A little about me…
Collaborative research projects that motivated this work
Animal movement model primer
Agent-Based Models with SMC
“The best thing about being a statistician is
that you get to play in everyone’s backyard.” - John Tukey
“Problem first, not solution backward” - Jeff
Leek
Understand and report uncertainty
Create software for implementation
\[\underline{z}_{t} = H_t \underline{s}_{t} + \underline{\epsilon}_t, \; \; \;\;\;\; \underline{\epsilon}_t \sim N(0,\sigma^2_{\epsilon} I)\] where \(\underline{z}_t\) is a matrix of the observed locations for the agents at time t, \(H_t\) is an incidence matrix to determine whether an agent is observed at time t, \(\underline{s}_t\) is the latent location at time, and \(\epsilon_t\) is the error process
\[\underline{s}_{i,t} = \underline{s}_{i,t-1} + u_{i,t}\underline{\delta}_{i,t} + \underline{\eta}_{i,t}\]
\[\left(\delta_{x,i,t}, \delta_{y,i,t}\right)' = \left( \cos(\theta_{i,t}), \sin(\theta_{i,t}) \right)\] \[\theta_{i,t} \sim VonMises(\mu, \kappa)\] >- \(\mu\) is an angular heading and \(\kappa\) is a concentration parameter
\[u_{i,t} = N_+(\mu_u, \sigma^2_u)\]
Mixture distribution on the angle heading
Spatial covariates (can influence step distance and heading)
State switching models (different movement during different behaviors)
Collective movement (animals influence other behavioral patterns)
Agent based models are a simulation based approach using a set of agents.
Each agent is given a relatively simple set of rules, which control interactions between agents.
The collective behavior of agents can model complex population level characteristics.
The agents have a set of characteristics:
Collective animal movement refers to a situation where animals influence each others behavior.
Commonly, there are three types of behaviors considered
The only rule of the model is: at each time step a given particle driven with a constant absolute velocity assumes the average direction of motion of the particles in its neighborhood of radius r with some random perturbation added.
Vicsek, et. al 1995.
The agents have a set of characteristics:
Complex, collective movement models where agents interact pose challenges for parameter estimation.
Parameters associated with the agent rules are estimated to allow for uncertainty in model parameters.
We want to estimate model parameters, denoted as \(\Theta\), such as those in the probability distributions for step size and turning angle.
However for each agent at a given time we also need to estimate a set of “state parameters”, denoted \(\mathcal{X}\):
We want the joint posterior distribution \(p(\Theta,\mathcal{X}|y)\).
For iter in 1:N
Particle-MCMC combines particle-based methods with MCMC.
A particle filter is used to propose \(p(\mathcal{X}|\Theta)\) in the MCMC framework
There are ways to carry out a Gibbs-sampler and iterate between sampling \(\mathcal{X}\) and \(\Theta\).