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CPS1846 Maryam I. et al.
predict the underlying process at an arbitrary location ∈ and to estimate
the uncertainty in the prediction. For ∈ ,

() = + ()+ ∈ ()

where is mean and ∈ is the error term. The unknown spatial process () is
assumed to be the sum of independent processes having different scales of
spatial dependence. Each process is alinear combination of basis functions
where () is the number of basis function at level .

()
() = ∑ () = ∑ ∑ ()

,

=1 =1 =1

The basis function ( ) are fixed. These are constructed at each level using
,
the unimodal and symmetric radial basis functions. Radial basis functions are
functions that depend only on the distance from the center. The inference
methodology of multi-resolution lattice kriging Nychka et al. (2015) is the
direct consequence of maximizing the likelihood function. This inference
framework does not account for the uncertainty in the parameters of multi-
resolution lattice kriging (Nychka et al., 2015). Here, a different approach is
proposed to estimate the multi-resolution lattice kriging model parameters.
This methodology makes use of the uncertainty in the parameters. For this, a
Bayesian framework is considered in which the posterior densities of the multi-
resolution lattice kriging parameters are estimated using Approximate
Bayesian Computation (ABC). In addition to this, it allows the spatial
predictions and quantification of standard errors in these predictions
accounting for the uncertainties in the multi-resolution lattice kriging model
parameters. The primary data used in this thesis are the respected HadCRUT4
(version 4.5.0.0) temperature anomalies (Morice et al., 2012). It is a
combination of global land surface air temperature (CRUTEM4) (Jones et al.,
2012) and sea surface monthly temperatures (HadSST3) (Kennedy et al., 2011b,
a; Kennedy, 2014). The HadCRUT4 database consists of temperature anomalies
with respect to the baseline (1961-1990). Monthly temperatures are provided
beginning from 1850 over a 5 x 5 grid. The average temperature anomalies
o
o
of the stations falling within each grid are provided (Morice et al., 2012).

3. Result
The ensemble temperature data set created by Ilyas et al. (2017) presumed
perfect knowledge of multi-resolution lattice kriging covariance parameters.
The approximate Bayesian computation based multi-resolution lattice kriging
developed in Section 2.1 is applied to the sparse HadCRUT4 ensemble data

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