Menu Close

Reply To: VDDT Probability Distributions

Leonardo FridLeonardo Frid

While it is true that outputs of VDDT and Path can exhibit Weibull distributed return intervals for transitions, neither model actually uses a Weibull distribution in it’s algorithm. For all the cells of the same class at any one time step there are two possible outcomes — undergoing a transition or not. The resulting probability distribution for the number of transitioning pixels of that class at any one time step is the binomial distribution. This process is analogous to tossing a coin and describing the number of heads vs. tails or tossing a dice and describing the number of times out of a fixed number of tosses that you get a specific outcome like rolling a 6.

If the probability of a transition is constant as a function of age or time since a previous transition then the return interval (or time since the last transition) follows the exponential distribution. The exponential distribution is a special case of the Weibull distribution. If the probability of a transition changes with the age of a cell or time since a previous transition, then return intervals for transitions exhibited in the model outputs will follow the Weibull distribution. The model does not sample from a binomial, Weibull or exponential distribution; rather the models exhibit these kinds of distributions in their outputs as a mathematical consequence of the simple binomial (yes/no) process used in the algorithm.

If a user would like to influence the kind of distribution followed by return intervals in the model then they need to modify the probabilities of transitions and how they change or stay constant over time. In this sense the state and transition model is quite transparent in terms of assumptions which only consist of the state classes defined by the user, the transitions between state classes and the probabilities assigned to those transitions.

There is a more detailed description of this in Appendix B of the VDDT user guide

Please let me know if any of this requires further clarification.