Precision matrix for an AR2 model.

Creates a precision matrix as a sparse matrix object considering the specification stated in Details.

Usage

ar2precision(n, a)

Arguments

n

the size of the model.

a

a length three vector with the coefficients. See details.

Value

the precision matrix as a sparse matrix object with edge correction.

Details

Let the second order auto-regression model be defined as

$$a_0 x_t + a_1 x_{t-1} + a_2 x_{t-2} = w_t, w_t ~ N(0, 1).$$

The n times n symmetric precision matrix Q for x_1, x_2, ..., x_n has the following non-zero elements:

\(Q_{1,1} = Q_{n,n} = a_0^2\)

\(Q_{2,2} = Q_{n-1,n-1} = a_0^2 + a_1^2\)

\(Q_{1,2} = Q_{2,1} = Q_{n-1,n} = Q_{n,n-1} = a_0 a_1\)

\(Q_{t,t} = q_0 = a_0^2 + a_1^2 + a_2^2, t = 3, 4, ..., n-2\)

\(Q_{t,t-1} = Q_{t-1,t} = q_1 = a_1(a_0 + a_2), t = 3, 4, ..., n-1\)

\(Q_{t,t-2} = Q_{t-2,t} = q_2 = a_2 a_0, t = 3, 4, ..., n\)

See also

ar2cov

Examples

ar2precision(7, c(1, -1.5, 0.9))
#> 7 x 7 sparse Matrix of class "dsTMatrix"
#>                                             
#> [1,]  1.0 -1.50  0.90  .     .     .     .  
#> [2,] -1.5  3.25 -2.85  0.90  .     .     .  
#> [3,]  0.9 -2.85  4.06 -2.85  0.90  .     .  
#> [4,]  .    0.90 -2.85  4.06 -2.85  0.90  .  
#> [5,]  .    .     0.90 -2.85  4.06 -2.85  0.9
#> [6,]  .    .     .     0.90 -2.85  3.25 -1.5
#> [7,]  .    .     .     .     0.90 -1.50  1.0