydim,
  resample.sch
)
MKF.Full.RB(
  MKFstep.Full.RB,
  nobs,
  yy,
  mm,
  par,
  II.init,
  mu.init,
  SS.init,
  xdim,
  ydim,
  resample.sch
)

Arguments

`MKFstep.Full.RB`	a function that performs one step propagation under mixture Kalman filter, with full information proposal distribution. Its input includes `(mm,II,mu,SS,logww,yyy,par,xdim,ydim)`, where `II`, `mu`, and `SS` are the indicators and its corresponding mean and variance matrix of the Kalman filter components in the last iterations. `logww` is the log weight of the last iteration. `yyy` is the observation at current time step. It should return the Rao-Blackwellization estimation of the mean and variance.
`nobs`	the number of observations `T`.
`yy`	the observations with `T` columns and `ydim` rows.
`mm`	the Monte Carlo sample size `m`.
`par`	a list of parameter values to pass to `Sstep`.
`II.init`	the initial indicators.
`mu.init`	the initial mean.
`SS.init`	the initial variance.
`xdim`	the dimension of the state varible `x_t`.
`ydim`	the dimension of the observation `y_t`.
`resample.sch`	a binary vector of length `nobs`, reflecting the resampling schedule. resample.sch[i]= 1 indicating resample should be carried out at step `i`.

Value

The function returns a list with components:

`xhat`	the fitted value.
`xhatRB`	the fitted value using Rao-Blackwellization.
`Iphat`	the estimated indicators.
`IphatRB`	the estimated indicators using Rao-Blackwellization.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

One Propagation Step under Mixture Kalman Filter for Fading Channels

Description

This function implements the one propagation step under mixture Kalman filter for fading channels.

Usage

MKFstep.fading(mm, II, mu, SS, logww, yyy, par, xdim, ydim, resample)
MKFstep.fading(mm, II, mu, SS, logww, yyy, par, xdim, ydim, resample)

Arguments

`mm`	the Monte Carlo sample size.
`II`	the indicators.
`mu`	the mean in the last iteration.
`SS`	the covariance matrix of the Kalman filter components in the last iteration.
`logww`	is the log weight of the last iteration.
`yyy`	the observations with `T` columns and `ydim` rows.
`par`	a list of parameter values. `HH` is the state coefficient matrix, `WWt(WW)` is the state innovation covariance matrix, `VVt(VV)` is the covariance matrix of the observation noise, `GG1` and `GG2` are the observation coefficient matrix.
`xdim`	the dimension of the state variable `x_t`.
`ydim`	the dimension of the observation `y_t`.
`resample`	a binary vector of length `obs`, reflecting the resampling schedule. resample.sch[i]= 1 indicating resample should be carried out at step `i`.

Value

The function returns a list with components:

`xhat`	the fitted value.
`xhatRB`	the fitted value using Rao-Blackwellization.
`Iphat`	the estimated indicators.
`IphatRB`	the estimated indicators using Rao-Blackwellization.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Fitting Univariate Autoregressive Markov Switching Models

Description

Fit autoregressive Markov switching models to a univariate time series using the package MSwM.

Usage

MSM.fit(y, p, nregime = 2, include.mean = T, sw = NULL)
MSM.fit(y, p, nregime = 2, include.mean = T, sw = NULL)

Arguments

`y`	a time series.
`p`	AR order.
`nregime`	the number of regimes.
`include.mean`	a logical value for including constant terms.
`sw`	logical values for whether coefficients are switching. The length of `sw` has to be equal to the number of coefficients in the model plus include.mean.

Value

MSM.fit returns an object of class codeMSM.lm or MSM.glm, depending on the input model.

Generate Univariate 2-regime Markov Switching Models

Description

Generate univariate 2-regime Markov switching models.

Usage

MSM.sim(
  nob,
  order = c(1, 1),
  phi1 = NULL,
  phi2 = NULL,
  epsilon = c(0.1, 0.1),
  sigma = c(1, 1),
  cnst = c(0, 0),
  ini = 500
)
MSM.sim(
  nob,
  order = c(1, 1),
  phi1 = NULL,
  phi2 = NULL,
  epsilon = c(0.1, 0.1),
  sigma = c(1, 1),
  cnst = c(0, 0),
  ini = 500
)

Arguments

`nob`	number of observations.
`order`	AR order for each regime.
`phi1`, `phi2`	AR coefficients.
`epsilon`	transition probabilities (switching out of regime 1 and 2).
`sigma`	standard errors for each regime.
`cnst`	constant term for each regime.
`ini`	burn-in period.

Value

MSM.sim returns a list with components:

`series`	a time series following SETAR model.
`at`	innovation of the time series.
`state`	states for the time series.
`epsilon`	transition probabilities (switching out of regime 1 and 2).
`sigma`	standard error for each regime.
`cnst`	constant terms.
`order`	AR-order for each regime.
`phi1`, `phi2`	the AR coefficients for two regimes.

Examples

y=MSM.sim(100,c(1,1),0.7,-0.5,c(0.5,0.6),c(1,1),c(0,0),500)
y=MSM.sim(100,c(1,1),0.7,-0.5,c(0.5,0.6),c(1,1),c(0,0),500)

Estimation of a Multivariate Two-Regime SETAR Model

Description

Estimation of a multivariate two-regime SETAR model, including threshold. The procedure of Li and Tong (2016) is used to search for the threshold.

Usage

mTAR(
  y,
  p1,
  p2,
  thr = NULL,
  thrV = NULL,
  delay = c(1, 1),
  Trim = c(0.1, 0.9),
  k0 = 300,
  include.mean = TRUE,
  score = "AIC"
)
mTAR(
  y,
  p1,
  p2,
  thr = NULL,
  thrV = NULL,
  delay = c(1, 1),
  Trim = c(0.1, 0.9),
  k0 = 300,
  include.mean = TRUE,
  score = "AIC"
)

Arguments

`y`	a (`nT`-by-`k`) data matrix of multivariate time series, where `nT` is the sample size and `k` is the dimension.
`p1`	AR-order of regime 1.
`p2`	AR-order of regime 2.
`thr`	threshold variable. Estimation is needed if `thr` = NULL.
`thrV`	vector of threshold variable. If it is not null, thrV must have the same sample size of that of y.
`delay`	two elements (i,d) with "i" being the component and "d" the delay for threshold variable.
`Trim`	lower and upper quantiles for possible threshold value.
`k0`	the maximum number of threshold values to be evaluated.
`include.mean`	logical values indicating whether constant terms are included.
`score`	the choice of criterion used in selection threshold, namely (AIC, det(RSS)).

Value

mTAR returns a list with the following components:

`data`	the data matrix, y.
`beta`	a (`p*k+1`)-by-(`2k`) matrices. The first `k` columns show the estimation results in regime 1, and the second `k` columns show these in regime 2.
`arorder`	AR orders of regimes 1 and 2.
`sigma`	estimated innovational covariance matrices of regimes 1 and 2.
`residuals`	estimated innovations.
`nobs`	numbers of observations in regimes 1 and 2.
`model1`, `model2`	estimated models of regimes 1 and 2.
`thr`	threshold value.
`delay`	two elements (`i`,`d`) with "`i`" being the component and "`d`" the delay for threshold variable.
`thrV`	vector of threshold variable.
`D`	a set of positive threshold values.
`RSS`	residual sum of squares.
`information`	overall information criteria.
`cnst`	logical values indicating whether the constant terms are included in regimes 1 and 2.
`sresi`	standardized residuals.

