Continuous-lag Markov Chains

Recent applications of the Markov chain theory in geology have introduced continuous-lag modelling of spatial variability (Carle and Fogg, 1997 Fogg et al., 1998). The continuous-lag approach extends the probability of state transitions recorded at fixed intervals (discrete-lag) to any desired interval by considering conditional rates of 

Application of continuous-lag Markov chains requires the knowledge of lag-dependent transition probabilities. Through Sylvester s theorem (Agterberg, 1974 Carle and Fogg, 1997) these can be derived from  

Correspondingly, the spectral component matrices may be obtained from P by  

The continuous-lag transition probabilities were obtained by solving Eqs. (6) and (7) on the basis of the transition probability matrix for the 20 cm interval. Eigenvalues and spectral components were inserted in Eq. (5) to yield transition probabilities for continuous lags. 

For all other transitions, the lag-dependent transition probability starts at zero for vanishing lag. Then, the probabilities increase with increasing lag, but are confined to maximal transition probabilities less than 1. This is because transition probabilities greater than zero for transitions of a CL on a different one presume the trivial fact that at least two contamination levels (categories) are present. Thus, a transition probability of one category to a different one can never approach one. 

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Carle, S. R Fogg, G. E. (1997) Modelling spatial variability with one and multidimensional continuous-lag Markov chains. Math. Geol. 29, 891-916. 

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