# Multivariate State-Space Models workshop

**Analysis of Stochastic Time Series Data using State-Space Models and Kalman Filters**

Ecological Society Meetings 2008, Milwaukee, WI

Elizabeth Holmes(organizer), Yasmin Lucero, Brice Semmens, Eric Ward

Time series data is common collected for many ecological experimental and observational studies. Often these data are fundamentally stochastic, that is the ecological processes being observed experience random temporal variability. Almost always the data are corrupted to some degree by measurement error. State-space modeling provides established and straight-forward method for maximum-likelihood estimation of stochastic time series models from data with measurement error. In addition, the state-space framework provides a clean framework for dealing with multiple types of measurements and multiple types of ‘unseen’ states. Some examples of the types of data and problems that state-space models are routinely used for include development of forecasting models for fisheries stocks when multiple data reflective of stock are to be combined, analysis of animal tracking data, and analysis of population or community time series data. This workshop session will provide an introduction to autoregressive models of the form X(t) = a X(t-1) + b X(t-2) … z X(t-k) + err1(1) and to the use of state-space models and Kalman filters for estimating autoregressive models from data that is overlaid with measurement error. The introductory lectures will be followed by four hands-on computer labs using examples where state-space models are commonly used in ecological analyses: estimation of forecasting models using multiple indicator time series, estimation of stochastic density-dependent population models from count data, and analysis of animal satellite tracking data. The material will be presented at an introductory level to the extent possible. Students need to have a basic understanding of likelihood inference (meaning for example that the terms likelihood function, likelihood surface, and probability density function are not completely unfamiliar concepts). The computer labs will be done in R, but knowledge of R is not necessary. Students must provide their own laptops.

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