Go Back
Spatial Conditional Independence Assumption
The conditional independence assumption of nearest spatial data within a neighborhood, as an extension of a cardinal-neighbor property of Pickard random fields (Pickard 1980) toward the sparse sample data situation, was presentated in both Li and Zhang (2008) and Li (2007, appendix).
Spatial conditional independence of nearest data neighbors in cardinal directions
If only considering cardinal directions, the spatial conditional independence assumption actually holds in a Pickard random field (PRF). Pickard (1980) proved the existence of a curious unilateral Markov field, which has a special property that for a generic clique
,
given any grid cell D among A, B, C and D within the clique its two diagonally adjacent cells B and C are conditionally independent. So in a PRF, we have
(1)
Based on this property, Pickard presented his model, which is still complex and involves three-cell cliques, for binary processes.
Elfeki and Dekking (2001) adopted a full independence assumption of two 1-D Markov chains so as to simplify their model and use only two-point transition probabilities. The CMC model (Elfeki and Dekking, 2001, p. 573) can be simply written as
(2)
which requires the two conditional probabilities Pr(D|B) and Pr(D|C) (described by two 1-D Markov chains here) be fully independent of each other. The consequence of such an assumption is the small class underestimation problem caused by exclusion of conflict transitions (Li, 2007).
The PRF was further adapted by others (e.g., Haslett, 1985; Idier, Goussard and Ridolfi, 2001; Fjortoft and others, 2003) for image processing, where they assume that given a pixel x (i.e., its state) all its four adjacent pixels y1, y2, y3 and y4 (i.e., the states of the upper, left, right, and underlying adjacent pixels) in a neighborhood like
are conditionally independent. This adaptation is straightforward and it still makes a PRF. To explain this, let’s first check the pixel y1. In a PRF, given x, we have y1 and y2 are conditionally independent, and we also have y1 and y3 are conditionally independent according to Equation (1). Looking at the direction from y4 to y1, we can find y4 is a past state of y1 beyond x. So given x, y1 is independent of y4. Finally, we have
(3)
The Pickard’s theorem is used for interactions of adjacent pixels or pixel blocks in image processing. Note that pixels in the generic clique need not be single pixels; they can be pixel blocks, as used in Derin and others (1984) and Rosholm (1997). Here we want to apply the conditional independence to sparse data for distant interactions.
Assume we have a general sparse-data structure in an underlying PRF (i.e., a finite stationary Markov field) as
where x1, x2, x3 and x4 are four nearest data neighbors of x along cardinal directions on a regular lattice. They may be distant from the pixel x with distances h1, h2, h3 and h4, respectively. Given the pixel x, they can still be regarded as being conditionally independent.
To prove this, let’s use E, F, G and H to represent the four pixel blocks respectively, we immediately have that given x the four pixel blocks in a PRF are conditionally independent, that is, for any pixel block of E we have
(4)
Since only one pixel is known in each of these pixel blocks, from Equation (4) we further have
(5)
Thus, it can be seen that sparse data with distant interactions along cardinal directions can be conditionally independent in an underlying PRF, that is, Equation (5) holds. Both (3) and (5) can be directly extended to three dimensions. And such a property should be applicable to point sample data.
Generalized spatial conditional independence assumption for nearest data within a neighborhood
Further relaxing (5) to include data in non-cardinal directions is practically feasible in some situations (Li, 2007). So we may generally write the spatial conditional independence assumption in a multi-dimensional space as
(6)
Compared to the “forced” independence of multiple chains in the CMC theory, this conditional independent assumption is much more reasonable.
While the conditional independence of nearest data in cardinal directions within a neighborhood is a property of an underlying Pickard random field, the generalized conditional independence assumption for nearest data within a neighborhood is just a practical assumption for model simplification, similar to the spatial stationarity assumption. It is not something that can be proved theoretically. Those who claimed they proved the spatial conditional independence assumption and completed the theoretical foundation of MCRF were misleading readers. The MCRF theory was proposed for spatial data. Ignoring the nature of spatial data to talk about the theoretical foundation of MCRF is misleading.References:
Elfeki, A.M., and F.M. Dekking. 2001. A Markov chain model for subsurface characterization: Theory and applications. Math. Geol., 33: 569–589.
Derin, H., Elliott, H., Cristi, R., and Geman, D., 1984, Bayes smoothing algorithms for segmentation of binary images modeled by Markov random fields: IEEE Trans. Pattern Analysis and Machine Intelligence, 6(6): 707-720.
Fjortoft, R., Delignon, Y., Pieczynski, W., Sigelle, M., and Tupin, F., 2003, Unsupervised classification of radar images using hidden Markov chains and hidden Markov random fields. IEEE Trans. Geosciences and Remote Sensing, 41(3): 675-686.
Haslett, J., 1985, Maximum likelihood discriminant analysis on the plane using a Markovian model of spatial context. Pattern Recognition, 18(3-4): 287-296.
Idier, J., Goussard, Y., and Ridolfi, A., 2001, Unsupervised image segmentation using a telegraph parameterization of Pickard pandom fields, In Moore, M., ed., Spatial Statistics. Methodological Aspects and Some Applications, vol. 159 of Lecture Notes in Statistics, Springer Verlag, New York, p. 115-140.
Li, W. 2007. Markov chain random fields for estimation of categorical variables. Math. Geol., 39(3): 321-335.
Li, W., Zhang, C. 2008. A single-chain-based multidimensional Markov chain model for subsurface characterization. Environmental and Ecological Statistics, 15(2): 157-174.
Pickard, D.K., 1980, Unilateral Markov fields: Adv. Appl. Probab., 12(3): 655-671.
Rosholm, A., 1997, Statistical Methods for Segmentation and Classification of Images: PhD Dissertation. Technical University of Denmark. Lyngby, Denmark, 187p.