Ngme2 random walk model

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.

library(ngme2)
#> This is ngme2 of version 0.6.0
#> - See our homepage: https://davidbolin.github.io/ngme2 for more details.

set.seed(16)
m1 <- f(rexp(5), model="rw1", noise = noise_normal())
m1$operator$K
#> 5 x 5 sparse Matrix of class "dgCMatrix"
#>                   
#> [1,]  1  .  .  . .
#> [2,] -1  1  .  . .
#> [3,]  . -1  1  . .
#> [4,]  .  . -1  1 .
#> [5,]  .  .  . -1 1

m2 <- f(rnorm(6), model="rw2", cyclic = TRUE)
m2$operator$K
#> 6 x 6 sparse Matrix of class "dgCMatrix"
#>                       
#> [1,]  1 -2  1  .  .  .
#> [2,]  .  1 -2  1  .  .
#> [3,]  .  .  1 -2  1  .
#> [4,]  .  .  .  1 -2  1
#> [5,]  1  .  .  .  1 -2
#> [6,] -2  1  .  .  .  1

Simulation

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

simulate(m1)
#>          sim_1
#> 1  0.123119811
#> 2 -0.118100772
#> 3  1.037880995
#> 4 -0.009440498
#> 5  0.021668447