Land change models commonly model the expected quantity of change as a Markov chain. Markov transition probabilities can be estimated by tabulating the relative frequency of change for all transitions between two dates. To estimate the appropriate transition probability matrix for any future date requires the determination of an annualized matrix through eigendecomposition followed by matrix powering. However, the technique yields multiple solutions, commonly with imaginary parts and negative transitions, and possibly with no non-negative real stochastic matrix solution. In addition, the computational burden of the procedure makes it infeasible for practical use with large problems. This paper describes a Regression-Based Markov (RBM) approximation technique based on quadratic regression of individual transitions that is shown to always yield stochastic matrices, with very low error characteristics. Using land cover data for the 48 conterminous US states, median errors in probability for the five states with the highest rates of transition were found to be less than 0.00001 and the maximum error of 0.006 was of the same order of magnitude experienced by the commonly used compromise of forcing small negative transitions estimated by eigendecomposition to 0. Additionally, the technique can solve land change modeling problems of any size with extremely high computational efficiency.

Many aspects of the earth system are known to have preferred patterns of variability, variously known in the atmospheric sciences as modes or teleconnections. Approaches to discovering these patterns have included principal components analysis and empirical orthogonal teleconnection (EOT) analysis. The latter is very effective but is computationally intensive. Here, we present a sequential autoencoder for teleconnection analysis (SATA). Like EOT, it discovers teleconnections sequentially, with subsequent analyses being based on residual series. However, unlike EOT, SATA uses a basic linear autoencoder as the primary tool for analysis. An autoencoder is an unsupervised neural network that learns an efficient neural representation of input data. With SATA, the input is an image time series and the neural representation is a unidimensional time series. SATA then locates the 0.5% of locations with the strongest correlation with the neural representation and averages their temporal vectors to characterize the teleconnection. Evaluation of the procedure showed that it is several orders of magnitude faster than other approaches to EOT, produces teleconnection patterns that are more strongly correlated to well-known teleconnections, and is particularly effective in finding teleconnections with multiple centers of action (such as dipoles).