---
title: "synthesis"
output: 
  rmarkdown::html_vignette:
    toc: true
    number_sections: true
vignette: >
  %\VignetteIndexEntry{synthesis}
  %\VignetteEncoding{UTF-8}
  %\VignetteEngine{knitr::rmarkdown}
bibliography: ../inst/synthesis.bib
editor_options: 
  chunk_output_type: console
---

```{r, include = FALSE}
knitr::opts_chunk$set(
                      collapse = TRUE,
                      comment = "#>"
)
```

```{r setup}
op <- par(no.readonly = TRUE)
require(zoo)
library(synthesis)
```

# Introduction

Synthetic data generation has been widely used not only for its usage in privacy preserving data publishing but also for its capability to support testing of new algorithms or methods. The package of **synthesis** generates synthetic time series from commonly used statistical models, including linear, nonlinear and chaotic systems. So far, **synthesis** consists of five groups of statistical models, including linear, nonlinear, dynamical, classification, and state-space systems. The usage of the **synthesis** package covers the synthetic data generation for variable selection, prediction, and classification and clustering based problems.

# Overview

## Linear models

### Random walk model

The base model of the random walk with drift model [@Shumway2011] 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 ${{w}_{t}}$ [@Jiang2019]. 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.

### 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 [@Cryer2008]: 
\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 @Sharma2000 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 [@Galelli2014;@Jiang2019;@Jiang2020a]. 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 [@R2020]. 

### Threshold autoregressive model

Here, we show two examples of the two-regime threshold autoregressive (TAR) process[@Cryer2008] given by @Sharma2000:
\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.

```{r mod1, fig.cap=c('Example of Random walk model','Example of Autoregressive models','Example of Threshold autoregressive models'), fig.height=c(5,7,7), fig.width=c(5,9,9), out.width= c('80%','100%','100%')}
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)

###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)

###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)
```

### Sinusoidal model

A general form of sinusoidal models can be written as [@Shumway2011]: 

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

```{r mod6, fig.cap=c('Example of Sinusoidal model'), fig.height=5, fig.width=9, out.width= c('100%')}
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")
```

## Nonlinear systems

### 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 @Jiang2020a is shown in the figure below.

### 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 @Fackler1999. In each generated dataset, additional 500 data points were discarded so as to reduce the effect of an arbitrary initialization. 

```{r mod2, fig.cap=c('Example of Hysteresis loop', 'Example of Friedman with independent and correlated uniform variates'), fig.height=c(4,7), fig.width=c(4,9), out.width= c('80%','100%')}
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)
###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"))
```

## Dynamic systems

### Hénon map 

The Hénon map devised by @Henon1976 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. 

### Logistic  map 

The Logistic map can be given as [@Strogatz2000]: 
\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. 

### Duffing  map 

A Duffing map, also known as Holmes map, can be expressed as [@Dignowity2013]:
\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}

```{r mod3, fig.cap=c('Example of Hénon map','Example of Logistic map','Example of Duffing map'), fig.height=4, fig.width=9, out.width= '100%'}
###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)
```

### 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 [@Harrington2017]. 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.

```{r mod4, fig.cap='Example of Rossler system: Phase portraits in a 2D projection of its state space', fig.height=5, fig.width=9, out.width= '100%'}
###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")

```

### 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.

```{r mod5, fig.cap='Example of Lorenz system: Phase portraits in a 2D projection of its state space', fig.height=5, fig.width=9, out.width= '100%'}
###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")
```

## Classification systems

### 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. 

### 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.

### 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.

```{r class, fig.cap='Example of Classification system: Blobs, Circles and Spirals', fig.height=4, fig.width=9, out.width= '100%'}
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)
```

## State-space model

A state-space model or the dynamic linear model, which was introduced in @Kalman1960 and @Kalman1961. The model arose in the space tracking setting, where the state equation defines the motion
equations for the position or state of a spacecraft. 

### 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. $N(0,1)$.

```{r ss, fig.cap='Example of linear Gaussian state-space model', fig.height=7, fig.width=9, out.width= '100%'}
###Linear Gaussian state-space model

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

```

## Affine error model

An affine error model relating measurements to a (geophysical) variable, a standard form used in the triple collocation literature [@Zwieback2012]:

```{r affine, fig.cap='Example of the affine error model', fig.height=7, fig.width=9, out.width= '100%'}
# Affine error model with 1 true observation and 3 dummy variables
data.affine<-data.gen.affine(500)

plot.ts(cbind(data.affine$x,data.affine$dp), main="Affine error model")

```

## Brownian motion models

Brownian motion is the random motion of particles suspended in a medium (a liquid or a gas).

A geometric Brownian motion (GBM) (also known as exponential Brownian motion) is a continuous-time stochastic process in which the logarithm of the randomly varying quantity follows a Brownian motion (also called a Wiener process) with drift.

A fractional Brownian motion (fBm) is a generalization of Brownian motion. Unlike classical Brownian motion, the increments of fBm need not be independent. 

```{r brown, fig.cap='Example of Brownian motion models', fig.height=4, fig.width=9, out.width= '100%'}
# Brownian motion models
set.seed(100)
sample <- 500

par(mfrow=c(1,3), ps=10, cex.lab=1.5, pty="s")
data.bm <- data.gen.bm(do.plot = TRUE)
data.gbm <- data.gen.gbm(do.plot = TRUE)
data.fbm <- data.gen.fbm(do.plot = TRUE)
par(op)
```

## Build-up and wash-off models

In water quality modelling, particulate pollutant buildup and loss (e.g., TSS) from urban and suburban catchments representing a mix of pervious and impervious surfaces has been calculated by using more sophisticated buildup/wash-off (BUWO) models [@Wu2019].

```{r wq, fig.cap='Example of build-up and wash-off models', fig.height=4, fig.width=9, out.width= '100%'}
# Build up and wash off model
set.seed(101)
sample = 500

#create a gamma shape storm event
q<- seq(0,20, length.out=sample)
p <- pgamma(q, shape=9, rate =2, lower.tail = T)
p <- c(p[1],p[2:sample]-p[1:(sample-1)])

data.tss<-data.gen.BUWO(sample, k=0.5, a=5, m0=10, q=p)
plot.zoo(cbind(p, data.tss$x, data.tss$y), xlab=NA,
         ylab=c("Q","Bulid-up","Wash-off"), main="TSS")

```

