Statistical methods for movement ecology

# Statistical methods for movement ecology
### Marie-Pierre Etienne
### Ecodep
### 2022

---

# Movement ecology paradigm
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

##  <a name=cite-nathan2008movement></a>([Nathan, Getz, Revilla, et al., 2008](#bib-nathan2008movement)) presents individual movement as the results of

.pull-left[
<figure>
  <img src="nathan_fig.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
</figure>
.legend[  Movement drivers by [Nathan, Getz, Revilla, et al. (2008)](#bib-nathan2008movement) ]
]

* Motion capacities
* Internal state
* Environment
]
  
--

---
name: move2data
# From Movement to Movement data

.pull-left[
<figure>
  <img src="path_1.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
</figure>
]

---
template: move2data
count: false

.pull-left[
<figure>
  <img src="path_2.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
</figure>
]

---
template: move2data
count: false

.pull-left[
<figure>
  <img src="path_3.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
</figure>
]

---
template: move2data
count: false

.pull-left[
<figure>
  <img src="path_p3.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
</figure>
]

Time series with values in `$\mathbb{R}^2$` (on earth ...).

`$$\begin{array}\\ 
 \mbox{Time} & \mbox{Location} & \mbox{Turning angle} & \mbox{Speed}\\
t_{0} & (x_0, y_0) & NA & NA\\
t_{1} & (x_1, y_1) & NA & sp_1\\
t_{2} & (x_2, y_2) & ang_2 & sp_2\\
\vdots & \vdots& \vdots& \vdots \\
t_{n} & (x_n, y_n) & ang_n & sp_n\\\\
\end{array}$$`
]

---
# Ecological questions

## Behavioural ecology

<figure>
  <img src="traj_seg_booby_black.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
.legend[Peruvian booby  data courtesy of Sophie Bertrand]
</figure>
]

---
# Ecological questions

## Behavioural ecology

<figure>
  <img src="traj_seg_booby.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
.legend[Peruvian booby  data courtesy of Sophie Bertrand]
</figure>
]

## Habitat preference

<figure>
  <img src="SSL_covs.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
.legend[Sea Lions habitat description]
</figure>
]

---
name: data2model
# From Movement Data to Movement Model

Often analysed using discrete time model  <a name=cite-mcclintock2014discrete></a>([McClintock, Johnson, Hooten, et al., 2014](#bib-mcclintock2014discrete))

<figure>
  <img src="move_decomposition.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
.legend[Movement decomposition]
</figure>

---
name:signal

# Signal processing methods 
--

##  Heterogeneity in movement pattern interpretated as different internal states

.pull-left[
<figure>
  <img src="traj_seg_booby.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
.legend[Peruvian booby  data courtesy of Sophie Bertrand]
</figure>
]

* Classically addressed with Hidden Markov Model <a name=cite-morales2004extracting></a><a name=cite-walker2010pioneer></a><a name=cite-gurarie2009novel></a><a name=cite-michelot2016movehmm></a>([Morales, Haydon, Frair, et al., 2004](#bib-morales2004extracting); [Walker and Bez, 2010](#bib-walker2010pioneer); [Gurarie, Andrews, and Laidre, 2009](#bib-gurarie2009novel); [Michelot, Langrock, and Patterson, 2016](#bib-michelot2016movehmm))

* or change point detection approach. <a name=cite-patin2019identifying></a>([Patin, Etienne, Lebarbier, et al., 2019](#bib-patin2019identifying)) based on <a name=cite-picard2007segmentation></a>([Picard, Robin, Lebarbier, et al., 2007](#bib-picard2007segmentation))
]

---

.pull-left[
<figure>
  <img src="segmentation.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
.legend[Change point detection ([Patin, Etienne, Lebarbier, et al., 2019](#bib-patin2019identifying))]
</figure>
]

--
.pull-right[
<figure>
  <img src="seg_classification.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
 .legend[Change point detection and classification ([Patin, Etienne, Lebarbier, et al., 2019](#bib-patin2019identifying))]
</figure>
]

In  ([Patin, Etienne, Lebarbier, et al., 2019](#bib-patin2019identifying)) : a direct extension to simultaneous segmentation for home range shift.
--

Pros and cons of  Discrete time versus continuous time movement models  discussed in [McClintock, Johnson, Hooten, et al. (2014)](#bib-mcclintock2014discrete)

---

# Potential based Diffusions as continuous time movement model

Let  `$(X_s)_{s\geq0}\in\mathbb{R}^2$` denote the position at time `$s$`.

