These vignettes are short notes on various technical topics that I believe are not widely understood. They have arisen from my own efforts to understand the question under consideration and represent my understanding of the issue at the time of posting. My goal is to provide technical details about these topics, with an audience of both statisticians and researchers using statistical methods. Most of these vignettes are in the area of spatial statistics, but I may post others as topics arise. Please let me know if you have comments, questions, suggestions, or corrections(!).

### 6.) Solving systems of equations, generating multivariate normal draws, and inverting matrices efficiently in R (January 2012)

A tutorial on efficient use of the Cholesky decomposition for solving systems of equations given a positive definite matrix. The ideas are useful outside of the R environment but the code is given in R. Calculation with sparse matrices is discussed.

### 5.) Understanding intrinsic Gaussian Markov random field spatial models, including intrinsic conditional autoregressive models (December 2009)

A tutorial on understanding improper MRF priors for spatial fields, with a discussion of standard conditional autoregressive (CAR) models based on neighbor adjacencies, as well as MRF approximations to thin plate splines. In particular I focus on the MRF prior as a proper prior in a reduced-dimension space.

### 4.) When can we ignore temporal correlation in space-time data? (January 2008)

Statisticians tend to want to account for the structure in the data to the extent possible, but part of our job is to determine when complexity can be ignored in favor of simpler specifications. With space-time data, under a specific set of conditions, one can show that space-time kriging does not do any smoothing in time. This helps to justify simple spatial smoothing in some contexts.

### 3.) Kriging, interpolation, and uncertainty (January 2008)

My sense is that there is misunderstanding as to whether kriging interpolates (exactly goes through the data points, i.e., honors the data) or smooths. The answer is actually somewhat subtle. In the vignette I describe the situations under which kriging interpolates and those under which it smooths.

### 2.) Smoothing characteristics of CAR models (January 2008)

I provide some information on the smoothing kernel induced by a CAR model with 0-1 weights and how this causes the model to give prediction surfaces with bulls-eyes. The implication is that the CAR model with 0-1 weights may not be a good choice unless the spatial process is not very smooth, and even in that case, it's not clear if the 0-1 model will have attractive features. Instead, more carefully chosen weights in larger neighborhoods can give more pleasing smoothing.

### 1.) Why kriging in ArcGIS may be a bad idea: A statistician's perspective (January 2008)

In this vignette, I discuss some drawbacks to kriging in ArcGIS compared to kriging based on maximum likelihood or other spatial smoothing approaches. However, if you do wish to take this approach, I also provide details on how to carry out kriging in ArcGIS.

Last updated: January 2012.