As I read it,
spatial covariance matrices are important to the analysis, where perhaps strictly it is the full lagged covariances that matter.
Steig et al. (2009) presented a reconstruction (referred to as STEIGv1) using a regularized expectation maximization algorithm (Schneider 2001) to assess
the spatial covariance of temperature anomalies in the AVHRR satellite observations.
A third version of the Steig et al. (2009) reconstruction (referred to as STEIGv2) used linearly detrended AVHRR data to define
the spatial covariance patterns.
However, we typically use temporal persistence from 1 month to the next as well as
spatial covariance (as carried out using all the mapping methods cited above).
We consider the temporal variance to be the statistical variance of the values over intervals of time close to the defined interval, and
the spatial covariance to be the covariance over regions close in location, shape, and size to the defined region.
Much of this debate seems to be confusing «measurement error» with
a spatial covariance of land - use and temperature trends (someone with a Phd: is «heteroscadasticity» a correct term for this?).
[Response: The AWS records useful because they provide a totally independent estimate of
the spatial covariance pattern in the temperature field (which we primarily get from satellites).
Not exact matches
Longitudinal mixed models were also used to estimate the effect of vaccine dose on mean log - transformed antibody levels over time, using a
spatial exponential
covariance structure to model the correlation between measurements from the same individual while taking into account the number of study days between measurements.
We checked the validity of the assumed
covariance model for
spatial correlation using the Monte Carlo algorithm and empirical semi-variogram as described in Supplementary File 1.
We concluded that the adopted
covariance model was compatible with the data, as the empirical semi-variogram fell within the 95 % tolerance intervals computed via Monte Carlo simulation and a
spatial correlation test of residuals.
In particular, although they have considered
spatial dependencies in their
covariance (Supporting Online Material or «SOM», section 6.3), they do insist upon a Gaussian shape.
Aquatic GHG fluxes are measured using a variety of techniques (e.g., floating chambers, thin boundary methods, eddy
covariance towers, acoustic methods, and funnels; supplemental figure S1) that provide varying degrees of
spatial and temporal coverage and accuracy (St. Louis et al. 2000).
Although we did not explicitly address the temporal or
spatial resolution of emission data from each system, it is notable that the few published acoustic and eddy
covariance - based reservoir CH4 flux estimates are quite high compared to the median CH4 flux estimates from less temporally and / or spatially integrated measurement techniques (figure 1).
This is likely because we maximized the
covariance between the sea ice field and the atmospheric circulation by restricting our time averaging to the seasonal mean and restricting our
spatial domain to 0 ° — 180 ° W and 30 ° — 75 ° S. Inevitably, indices calculated from local data will explain more local variance than those based on remote data.
In contrast to the distance - weighted approach, Steig et al. (2009) and Monaghan et al. (2008) used
covariance methods to establish
spatial relationships among the observing stations and the regions to be infilled.
However, because only large - scale, first - order patterns are reconstructed, similar patterns of
spatial / temporal
covariance are found in the station data alone (otherwise, there would be very little skill in the reconstruction).
Kato, T. & Tang, Y.
Spatial variability and major controlling factors of CO2 sink strength in Asian terrestrial ecosystems: Evidence from eddy
covariance data.