References

Li, D., and Tong. H. (2016) Nested sub-sample search algorithm for estimation of threshold models. Statisitca Sinica, 1543-1554.

Examples

phi1=matrix(c(0.5,0.7,0.3,0.2),2,2)
phi2=matrix(c(0.4,0.6,0.5,-0.5),2,2)
sigma1=matrix(c(1,0,0,1),2,2)
sigma2=matrix(c(1,0,0,1),2,2)
c1=c(0,0)
c2=c(0,0)
delay=c(1,1)
Trim=c(0.2,0.8)
include.mean=TRUE
y=mTAR.sim(1000,0,phi1,phi2,sigma1,sigma2,c1,c2,delay,ini=500)
est=mTAR(y$series,1,1,0,y$series,delay,Trim,300,include.mean,"AIC")
est2=mTAR(y$series,1,1,NULL,y$series,delay,Trim,300,include.mean,"AIC")
phi1=matrix(c(0.5,0.7,0.3,0.2),2,2)
phi2=matrix(c(0.4,0.6,0.5,-0.5),2,2)
sigma1=matrix(c(1,0,0,1),2,2)
sigma2=matrix(c(1,0,0,1),2,2)
c1=c(0,0)
c2=c(0,0)
delay=c(1,1)
Trim=c(0.2,0.8)
include.mean=TRUE
y=mTAR.sim(1000,0,phi1,phi2,sigma1,sigma2,c1,c2,delay,ini=500)
est=mTAR(y$series,1,1,0,y$series,delay,Trim,300,include.mean,"AIC")
est2=mTAR(y$series,1,1,NULL,y$series,delay,Trim,300,include.mean,"AIC")

Estimation of Multivariate TAR Models

Description

Estimation of multivariate TAR models with given thresholds. It can handle multiple regimes.

Usage

mTAR.est(
  y,
  arorder = c(1, 1),
  thr = c(0),
  delay = c(1, 1),
  thrV = NULL,
  include.mean = c(TRUE, TRUE),
  output = TRUE
)
mTAR.est(
  y,
  arorder = c(1, 1),
  thr = c(0),
  delay = c(1, 1),
  thrV = NULL,
  include.mean = c(TRUE, TRUE),
  output = TRUE
)

Arguments

`y`	vector time series.
`arorder`	AR order of each regime. The number of regime is length of arorder.
`thr`	threshold value(s). There are k-1 threshold for a k-regime model.
`delay`	two elements (i,d) with "i" being the component and "d" the delay for threshold variable.
`thrV`	external threshold variable if any. If thrV is not null, it must have the same number of observations as y-series.
`include.mean`	logical values indicating whether constant terms are included. Default is TRUE for all.
`output`	a logical value indicating four output. Default is TRUE.

Value

mTAR.est returns a list with the following components:

`data`	the data matrix, `y`.
`k`	the dimension of `y`.
`arorder`	AR orders of regimes 1 and 2.
`beta`	a (`p*k+1`)-by-(`2k`) matrices. The first `k` columns show the estimation results in regime 1, and the second `k` columns show these in regime 2.
`sigma`	estimated innovational covariance matrices of regimes 1 and 2.
`thr`	threshold value.
`residuals`	estimated innovations.
`sresi`	standardized residuals.
`nobs`	numbers of observations in different regimes.
`cnst`	logical values indicating whether the constant terms are included in different regimes.
`AIC`	AIC value.
`delay`	two elements (`i,d`) with "`i`" being the component and "`d`" the delay for threshold variable.
`thrV`	values of threshold variable.

Examples

phi1=matrix(c(0.5,0.7,0.3,0.2),2,2)
phi2=matrix(c(0.4,0.6,0.5,-0.5),2,2)
sigma1=matrix(c(1,0,0,1),2,2)
sigma2=matrix(c(1,0,0,1),2,2)
c1=c(0,0)
c2=c(0,0)
delay=c(1,1)
y=mTAR.sim(100,0,phi1,phi2,sigma1,sigma2,c1,c2,delay,ini=500)
est=mTAR.est(y$series,c(1,1),0,delay)
phi1=matrix(c(0.5,0.7,0.3,0.2),2,2)
phi2=matrix(c(0.4,0.6,0.5,-0.5),2,2)
sigma1=matrix(c(1,0,0,1),2,2)
sigma2=matrix(c(1,0,0,1),2,2)
c1=c(0,0)
c2=c(0,0)
delay=c(1,1)
y=mTAR.sim(100,0,phi1,phi2,sigma1,sigma2,c1,c2,delay,ini=500)
est=mTAR.est(y$series,c(1,1),0,delay)

Prediction of A Fitted Multivariate TAR Model

Description

Prediction of a fitted multivariate TAR model.

Usage

mTAR.pred(model, orig, h = 1, iterations = 3000, ci = 0.95, output = TRUE)
mTAR.pred(model, orig, h = 1, iterations = 3000, ci = 0.95, output = TRUE)

Arguments

`model`	multivariate TAR model.
`orig`	forecast origin.
`h`	forecast horizon.
`iterations`	number of iterations.
`ci`	confidence level.
`output`	a logical value for output.

Value

mTAR.pred returns a list with components:

`model`	the multivariate TAR model.
`pred`	prediction.
`Ysim`	fitted `y`.

Examples

phi1=matrix(c(0.5,0.7,0.3,0.2),2,2)
phi2=matrix(c(0.4,0.6,0.5,-0.5),2,2)
sigma1=matrix(c(1,0,0,1),2,2)
sigma2=matrix(c(1,0,0,1),2,2)
c1=c(0,0)
c2=c(0,0)
delay=c(1,1)
y=mTAR.sim(100,0,phi1,phi2,sigma1,sigma2,c1,c2,delay,ini=500)
est=mTAR.est(y$series,c(1,1),0,delay)
pred=mTAR.pred(est,100,1,300,0.90,TRUE)
phi1=matrix(c(0.5,0.7,0.3,0.2),2,2)
phi2=matrix(c(0.4,0.6,0.5,-0.5),2,2)
sigma1=matrix(c(1,0,0,1),2,2)
sigma2=matrix(c(1,0,0,1),2,2)
c1=c(0,0)
c2=c(0,0)
delay=c(1,1)
y=mTAR.sim(100,0,phi1,phi2,sigma1,sigma2,c1,c2,delay,ini=500)
est=mTAR.est(y$series,c(1,1),0,delay)
pred=mTAR.pred(est,100,1,300,0.90,TRUE)

Generate Two-Regime (TAR) Models

Description

Generates multivariate two-regime threshold autoregressive models.

Usage

mTAR.sim(
  nob,
  thr,
  phi1,
  phi2,
  sigma1,
  sigma2 = NULL,
  c1 = NULL,
  c2 = NULL,
  delay = c(1, 1),
  ini = 500
)
mTAR.sim(
  nob,
  thr,
  phi1,
  phi2,
  sigma1,
  sigma2 = NULL,
  c1 = NULL,
  c2 = NULL,
  delay = c(1, 1),
  ini = 500
)

Arguments

`nob`	number of observations.
`thr`	threshold value.
`phi1`	VAR coefficient matrix of regime 1.
`phi2`	VAR coefficient matrix of regime 2.
`sigma1`	innovational covariance matrix of regime 1.
`sigma2`	innovational covariance matrix of regime 2.
`c1`	constant vector of regime 1.
`c2`	constant vector of regime 2.
`delay`	two elements (i,d) with "i" being the component index and "d" the delay for threshold variable.
`ini`	burn-in period.

