2.1.1 Random walk model

The base model of the random walk with drift model (Shumway and Stoffer 2011) is given by, \[\begin{equation} \label{eq:2} {{x}_{t}}\text{ =}\delta \text{ +}{{x}_{t-1}}\text{ +}{{w}_{t}} \end{equation}\] where \(t= 1, 2, ..., n\) and \({{w}_{t}}\) is Gaussian white noise, \({{w}_{t}}\sim N(0,\sigma_w^{2})\). The constant \(\delta\) is known as the drift, and when \(\delta =0\), it is regarded as a random walk model. The term random walk origins from the fact that when \(\delta =0\), the value of the time series at time \(t\) is the value of the series at time \(t-1\) plus a completely random movement determined by \({{w}_{t}}\) (Jiang, Sharma, and Johnson 2019). Note that the equation can be formulated as a cumulative sum of white noise variates as follows: \[\begin{equation} \label{eq:3} {{x}_{t}}\text{ = }\delta t\text{ + }\sum\limits_{j=1}^{t}{{{w}_{t}}} \end{equation}\] where the drift \(\delta\) in the model can be seen as the trend of the time series. Therefore, this model is a good proxy for simulating trend, for example, the global temperature data. In the figure below, we show two synthetic time series with and without drift.

2.1.2 Autoregressive model

Autoregressive model (AR), as its name suggests, is a regression with lagged values of the variable itself. The general expression of a \(p\)th-order autoregressive process can be written as (Cryer and Chan 2008): \[\begin{equation} {{x}_{t}}=c+{{\phi }_{1}}{{x}_{t-1}}+{{\phi }_{2}}{{x}_{t-2}}+\cdots +{{\phi }_{p}}{{x}_{t-p}}+{{\varepsilon }_{t}} \end{equation}\]

The following linear AR models are given by Sharma (2000) and included in this package: \[\begin{align} {{x}_{t}}=0.9{{x}_{t-1}}+0.866{{\varepsilon }_{t}}\\ {{x}_{t}}=0.6{{x}_{t-1}}-0.4{{x}_{t-4}}+{{\varepsilon }_{t}}\\ {{x}_{t}}=0.3{{x}_{t-1}}-0.6{{x}_{t-4}}-0.5{{x}_{t-9}}+{{\varepsilon }_{t}} \end{align}\]

where \(\epsilon\) is random Gaussian noise with zero mean and unit standard deviation. For each model, \({{x}_{t}}\) was arbitrarily initialized and a total number of \(N+500\) data points were generated. The first 500 points were discarded to reduce any effects from the arbitrary initialization. These AR models are well studied particularly for validating variable selection algorithms (Galelli et al. 2014; Jiang, Sharma, and Johnson 2019, 2020). It should be noted that variants of the random walk and autoregressive models introduced here can also be generated by the arima.sim() from the stats package (R Core Team 2020).

2.1.3 Threshold autoregressive model

Here, we show two examples of the two-regime threshold autoregressive (TAR) process(Cryer and Chan 2008) given by Sharma (2000): \[\begin{equation} {{x}_{t}}= \begin{cases} -0.9{{x}_{t-3}}+0.1{{\varepsilon }_{t}} & if\text{ }{{x}_{t-3}}\le 0 \\ 0.4{{x}_{t-3}}+0.1{{\varepsilon }_{t}} & if\text{ }{{x}_{t-3}}>0 \end{cases} \end{equation}\]

\[\begin{equation} {{x}_{t}}= \begin{cases} 0.5{{x}_{t-6}}-0.5{{x}_{t-10}}+0.1{{\varepsilon }_{t}} & if\text{ }{{x}_{t-6}}\le 0 \\ 0.8{{x}_{t-10}}+0.1{{\varepsilon }_{t}} & if\text{ }{{x}_{t-6}}>0 \end{cases} \end{equation}\] where \(\epsilon\) is random Gaussian noise with zero mean and unit standard deviation. Similarly, for each model, \({{x}_{t}}\) was arbitrarily initialized and a total number of \(N+500\) data points were generated. The first 500 points were discarded to reduce any effects from the arbitrary initialization.

