In mbjoseph/neuralecology: Fit Neural Hierarchical Models of Ecological Populations

knitr::opts_chunk$set(echo = FALSE)

If you only remember one thing...

\begin{center} \scalebox{2}{% $X \beta \rightarrow NN_{\theta}(X)$% } \end{center}

You can replace $X \beta$ with a neural network.

Hierarchical models predict states from data \footnote[frame]{\tiny{Berliner LM. Hierarchical Bayesian time series models. Maximum entropy and Bayesian methods 1996: 15-22}}

\begin{center} \begin{tikzpicture} \node (states) [par] {States}; \node (parameters) [par, left of=states, xshift=-2cm, yshift=2cm]{Parameters}; \node (observations) [whitebox, right of=states, above of=states, xshift=2cm, yshift=1cm] {Observations}; \draw [arrow] (states) -- (observations); \draw [arrow] (parameters) -- (observations); \draw [arrow] (parameters) -- (states); \end{tikzpicture} \end{center}

Occupancy models are hierarchical\footnote[frame]{\tiny{MacKenzie, Darryl I., et al. "Estimating site occupancy rates when detection probabilities are less than one." Ecology 83.8 (2002): 2248-2255.}}

\begin{center} \begin{tikzpicture} \node (process) [par] {Occupancy state $z$}; \node (data) [whitebox, right of=process, above of=process, xshift=2cm, yshift=1cm] {Detection data $y$}; \node (p) [par, left of=process, xshift=-2.5cm, yshift=2cm]{$p$}; \node (psi) [par, left of=process, xshift=-2.5cm]{$\psi$}; \draw [arrow] (p) -- (data); \draw [arrow] (process) -- (data); \draw [arrow] (psi) -- (process); \end{tikzpicture} \end{center}

$$y \sim \text{Bernoulli}(z \times p)$$

$$z \sim \text{Bernoulli}(\psi)$$

Parameters can depend on covariates

$$\psi = f(X; \theta).$$

Often we assume linear relationships

$$\psi = \text{logit}^{-1}(X \beta).$$ $\quad$

\begin{center} \begin{tikzpicture} % linear mapping \tikzstyle{every pin edge}=[<-,shorten <=1pt] \tikzstyle{input neuron}=[neuron, draw=black, fill=white]; \tikzstyle{output neuron}=[neuron, draw=black, fill=AlertOrange!50!white]; \tikzstyle{annot} = [text width=4em, text centered]

% Draw the input layer nodes \foreach \name / \y in {1,...,4} % This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4} \node[input neuron] (I-\name) at (0,-\y) {$x_{\y}$};

% Draw the output layer node \node[output neuron, right of=I-2, above of=I-3, xshift=1cm, yshift=-0.45cm] (O) {$\psi$};

% Connect every node in the hidden layer with the output layer \foreach \source in {1,...,4} \draw (I-\source) edge (O);

\end{tikzpicture} \end{center}

We can accommodate nonlinearity

• Polynomials
• Splines
• Random fields
• Gaussian processes
• Tree-based methods (random forest, XGBoost)
• Neural networks?

Related work: tree-based function approximators\footnote[frame]{\tiny{Hutchinson, Rebecca, Li-Ping Liu, and Thomas Dietterich. "Incorporating boosted regression trees into ecological latent variable models." Proceedings of the AAAI Conference on Artificial Intelligence. Vol. 25. No. 1. 2011.}}

\begin{center} \scalebox{2}{% $X \beta \rightarrow f_{\theta}(X),$% } \end{center}

where $f$ is a boosted regression tree.

(speculation: better for tabular features?)\footnote[frame]{\tiny{Grinsztajn, Leo, Edouard Oyallon, and Gael Varoquaux. "Why do tree-based models still outperform deep learning on typical tabular data?." Thirty-sixth Conference on Neural Information Processing Systems Datasets and Benchmarks Track. 2022.}}

Neural networks approximate functions

\begin{center} \begin{tikzpicture} \tikzstyle{every pin edge}=[<-,shorten <=1pt] \tikzstyle{input neuron}=[neuron, draw=black, fill=white]; \tikzstyle{output neuron}=[neuron, draw=black, fill=white]; \tikzstyle{hidden neuron}=[neuron, draw=black, fill=UniBlue!50!white]; \tikzstyle{annot} = [text width=4em, text centered]

% Draw the input layer nodes \foreach \name / \y in {1,...,4} % This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4} \node[input neuron] (I-\name) at (0,-\y) {$x_{\y}$};

% Draw the hidden layer nodes \foreach \name / \y in {1,...,5} \path[yshift=0.5cm] node[hidden neuron] (H-\name) at (\layersep,-\y cm) {$h_{\y}$};

% Draw the output layer node \node[output neuron, right of=H-3, xshift=1cm] (O) {$\hat y$};

% Connect every node in the input layer with every node in the % hidden layer. \foreach \source in {1,...,4} \foreach \dest in {1,...,5} \path (I-\source) edge (H-\dest);

% Connect every node in the hidden layer with the output layer \foreach \source in {1,...,5} \path (H-\source) edge (O);

\end{tikzpicture} \end{center}

$\hat y:$ observable data

e.g., bounding boxes, species identity, key points, a number, words

Neural networks can do more than just predict data

$\;$

\begin{center} \begin{tikzpicture} \tikzstyle{every pin edge}=[<-,shorten <=1pt] \tikzstyle{input neuron}=[neuron, draw=black, fill=white]; \tikzstyle{output neuron}=[neuron, draw=black, fill=AlertOrange!50!white]; \tikzstyle{hidden neuron}=[neuron, draw=black, fill=UniBlue!50!white]; \tikzstyle{annot} = [text width=4em, text centered]