Value

mTAR.sim returns a list with following components:

`series`	a time series following the multivariate two-regime VAR model.
`at`	innovation of the time series.
`threshold`	threshold value.
`delay`	two elements (i,d) with "i" being the component index and "d" the delay for threshold variable.
`n1`	number of observations in regime 1.
`n2`	number of observations in regime 2.

Examples

phi1=matrix(c(0.5,0.7,0.3,0.2),2,2)
phi2=matrix(c(0.4,0.6,0.5,-0.5),2,2)
sigma1=matrix(c(1,0,0,1),2,2)
sigma2=matrix(c(1,0,0,1),2,2)
c1=c(0,0)
c2=c(0,0)
delay=c(1,1)
y=mTAR.sim(100,0,phi1,phi2,sigma1,sigma2,c1,c2,delay,ini=500)
phi1=matrix(c(0.5,0.7,0.3,0.2),2,2)
phi2=matrix(c(0.4,0.6,0.5,-0.5),2,2)
sigma1=matrix(c(1,0,0,1),2,2)
sigma2=matrix(c(1,0,0,1),2,2)
c1=c(0,0)
c2=c(0,0)
delay=c(1,1)
y=mTAR.sim(100,0,phi1,phi2,sigma1,sigma2,c1,c2,delay,ini=500)

Setting Up The Predictor Matrix in A Neural Network for Time Series Data

Description

The function sets up the predictor matrix in a neural network for time series data.

Usage

NNsetting(zt, locY = 1, nfore = 0, lags = c(1:5), include.lagY = TRUE)
NNsetting(zt, locY = 1, nfore = 0, lags = c(1:5), include.lagY = TRUE)

Arguments

`zt`	data matrix, including the dependent variable `Y(t)`.
`locY`	location of the dependent variable (column number).
`nfore`	number of out-of-sample prediction (1-step ahead).
`lags`	a vector containing the lagged variables used to form the x-matrix.
`include.lagY`	indicator for including lagged `Y(t)` in the predictor matrix.

Value

The function returns a list with following components.

`X`	`x`-matrix for training a neural network.
`y`	`y`-output for training a neural network.
`predX`	`x`-matrix for the prediction subsample.
`predY`	`y`-output for the prediction subsample.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Prediction of CFAR Processes

Description

Prediction of CFAR processes.

Usage

p_cfar(model, f, m = 3)
p_cfar(model, f, m = 3)

Arguments

`model`	CFAR model.
`f`	the functional time series data.
`m`	the forecast horizon.

Value

The function returns a prediction of the CFAR process.

References

Liu, X., Xiao, H., and Chen, R. (2016) Convolutional autoregressive models for functional time series. Journal of Econometrics, 194, 263-282.

Examples

phi_func= function(x)
{
 	return(dnorm(x,mean=0,sd=0.1))
}
y=g_cfar1(100,5,phi_func)
f_grid=y$cfar
index=seq(1,1001,by=50)
f=f_grid[,index]
est=est_cfar(f,1)
pred=p_cfar(est,f,1)
phi_func= function(x)
{
 	return(dnorm(x,mean=0,sd=0.1))
}
y=g_cfar1(100,5,phi_func)
f_grid=y$cfar
index=seq(1,1001,by=50)
f=f_grid[,index]
est=est_cfar(f,1)
pred=p_cfar(est,f,1)

Partial Curve Prediction of CFAR Processes

Description

Partial prediction for CFAR processes. t curves are given and we want to predit the curve at time t+1, but we know the first n observations in the curve, to predict the n+1 observation.

Usage

p_cfar_part(model, f, new.obs)
p_cfar_part(model, f, new.obs)

Arguments

`model`	CFAR model.
`f`	the functional time series data.
`new.obs`	the given first `n` observations.

Value

The function returns a prediction of the CFAR process.

References

Liu, X., Xiao, H., and Chen, R. (2016) Convolutional autoregressive models for functional time series. Journal of Econometrics, 194, 263-282.

ND Test

Description

Compute the ND test statistic of Pena and Rodriguez (2006, JSPI).

Usage

PRnd(x, m = 10, p = 0, q = 0)
PRnd(x, m = 10, p = 0, q = 0)

Arguments

`x`	time series.
`m`	the maximum number of lag of correlation to test.
`p`	AR order.
`q`	MA order.

Value

PRnd function outputs the ND test statistic and its p-value.

References

Pena, D., and Rodriguez, J. (2006) A powerful Portmanteau test of lack of fit for time series. series. Journal of American Statistical Association, 97, 601-610.

Examples

y=arima.sim(n=500,list(ar=c(0.8,-0.6,0.7)))
PRnd(y,10,3,0)
y=arima.sim(n=500,list(ar=c(0.8,-0.6,0.7)))
PRnd(y,10,3,0)

Rank-Based Portmanteau Tests

Description

Performs rank-based portmanteau statistics.

Usage

rankQ(zt, lag = 10, output = TRUE)
rankQ(zt, lag = 10, output = TRUE)

Arguments

`zt`	time series.
`lag`	the maximum lag to calculate the test statistic.
`output`	a logical value for output. Default is TRUE.

Value

rankQ function outputs the test statistics and p-values for Portmanteau tests, and returns a list with components:

`Qstat`	test statistics.
`pv`	p-values.

Examples

phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=2000, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1),sigma=c(1, 1))
rankQ(y$series,10,output=TRUE)
phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=2000, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1),sigma=c(1, 1))
rankQ(y$series,10,output=TRUE)

Estimating of Random-Coefficient AR Models

Description

Estimate random-coefficient AR models.

Usage

rcAR(x, lags = c(1), include.mean = TRUE)
rcAR(x, lags = c(1), include.mean = TRUE)

Arguments

`x`	a time series of data.
`lags`	the lag of AR models. This is more flexible than using order. It can skip unnecessary lags.
`include.mean`	a logical value indicating whether the constant terms are included.

Value

rcAR function returns a list with following components:

`par`	estimated parameters.
`se.est`	standard errors.
`residuals`	residuals.
`sresiduals`	standardized residuals.

Examples

t=50
x=rnorm(t)
phi1=matrix(0.4,t,1)
for (i in 2:t){
   phi1[i]=0.7*phi1[i-1]+rnorm(1,0,0.1)
	x[i]=phi1[i]*x[i-1]+rnorm(1)
}
est=rcAR(x,1,FALSE)
t=50
x=rnorm(t)
phi1=matrix(0.4,t,1)
for (i in 2:t){
   phi1[i]=0.7*phi1[i-1]+rnorm(1,0,0.1)
	x[i]=phi1[i]*x[i-1]+rnorm(1)
}
est=rcAR(x,1,FALSE)

Refine A Fitted 2-Regime Multivariate TAR Model

Description

Refine a fitted 2-regime multivariate TAR model using "thres" as threshold for t-ratios.

Usage

ref.mTAR(m1, thres = 1)
ref.mTAR(m1, thres = 1)

Arguments

`m1`	a fitted mTAR object.
`thres`	threshold value.

Value

ref.mTAR returns a list with following components:

`data`	data matrix, `y`.
`arorder`	AR orders of regimes 1 and 2.
`sigma`	estimated innovational covariance matrices of regimes 1 and 2.
`beta`	a (`p*k+1`)-by-(`2k`) matrices. The first `k` columns show the estimation results in regime 1, and the second `k` columns shows these in regime 2.
`residuals`	estimated innovations.
`sresi`	standard residuals.
`criteria`	overall information criteria.