set.seed(2021)
sample=500

###Synthetic example - RW model
data.rw <- data.gen.rw(nobs=sample,drift=0.1,sd=1)

plot.ts(data.rw$xd, ylim=c(-35,55), main="Random walk", xlab=NA, ylab=NA, cex.axis=1.5)
lines(data.rw$x, col=4); abline(h=0, col=4, lty=2); abline(a=0, b=.1, lty=2)

Example of Random walk model

###Synthetic example - AR models
data.ar1 <- data.gen.ar1(nobs=sample)
data.ar4 <- data.gen.ar4(nobs=sample)
data.ar9 <- data.gen.ar9(nobs=sample)

plot.zoo(cbind(data.ar1$x,data.ar4$x,data.ar9$x), col=c("black","red","blue"),
         ylab=c("AR1","AR4","AR9"),main=NA, xlab=NA, cex.axis=1.5)

Example of Autoregressive models

###Synthetic example - TAR models
# Two TAR models in Sharma (2000)
tar1 <- data.gen.tar1(nobs=1000)$x #TAR in Equation (8)
tar2 <- data.gen.tar2(nobs=1000)$x #TAR in Equation (9)

# Generalized TAR, an example in Jiang et al. (2020)
tar <- data.gen.tar(nobs=1000,ndim=9,phi1=c(0.6,-0.1),
                     phi2=c(-1.1,0),theta=0,d=2,p=2,noise=0.1)$x 

plot.zoo(cbind(tar1,tar2,tar), col=c("black","red","blue"), ylab=c("TAR1","TAR2","TAR"),
              main=NA, xlab=NA, cex.axis=1.5)

Example of Threshold autoregressive models

2.1.4 Sinusoidal model

A general form of sinusoidal models can be written as (Shumway and Stoffer 2011):

\[\begin{equation} x_t = Acos(2\pi ft + \phi) \end{equation}\] where A is the amplitude, is the phase, and f is the frequency.

set.seed(2021)
sample=500

sw <- synthesis::data.gen.SW(nobs=sample, freq=25, A=2, phi=0.6*pi, mu=0, sd=0.1)
plot(sw$t,sw$x, type='o', ylab='Cosines', xlab="t")

Example of Sinusoidal model

2.2.1 Hysteresis loop

\[\begin{equation} \begin{cases} {{x}_{t}}=a\cos (2\pi ft)+s{{\varepsilon }_{t}} \\ {{y}_{t}}=b\cos {{(2\pi ft)}^{m}}-c\sin {{(2\pi ft)}^{n}}+s{{\varepsilon }_{t}} \end{cases} \end{equation}\] where \(a\), \(b\) and \(c\) are parameters, \(m\) and \(n\) are integer numbers, and s is a scaling factor used to alter the levle of noise in the output, which all together specify the classical hysteresis loop (HL). The default HL model datasets was generated from \(y_t\) with \(f = 25Hz\), and additional nine candidate predictors were generated with various frequencies. The default values of the system parameters are \(a = 0.8\), \(b = 0.6\), and \(c = 0.2\), which is known to produce a typical form of hysteresis system. As an example, the formulation of the synthetic data from Jiang, Sharma, and Johnson (2020) is shown in the figure below.

2.2.2 Friedman

\[\begin{equation} y=10\sin (\pi {{x}_{1}}{{x}_{2}})+20\sin {{({{x}_{3}}-0.5)}^{2}}+10{{x}_{4}}+5{{x}_{5}}+s\varepsilon \end{equation}\] where \(s\) is a scaling factor used to alter the level of noise in the output. Variate, \(x_i\), is sampled from a uniform distribution, \(x\sim U(0,1)\), for all \(i = 1, ..., 5\). In the original formulation, 10-dimension inputs \(x\) are generated while only first five inputs are relevant with the response. Additionally, datasets can be generated with both zero and various degrees of collinearity. The 10 candidate inputs were generated from correlated uniform variates according to the method by Fackler (1999). In each generated dataset, additional 500 data points were discarded so as to reduce the effect of an arbitrary initialization.

