Using Random Walks for Time Varying Parameters: Some Positives, Drawbacks, and Alternatives

Many assessment models assume that estimated quantities such as catchability or selectivity are constant.  Often such assumed constant values are changing over time, and not allowing for change can cause bias in model estimates.  While several mechanisms for time-varying processes are understood in general, causes for changes in specific cases are often not.  Random walks can account for time changing processes, without specification of the underlying mechanism.  We will review the strengths and limitations of modeling temporally changing quantities in assessments using random walks.  A quantity following a random walk is assumed to equal the quantity from the previous time step, plus a perturbation (e.g., q_t+1=q_t+e_t).  Random walks are based on the premise that large changes over a time step are uncommon, but that substantial drift can occur over time.  For a frequentist, the parameters being estimated are the initial value at the start of the process, and the variance for the perturbations.  The perturbations are random effects.  For a Bayesian the perturbations are parameters drawn from an informative prior.  In many assessment models estimation is by penalized likelihood (or for a Bayesian the highest posterior density), which often require the assumption that the process variance is known.    Random walks allow for some sharing of information over time, and this also facilitates filling in of gaps in time-series.  They have been used to allow for temporal variation in fishery and survey catchability and selectivity, parameters of size at age models, mortality rates, and recruitment.  Simulation results have shown that random walks can improve assessment results in the face of a range of temporal variation, and typically do not cause harm if used when such temporal variation is absent.  On the other hand, simulations and other work have shown that random walks can be improved upon when there is knowledge of mechanisms causing changes.  Some issues with use of random walks include the need to specify known variances or informative priors for variances when used with a penalized likelihood approach, and lack of realism as a basis for forward simulations.  We will discuss some future directions to address such concerns including use of other Markov processes, and the practicality of estimation of process error variances.