Examples

phi1=matrix(c(0.5,0.7,0.3,0.2),2,2)
phi2=matrix(c(0.4,0.6,0.5,-0.5),2,2)
sigma1=matrix(c(1,0,0,1),2,2)
sigma2=matrix(c(1,0,0,1),2,2)
c1=c(0,0)
c2=c(0,0)
delay=c(1,1)
y=mTAR.sim(100,0,phi1,phi2,sigma1,sigma2,c1,c2,delay,ini=500)
est=mTAR.est(y$series,c(1,1),0,delay)
ref.mTAR(est,0)
phi1=matrix(c(0.5,0.7,0.3,0.2),2,2)
phi2=matrix(c(0.4,0.6,0.5,-0.5),2,2)
sigma1=matrix(c(1,0,0,1),2,2)
sigma2=matrix(c(1,0,0,1),2,2)
c1=c(0,0)
c2=c(0,0)
delay=c(1,1)
y=mTAR.sim(100,0,phi1,phi2,sigma1,sigma2,c1,c2,delay,ini=500)
est=mTAR.est(y$series,c(1,1),0,delay)
ref.mTAR(est,0)

Simulate A Sample Trajectory

Description

The function generates a sample trajectory of the target and the corresponding observations with sensor locations at (0,0) and (20,0).

Usage

simPassiveSonar(nn = 200, q, r, start, seed)
simPassiveSonar(nn = 200, q, r, start, seed)

Arguments

`nn`	sample size.
`q`	contains the information about the covariance of the noise.
`r`	contains the information about `V`, where `V*t(V)` is the covariance matrix of the observation noise.
`start`	the initial value.
`seed`	the seed of random number generator.

Value

The function returns a list with components:

`xx`	the state data.
`yy`	the observed data.
`H`	the state coefficient matrix.
`W`	`W*t(W)` is the state innovation covariance matrix.
`V`	`V*t(V)` is the observation noise covariance matrix.

Examples

s2 <- 20 #second sonar location at (s2,0)
q <- c(0.03,0.03)
r <- c(0.02,0.02)
nobs <- 200
start <- c(10,10,0.01,0.01)
H <- c(1,0,1,0,0,1,0,1,0,0,1,0,0,0,0,1)
H <- matrix(H,ncol=4,nrow=4,byrow=TRUE)
W <- c(0.5*q[1], 0,0, 0.5*q[2],q[1],0,0,q[2])
W <- matrix(W,ncol=2,nrow=4,byrow=TRUE)
V <- diag(r)
mu0 <- start
SS0 <- diag(c(1,1,1,1))*0.01
simu_out <- simPassiveSonar(nobs,q,r,start,seed=20)
yy<- simu_out$yy
tt<- 100:200
plot(simu_out$xx[1,tt],simu_out$xx[2,tt],xlab='x',ylab='y')
s2 <- 20 #second sonar location at (s2,0)
q <- c(0.03,0.03)
r <- c(0.02,0.02)
nobs <- 200
start <- c(10,10,0.01,0.01)
H <- c(1,0,1,0,0,1,0,1,0,0,1,0,0,0,0,1)
H <- matrix(H,ncol=4,nrow=4,byrow=TRUE)
W <- c(0.5*q[1], 0,0, 0.5*q[2],q[1],0,0,q[2])
W <- matrix(W,ncol=2,nrow=4,byrow=TRUE)
V <- diag(r)
mu0 <- start
SS0 <- diag(c(1,1,1,1))*0.01
simu_out <- simPassiveSonar(nobs,q,r,start,seed=20)
yy<- simu_out$yy
tt<- 100:200
plot(simu_out$xx[1,tt],simu_out$xx[2,tt],xlab='x',ylab='y')

Simulate Signals from A System with Rayleigh Flat-Fading Channels

Description

The function generates a sample from a system with Rayleigh flat-fading channels.

Usage

simu_fading(nobs, par)
simu_fading(nobs, par)

Arguments

`nobs`	sample size.
`par`	a list with following components: `HH` is the state coefficient matrix; `WW`, `WWt(WW)` is the state innovation covariance matrix; `VV`, `VVt(VV)` is the observation noise covariance matrix; `GG` is the observation model.

Examples

HH <- matrix(c(2.37409, -1.92936, 0.53028,0,1,0,0,0,0,1,0,0,0,0,1,0),ncol=4,byrow=TRUE)
WW <- matrix(c(1,0,0,0),nrow=4)
GG <- matrix(0.01*c(0.89409,2.68227,2.68227,0.89409),nrow=1)
VV <- 1.3**15*0.0001
par <- list(HH=HH,WW=WW,GG=GG,VV=VV)
set.seed(1)
simu <- simu_fading(200,par)
HH <- matrix(c(2.37409, -1.92936, 0.53028,0,1,0,0,0,0,1,0,0,0,0,1,0),ncol=4,byrow=TRUE)
WW <- matrix(c(1,0,0,0),nrow=4)
GG <- matrix(0.01*c(0.89409,2.68227,2.68227,0.89409),nrow=1)
VV <- 1.3**15*0.0001
par <- list(HH=HH,WW=WW,GG=GG,VV=VV)
set.seed(1)
simu <- simu_fading(200,par)

Simulate A Moving Target in Clutter

Description

The function simulates a target signal under clutter environment.

Usage

simuTargetClutter(nobs, pd, ssw, ssv, xx0, ss0, nyy, yrange)
simuTargetClutter(nobs, pd, ssw, ssv, xx0, ss0, nyy, yrange)

Arguments

`nobs`	the number observations.
`pd`	the probability to observe the true signal.
`ssw`	the standard deviation in the state equation.
`ssv`	the standard deviation for the observation noise.
`xx0`	the initial location.
`ss0`	the initial speed.
`nyy`	the dimension of the data.
`yrange`	the range of data.

Value

The function returns a list with components:

`xx`	the location.
`ss`	the speed.
`ii`	the indicators for whether the observation is the true signal.
`yy`	the data.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Examples

data=simuTargetClutter(30,0.5,0.5,0.5,0,0.3,3,c(-30,30))
data=simuTargetClutter(30,0.5,0.5,0.5,0,0.3,3,c(-30,30))

Sequential Importance Sampling Step for Fading Channels

Description

This function implements one step of the sequential importance sampling method for fading channels.

Usage

SISstep.fading(mm, xx, logww, yyy, par, xdim2, ydim)
SISstep.fading(mm, xx, logww, yyy, par, xdim2, ydim)

Arguments

`mm`	the Monte Carlo sample size `m`.
`xx`	the sample in the last iteration.
`logww`	the log weight in the last iteration.
`yyy`	the observations with `T` columns and `ydim` rows.
`par`	a list of parameter values. `HH` is the state coefficient model, `WWt(WW)` is the state innovation covariance matrix, `VVt(VV)` is the covariance of the observation noise, `GG` is the observation model.
`xdim2`	the dimension of the state variable `x_t`.
`ydim`	the dimension of the observation `y_t`.

Value

The function returns a list with the following components:

`xx`	the new sample.
`logww`	the log weights.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Generic Sequential Monte Carlo Method

Description

Function of generic sequential Monte Carlo method with delay weighting not using full information proposal distribution.

