Help for package NTS

Type:

Package

Title:

Nonlinear Time Series Analysis

Version:

1.1.3

Author:

Ruey Tsay [aut], Rong Chen [aut], Xialu Liu [aut, cre]

Maintainer:

Xialu Liu <xialu.liu@sdsu.edu>

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).

Depends:

R (≥ 3.6.0)

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

Encoding:

UTF-8

Imports:

base,dlm,graphics,MASS,MSwM,Rdpack,parallel,splines,stats,tensor

RdMacros:

Rdpack

RoxygenNote:

7.2.3

Suggests:

testthat

NeedsCompilation:

Packaged:

2023-09-24 20:11:22 UTC; Xialu

Repository:

CRAN

Date/Publication:

2023-09-24 21:10:06 UTC

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)

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")

F Test for Nonlinearity

Description

Compute the F-test statistic for nonlinearity

Usage

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)

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)

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
)

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.

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
)

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)

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, WW*t(WW) is the state innovation covariance matrix, VV*t(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)

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
)

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)

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)

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.

ND Test

Description

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

Usage

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)

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)

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, WW*t(WW) is the state innovation covariance matrix, VV*t(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
)

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)

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
)

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

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
)

Arguments

SISstep.Full.RB

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
)

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)

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)

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)

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

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)

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

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)

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)

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, W*t(W) is the state innovation covariance matrix, V*t(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.

Tsay Test for Nonlinearity

Description

Perform Tsay (1986) nonlinearity test.

Usage

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)

Backtest for Univariate TAR Models

Description

Perform back-test of a univariate SETAR model.

Usage

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)

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)

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)

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)

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)

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)

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
)

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.

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
)

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
)

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)

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
)

Arguments

tmax

length of time.

rho

parameter for O-U process (noise process).

phi_func1

the first convolutional function. Default is 0.5*x^2+0.5*x+0.13.

phi_func2

the second convolutional function. Default is 0.7*x^4-0.1*x^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)

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
)

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 0.5*x^2+0.5*x+0.13.

phi_func2

the second convolutional function. Default is 0.7*x^4-0.1*x^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)

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)

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)

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"
)

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")

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
)

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)

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)

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)

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
)

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)

Prediction of CFAR Processes

Description

Prediction of CFAR processes.

Usage

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)

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)

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.

Rank-Based Portmanteau Tests

Description

Performs rank-based portmanteau statistics.

Usage

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)

Estimating of Random-Coefficient AR Models

Description

Estimate random-coefficient AR models.

Usage

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)

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)

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)

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)

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')

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)

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))

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)

Arguments

nobs

sample size.

par

a list with following components: HH is the state coefficient matrix; WW, WW*t(WW) is the state innovation covariance matrix; VV, VV*t(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)

Threshold Nonlinearity Test

Description

Threshold nonlinearity test.

Usage

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)

Estimate Time-Varying Coefficient AR Models

Description

Estimate time-varying coefficient AR models.

Usage

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)

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)

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)

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
)

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(nT*0.5). The default is k0=300. But k0 = floor(nT*0.8) 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)

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
)

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)

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)

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)

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
)

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)

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)

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.