% Draw the input layer nodes \foreach \name / \y in {1,...,4} % This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4} \node[input neuron] (I-\name) at (0,-\y) {$x_{\y}$};

% Draw the hidden layer nodes \foreach \name / \y in {1,...,5} \path[yshift=0.5cm] node[hidden neuron] (H-\name) at (\layersep,-\y cm) {$h_{\y}$};

% Draw the output layer node \node[output neuron, right of=H-3, xshift=1cm] (O) {$\theta$};

% Connect every node in the input layer with every node in the % hidden layer. \foreach \source in {1,...,4} \foreach \dest in {1,...,5} \path (I-\source) edge (H-\dest);

% Connect every node in the hidden layer with the output layer \foreach \source in {1,...,5} \path (H-\source) edge (O);

\end{tikzpicture} \end{center}

e.g., neural network in an occupancy model\footnote[frame]{\tiny{Joseph MB. 2020. \textit{Neural hierarchical models of ecological populations}. Ecology Letters, 23(4) and Seo E et al. \textit{StatEcoNet: Statistical Ecology Neural Networks for Species Distribution Modeling}. Proceedings of the AAAI Conference on Artificial Intelligence, 35(1).}}

$\;$

\begin{center} \begin{tikzpicture} \tikzstyle{every pin edge}=[<-,shorten <=1pt] \tikzstyle{input neuron}=[neuron, draw=black, fill=white]; \tikzstyle{output neuron}=[neuron, draw=black, fill=AlertOrange!50!white]; \tikzstyle{hidden neuron}=[neuron, draw=black, fill=UniBlue!50!white]; \tikzstyle{annot} = [text width=4em, text centered]

% Draw the input layer nodes \foreach \name / \y in {1,...,4} % This is the same as writing \foreach \name / \y in {1/1,2/2,3/3,4/4} \node[input neuron] (I-\name) at (0,-\y) {$x_{\y}$};

% Draw the hidden layer nodes \foreach \name / \y in {1,...,5} \path[yshift=0.5cm] node[hidden neuron] (H-\name) at (\layersep,-\y cm) {$h_{\y}$};

% Draw the output layer node \node[output neuron, right of=H-3, xshift=1cm] (O) {$\psi$};

% Draw other hierarchical model components \node[output neuron, above of=O] (p) {$p$}; \node[output neuron,label=below:\tiny Occupancy, right of=O] (z) {$z$}; \node[input neuron,label=above:\tiny Detection data, above of=z, right of=z] (y) {$y$};

\path (O) edge (z); \path (z) edge (y); \path (p) edge (y);

% Connect every node in the input layer with every node in the % hidden layer. \foreach \source in {1,...,4} \foreach \dest in {1,...,5} \path (I-\source) edge (H-\dest);

% Connect every node in the hidden layer with the output layer \foreach \source in {1,...,5} \path (H-\source) edge (O);

\end{tikzpicture} \end{center}

Imperfect data motivate hierarchical models

• Occupancy (nondetection $\neq$ absence)
• N-mixture (count $\neq$ abundance)
• Capture-recapture (not captured $\neq$ dead)
• Animal movement (location $\neq$ behavioral state)

Example: convolutional animal movement model

Parameterizing a hidden Markov model \newline with a convolutional neural network

Test set predictions

Parameter estimation

We observe $y_{1:N}$ and minimize loss:

$$\text{Loss}(\theta) = - \sum_{i} \underbrace{\ell(y_i, \theta)}_{\substack{\text{Log} \ \text{probability}}}$$

• Stochastic minibatch optimization

Trade-offs: why you should (not) do this

Good

• feature extraction from images & time series
• scalable

Not so good

• interpretability
• implementation
• tricky posteriors

What's the observation model for classifier output?

knitr::include_graphics("figs/animal-detection.png")

\tiny Norouzzadeh, Mohammad Sadegh, et al. "A deep active learning system for species identification and counting in camera trap images." Methods in ecology and evolution 12.1 (2021): 150-161.

Example observation model for classifier output\footnote[frame]{\tiny{Spiers, Anna I., et al. "Estimating species misclassification with occupancy dynamics and encounter rates: A semi‐supervised, individual‐level approach." Methods in Ecology and Evolution (2022).}}

We encounter $L_k$ individuals for species $k=1, ..., K$

Observe total encounters: $N = \sum_k L_k$

$$N \sim \text{Poisson}\big( \sum_k \lambda_k z_k \big)$$

Observe classifier predicted species for each encounter $i=1, ..., N$:

$$y_i \sim \text{Categorical}(\pmb \theta_{k[i]})$$

Partially observe (label) true species identities $k[i]$:

$$k[i] \sim \text{Categorical}\bigg( \dfrac{z_k \lambda_k}{\sum_k z_k \lambda_k} \bigg)$$

What's the observation model for classifier output?

Considerations

• Out-of-distribution samples
• Hard vs. soft labels
• Correlated errors
• Calibration

Practical questions: model output as data

• Optimal labeling
• Two stage approach (classifier + model) vs. one integrated model

Imperfect data motivate hierarchical models

• Occupancy (nondetection $\neq$ absence)
• N-mixture (count $\neq$ abundance)
• Capture-recapture (not captured $\neq$ dead)
• Animal movement (location $\neq$ behavioral state)
• Coupled classification (classifier output $\neq$ species ID)

Thank you

\small Get in touch: [e-mail], @mxwlj

\normalsize

Code

https://github.com/mbjoseph/neuralecology

Paper

Joseph MB. 2020. Neural hierarchical models of ecological populations. Ecology Letters, 23(4).

mbjoseph/neuralecology documentation built on Nov. 6, 2022, 3:48 p.m.