Satellite based PM concentrations and application to COPD in Cleveland
Supplementary Online Material on Local time-space Kriging
Theoretical Model - Local Time Space Kriging
Large environmental process tends to exhibit non-stationary behavior (Liang and Kumar 2013). For
example, air pollutants from static emission sources are transported by winds, leading to spatial,
seasonal and diurnal variability (Kumar, Chu, Foster, Peters and Willis 2011). Similar interactions
between emission sources and meteorological conditions generate processes that are non-stationary
both in the mean and covariance structure. The usual assumption of stationary and separable
spatiotemporal covariance is not flexible enough for large datasets both in terms of computation and
modeling. Local time space Kriging (LTSK) using a neighborhood has been proposed to address both
problems (Haas 1995, Gething, et al. 2007). We generalize the local Kriging approach with the nonseparable product-sum model (De Iaco, Myers and Posa 2001) and implement this method specifically
for large data sets.
Let
denote the observed Gaussian process defined over
denotes the spatial domain and t indexes discrete timestamps. Let
where
denote the
spatiotemporal locations where predictions are needed. We term
as query point. The
implementation of local Kriging requires a location specific neighborhood denoted by
. We assume
that within the local neighborhood, data are second order stationary with a non-separable
spatiotemporal covariance function specified using the product sum approach (De Iaco, et al. 2001) as
where
for all
and
for all
isotrpoic variogram models with finite sills(De Iaco, et al. 2001) and
parameter and
;
denote
denote a strictly positive
where
sill and range parameter for spatial variogram and
and
denote the nugget , partial
;
parameters are defined
similarly.
We incorporate substantive knowledge of the underlying process to specify the local neighborhood
around each query point. Let H and U denote the distance and time thresholds where we expect for the
upper limit of local spatial and temporal ranges, respectively. In case of aerosol optical depth (AOD)
data, such knowledge can be based on the life cycle of the aerosol and the underlying spatial resolution
of the satellite data. As a result, define a cylinder
around each query point
The strength of the underlying spatiotemporal correlations is allowed to vary both spatially and
temporally over query points. We thus estimate a location specific variogram
with the
procedure described below. Given the estimated spatiotemporal range parameters, denoted by
and
respectively, we define the local neighborhood as a subset of
We thus identify a local spatiotemporal domain where the process at observed location are highly
correlated with the process realized at query point j, conditioning on the estimated variogram.
In practice the implementation of the neighborhood specification faces two challenges: sparse data and
computational issues for large data. In presence of missing data, narrow spatiotemporal lags can result
in very few data points to accurately estimate time-space covariance function (Kyriakidis and Journel
1999). Consequently, it will result in discontinuity in the prediction across time and space. For example,
in the satellite based air pollution data, there are systematic gaps (due to cloud cover and/or data
contamination) and sufficient data points may not be available if small time-space lags are used to
define the neighborhood. To ensure adequate sample size, we can divide the local neighborhood into
non-overlapping spatiotemporal voxels, termed as cubes hereafter. Let
and
denote the
number of distinct cubes across space and time, we proceed with the following procedure only when
and
for a given lower limits. For the demonstration purposes we set
.
A wider specification of spatiotemporal lags can result in too many data points within the local
neighborhood, especially for satellite data. The computation of empirical variogram described below
and the Kriging operation involves working with a dense
matrix where denotes the number of
neighbors to query point j. If
is large, repeatitive inversion requires
computation and the
implemeantion becomes computationally expencsive for large number of
Dimension reduction and
subsampling have been utilized to address the Kriging for large data sets (Vecchia 1988, Rennen 2009,
Cressie, Shi and Kang 2010). Utilizing the spatial sampling design methods, we reduce the number of
neighborhood by sub-sampling within the cylinder . This requires setting up a upper limit of data
points (Ln). If
exceeds Ln, we sample Ln data points from the total possible
neighbors in a
spatiotemporally balanced way. Balance sampling achieves good predictive performance of the
underlying process(Stevens and Olsen 2004). Specifically we divide the space and time domain into
non-overlapping cubes and sample from each cube. Let denote the sampling probability of cube
, we assign
the number of points in each local spatiotemporal neighborhood. Even
though this approach discards some data points, the loss of information is relatively small because of
strong spatiotemporal autocorrelation within the neighborhood. The Kriging operation requires at most
computatiaion, which only depends upon the number of query points.
We estimate empirical variogram using classical method of moments. Suppress the dependency upon
query point j for brevity of notation. We define the empirical variogram as
where
and
and
denote the distance and time bin. We consider distance bins up to
of the
maximum distance and time lag. The number of bins must be sufficient to estimate the empirical
variogram. We use
bins for distance and a single time point as the time bin.
With thousands of query point, manual estimation of the variogram is challenging. We propose an
automatic way to estimate the variogram. Let
and
denote the smallest distance and time lags. We
estimate the spatial and temporal variograms
and
from
and
respectively(De Cesare, Myers and Posa 2002). We propose four candidate variogram with finite sills:
exponential, spherical , Gaussian and Matern. The range parameters are estimated using the initial
distance lag when the empirical variogram exceeds 80% of the maxim. This rough estimates in general
do not affect the predictive performance of Kriging(Zhang and Wang 2010). The nugget and sill
parameter are estimated using least square methods, with weights proportional to the number of pairs
of data in each lag. The variogram model with the smallest square deviation between the estimates and
fitted values is chosen for and respectively. Let
and
denote the estimated sill parameters.
We estimate the global sill based on sample variance of data points, plus a method to correct for
the postive biase due to the correlation between these data points(Cressie 1988)
These preliminary estimates are adjusted because they may produce an invalid variogram model.
Specifically, we adjust so that
. The parameter
. The resulting covariance function is
It follows from ordinary Kriging method to predict
using all data from
with the estimated
covariance funtion. In case the prediction of the observed process over a region is desired, we utilize the
above covariance estimates with block Kriging to adjust for the point to areal misalignment(Cressie
1993).
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