In this vignette, we will briefly introduce the random walk model in
ngme2.
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 is constructed assuming independent increments given data \(\mathbf{x}\) (use as location, sorted) of length \(n\):
When \(\epsilon\) follows the NIG distribution: \[ \begin{align} \Delta w_i &= -\mu + \mu V_i + \sigma \sqrt{V_i} Z , \; n=1, \dots, n-1 \\ V_i &\sim IG(\nu, \nu (\Delta x_i)^2), \\ Z_i &\sim N(0, 1), \end{align} \]
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
\[ K = \begin{bmatrix} -1 & 1 \\ & \ddots & \ddots \\ & & -1 & 1 \\ 1 & & & -1 \end{bmatrix}. \]
Similarily, the second-order random walk is constructed assuming the second order difference is independent:
When \(\epsilon\) follows the NIG distribution: \[ \begin{align} \Delta^2 w_i &= -\mu + \mu V_i + \sigma \sqrt{V_i} Z , \; n=1, \dots, n-2 \\ V_i &\sim IG(\nu, \nu (\Delta x_i)^2), \\ Z_i &\sim N(0, 1), \end{align} \]
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
\[ K = \begin{bmatrix} 1 & -2 & 1 \\ & 1 & -2 & 1 \\ & & \ddots & \ddots & \ddots \\ 1 & & & 1 & -2 \\ -2 & 1 & & & 1 \end{bmatrix}. \]
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 . . . 1Doing the simulation is simple, just pass the corresponding model
into simulate function.
simulate(m1)
#> sim_1
#> 1 -0.36664906
#> 2 -0.72728336
#> 3 -1.34137358
#> 4 -0.42978618
#> 5 -0.06984852