Usage

SMC(
  Sstep,
  nobs,
  yy,
  mm,
  par,
  xx.init,
  xdim,
  ydim,
  resample.sch,
  delay = 0,
  funH = identity
)
SMC(
  Sstep,
  nobs,
  yy,
  mm,
  par,
  xx.init,
  xdim,
  ydim,
  resample.sch,
  delay = 0,
  funH = identity
)

Arguments

`Sstep`	a function that performs one step propagation using a proposal distribution. Its input includes `(mm,xx,logww,yyy,par,xdim,ydim)`, where `xx` and `logww` are the last iteration samples and log weight. `yyy` is the observation at current time step. It should return `xx` (the samples xt) and `logww` (their corresponding log weight).
`nobs`	the number of observations `T`.
`yy`	the observations with `T` columns and `ydim` rows.
`mm`	the Monte Carlo sample size.
`par`	a list of parameter values to pass to `Sstep`.
`xx.init`	the initial samples of `x_0`.
`xdim`	the dimension of the state variable `x_t`.
`ydim`	the dimension of the observation `y_t`.
`resample.sch`	a binary vector of length `nobs`, reflecting the resampling schedule. resample.sch[i]= 1 indicating resample should be carried out at step `i`.
`delay`	the maximum delay lag for delayed weighting estimation. Default is zero.
`funH`	a user supplied function `h()` for estimation `E(h(x_t) \| y_t+d`). Default is identity for estimating the mean. The function should be able to take vector or matrix as input and operates on each element of the input.

Value

The function returns xhat, an array with dimensions (xdim; nobs; delay+1), and the scaled log-likelihood value loglike. If loglike is needed, the log weight calculation in the Sstep function should retain all constants that are related to the parameters involved. Otherwise, Sstep function may remove all constants that are common to all the Monte Carlo samples. It needs a utility function circular2ordinal, also included in the NTS package, for efficient memory management.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Examples

nobs= 100; pd= 0.95; ssw= 0.1; ssv= 0.5;
xx0= 0; ss0= 0.1; nyy= 50;
yrange= c(-80,80); xdim= 2; ydim= nyy;
mm= 10000
yr=yrange[2]-yrange[1]
par=list(ssw=ssw,ssv=ssv,nyy=nyy,pd=pd,yr=yr)
simu=simuTargetClutter(nobs,pd,ssw,ssv,xx0,ss0,nyy,yrange)
xx.init=matrix(nrow=2,ncol=mm)
xx.init[1,]=yrange[1]+runif(mm)*yr
xx.init[2,]=rep(0.1,mm)
resample.sch=rep.int(1,nobs)
out= SMC(Sstep.Clutter,nobs,simu$yy,mm,par,xx.init,xdim,ydim,resample.sch)
nobs= 100; pd= 0.95; ssw= 0.1; ssv= 0.5;
xx0= 0; ss0= 0.1; nyy= 50;
yrange= c(-80,80); xdim= 2; ydim= nyy;
mm= 10000
yr=yrange[2]-yrange[1]
par=list(ssw=ssw,ssv=ssv,nyy=nyy,pd=pd,yr=yr)
simu=simuTargetClutter(nobs,pd,ssw,ssv,xx0,ss0,nyy,yrange)
xx.init=matrix(nrow=2,ncol=mm)
xx.init[1,]=yrange[1]+runif(mm)*yr
xx.init[2,]=rep(0.1,mm)
resample.sch=rep.int(1,nobs)
out= SMC(Sstep.Clutter,nobs,simu$yy,mm,par,xx.init,xdim,ydim,resample.sch)

Generic Sequential Monte Carlo Using Full Information Proposal Distribution

Description

Generic sequential Monte Carlo using full information proposal distribution.

Usage

SMC.Full(
  SISstep.Full,
  nobs,
  yy,
  mm,
  par,
  xx.init,
  xdim,
  ydim,
  resample.sch,
  delay = 0,
  funH = identity
)
SMC.Full(
  SISstep.Full,
  nobs,
  yy,
  mm,
  par,
  xx.init,
  xdim,
  ydim,
  resample.sch,
  delay = 0,
  funH = identity
)

Arguments

`SISstep.Full`	a function that performs one step propagation using a proposal distribution. Its input includes `(mm,xx,logww,yyy,par,xdim,ydim,resample)`, where `xx` and `logww` are the last iteration samples and log weight. `yyy` is the observation at current time step. It should return `xx` (the samples xt) and `logww` (their corresponding log weight), `resample` is a binary value for resampling.
`nobs`	the number of observations `T`.
`yy`	the observations with `T` columns and `ydim` rows.
`mm`	the Monte Carlo sample size `m`.
`par`	a list of parameter values to pass to `Sstep`.
`xx.init`	the initial samples of `x_0`.
`xdim`	the dimension of the state varible `x_t`.
`ydim`	the dimension of the observation `y_t`.
`resample.sch`	a binary vector of length `nobs`, reflecting the resampling schedule. resample.sch[i]= 1 indicating resample should be carried out at step `i`.
`delay`	the maximum delay lag for delayed weighting estimation. Default is zero.
`funH`	a user supplied function `h()` for estimation `E(h(x_t) \| y_t+d`). Default is identity for estimating the mean. The function should be able to take vector or matrix as input and operates on each element of the input.

Value

The function returns a list with the following components:

`xhat`	the fitted values.
`loglike`	the log-likelihood.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Generic Sequential Monte Carlo Using Full Information Proposal Distribution and Rao-Blackwellization

Description

Generic sequential Monte Carlo using full information proposal distribution with Rao-Blackwellization estimate, and delay is 0.

Usage

SMC.Full.RB(
  SISstep.Full.RB,
  nobs,
  yy,
  mm,
  par,
  xx.init,
  xdim,
  ydim,
  resample.sch
)
SMC.Full.RB(
  SISstep.Full.RB,
  nobs,
  yy,
  mm,
  par,
  xx.init,
  xdim,
  ydim,
  resample.sch
)

Arguments

`SISstep.Full.RB`	a function that performs one step propagation using a proposal distribution. Its input includes `(mm,xx,logww,yyy,par,xdim,ydim,resample)`, where `xx` and `logww` are the last iteration samples and log weight. `yyy` is the observation at current time step. It should return `xx` (the samples xt) and `logww` (their corresponding log weight), `resample` is a binary value for resampling.
`nobs`	the number of observations `T`.
`yy`	the observations with `T` columns and `ydim` rows.
`mm`	the Monte Carlo sample size `m`.
`par`	a list of parameter values to pass to `Sstep`.
`xx.init`	the initial samples of `x_0`.
`xdim`	the dimension of the state varible `x_t`.
`ydim`	the dimension of the observation `y_t`.
`resample.sch`	a binary vector of length `nobs`, reflecting the resampling schedule. resample.sch[i]= 1 indicating resample should be carried out at step `i`.

Value

The function returns a list with the following components:

`xhat`	the fitted values.
`xhatRB`	the fitted values using Rao-Blackwellization.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Generic Sequential Monte Carlo Smoothing with Marginal Weights

Description

Generic sequential Monte Carlo smoothing with marginal weights.

Usage

SMC.Smooth(
  SISstep,
  SISstep.Smooth,
  nobs,
  yy,
  mm,
  par,
  xx.init,
  xdim,
  ydim,
  resample.sch,
  funH = identity
)
SMC.Smooth(
  SISstep,
  SISstep.Smooth,
  nobs,
  yy,
  mm,
  par,
  xx.init,
  xdim,
  ydim,
  resample.sch,
  funH = identity
)

Arguments

`SISstep`	a function that performs one propagation step using a proposal distribution. Its input includes `(mm,xx,logww,yyy,par,xdim,ydim)`, where `xx` and `logww` are the last iteration samples and log weight. `yyy` is the observation at current time step. It should return xx (the samples xt) and logww (their corresponding log weight).
`SISstep.Smooth`	the function for backward smoothing step.
`nobs`	the number of observations `T`.
`yy`	the observations with `T` columns and `ydim` rows.
`mm`	the Monte Carlo sample size `m`.
`par`	a list of parameter values.
`xx.init`	the initial samples of `x_0`.
`xdim`	the dimension of the state variable `x_t`.
`ydim`	the dimension of the observation `y_t`.
`resample.sch`	a binary vector of length `nobs`, reflecting the resampling schedule. resample.sch[i]= 1 indicating resample should be carried out at step `i`.
`funH`	a user supplied function `h()` for estimation `E(h(x_t) \| y_1,...,y_T`). Default is identity for estimating the mean. The function should be able to take vector or matrix as input and operates on each element of the input.

