Kalman filtering the delay-difference equation: Practical approaches and simulations

• Published in: Fishery bulletin (Washington, D.C.) Vol. 94; no. 4; pp. 678 - 691
• Main Authors: Kimura, D K, Balsiger, J W, Ito, D H
• Format: Journal Article
• Language: English
• Published: 01-10-1996
• Subjects: Marine; Thunnus albacares; ISE, Tropical Pacific
• ISSN: 0090-0656

Recently, J. J. Pella showed how the Kalman filter could be applied to production modeling to estimate the size and productivity of fish stocks from a time series of catches and relative abundance indices. We apply these methods to the Deriso-Schnute delay-difference equation. The Kalman filter approach incorporates process and measurement error naturally in the model description. When the production model is the delay-difference equation, the error structure is particularly attractive because process error can be interpreted as simply the variance of recruitment, and measurement error as the variance of the relative abundance estimates. We derived prior distributions of initial biomass in order to begin the Kalman filter calculations. Reanalysis of the data from the eastern tropical Pacific for yellowfin tuna, Thunnus albacares, shows that modeling results can differ greatly depending on whether error is interpreted to be process error or measurement error. Simulation results show that nonlinear least squares and Kalman filter estimates agree well if data contain only measurement error. In contrast, the Kalman filter was clearly superior if simulated data contained significant amounts of process error. The presence of process error positively biased biomass estimates from both the nonlinear least-squares and Kalman filter methods. The Kalman filter performed well with Schnute's form of the delay-difference equation, even though this model violates the assumption of independent process error vectors. The Kalman filter also performed well when the variance ratio r was assumed known and individual variances were estimated from the data. However, it appeared difficult to estimate r as a parameter in the maximum-likelihood estimation.