RW-model.Rmd

---
title: "Ngme2 random walk model"
output: rmarkdown::html_vignette
vignette: >
  %\VignetteIndexEntry{Ngme2 random walk model}
  %\VignetteEngine{knitr::rmarkdown}
  %\VignetteEncoding{UTF-8}
---

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

In this vignette, we will briefly introduce the random walk model in `ngme2`.

## Description

In `ngme2`, we currently support random walk model of the first-order and the second order.

Remeber in `ngme2`, all the models has the same form:
$$K \mathbf{W} = \boldsymbol{\epsilon},$$
where $K$ is the operator matrix, $\epsilon$ is the random noise (maybe non-Gaussian),
$\mathbf{W}$ is the random vector we want to model.

### The first-order random walk (RW(1))

The **first-order random walk** is constructed assuming independent increments given data
$\mathbf{x}$ (use as location, sorted) of length $n$:

1. When $\epsilon$ follows the NIG distribution:
$$
\begin{align}
\Delta w_i &= -\mu + \mu V_i + \sigma \sqrt{V_i} Z , \; n=1, \dots, n-1 \\
V_i &\sim IG(\nu, \nu (\Delta x_i)^2), \\
Z_i &\sim N(0, 1),
\end{align}
$$

2. When $\epsilon$ follows the normal distribution:
$$
\begin{align}
\Delta w_i &= \epsilon_i , \; n=1, \dots, n-1 \\
\epsilon_i &\sim N(0, \sigma^2 \Delta x_i), \\
\end{align}
$$

where $\Delta w_i := w_{i+1} - w_{i}$, $\Delta x_i := x_{i+1} - x_{i}$.

The operator matrix $K$ of dimension $(n-1 \times n)$ is

$$
K =
  \begin{bmatrix}
    -1 & 1 \\
      & -1 & 1 \\
      & & \ddots & \ddots \\
      & & & -1 & 1
  \end{bmatrix}.
$$

We also provide the special case of **circular random walk**, which the 1st element and n-th element is connected.
In the **circular** RW(1) case, the operator matrix $K$ of dimension $(n-1 \times n-1)$ is

$$
K =
  \begin{bmatrix}
    -1 & 1 \\
      & \ddots & \ddots \\
      &  & -1 & 1 \\
       1 & & & -1
  \end{bmatrix}.
$$

### The second-order random walk (RW(2))

Similarily, the **second-order random walk** is constructed assuming the second order difference is independent:

1. When $\epsilon$ follows the NIG distribution:
$$
\begin{align}
\Delta^2 w_i &= -\mu + \mu V_i + \sigma \sqrt{V_i} Z , \; n=1, \dots, n-2 \\
V_i &\sim IG(\nu, \nu (\Delta x_i)^2), \\
Z_i &\sim N(0, 1),
\end{align}
$$

2. When $\epsilon$ follows the normal distribution:
$$
\begin{align}
\Delta^2 w_i &= \epsilon_i , \; n=1, \dots, n-2 \\
\epsilon_i &\sim N(0, \sigma^2 \Delta x_i), \\
\end{align}
$$

where $\Delta^2 w_i := w_{i+2} - 2w_{i+1} - w_{i}$, $\Delta x_i := x_{i+1} - x_{i}$.

The operator matrix $K$ of dimension $(n-2 \times n)$ is

$$
K =
  \begin{bmatrix}
    1 & -2 & 1 \\
      & 1 & -2 & 1 \\
      & & \ddots & \ddots & \ddots \\
      & & & 1 & -2 & 1
  \end{bmatrix}.
$$

In the **circular** RW(2) case, the operator matrix $K$ of dimension $(n-2 \times n-2)$ is

$$
K =
  \begin{bmatrix}
    1 & -2 & 1 \\
      & 1 & -2 & 1 \\
      & & \ddots & \ddots & \ddots \\
      1 & & & 1 & -2 \\
      -2 & 1 & & & 1
  \end{bmatrix}.
$$


## Usage

Use the `f(model = "rw1")` or `f(model = "rw2")` (in formula) to specify a random walk model.

```{r rw-usage}
library(ngme2)
set.seed(16)
m1 <- f(rexp(5), model="rw1", noise = noise_normal())
m1$operator$K

m2 <- f(rnorm(6), model="rw2", cyclic = TRUE)
m2$operator$K
```

## Simulation

Doing the simulation is simple, just pass the corresponding model into `simulate` function.

```{r rw-simulate}
simulate(m1)
```