Value

The function returns the smoothed values.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Sequential Monte Carlo for A Moving Target under Clutter Environment

Description

The function performs one step propagation using the sequential Monte Carlo method with partial state proposal for tracking in clutter problem.

Usage

Sstep.Clutter(mm, xx, logww, yyy, par, xdim, ydim)
Sstep.Clutter(mm, xx, logww, yyy, par, xdim, ydim)

Arguments

`mm`	the Monte Carlo sample size `m`.
`xx`	the sample in the last iteration.
`logww`	the log weight in the last iteration.
`yyy`	the observations.
`par`	a list of parameter values `(ssw,ssv,pd,nyy,yr)`, where `ssw` is the standard deviation in the state equation, `ssv` is the standard deviation for the observation noise, `pd` is the probability to observe the true signal, `nyy` the dimension of the data, and `yr` is the range of the data.
`xdim`	the dimension of the state varible.
`ydim`	the dimension of the observation.

Value

The function returns a list with the following components:

`xx`	the new sample.
`logww`	the log weights.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Examples

nobs <- 100; pd <- 0.95; ssw <- 0.1; ssv <- 0.5;
xx0 <- 0; ss0 <- 0.1; nyy <- 50;
yrange <- c(-80,80); xdim <- 2; ydim <- nyy;
simu <- simuTargetClutter(nobs,pd,ssw,ssv,xx0,ss0,nyy,yrange)
resample.sch <- rep(1,nobs)
mm <- 10000
yr <- yrange[2]-yrange[1]
par <- list(ssw=ssw,ssv=ssv,nyy=nyy,pd=pd,yr=yr)
yr<- yrange[2]-yrange[1]
xx.init <- matrix(nrow=2,ncol=mm)
xx.init[1,] <- yrange[1]+runif(mm)*yr
xx.init[2,] <- rep(0.1,mm)
out <- SMC(Sstep.Clutter,nobs,simu$yy,mm,par,xx.init,xdim,ydim,resample.sch)
nobs <- 100; pd <- 0.95; ssw <- 0.1; ssv <- 0.5;
xx0 <- 0; ss0 <- 0.1; nyy <- 50;
yrange <- c(-80,80); xdim <- 2; ydim <- nyy;
simu <- simuTargetClutter(nobs,pd,ssw,ssv,xx0,ss0,nyy,yrange)
resample.sch <- rep(1,nobs)
mm <- 10000
yr <- yrange[2]-yrange[1]
par <- list(ssw=ssw,ssv=ssv,nyy=nyy,pd=pd,yr=yr)
yr<- yrange[2]-yrange[1]
xx.init <- matrix(nrow=2,ncol=mm)
xx.init[1,] <- yrange[1]+runif(mm)*yr
xx.init[2,] <- rep(0.1,mm)
out <- SMC(Sstep.Clutter,nobs,simu$yy,mm,par,xx.init,xdim,ydim,resample.sch)

Sequential Importance Sampling under Clutter Environment

Description

This function performs one step propagation using the sequential importance sampling with full information proposal distribution under clutter environment.

Usage

Sstep.Clutter.Full(mm, xx, logww, yyy, par, xdim, ydim, resample.sch)
Sstep.Clutter.Full(mm, xx, logww, yyy, par, xdim, ydim, resample.sch)

Arguments

`mm`	the Monte Carlo sample size `m`.
`xx`	the samples in the last iteration.
`logww`	the log weight in the last iteration.
`yyy`	the observations.
`par`	a list of parameter values `(ssw,ssv,pd,nyy,yr)`, where `ssw` is the standard deviation in the state equation, `ssv` is the standard deviation for the observation noise, `pd` is the probability to observe the true signal, `nyy` the dimension of the data, and `yr` is the range of the data.
`xdim`	the dimension of the state variable `x_t`.
`ydim`	the dimension of the observation `y_t`.
`resample.sch`	a binary vector of length `obs`, reflecting the resampling schedule. resample.sch[i]= 1 indicating resample should be carried out at step `i`.

Value

The function returns a list with the following components:

`xx`	the new sample.
`logww`	the log weights.
`r.index`	resample index, if `resample.sch=1`.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Sequential Importance Sampling under Clutter Environment

Description

This function performs one step propagation using the sequential importance sampling with full information proposal distribution and returns Rao-Blackwellization estimate of mean under clutter environment.

Usage

Sstep.Clutter.Full.RB(mm, xx, logww, yyy, par, xdim, ydim, resample.sch)
Sstep.Clutter.Full.RB(mm, xx, logww, yyy, par, xdim, ydim, resample.sch)

Arguments

`mm`	the Monte Carlo sample size `m`.
`xx`	the samples in the last iteration.
`logww`	the log weight in the last iteration.
`yyy`	the observations.
`par`	a list of parameter values `(ssw,ssv,pd,nyy,yr)`, where `ssw` is the standard deviation in the state equation, `ssv` is the standard deviation for the observation noise, `pd` is the probability to observe the true signal, `nyy` the dimension of the data, and `yr` is the range of the data.
`xdim`	the dimension of the state variable `x_t`.
`ydim`	the dimension of the observation `y_t`.
`resample.sch`	a binary vector of length `obs`, reflecting the resampling schedule. resample.sch[i]= 1 indicating resample should be carried out at step `i`.

Value

The function returns a list with the following components:

`xx`	the new sample.
`logww`	the log weights.
`xhat`	the fitted vlaues.
`xhatRB`	the fitted values using Rao-Blackwellization.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Sequential Importance Sampling for A Target with Passive Sonar

Description

This function uses the sequential importance sampling method to deal with a target with passive sonar for smoothing.

Usage

Sstep.Smooth.Sonar(mm, xxt, xxt1, ww, vv, par)
Sstep.Smooth.Sonar(mm, xxt, xxt1, ww, vv, par)

Arguments

`mm`	the Monte Carlo sample size `m`.
`xxt`	the sample in the last iteration.
`xxt1`	the sample in the next iteration.
`ww`	the forward filtering weight.
`vv`	the backward smoothing weight.
`par`	a list of parameter values. `H` is the state coefficient matrix, and `W*t(W)` is the state innovation covariance matrix.

Value

The function returns a list with the following components:

`xx`	the new sample.
`logww`	the log weights.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Sequential Importance Sampling Step for A Target with Passive Sonar

Description

This function implements one step of the sequential importance sampling method for a target with passive sonar.

Usage

Sstep.Sonar(mm, xx, logww, yy, par, xdim = 1, ydim = 1)
Sstep.Sonar(mm, xx, logww, yy, par, xdim = 1, ydim = 1)

Arguments

`mm`	the Monte Carlo sample size `m`.
`xx`	the sample in the last iteration.
`logww`	the log weight in the last iteration.
`yy`	the observations with `T` columns and `ydim` rows.
`par`	a list of parameter values. `H` is the state coefficient matrix, `Wt(W)` is the state innovation covariance matrix, `Vt(V)` is the covariance matrix of the observation noise, `s2` is the second sonar location.
`xdim`	the dimension of the state variable `x_t`.
`ydim`	the dimension of the observation `y_t`.

