Constructs a first-order random walk model for spatial or temporal processes. The RW1 model assumes that first-order differences \(\Delta W_i = W_i - W_{i-1}\) are independent and identically distributed Gaussian variables.
rw1(mesh, cyclic = FALSE, constr = TRUE, ...)numerical vector or inla.mesh.1d object, locations to build the mesh. For numerical vectors, assumes equally spaced locations.
logical, whether the mesh is circular. If TRUE, the first and last locations are treated as neighbors. Cannot be FALSE when constr = TRUE.
logical, whether to enforce the sum-to-zero constraint \(\sum_{i=1}^n h_i W_i = 0\). If FALSE, fixes the first element \(W_1 = 0\). Must be TRUE for cyclic models.
additional arguments (currently unused)
An `ngme_operator` object containing the precision matrix and related components for the RW1 model.
The RW1 model is defined by the precision matrix \(K\) that penalizes first-order differences. The model has different structures depending on the constraints:
**Non-cyclic, constrained (default)**: The first row of \(K\) enforces the constraint \(\sum_{i=1}^n h_i W_i = 0\), where \(h_i\) are the mesh weights.
**Non-cyclic, unconstrained**: The first element is fixed at 0 (\(W_1 = 0\)).
**Cyclic**: Treats the domain as circular, connecting the first and last locations as neighbors. The constraint \(\sum_{i=1}^n h_i W_i = 0\) is always enforced. The precision matrix is expanded to size \((n+1) \times (n+1)\) to handle the constraint properly.
# Non-cyclic constrained RW1 (default)
rw1_default <- rw1(1:5)
print(rw1_default$K)
#> 5 x 5 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 0.5 1 1 1 0.5
#> [2,] -1.0 1 . . .
#> [3,] . -1 1 . .
#> [4,] . . -1 1 .
#> [5,] . . . -1 1.0
# Non-cyclic unconstrained RW1 (fixes first element to 0)
rw1_fixed <- rw1(1:5, constr = FALSE)
print(rw1_fixed$K)
#> 5 x 5 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 1 . . . .
#> [2,] -1 1 . . .
#> [3,] . -1 1 . .
#> [4,] . . -1 1 .
#> [5,] . . . -1 1
# Cyclic RW1 (connects first and last locations)
rw1_cyclic <- rw1(1:5, cyclic = TRUE)
print(rw1_cyclic$K)
#> 6 x 6 sparse Matrix of class "dgCMatrix"
#>
#> [1,] . 0.5 1 1 1 0.5
#> [2,] 0.5 1.0 -1 . . .
#> [3,] 1.0 . 1 -1 . .
#> [4,] 1.0 . . 1 -1 .
#> [5,] 1.0 . . . 1 -1.0
#> [6,] 0.5 -1.0 . . . 1.0
# Using with unequally spaced locations
locations <- c(0, 1, 3, 6, 10)
rw1_unequal <- rw1(locations)
print(rw1_unequal$K)
#> 5 x 5 sparse Matrix of class "dgCMatrix"
#>
#> [1,] 0.5 1.5 2.5 3.5 2
#> [2,] -1.0 1.0 . . .
#> [3,] . -1.0 1.0 . .
#> [4,] . . -1.0 1.0 .
#> [5,] . . . -1.0 1