.pull-left[
* Brownian motion: a pure diffusion model
`$$dX_s = dW_s, \quad X_0=x_0.$$`
<figure>
  <img src="bm.png" alt="at the beginning is" style="width:60%" class = "centerimg" />
</figure>
]

--
.pull-left[
* Ornstein Uhlenbeck process: central place behavior
`$$dX_s = -B (X_s- \mu) ds + dW_s, \quad X_0=x_0.$$`

<figure>
  <img src="ou.png" alt="at the beginning is" style="width:60%" class = "centerimg" />
</figure>
]

Popular models as  Brownian Motion and Ornstein Uhlenbeck have known transition densities 
`$q(x_t, x_{t+s})$` which is not the case in general.

---

# Potential based Diffusions as continuous time movement model

<a name=cite-brillinger2002employing></a>[Brillinger, Preisler, Ager, et al. (2002)](#bib-brillinger2002employing) propose a flexible framework
`$$dX_s = -\nabla H(X_s) ds + \gamma dW_s, \quad X_0=x_0.$$`

but no explicit transitions `$q(x_t, x_{t+s})$`

In <a name=cite-Gloaguen2018stochastic></a>[Gloaguen, Etienne, and Le Corff (2018a)](#bib-Gloaguen2018stochastic), as part  of *P. Gloaguen's PhD*, explore `$H(X_s) = \sum_{k=1}^K \pi_k \varphi_k(X_s),$`

.pull-left[
<figure>
  <img src="map2.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
</figure>
]

.pull-right[
* Euler approximation : biased estimates with low frequency data
* <a name=cite-ozaki1992bridge></a>([Ozaki, 1992](#bib-ozaki1992bridge)) and <a name=cite-kessler1997estimation></a>[Kessler (1997)](#bib-kessler1997estimation) same results than
* MCEM based on exact simulation <a name=cite-beskos2006exact></a>([Beskos, Papaspiliopoulos, Roberts, et al., 2006](#bib-beskos2006exact)) limits the flexibility of the SDE.
]

---

# Potential based Diffusions as continuous time movement model 
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## Partially observed SDE 
.pull-lefts[
  <img src="path_4.png" alt="at the beginning is" style="width:70%" class = "centerimg" />
]

.pull-rights[
Let  `$Y_k$` be the recorded position `$s_k$`, a noisy observation of the true position `$X_k$`:

`$$dX_s = b(X_s) ds + \gamma dW_s, \quad X_0=x_0; \quad Y_k \overset{ind}{\sim} \mathcal{L} (X_k, \theta{o}).$$`
]

### Additive smoothing distributions for the E Step

`$$\sum_{k=0}^{n-1}\mathbb{E}( h(X_k, X_{k+1}) \vert Y_{0:n})$$`

--
* The particle-based, rapid incremental smoother (PaRIS)  algorithm <a name=cite-olsson2017efficient></a>([Olsson and Westerborn, 2017](#bib-olsson2017efficient)) provides  an online smoother using a rewriting of the Backward weight and an acceptation/rejection mechanism but depends on `$q(\xi_{k-1}, \xi_{k})$`

* The generalized random PaRIS algorithm, in  <a name=cite-gloaguen2018online></a>([Gloaguen, Etienne, and Le Corff, 2018b](#bib-gloaguen2018online)), uses simple Euler approximation to propose the particles and uses a General Poisson Estimator to replace `$q(\xi_{k-1}, \xi_{k})$` with an unbiased estimator.

.care[Restrictive constraints on the drift and the diffusion term], are relaxed in <a name=cite-martin2021backward></a>([Martin, Etienne, Gloaguen, et al., 2021](#bib-martin2021backward)).

---

# Potential based Diffusions as continuous time movement model

## Flexible movement model for habitat preference

---

# Potential based Diffusions as continuous time movement model

## Flexible movement model for habitat preference

<figure>
  <img src="DAG2.png" alt="at the beginning is" style="width:80%" class = "centerimg" />
   .legend[Flexible movement model which accounts for environment]
</figure>

---

# Potential based Diffusions as continuous time movement model

## Flexible movement model for habitat preference

A classical choice of **resource selection function**, (i.e stationary distribution including covariates)
$$\pi{\left(x \vert \beta\right)} \varpropto  \exp\left(\sum_{j=1}^J \beta_j c_j (x) \right). $$

Diffusion, under regularity condition admits a stationary  distribution.

