# Spatial Class Reading Notes

Note: This is for point-referenced data/Gaussian Process models only.

My interests in Spatial Stats is one of the biggest reasons I came to UCSC. As part of my effort to read more, I used this page to summarize the readings assigned for the class. Slides for the STAT 226 class are very minimal and leaves out a lot of detail, so it’s almost required to do the readings to understand the material.

- Statistical Methods for Spatial Data Analysis, O. Schabenberger and C.A. Gotway. Chapman and Hall.
- Hierarchical Modeling and Analysis for Spatial Data, S. Banerjee and A.E. Gelfand. Chapman and Hall. 2nd edition
- Model-based Geostatistics, P.J. Diggle and P.J. Ribeiro. Springer.

None of the textbooks are completely comprehensive/helpful. That’s why I put this guide together. Note that the first edition of the Banerjee and Gelfand book can be found online. They have different chapter numbers, however.

Table of Contents:

## Suggested Reading Order

- Overview of Theory: Schabenberger & Gotway Ch 2 + Ch 4.1-4.3
- K-L Expansion: Xia Thesis
- Classical Parameter Estimation: Diggle & Ribeiro Ch 5.1-5.5
- Kriging: Banerjee & Gelfand Ch 2.4
- Bayesian Hierarchical Modeling: Banerjee & Gelfand Ch 6

Refer to other references as needed. Here are brief summaries with highly recommended reading in **bold**.

## Textbook Chapter Summaries

### Schabenberger & Gotway Ch 2

Overall a really good introduction to spatial processes / geostatistics for statistics students.

I would basically read the whole thing except 2.4.

**2.1**: overview and motivations for the fundamentals of random fields.**2.2**: goes over stationarity, isotropy, instrinsic stationarity (variogram).**2.3**: spatial continuity & differentiability. more detailed than class slides.- 2.4: Goes over the basics of modeling. 2.4.2 (Covolultion Rep.) is covered in class mostly as a way to model non-stationary processes. Not every proceess can be represented this way, which is part of the motivation of the Karhunen-Loeve Representation.
**2.5**: excellent introduction to spectral analysis. You can skip 2.5.6.

### Schabenberger & Gotway Ch 4

Detailed explanations on covariance estimation. Most explanation depends heavily on semi-variograms. I think this is to make things more general for instrinsically stationary processes (not just stuff for weakly stationary processes.) IMO Diggle and Ribeiro is a better reference for estimation content.

- 4.1: overview and motivations for the estimation of covariances
**4.2**: explains the basics of a covariance model. Sill, practical range, nugget effect.- 4.3: goes into the Matern family of covariance functions. I didn’t come across this until later so perhaps it would have been useful for HW 2. Omits discussion about the spectral aspect of the Matern.
- 4.4: Skipped
- 4.5: Again, based on Semi-Variograms. So it’s better to use the Ribeiro book

### Diggle & Ribeiro Ch 5.1-5.5

Extremely helpful chapter on all things variograms/inference related.

Definitely recommend reading before attempting actual spatial data analysis problems like in HW 3.

### Banerjee & Gelfand Ch2.

A decent reference, explanations are not thorough. 2.4 is helpful.

- 2.1: overview of different variograms.
- 2.2: really brief treatment of anisotropy
- 2.3: EDA of spatial data. Outside of 2.3.1, it is not really worth reading.
**2.4**: Very helpful and thorough chapter on Kirging. Compares Kriging with a conditional expectation of a multivariate normal, which is what most statisticians are used to.- 2.5: Helpful tutorial on EDA. Listed under resources.

### Banerjee & Gelfand Ch3.

Short chapter, some helpful clarifications.

- 3.1.1: Too theoretical for me. Skipped.
- 3.1.2: Useful overview on Bochner’s Theorem, spectra, valid covariance functions.
- 3.1.3: Notes on covariance tapering. Skipped
- 3.1.4: Notes on smoothness of the Matern family. Somewhat helpful and more complex than other explanations I’ve seen.
- 3.2: on non-stationary spatial models. Interesting but skipped.

### Banerjee & Gelfand Ch 6.

Seems to be the only real reference for fitting Bayesian Hierarchical Models for Spatial Data. Since it’s the focus of the book, it is a very good chapter.

## Other Important Resources

#### On Bessel Functions

- Some basics on Bessel Functions
- Inequalities and identities for Modified Bessel Functions
- Abramowitz and Stegun. Handbook of Mathematical Functions.

#### Review of Characteristic Functions

Allan Gut, Intermediate Probability Chapter 4. (PDFs available online) Theorem 4.8 and examples are helpful.

#### Karhunen-Loeve Expansion

- Thesis by Gangqiang Xia is very helpful and goes over the motivations of the K-L Expansion and the approximation based on spectral densities. I should’ve read this before doing HW2. Bruno will provide a copy for the class.
- Notes on Karhunen-Loeve Expansion by Y. Lee (also helpful)
- General Notes on Eigen values of Integrals First 3 paragraphs is all you need.

#### Spatial EDA

- Banerjee & Gelfand Ch 2.5: Tutorial section.

#### Software

- The
`geoR`

tutorial. Super helpful for Empricial Variogram/Maximum Likelihood Inference - If you need to fit Process Convolutions with a spherical Bezier kernels and a MRF prior, here’s existing code from the paper by Lemos and Sanso