Hierarchical Species Distribution Modeling

Inhomogeneous Poisson Point Process

$$\log L (\lambda; \mathbf{U}) = \sum\limits_{i=1}^n \log \lambda (\mathbf{s}_i, t_i) - \int\limits_{\mathcal{A}}\int\limits_0^T \lambda(\mathbf{s},t) dtd\mathbf{s} - \log(n!)$$

The probability that the number of points $> 0 \sim$ Bernoulli$(1-e^{-\bar{\lambda}})$

Case Study: Oceanic Whitetip Shark Distribution

Shark Presence Data

Ocean Climate Data

GLM

Maxent

Metropolis-Hastings

Hidden Markov Models

Hidden Markov Models

Hidden Markov Models

Definition

1. Number of states, $K$, i.e. latent patterns of interest.
2. Transition probability matrix $\Gamma$, for the hidden states $X$: $P(X_t = j | X_{t-1} = i)$ for $i,j \in \{1, \dots, K\}$.
3. State-dependent distributions for the observations $Y$ that stem from the latent states: $f(Y_t|X_t = k) \Rightarrow$ any valid distribution function.
4. Initial state distribution, $\delta$, i.e. the first pattern exhibited: $P(X_1 = k)$.

Case Study: Ontogeny of Bald Eagle Behavior

Telemetry Data

Data from 10 tagged bald eagles from the Great Plains of the United States, followed from 2016-2020.

Dataset contains 3,313,548 position readings.

Variables: animal id, date, latitude, longitude, speed, heading, altitude, battery power, differences in time, distance and altitude from previous position fix, and binned fix rate.

Question: Given these data, can we detect how the eagles’ behaviors differ based on their age?

State-Dependent Distributions

Covariates and Transition Probabilities

Covariates, such as environmental factors, tree cover, distance to water bodies - or in the case my model at this time speed in km per hour, altitude and heading - are incorporated into the transition probabilities $\gamma^{(t)}_{ij}$ via the multinomial logistic link function:

$$\begin{align*}\gamma^{(t)}_{ij} &= \frac{\exp(\eta_{ij})}{\sum_{k=1}^N\exp(\eta_{ik})}, \\ \\ \mathrm{where} \ \eta_{ij} &= \begin{cases}\beta_0^{(ij)} + \sum_{l=1}^p \beta_l^{(ij)}\omega_l^t & \mathrm{if} \ i \neq j \\ 0 & \mathrm{otherwise}\end{cases}.\end{align*}$$

Stats Cauldron

RStan Stove

Results

Age Differences (yet to be explored...)

Conclusion and Next Steps

1. Continue work with the bald eagle data and identify 4-5 flight modes.
2. Use of environmental and landscape variables as covariates.
3. Hyperparameters to model differences between birds.
4. State decoding in order to determine the underlying state sequence that led to the observed data.

Hierarchical Hidden Markov Model

Topological Data Analysis

Topology, the study of shape, can produce more explainable models: data has shape.

For example, an atmospheric river is a long narrow high-moisture filament, resembling a river in many ways including their shape.

Topological data analysis and machine learning for recognizing atmospheric river patterns in large climate datasets by Grzegorz Muszynski et al (2018)

Thank you! Questions?