Combining the ideas of <a name=cite-michelot2019linking></a>([Michelot, Blackwell, and Matthiopoulos, 2019](#bib-michelot2019linking)), and  ([Brillinger, Preisler, Ager, et al., 2002](#bib-brillinger2002employing)) lead to the Langevin diffusion as movement model, 
$$
	d X_t = \frac{\gamma^2}{2} \nabla \log \pi{\left(X_t\right)} \, d t + \gamma \,d W_t,\quad X_0 =x_0.
$$
--

Using Euler approximation

`$$	X_{i+1} \vert \lbrace X_i = x_i \rbrace = 
    	x_i + \frac{\gamma^2 \Delta_i}{2} \sum_{j=1}^J \beta_j \nabla c_j(x_i) + \sqrt{\Delta_i}  \varepsilon_{i+1},\quad 
    	\varepsilon_{i+1} \overset{ind}{\sim} N \left( {0} , \gamma^2 \boldsymbol{I}_d \right),$$`

leads to a simple linear model published in <a name=cite-michelot2019langevin></a>[Michelot, Etienne, Blackwell, et al. (2019)](#bib-michelot2019langevin).
    	
    	
    	
---

# Potential based Diffusions as continuous time movement model

## Flexible movement model for habitat preference

.pull-left[
<figure>
  <img src="SSL_covs.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
.legend[Sea Lions habitat description]
</figure>
]

<figure>
  <img src="SSL_logUD.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
.legend[Sea Lions habitat description]
</figure>
]

.small[ On going work on Sea birds in the Mediteranean sea and Roe deer next to Toulouse]
---
# On going work

### Some natural extension of the Langevin model useful in movement ecology

* Coupling Hidden Markov model or change point detection model with a Langevin distribution:
<figure>
  <img src="DAG4.png" alt="at the beginning is" style="width:60%" class = "centerimg" />
</figure>

* Change the Langevin model for an hypoellitptic approach
`$$d X_t = V_t dt; X_0 =x_0.$$`

$$d V_t = - \nabla \log \pi{\left(X_t\right)} \, d t  -\frac{\nu^2}{2} V_t + \nu \,d W_t,\quad $$

---
# Perspectives

### Some natural extension of the Langevin model useful in movement ecology

* Introduction of an individual random effect,

###  Handling categorical covariates
<a name=cite-lejay2018maximum></a>[Lejay and Pigato (2018)](#bib-lejay2018maximum) define a threshold diffusion and proposes an ML estimation method. Explore the generalisation to `$\mathbb{R}^2$`.

### Longer term perspective
* Collective movement: Cheaper GPS imply  massive deployment. Invest the area of collective movement analysis.
* Combined sound monitoring

---
# A few words to finish

* Close interaction with biologist,
  - it's helpful,
  - provides exciting statistical problems,
  - experiment a large diversity of approaches.
* Hidden variables approach
  - provide flexible models (spatial abundance, classification in time series, Partially observed SDE)
  - popularized in Ecology in a Bayesian setting but frequentist approach is also possible
* The Markovian property
  - not so realistic,
  - but a key component in both theoretical approach and computational aspects

---
class: biblio
# Bibliography
<img class="logopos_right" src="article.png" style="height:6%">

<a name=bib-beskos2006exact></a>[Beskos, A., O. Papaspiliopoulos, G. O.
Roberts, et al.](#cite-beskos2006exact) (2006). "Exact and
computationally efficient likelihood-based estimation for discretely
observed diffusion processes (with discussion)". In: _Journal of the
Royal Statistical Society: Series B (Statistical Methodology)_ 68.3,
pp. 333-382.

<a name=bib-brillinger2002employing></a>[Brillinger, D. R., H. K.
Preisler, A. A. Ager, et al.](#cite-brillinger2002employing) (2002).
"Employing stochastic differential equations to model wildlife motion".
In: _Bulletin of the Brazilian Mathematical Society_ 33.3, pp. 385-408.

<a name=bib-gloaguen2018online></a>[Gloaguen, P., M. Etienne, and S. Le
Corff](#cite-gloaguen2018online) (2018b). "Online sequential Monte
Carlo smoother for partially observed diffusion processes". In:
_EURASIP Journal on Advances in Signal Processing_ 2018.1, p. 9.

