ar2precision {INLAspacetime}R Documentation

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

a0xt+a1xt1+a2xt2=wt,wt N(0,1).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:

Q1,1=Qn,n=a02Q_{1,1} = Q_{n,n} = a_0^2

Q2,2=Qn1,n1=a02+a12Q_{2,2} = Q_{n-1,n-1} = a_0^2 + a_1^2

Q1,2=Q2,1=Qn1,n=Qn,n1=a0a1Q_{1,2} = Q_{2,1} = Q_{n-1,n} = Q_{n,n-1} = a_0 a_1

Qt,t=q0=a02+a12+a22,t=3,4,...,n2Q_{t,t} = q_0 = a_0^2 + a_1^2 + a_2^2, t = 3, 4, ..., n-2

Qt,t1=Qt1,t=q1=a1(a0+a2),t=3,4,...,n1Q_{t,t-1} = Q_{t-1,t} = q_1 = a_1(a_0 + a_2), t = 3, 4, ..., n-1

Qt,t2=Qt2,t=q2=a2a0,t=3,4,...,nQ_{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))

[Package INLAspacetime version 0.1.8 Index]