sample=1000
###synthetic example - Hysteresis loop
#Frequency, sampled from a given range
fd <- c(3,5,10,15,25,30,55,70,95)
data.HL <- data.gen.HL(nobs=sample,m=3,n=5,fp=25,fd=fd, sd.x=0, sd.y=0)

plot(data.HL$x,data.HL$dp[,data.HL$true.cpy], xlab="x", ylab = "y", type = "p", cex.axis=1.5,cex.lab=1.5)

Example of Hysteresis loop

###synthetic example - Friedman
#Friedman with independent uniform variates
data.fm1 <- data.gen.fm1(nobs=sample, ndim = 9, noise = 0)
#Friedman with correlated uniform variates
data.fm2 <- data.gen.fm2(nobs=sample, ndim = 9, r = 0.6, noise = 0) 

plot.zoo(cbind(data.fm1$x,data.fm2$x), col=c("red","blue"), main=NA, xlab=NA,
              ylab=c("Friedman with \n independent uniform variates",
                     "Friedman with \n correlated uniform variates"))

Example of Friedman with independent and correlated uniform variates

2.3.1 Hénon map

The Hénon map devised by Hénon (1976) is a two-dimensional map used to illuminate microstructure of strange attractors. \[\begin{equation} \begin{cases} {{x}_{n+1}}=1-ax_{n}^{2}+{{\theta }_{n}} \\ {{\theta }_{n+1}}=b{{x}_{n}} \end{cases} \end{equation}\] where \(a\) and \(b\) are two parameters. One type of chaotic Hénon maps has values of \(a = 1.4\) and \(b = 0.3\). With varying values of \(a\) and \(b\), the map may be chaotic, intermittent, or converging to a periodic orbit. The initial condition of this system is randomly generated from a uniform distribution ranging from \((-0.5, 0.5)\), and similarly the first 500 points were discarded so as to reduce the effect of an arbitrary initialization.

2.3.2 Logistic map

The Logistic map can be given as (Strogatz 2000): \[\begin{equation} {{x}_{n+1}}=r{{x}_{n}}(1-{{x}_{n}}) \end{equation}\] where \(r\) represents the growth rate and the value of the control parameter \(r\) is restricted to the range of \([0, 4]\). The initial condition of this system is randomly generated from a uniform distribution ranging from \((0, 1)\), and the first 500 points were discarded in order to reduce the effect of an arbitrary initialization.

2.3.3 Duffing map

A Duffing map, also known as Holmes map, can be expressed as (Dignowity et al. 2013): \[\begin{equation} \begin{cases} {{x}_{n+1}}={{\theta }_{n}} \\ {{\theta }_{n+1}}=-\beta {{x}_{n}}+\alpha {{\theta }_{n}}-\theta _{n}^{3} \end{cases} \end{equation}\]

###Synthetic example - Iterated mappings
set.seed(2021)
par(mfrow=c(1,3), ps=12, cex.lab=1.5, pty="s")
sample <- 1000
Henon.map <- data.gen.Henon(nobs = sample, do.plot=TRUE)
Logistic.map <- data.gen.Logistic(nobs = sample, do.plot=TRUE)
Duffing.map <- data.gen.Duffing(nobs = sample, do.plot=TRUE)

Example of Hénon map

2.3.4 Rössler system

\[\begin{equation} \begin{cases} \dot{x}=-y-z, \\ \dot{y}=x+ay, \\ \dot{z}=b+z(x-c). \end{cases} \end{equation}\] The chaotic system depends on three parameters, \(a\), \(b\) and \(c\).The system parameters with values of \(a = 0.2\), \(b = 0.2\), and \(c = 5.7\) can produce a deterministic chaotic time series (Harrington and Van Gorder 2017). In default setting, the time range is fixed from 0 to 50, and a total number of \(N=1000\) paired observations \((x_t, y_t, z_t)\) were generated from an initial condition of \((-2, -10, 0.2)\) with no white noise.