<a name=bib-Gloaguen2018stochastic></a>[Gloaguen, P., M. Etienne, and
S. Le Corff](#cite-Gloaguen2018stochastic) (2018a). "Stochastic
differential equation based on a multimodal potential to model movement
data in ecology". In: _Journal of the Royal Statistical Society: Series
C (Applied Statistics)_ 67.3, pp. 599-619. DOI:
[10.1111/rssc.12251](https://doi.org/10.1111%2Frssc.12251).

<a name=bib-gurarie2009novel></a>[Gurarie, E., R. D. Andrews, and K. L.
Laidre](#cite-gurarie2009novel) (2009). "A novel method for identifying
behavioural changes in animal movement data". In: _Ecology Letters_
12.5, pp. 395-408.

<a name=bib-kessler1997estimation></a>[Kessler,
M.](#cite-kessler1997estimation) (1997). "Estimation of an ergodic
diffusion from discrete observations". In: _Scandinavian Journal of
Statistics_ 24.2, pp. 211-229.

<a name=bib-lejay2018maximum></a>[Lejay, A. and P.
Pigato](#cite-lejay2018maximum) (2018). "Maximum likelihood drift
estimation for a threshold diffusion". In: _Scandinavian Journal of
Statistics_.

---
class: biblio
count: false
# Bibliography
<img class="logopos_right" src="article.png" style="height:6%">

<a name=bib-martin2021backward></a>[Martin, A., M. Etienne, P.
Gloaguen, et al.](#cite-martin2021backward) (2021). "Backward
importance sampling for online estimation of state space models".
working paper or preprint. URL:
[https://hal.archives-ouvertes.fr/hal-02476102](https://hal.archives-ouvertes.fr/hal-02476102).

<a name=bib-mcclintock2014discrete></a>[McClintock, B. T., D. S.
Johnson, M. B. Hooten, et al.](#cite-mcclintock2014discrete) (2014).
"When to be discrete: the importance of time formulation in
understanding animal movement". In: _Movement Ecology_ 2.1, p. 21.

<a name=bib-michelot2019linking></a>[Michelot, T., P. G. Blackwell, and
J. Matthiopoulos](#cite-michelot2019linking) (2019). "Linking resource
selection and step selection models for habitat preferences in
animals". In: _Ecology_ 100.1.

<a name=bib-michelot2019langevin></a>[Michelot, T., M. Etienne, P.
Blackwell, et al.](#cite-michelot2019langevin) (2019). "The Langevin
diffusion as a continuous-time model of animal movement and habitat
selection". In: _Methods in Ecology and Evolution_.

<a name=bib-michelot2016movehmm></a>[Michelot, T., R. Langrock, and T.
A. Patterson](#cite-michelot2016movehmm) (2016). "moveHMM: an R package
for the statistical modelling of animal movement data using hidden
Markov models". In: _Methods in Ecology and Evolution_ 7.11, pp.
1308-1315.

<a name=bib-morales2004extracting></a>[Morales, J. M., D. T. Haydon, J.
Frair, et al.](#cite-morales2004extracting) (2004). "Extracting more
out of relocation data: building movement models as mixtures of random
walks". In: _Ecology_ 85.9, pp. 2436-2445.

<a name=bib-nathan2008movement></a>[Nathan, R., W. M. Getz, E. Revilla,
et al.](#cite-nathan2008movement) (2008). "A movement ecology paradigm
for unifying organismal movement research". In: _Proceedings of the
National Academy of Sciences_ 105.49, pp. 19052-19059.

---
class: biblio
count: false
# Bibliography
<img class="logopos_right" src="article.png" style="height:6%">

<a name=bib-olsson2017efficient></a>[Olsson, J. and J.
Westerborn](#cite-olsson2017efficient) (2017). "Efficient
particle-based online smoothing in general hidden Markov models: the
PaRIS algorithm". In: _Bernoulli_ 23.3, pp. 1951-1996.

<a name=bib-ozaki1992bridge></a>[Ozaki, T.](#cite-ozaki1992bridge)
(1992). "A bridge between nonlinear time series models and nonlinear
stochastic dynamical systems: a local linearization approach". In:
_Statistica Sinica_, pp. 113-135.