### Markov chain geostatistics

**Markov chain geostatistics** is also called "the Bayesian **Markov chain random field** (MCRF) approach" or simply
"the **MCRF** approach". It includes the MCRF theory, specific MCRF/coMCRF models, simulation algorithms, and transiogram modeling
methods. This Bayesian geospatial statistical approach was initially proposed in Li and Zhang (2008) and Li (2007a), and then gradually
developed and extended during last ten years. It was proposed mainly for modeling categorical/discrete fields (e.g. various landscape classes)
in multiple (two, three or spatiotemporal) dimensions. Our research in multi-D Markov chain modeling started in 1990s. The long-time effort
gradually reached a new generalized fundamental spatial statistical model at neighborhood nearest data level - **the MCRF model**,
and a **MCRF-based geostatistical approach**. The whole development process of the MCRF approach is a **bottom-up** process, that
is, we started from using 1-D Markov chain models (e.g., 1-D Markov chain modeling of soil texture layer sequences conducted in early 1990s)
and encountering scientific issues in 2-D Markov chain modeling (e.g., a 2-D coupled Markov chain model we used during 1998-1999 showed some
significant defects), then we explored the scientific issues and gradually solved them, and finally we formulated a generalized theoretical
framework based on the pieces of progress we had made. All of the related researches from 1990s to now were conducted using computer programs
developed by oureselves. The core ideas of this approach include: (1) **single-Markov-chain random-field**, which extends 1-D Markov
chains (first a 1-D step-by-step moving and then a randomly-jumping Markov chain) into a locally-conditioned Markov chain (moving or jumping
in a space); (2) **spatial sequential Bayesian updating over nearest data within a neighborhood** for local conditioning, which
makes the locally-conditioned Markov chain essentially a special spatial Bayesian network over a neighborhood; (3) **spatial conditional
independence assumption of nearest data within a neighborhood** by extending the cardinal-neighbor conditional independence property of
Pickard random fields, which simplifies the MCRF general full solution (with multi-point likelihoods) into a simplified solution (with only
spatial transition probabilities); and (4) **transiogram concepts and methods** for transition probability model estimation from
sample data and convenience of description, based on properties of transition probabilities and 1-D stationary Markov chain theory as well as
pioneer studies and the variogram. Therefore, the MCRF theory formally extended the 1-D Markov chain model into a multi-D spatial model, which
has proven to be practical in application studies. Although the name of MCRF may sound like a Markov random field (MRF) model, it is neither
a conventional MRF model nor derived from the MRF model or Gibbs distribution. It is a spatial model with sequential Bayesian updating on
nearest data within a neighborhood, and can be visualized as a *probabilistic directed acyclic graph*. Thus the MCRF model can be
regarded as a *special Bayesian Network over a neighborhood of spatial data* (or a *neighborhood-based spatial Bayesian
network*). This does not mean that the MCRF model had existed previously due to the existence of
Bayesian networks. In essence, the relationship between the MCRF model and Bayesian networks (including Naive Bayes) is just like the
relationship between kriging and multiple linear regression. In this website, we introduce this approach with as more details as possible, so
that readers can understand it without confusions.

Comments are welcome and may be sent to W. Li (weidong.li@uconn.edu).

**A Note about the MCRF approach**