###Synthetic example - Rossler
ts.r <- synthesis::data.gen.Rossler(a = 0.2, b = 0.2, w = 5.7, start=c(-2, -10, 0.2), 
                         time = seq(0, 50, length.out = 5000), s=0)

par(mfrow=c(1,2), ps=12, cex.lab=1.5)
plot(ts.r$x,ts.r$y, xlab="x",ylab = "y", type = "l")
plot(ts.r$x,ts.r$z, xlab="x",ylab = "z", type = "l")

Example of Rossler system: Phase portraits in a 2D projection of its state space

2.3.5 Lorenz system

\[\begin{equation} \begin{cases} \dot{x}=\sigma (y-x), \\ \dot{y}=x(\rho -z)-y, \\ \dot{z}=xy-\beta z. \end{cases} \end{equation}\] where \(\sigma\), \(\rho\), \(\beta\)>0 are parameters. The default values for the system parameters are \(\sigma=10\), \(\rho=28\), \(\beta=8/3\) and the time range is fixed from 0 to 50, and a total number of \(N=1000\) paired observations \((x_t, y_t, z_t)\) were generated from an initial condition of \((-13, -14, 47)\) with no white noise.

###Synthetic example - Lorenz
ts.l <- data.gen.Lorenz(sigma = 10, beta = 8/3, rho = 28, start = c(-13, -14, 47), 
                        time = seq(0, 50, length.out = 5000), s=0)

par(mfrow=c(1,2), ps=12, cex.lab=1.5)
plot(ts.l$x,ts.l$y, xlab="x",ylab = "y", type = "l")
plot(ts.l$x,ts.l$z, xlab="x",ylab = "z", type = "l")

Example of Lorenz system: Phase portraits in a 2D projection of its state space

2.4.1 Blobs

\[\begin{equation} p(\mathbf{x}|{{\mu }_{i}},{{\mathbf{\Sigma }}_{i}})=\frac{1}{{{(2\pi )}^{D/2}}|{{\mathbf{\Sigma}}_{i}}{{|}^{1/2}}}\exp \left\{ -\frac{1}{2}{{(\mathbf{x}-{{\mu }_{i}})}^{T}}\;{{\mathbf{\Sigma}}_{i}^{-1}}\;(\mathbf{x}-{{\mu }_{i}}) \right\} \end{equation}\] where \(\mu_i\) is the mean vector and \(\Sigma_i\) is covariance matrix. The idea is to place “blobs” of probability mass in the space to cover the data well.

2.4.2 Circles

\[\begin{equation} \begin{matrix} x=a+r\cos t \\ y=b+r\sin t \end{matrix} \end{equation}\] where \(t\) is a parametric variable in the range of 0 to \(2\pi\). Gaussian noise can be added to each data point generated by the function of the synthesis package.

2.4.3 Spirals

\[\begin{equation} \begin{matrix} x=r(\varphi )\cos \varphi \\ y=r(\varphi )\sin \varphi \end{matrix} \end{equation}\] where \(r\) is a monotonic continuous function of angle \(\varphi\).

Examples of datasets generated by these three classification-based functions in the synthesis package are provided below.

set.seed(2021)
sample=500

par(mfrow=c(1,3), ps=12, cex.lab=1.5, pty="s")
Blobs=data.gen.blobs(nobs=sample, features=2, centers=5, sd=1, bbox=c(-10,10), do.plot=TRUE)
Circles=data.gen.circles(n = sample, r_vec=c(1,1.5), start=runif(1,-1,1), s=0.1, do.plot=TRUE)
Spirals=data.gen.spirals(n = sample, cycles=3, s=0.01, do.plot=TRUE)

Example of Classification system: Blobs, Circles and Spirals

2.5.1 Linear Gaussian state-space model

\[\begin{equation} \begin{matrix} x_t = & \phi x_{t-1} + \sigma_v v_t \\ y_t = & x_t + \sigma_e e_t \end{matrix} \end{equation}\] where \(v_t\) and \(e_t\) denote independent standard Gaussian random variables, i.e. \(N(0,1)\).

###Linear Gaussian state-space model

data.LGSS <- data.gen.LGSS(theta=c(0.75,1.00,0.10), nobs=500, do.plot = TRUE)

Example of linear Gaussian state-space model