APPENDIX
Spatio-Temporal Analysis and Incidence Mapping
The BME analysis is modeled over a space/time random field (S/TRF), which estimates
the distribution of mortality rates over space and time as a function of possible field moments.
BME utilizes random field theory to create incidence estimates where the mean square error is
minimized at nodes on an estimation grid. The BME framework allows for the incorporation of
data modeled by a distribution, such as Uniform Method Bayesian Maximum Entropy
(UMBME) and Poisson Kriging.
The BME analysis can be broken into three main steps. The first step is to examine the
data to obtain a prior probability density function (PDF) of the S/TRF for pedestrian mortality.
Second, Bayesian conditionalization is used to find a posterior PDF for Cali, Colombia. Third,
the posterior PDF is used to isolate the incidence to derive space-time maps of Cali mortality
incidence from 2008-2010, represented as spatial random fields. The inputs for the spatial
temporal model are: the mean trend for the data, a covariance model, and the calculation
parameters: 1) rates; 2) maximum number of data points contributing to an estimate; 3) spatial
search radius; and 4) coordinates of the estimate point. The outputs for the BME analysis are the
moments of the BME posterior PDF, specifically the expected value for the estimate point and
the variance of the moment (Law, Serre, Christakos et al. 2004; Serre, Christakos and Lee 2004;
Akita, Carter and Serre 2007; Allshouse, Pleil, Rappaport et al. 2009; de Nazelle, Arunachalam
and Serre 2010; Hampton, Serre, Gesink et al. 2011). The numerical processing of these data was
performed using BMElib and MATLAB 7.8 (Mathworks 2010) and the BMElib package (Serre
2002).
The mean trend is considered to be a deterministic function. The residual, S/TRF models
the uncertainties and variability’s associated with the dataset over time (Serre, Christakos and
Lee 2004). The model for the mean trend is: mZ(s,t) = ms(s) + mt(t), where: ms(s) is the spatial
component of the mean trend and mt(t) is the temporal component of the mean trend using an
exponential kernel smoothing function to obtain geographic and temporal averages (Choi, Serre
and Christakos 2003; Serre, Christakos and Lee 2004; Akita, Carter and Serre 2007). To
determine the appropriate kernel smoothing parameters, an exploratory analysis was conducted.
Prior to the BME analysis, the mean trend was removed from the dataset, smoothing the spatiotemporal fluctuations and resulting in a residual field that is homogenous in space and stationary
in time. The mean trend illustrates systematic trends within the dataset. The exponential kernel
smoothing parameters used to obtain our mean trend model were 0.25km for the spatial mean
trend component ms(s) and 3 months for the temporal mean trend component mt(t).
The space time variability of the field is described in terms of a covariance function
Cx(r,t)=E[(X(s,t)-mx(s,t))(X(s’,t’)-mx(s’,t’))], where r=||s-s’|| and t=|t-t’| are the spatial and
temporal lags (Serre, Christakos and Lee 2004). Covariance functions present a quantitative
description of the correlation between pairs of observations as a function of the inter-pair
distances. The general spatial variability is shown in the sill (the covariance at distance 0) and
by the slope of the model near the origin: The larger the sill, and the steeper the slope, the greater
the spatial variability. The covariance range, the distance from the sill to the point where the
curve becomes asymptotic, or loses 95% of the inter-pair correlation, identifies the area in which
neighboring observations influence the rate at a location. The overall event patterns are
illustrated by the nature of the model as the distance from the sill increases. An outcome that is
spatially homogeneous will have an asymptotic curve, and non-asymptotic curves imply the
outcome trends at a distance (Law, Serre, Christakos et al. 2004; de Nazelle, Arunachalam and
Serre 2010).
We calculated the experimental covariances of the residual space-time mortality field
and a covariance function, which models the experimental values. For this data set, the
covariance model is a non-separable model with the superposition of three exponential models
with varying spatial and temporal scales, as shown in the following equation: Cx(r,t) = c1 exp (3r/ar1 ) (-3 t/at1 ) + c2 exp (-3r/ar2 ) (3 t/at2 ) + c3 exp (-3r/ar3 ) (-3 t/at3) , where c1=
2(deaths/10,000person-yrs)2, ar1= 0.15km, at1= 35months, c2= 8(deaths/10,000person-yrs)2, ar2=
.28km, at2=30months and c3= 7.8(deaths/10,000person-yrs)2, ar3= 1.1km, at3 = 8 months.