Value

The function returns a list with the following components:

`xx`	the new sample.
`logww`	the log weights.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Threshold Nonlinearity Test

Description

Threshold nonlinearity test.

Usage

thr.test(y, p = 1, d = 1, thrV = NULL, ini = 40, include.mean = T)
thr.test(y, p = 1, d = 1, thrV = NULL, ini = 40, include.mean = T)

Arguments

`y`	a time series.
`p`	AR order.
`d`	delay for the threshold variable.
`thrV`	threshold variable.
`ini`	initial number of data to start RLS estimation.
`include.mean`	a logical value for including constant terms.

Value

thr.test returns a list with components:

`F-ratio`	F statistic.
`df`	the numerator and denominator degrees of freedom.
`ini`	initial number of data to start RLS estimation.

References

Tsay, R. (1989) Testing and Modeling Threshold Autoregressive Processes. Journal of the American Statistical Associations 84(405), 231-240.

Examples

phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=2000, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1),sigma=c(1, 1))
thr.test(y$series,p=2,d=2,ini=40,include.mean=TRUE)
phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=2000, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1),sigma=c(1, 1))
thr.test(y$series,p=2,d=2,ini=40,include.mean=TRUE)

Tsay Test for Nonlinearity

Description

Perform Tsay (1986) nonlinearity test.

Usage

Tsay(y, p = 1)
Tsay(y, p = 1)

Arguments

`y`	time series.
`p`	AR order.

Value

The function outputs the F statistic, p value, and the degrees of freedom. The null hypothesis is there is no nonlinearity.

References

Tsay, R. (1986) Nonlinearity tests for time series. Biometrika 73(2), 461-466.

Examples

phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=2000, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1),sigma=c(1, 1))
Tsay(y$series,2)
phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=2000, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1),sigma=c(1, 1))
Tsay(y$series,2)

Estimate Time-Varying Coefficient AR Models

Description

Estimate time-varying coefficient AR models.

Usage

tvAR(x, lags = c(1), include.mean = TRUE)
tvAR(x, lags = c(1), include.mean = TRUE)

Arguments

`x`	a time series of data.
`lags`	the lagged variables used, e.g. lags=c(1,3) means lag-1 and lag-3 are used as regressors. It is more flexible than specifying an order.
`include.mean`	a logical value indicating whether the constant terms are included.

Value

trAR function returns the value from function dlmMLE.

Examples

t=50
x=rnorm(t)
phi1=matrix(0.4,t,1)
for (i in 2:t){
   phi1[i]=0.7*phi1[i-1]+rnorm(1,0,0.1)
	x[i]=phi1[i]*x[i-1]+rnorm(1)
}
est=tvAR(x,1)
t=50
x=rnorm(t)
phi1=matrix(0.4,t,1)
for (i in 2:t){
   phi1[i]=0.7*phi1[i-1]+rnorm(1,0,0.1)
	x[i]=phi1[i]*x[i-1]+rnorm(1)
}
est=tvAR(x,1)

Filtering and Smoothing for Time-Varying AR Models

Description

This function performs forward filtering and backward smoothing for a fitted time-varying AR model with parameters in 'par'.

Usage

tvARFiSm(x, lags = c(1), include.mean = TRUE, par)
tvARFiSm(x, lags = c(1), include.mean = TRUE, par)

Arguments

`x`	a time series of data.
`lags`	the lag of AR order.
`include.mean`	a logical value indicating whether the constant terms are included.
`par`	the fitted time-varying AR models. It can be an object returned by function. `tvAR`.

Value

trARFiSm function return values returned by function dlmFilter and dlmSmooth.

Examples

t=50
x=rnorm(t)
phi1=matrix(0.4,t,1)
for (i in 2:t){
   phi1[i]=0.7*phi1[i-1]+rnorm(1,0,0.1)
	x[i]=phi1[i]*x[i-1]+rnorm(1)
}
est=tvAR(x,1)
tvARFiSm(x,1,FALSE,est$par)
t=50
x=rnorm(t)
phi1=matrix(0.4,t,1)
for (i in 2:t){
   phi1[i]=0.7*phi1[i-1]+rnorm(1,0,0.1)
	x[i]=phi1[i]*x[i-1]+rnorm(1)
}
est=tvAR(x,1)
tvARFiSm(x,1,FALSE,est$par)

Estimation of a Univariate Two-Regime SETAR Model

Description

Estimation of a univariate two-regime SETAR model, including threshold value, performing recursive least squares method or nested sub-sample search algorithm. The procedure of Li and Tong (2016) is used to search for the threshold.

Usage

uTAR(
  y,
  p1,
  p2,
  d = 1,
  thrV = NULL,
  thrQ = c(0, 1),
  Trim = c(0.1, 0.9),
  include.mean = TRUE,
  method = "RLS",
  k0 = 300
)
uTAR(
  y,
  p1,
  p2,
  d = 1,
  thrV = NULL,
  thrQ = c(0, 1),
  Trim = c(0.1, 0.9),
  include.mean = TRUE,
  method = "RLS",
  k0 = 300
)

Arguments

`y`	a vector of time series.
`p1`, `p2`	AR-orders of regime 1 and regime 2.
`d`	delay for threshold variable, default is 1.
`thrV`	threshold variable. If thrV is not null, it must have the same length as that of y.
`thrQ`	lower and upper quantiles to search for threshold value.
`Trim`	lower and upper quantiles for possible threshold values.
`include.mean`	a logical value indicating whether constant terms are included.
`method`	"RLS": estimate the model by conditional least squares method implemented by recursive least squares; "NeSS": estimate the model by conditional least squares method implemented by Nested sub-sample search (NeSS) algorithm.
`k0`	the maximum number of threshold values to be evaluated, when the nested sub-sample search (NeSS) method is used. If the sample size is large (> 3000), then k0 = floor(nT0.5). The default is k0=300. But k0 = floor(nT0.8) if nT < 300.

Value

uTAR returns a list with components:

`data`	the data matrix, y.
`arorder`	AR orders of regimes 1 and 2.
`delay`	the delay for threshold variable.
`residuals`	estimated innovations.
`sresi`	standardized residuals.
`coef`	a 2-by-(p+1) matrices. The first row shows the estimation results in regime 1, and the second row shows these in regime 2.
`sigma`	estimated innovational covariance matrices of regimes 1 and 2.
`nobs`	numbers of observations in regimes 1 and 2.
`model1`, `model2`	estimated models of regimes 1 and 2.
`thr`	threshold value.
`D`	a set of threshold values.
`RSS`	RSS
`AIC`	AIC value
`cnst`	logical values indicating whether the constant terms are included in regimes 1 and 2.

References

Li, D., and Tong. H. (2016) Nested sub-sample search algorithm for estimation of threshold models. Statisitca Sinica, 1543-1554.

Examples

phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=2000, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1),sigma=c(1, 1))$series
est=uTAR(y=y,p1=2,p2=2,d=2,thrQ=c(0,1),Trim=c(0.1,0.9),include.mean=TRUE,method="NeSS",k0=50)
phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=2000, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1),sigma=c(1, 1))$series
est=uTAR(y=y,p1=2,p2=2,d=2,thrQ=c(0,1),Trim=c(0.1,0.9),include.mean=TRUE,method="NeSS",k0=50)

General Estimation of TAR Models

Description

General estimation of TAR models with known threshold values. It perform LS estimation of a univariate TAR model, and can handle multiple regimes.

