Explore a regional streamflow model

Select a target location you want to simulate using a regional donor streamflow monitoring station. Set the quantization bits, and the length of record appropriate for a question of interest, and see how unpredicted states are penalized.

Select a target station and a donor station. Change the quantization and prior controls to see how missing states and prior strength affect the divergence between the observed and model distributions.

The explorer demonstrates sensitivity to the prior. It does not select a universally correct prior or replace a hydrologic review of station suitability and record quality.

The consequence of assigning zero probability

The KL divergence is a measure of agreement between probability distributions. When computed using the $\log$ base 2, the quantity represents the number of extra bits per sample required to encode a sequence of values when the dictionary of symbols (states) is approximated using distribution Q instead of the true (observed) distribution P.

When a model does not simulate the full observed state space, measuring the divergence of distributions by the KL Divergence is unbounded since the unobserved state probability is in the denominator. If the state with nonzero probability in the observation is simply neglected, the remaining probabilities sum to less than 1, which suppresses the KL divergence since the measure is evaluated on proper distributions.

Instead of defining an observed state as impossible, which is a statement of certainty, we can set a prior probability and use the resulting posterior distribution to compute KL divergence.

The question is, what is an appropriate prior to use?

Notes

Letting $Q_c$ be the simulated data counts (Dirichlet prior) by quantization bin, then the posterior observed distribution is $\hat Q_c = Q_c + \alpha$ where $\alpha$ is the prior strength, giving $\hat Q_i > 0 $ for all $i \in {1, 2, \dots, 2^\text{bitrate}}$. The posterior distribution is normalized to sum to 1.