varycoef | R Documentation |

This package offers functions to estimate and predict Gaussian process-based spatially varying coefficient (SVC) models. Briefly described, one generalizes a linear regression equation such that the coefficients are no longer constant, but have the possibility to vary spatially. This is enabled by modeling the coefficients using Gaussian processes with (currently) either an exponential or spherical covariance function. The advantages of such SVC models are that they are usually quite easy to interpret, yet they offer a very high level of flexibility.

The ensemble of the function `SVC_mle`

and the method
`predict`

estimates the defined SVC model and gives predictions of the
SVC as well as the response for some pre-defined locations. This concept
should be rather familiar as it is the same for the classical regression
(`lm`

) or local polynomial regression (`loess`

),
to name a couple. As the name suggests, we are using a *maximum
likelihood estimation* (MLE) approach in order to estimate the model. The
predictor is obtained by the empirical best linear unbiased predictor.
to give location-specific predictions. A detailed tutorial with examples is
given in a vignette; call `vignette("example", package = "varycoef")`

.
We also refer to the original article Dambon et al. (2021) which lays the
methodological foundation of this package.

With the before mentioned `SVC_mle`

function one gets an object
of class `SVC_mle`

. And like the method `predict`

for
predictions, there are several more methods in order to diagnose the model,
see `methods(class = "SVC_mle")`

.

As of version 0.3.0 of `varycoef`

, a joint variable selection of both
fixed and random effect of the Gaussian process-based SVC model is
implemented. It uses a *penalized maximum likelihood estimation* (PMLE)
which is implemented via a gradient descent. The estimation of the shrinkage
parameter is available using a *model-based optimization* (MBO). Here,
we use the framework by Bischl et al. (2017). The methodological foundation
of the PMLE is described in Dambon et al. (2022).

Jakob Dambon

Bischl, B., Richter, J., Bossek, J., Horn, D., Thomas, J.,
Lang, M. (2017). *mlrMBO: A Modular Framework for Model-Based
Optimization of Expensive Black-Box Functions*,
ArXiv preprint https://arxiv.org/abs/1703.03373

Dambon, J. A., Sigrist, F., Furrer, R. (2021).
*Maximum likelihood estimation of spatially varying coefficient
models for large data with an application to real estate price prediction*,
Spatial Statistics 41 100470 doi: 10.1016/j.spasta.2020.100470

Dambon, J. A., Sigrist, F., Furrer, R. (2022).
*Joint Variable Selection of both Fixed and Random Effects for
Gaussian Process-based Spatially Varying Coefficient Models*,
International Journal of Geographical Information Science
doi: 10.1080/13658816.2022.2097684

vignette("manual", package = "varycoef") methods(class = "SVC_mle")

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