Usage

uTAR.est(
  y,
  arorder = c(1, 1),
  thr = c(0),
  d = 1,
  thrV = NULL,
  include.mean = c(TRUE, TRUE),
  output = TRUE
)
uTAR.est(
  y,
  arorder = c(1, 1),
  thr = c(0),
  d = 1,
  thrV = NULL,
  include.mean = c(TRUE, TRUE),
  output = TRUE
)

Arguments

`y`	time series.
`arorder`	AR order of each regime. The number of regime is the length of arorder.
`thr`	given threshold(s). There are k-1 threshold for a k-regime model.
`d`	delay for threshold variable, default is 1.
`thrV`	external threshold variable if any. If it is not NULL, thrV must have the same length as that of y.
`include.mean`	a logical value indicating whether constant terms are included. Default is TRUE.
`output`	a logical value for output. Default is TRUE.

Value

uTAR.est returns a list with components:

`data`	the data matrix, y.
`k`	the number of regimes.
`arorder`	AR orders of regimes 1 and 2.
`coefs`	a k-by-(p+1) matrices, where `k` is the number of regimes. The i-th row shows the estimation results in regime i.
`sigma`	estimated innovational covariances for all the regimes.
`thr`	threshold value.
`residuals`	estimated innovations.
`sresi`	standardized residuals.
`nobs`	numbers of observations in different regimes.
`delay`	delay for threshold variable.
`cnst`	logical values indicating whether the constant terms are included in different regimes.
`AIC`	AIC value.

Examples

phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=200, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1),sigma=c(1, 1))
thr.est=uTAR(y=y$series, p1=2, p2=2, d=2, thrQ=c(0,1),Trim=c(0.1,0.9), method="RLS")
est=uTAR.est(y=y$series, arorder=c(2,2), thr=thr.est$thr, d=2)
phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=200, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1),sigma=c(1, 1))
thr.est=uTAR(y=y$series, p1=2, p2=2, d=2, thrQ=c(0,1),Trim=c(0.1,0.9), method="RLS")
est=uTAR.est(y=y$series, arorder=c(2,2), thr=thr.est$thr, d=2)

Prediction of A Fitted Univariate TAR Model

Description

Prediction of a fitted univariate TAR model.

Usage

uTAR.pred(model, orig, h = 1, iterations = 3000, ci = 0.95, output = TRUE)
uTAR.pred(model, orig, h = 1, iterations = 3000, ci = 0.95, output = TRUE)

Arguments

`model`	univariate TAR model.
`orig`	forecast origin.
`h`	forecast horizon.
`iterations`	number of iterations.
`ci`	confidence level.
`output`	a logical value for output, default is TRUE.

Value

uTAR.pred returns a list with components:

`model`	univariate TAR model.
`pred`	prediction.
`Ysim`	fitted y.

Examples

phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=2000, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1), sigma=c(1, 1))
thr.est=uTAR(y=y$series, p1=2, p2=2, d=2, thrQ=c(0,1), Trim=c(0.1,0.9), method="RLS")
est=uTAR.est(y=y$series, arorder=c(2,2), thr=thr.est$thr, d=2)
uTAR.pred(mode=est, orig=2000,h=1,iteration=100,ci=0.95,output=TRUE)
phi=t(matrix(c(-0.3, 0.5,0.6,-0.3),2,2))
y=uTAR.sim(nob=2000, arorder=c(2,2), phi=phi, d=2, thr=0.2, cnst=c(1,-1), sigma=c(1, 1))
thr.est=uTAR(y=y$series, p1=2, p2=2, d=2, thrQ=c(0,1), Trim=c(0.1,0.9), method="RLS")
est=uTAR.est(y=y$series, arorder=c(2,2), thr=thr.est$thr, d=2)
uTAR.pred(mode=est, orig=2000,h=1,iteration=100,ci=0.95,output=TRUE)

Generate Univariate SETAR Models

Description

Generate univariate SETAR model for up to 3 regimes.

Usage

uTAR.sim(
  nob,
  arorder,
  phi,
  d = 1,
  thr = c(0, 0),
  sigma = c(1, 1, 1),
  cnst = rep(0, 3),
  ini = 500
)
uTAR.sim(
  nob,
  arorder,
  phi,
  d = 1,
  thr = c(0, 0),
  sigma = c(1, 1, 1),
  cnst = rep(0, 3),
  ini = 500
)

Arguments

`nob`	number of observations.
`arorder`	AR-order for each regime. The length of arorder controls the number of regimes.
`phi`	a 3-by-p matrix. Each row contains the AR coefficients for a regime.
`d`	delay for threshold variable.
`thr`	threshold values.
`sigma`	standard error for each regime.
`cnst`	constant terms.
`ini`	burn-in period.

Value

uTAR.sim returns a list with components:

`series`	a time series following SETAR model.
`at`	innovation of the time seres.
`arorder`	AR-order for each regime.
`thr`	threshold value.
`phi`	a 3-by-p matrix. Each row contains the AR coefficients for a regime.
`cnst`	constant terms
`sigma`	standard error for each regime.

Examples

arorder=rep(1,2)
ar.coef=matrix(c(0.7,-0.8),2,1)
y=uTAR.sim(100,arorder,ar.coef,1,0)
arorder=rep(1,2)
ar.coef=matrix(c(0.7,-0.8),2,1)
y=uTAR.sim(100,arorder,ar.coef,1,0)

Sequential Monte Carlo Using Sequential Importance Sampling for Stochastic Volatility Models

Description

The function implements the sequential Monte Carlo method using sequential importance sampling for stochastic volatility models.

Usage

wrap.SMC(par.natural, yy, mm, setseed = T, resample = T)
wrap.SMC(par.natural, yy, mm, setseed = T, resample = T)

Arguments

`par.natural`	contains three parameters in AR(1) model. The first one is the stationary mean, the second is the AR coefficient, and the third is stationary variance.
`yy`	the data.
`mm`	the Monte Carlo sample size.
`setseed`	the seed number.
`resample`	the logical value indicating for resampling.

Value

The function returns the log-likelihood of the data.

References

Tsay, R. and Chen, R. (2018). Nonlinear Time Series Analysis. John Wiley & Sons, New Jersey.

Package 'NTS'

Help Index

Estimation of Autoregressive Conditional Mean Models

Description

Usage

Arguments

Value

Examples

Backtest for Univariate TAR Models

Description

Usage

Arguments

Value

Backtest

Description

Usage

Arguments

Value

Examples

Kalman Filter for Tracking in Clutter

Description

Usage

Arguments

Value

References

Examples

Check linear models with cross validation

Description

Usage

Arguments

Value

References

Estimation of a CFAR Process

Description

Usage

Arguments

Value

References

Estimation of a CFAR Process with Heteroscedasticity and Irregualar Observation Locations

Description

Usage

Arguments

Value

References

F Test for a CFAR Process

Description

Usage

Arguments

Value

References

F Test for a CFAR Process with Heteroscedasticity and Irregular Observation Locations

Description

Usage

Arguments

Value

References

F Test for Nonlinearity

Description

Usage

Arguments

Value

Examples

Generate a CFAR Process

Description

Usage

Arguments

Value

References

Generate a CFAR(1) Process

Description

Usage

Arguments

Value

References

Examples

Generate a CFAR(2) Process

Description

Usage

Arguments

Value