ar2precision {INLAspacetime}

Precision matrix for an AR2 model.

Description

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

ar2cor

Examples

ar2precision(7, c(1, -1.5, 0.9))