ts.sim: Time series simulation with changepoint effects

In changepointGA: Changepoint Detection via Modified Genetic Algorithm

Time series simulation with changepoint effects

Description

This is a function to simulate time series with changepoint effects. See details below.

Usage

ts.sim(
  beta,
  XMat,
  sigma,
  phi = NULL,
  theta = NULL,
  Delta = NULL,
  CpLoc = NULL,
  seed = NULL
)

Arguments

beta	A parameter vector contains other mean function parameters without changepoint parameters.
XMat	The covairates for time series mean function without changepoint indicators.
sigma	The standard deviation for time series residuals \epsilon_{t}.
phi	A vector for the autoregressive (AR) parameters for AR part.
theta	A vector for the moving average (MA) parameters for MA part.
Delta	The parameter vector contains the changepoint parameters for time series mean function.
CpLoc	A vector contains the changepoint locations range from 1\leq\tau\leq T_{s}.
seed	The random seed for simulation reproducibility.

Details

The simulated time series Z_{t}, t=1,\ldots,T_{s} is from a class of models,

Z_{t}=\mu_{t}+e_{t}.

• Time series observations are IID
  \mu_{t} is a constant and e_{t}'s are independent and identically distributed as N(0,\sigma).

• ARMA(p,q) model with constant mean
  \mu_{t} is a constant, and e_{t} follows an ARMA(p,q) process.

• ARMA(p,q) model with seasonality
  \mu_{t} is not a constant,

  \mu_{t} = ASin(\frac{2\pi t}{S})+BSin(\frac{2\pi t}{S}),

  and e_{t} follows an ARMA(p,q) process.

• ARMA(p,q) model with seasonality and trend
  \mu_{t} is not a constant,

  \mu_{t} = ASin(\frac{2\pi t}{S})+BSin(\frac{2\pi t}{S}) + \alpha t,

  and e_{t} follows an ARMA(p,q) process. The changepoint effects could be introduced through the \mu_{t} as

  \mu_{t} = \Delta_{1}I_{t>\tau_{1}} + \ldots + \Delta_{m}I_{t>\tau_{m}},

  where 1\leq\tau_{1}<\ldots<\tau_{m}\leq T_{s} are the changepoint locations and \Delta_{1},\ldots,\Delta_{m} are the changepoint parameter that need to be estimated.

Value

The simulated time series with attributes:

Z

The simulated time series.

Attributes

DesignX The covariates include all changepoint indicators. mu A vector includes the mean values for simulated time series sequences. theta The true parameter vector (including changepoint effects). CpEff The true changepoint magnitudes. CpLoc The true changepoint locations where we introduce the mean changing effects. arEff A vector givies the AR coefficients. maEff A vector givies the MA coefficients. seed The random seed used for this simulation.

Examples

##### M1: Time series observations are IID
Ts = 1000
betaT = c(0.5) # intercept
XMatT = matrix(1, nrow=Ts, ncol=1)
colnames(XMatT) = "intercept"
sigmaT = 1
DeltaT = c(2, -2)
Cp.prop = c(1/4, 3/4)
CpLocT = floor(Ts*Cp.prop)

myts = ts.sim(beta=betaT, XMat=XMatT, sigma=sigmaT,
              Delta=DeltaT, CpLoc=CpLocT, seed=1234)
TsPlotCheck(Y=myts, tau=CpLocT)

##### M2: ARMA(2,1) model with constant mean
Ts = 1000
betaT = c(0.5) # intercept
XMatT = matrix(1, nrow=Ts, ncol=1)
colnames(XMatT) = "intercept"
sigmaT = 1
phiT = c(0.5, -0.5)
thetaT = c(0.8)
DeltaT = c(2, -2)
Cp.prop = c(1/4, 3/4)
CpLocT = floor(Ts*Cp.prop)

myts = ts.sim(beta=betaT, XMat=XMatT, sigma=sigmaT,
              phi=phiT, theta=thetaT, Delta=DeltaT, CpLoc=CpLocT, seed=1234)
TsPlotCheck(Y=myts, tau=CpLocT)

##### M3: ARMA(2,1) model with seasonality
Ts = 1000
betaT = c(0.5, -0.5, 0.3) # intercept, B, D
period = 30
XMatT = cbind(rep(1, Ts), cos(2*pi*(1:Ts)/period), sin(2*pi*(1:Ts)/period))
colnames(XMatT) = c("intercept", "Bvalue", "DValue")
sigmaT = 1
phiT = c(0.5, -0.5)
thetaT = c(0.8)
DeltaT = c(2, -2)
Cp.prop = c(1/4, 3/4)
CpLocT = floor(Ts*Cp.prop)

myts = ts.sim(beta=betaT, XMat=XMatT, sigma=sigmaT,
              phi=phiT, theta=thetaT, Delta=DeltaT, CpLoc=CpLocT, seed=1234)
TsPlotCheck(Y=myts, tau=CpLocT)


##### M4: ARMA(2,1) model with seasonality and trend
# scaled trend if large number of sample size
Ts = 1000
betaT = c(0.5, -0.5, 0.3, 0.01) # intercept, B, D, alpha
period = 30
XMatT = cbind(rep(1, Ts), cos(2*pi*(1:Ts)/period), sin(2*pi*(1:Ts)/period), 1:Ts)
colnames(XMatT) = c("intercept", "Bvalue", "DValue", "trend")
sigmaT = 1
phiT = c(0.5, -0.5)
thetaT = c(0.8)
DeltaT = c(2, -2)
Cp.prop = c(1/4, 3/4)
CpLocT = floor(Ts*Cp.prop)

myts = ts.sim(beta=betaT, XMat=XMatT, sigma=sigmaT,
              phi=phiT, theta=thetaT, Delta=DeltaT, CpLoc=CpLocT, seed=1234)
TsPlotCheck(Y=myts, tau=CpLocT)

changepointGA documentation built on April 4, 2025, 4:39 a.m.