Page 143 - Invited Paper Session (IPS) - Volume 2

IPS192 Hukum C. et al.
function (, , ) in (5) uses truncated polynomial spline basis functions.

{1, , … . , , ( − ) , … . , ( − ) }. Other basis functions, e.g. B-splines

1 +
+
(Eilers and Marx, 1996) or radial functions, can also be used. Using a large
number of knots in expression (5) can lead to an unstable fit. In order to
overcome this problem, a penalty is usually put on the magnitude of the spline
parameters γ.
When geographically referenced responses play a central role in the
analysis and need to be converted to maps, we can use bivariate smoothing,
( , ) = ( , , , ) where and are spatial coordinates. This is
1
2
1
2
1
2
usually the case with environmental, agricultural, public health and poverty
mapping applications. Consequently we assume the following model
(Opsomer et al., 2008)
( , , , ) = + + + , (6)

1
0
1 1
2
2 2
where is the -th row of the following × matrix

= [( − )] 1<< [( − ′)] −1/2 , (7)

1<<

1<<
Where () = ‖‖ log‖‖, = ( , ) and ( = 1, … , ) are knots. Note
2

1
2

that the C(t) function is defined so that when there is a knot at every
observation (that is, the full rank case) this model for bivariate smoothing leads
to a thin plate spline (Green and Silverman, 1994). The second matrix on the
right-hand side of (7) applies a linear transformation to the radial basis
functions defining the first matrix, and was recommended by Ruppert et al.
(2003) as a way of making the radial spline behave approximately like a thin
plate spline. More details on the Z matrix can be found in Ruppert et al. (2003,
chapter 13). As suggested by Ruppert et al. (2003), fitting the approximation
(5) can be simplified by treating the vector γ as a random-effect vector in a
mixed model specification, which allows estimation of β and γ by maximum
likelihood methods. Following Opsomer et al. (2008), Wand (2003) and
Ruppert et al. (2003, chapter 4), the spline approximation to (4), and to (6), can
be written as
() = ƞ = + + . (8)
If other covariates are available, they can be included in the model as
parametric terms, by being added to the matrix X. In this model is assumed
to be a Gaussian random vector of dimension . In particular, it is assumed
that ~ (0, ∑ = ), where denotes the identity matrix of dimension
2

. Here is the -vector of random area effects. As usual, it is assumed that
the area effects in are distributed independently of the spline effects in
with ~ (0, ∑ = ). Under (8), a plug-in spatially non-linear empirical
2

predictor (SNLEP) for the total count in area is given by

̂ = + ( − )̂ , (9)

̂

Where ̂ = expit( + ̂ + ̂) for binary data and ̂ =

̂

exp( + ̂ + ̂) for count data, and , and donate respectively

130 | I S I W S C 2 0 1 9