# Markov chain geostatistics

**Markov chain geostatistics** uses Markov chain spatial models, simulation algorithms and associated spatial correlation measures (e.g., transiogram) based on the Markov chain random field theory, which extends a single Markov chain into a multi-dimensional random field for geostatistical modeling. A Markov chain random field is still a single spatial Markov chain. The spatial Markov chain moves or jumps in a space and decides its state at any unobserved location through interactions with its nearest known neighbors in different directions. The data interaction process can be well explained as a local sequential Bayesian updating process within a neighborhood. Because single-step transition probability matrices are difficult to estimate from sparse sample data and are impractical in representing the complex spatial heterogeneity of states, the **transiogram**, which is defined as a transition probability function over the distance lag, is proposed as the accompanying spatial measure of Markov chain random fields.

## References

- Li, W. 2007. Markov chain random fields for estimation of categorical variables. Math. Geol., 39(3): 321–335.
- Li, W. et al. 2015. Bayesian Markov chain random field cosimulation for improving land cover classification accuracy. Math. Geosci., 47(2): 123–148.
- Li, W., and C. Zhang. 2019. Markov chain random fields in the perspective of spatial Bayesian networks and optimal neighborhoods for simulation of categorical fields. Computational Geosciences, 23(5): 1087-1106.
- http://gisweb.grove.ad.uconn.edu/weidong/Markov_chain_spatial_statistics.htm