The hidden part of Markovian stochastic processes for biology and ecology

class: center, middle, inverse, title-slide

# The hidden part of Markovian stochastic processes for biology and ecology
### Marie-Pierre Etienne
### HDR Defense
### October 2021

---

name: intro

# How biology feeds statistics ?

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<figure>
  <img src="overview1.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
</figure>

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# A hierarchical spatial model to map the fish density
<img class="logopos_right" src="model.png" style="width:6%">
<img class="logopos_left" src="fish_model.png" style="height:7%">

---
# A hierarchical spatial model to map the fish density
<img class="logopos_right" src="model.png" style="width:6%">
<img class="logopos_left" src="fish_model.png" style="height:7%">

## Marine ecology

* France commitment for 2022 : 30% of territorial waters classified as Marine Protected Area (MPA)
* Localization also matters
* Mapping the fish density to identify critical areas

### 2 data sources

.pull-left[ 
**Scientific data**: a yearly survey according to a standardized sampling plan.
<figure>
  <img src="map_sci.png" alt="Orhago campaign and commercial data" style="width:60%" class = "centerimg" />
  .legend[Common sole - ORHAGO survey (kg; 2018)]
</figure>
]
--

.pull-right[
**Commercial catch**: Catch geolocalized thanks to Vessel Monitoring System (VMS). Preferential sampling.
<figure>
  <img src="map_com.png" alt="Orhago campaign and commercial data" style="width:60%" class = "centerimg" />
  .legend[Common sole -  Otter trawls targeting demersal species(kg; 2018)]
</figure>
]

.question[How to integrate commercial data to improve the estimation of spatial fish density?]

---
# A hierarchical spatial model to map the fish density
<img class="logopos_right" src="model.png" style="width:6%">
<img class="logopos_left" src="fish_model.png" style="height:7%">

## A model should account for the specifities of catch data

.left-column[
<figure>
  <img src="model_baptiste_1.png" alt="DAG for Baptist's model" style="width:100%" class = "centerimg" />
</figure>
]

---
count: false

# A hierarchical spatial model to map the fish density
<img class="logopos_right" src="model.png" style="width:6%">
<img class="logopos_left" src="fish_model.png" style="height:7%">

## A model should account for the specifities of catch data

.left-column[
<figure>
  <img src="model_baptiste_2.png" alt="DAG for Baptist's model" style="width:100%" class = "centerimg" />
</figure>
]

.right-column[
* Observation process : Zero inflated data ( *Sophie Ancelet's PhD* )
   <a name=cite-aitchison1955distribution></a><a name=cite-ancelet2010modelling></a><a name=cite-foster2013poisson></a>([Aitchison, 1955](#bib-aitchison1955distribution); [Ancelet, Etienne, Benoît, et al., 2010](#bib-ancelet2010modelling); [Foster and Bravington, 2013](#bib-foster2013poisson)),]

---
count: false

# A hierarchical spatial model to map the fish density
<img class="logopos_right" src="model.png" style="width:6%">
<img class="logopos_left" src="fish_model.png" style="height:7%">

## A model should account for the specifities of catch data

.left-column[
<figure>
  <img src="model_baptiste_3.png" alt="DAG for Baptist's model" style="width:100%" class = "centerimg" />
</figure>
]

.right-column[
* Observation process : Zero inflated data ( *Sophie Ancelet's PhD* )
   ([Aitchison, 1955](#bib-aitchison1955distribution); [Ancelet, Etienne, Benoît, et al., 2010](#bib-ancelet2010modelling); [Foster and Bravington, 2013](#bib-foster2013poisson)),
* Hidden layer : Spatially structured ( *Jean-Baptiste Lecomte's PhD* ) <a name=cite-lecomte2013compound></a>([Lecomte, Benoît, Ancelet, et al., 2013](#bib-lecomte2013compound)),
]

---
count: false

# A hierarchical spatial model to map the fish density
<img class="logopos_right" src="model.png" style="width:6%">
<img class="logopos_left" src="fish_model.png" style="height:7%">

## A model should account for the specifities of catch data

.left-column[
<figure>
  <img src="model_baptiste_4.png" alt="DAG for Baptist's model" style="width:100%" class = "centerimg" />
</figure>
]

.right-column[
* Observation process : Zero inflated data ( *Sophie Ancelet's PhD* )
  ([Aitchison, 1955](#bib-aitchison1955distribution); [Ancelet, Etienne, Benoît, et al., 2010](#bib-ancelet2010modelling); [Foster and Bravington, 2013](#bib-foster2013poisson)),
* Hidden layer : Spatially structured ( *Jean-Baptiste Lecomte's PhD* ) ([Lecomte, Benoît, Ancelet, et al., 2013](#bib-lecomte2013compound)),
* Integrate Commercial data
]

---
count: false

# A hierarchical spatial model to map the fish density
<img class="logopos_right" src="model.png" style="width:6%">
<img class="logopos_left" src="fish_model.png" style="height:7%">

## A model should account for the specifities of catch data

.left-column[
<figure>
  <img src="model_baptiste_full.png" alt="DAG for Baptist's model" style="width:100%" class = "centerimg" />
</figure>
]

.right-column[
* Observation process : Zero inflated data ( *Sophie Ancelet's PhD* )
  ([Aitchison, 1955](#bib-aitchison1955distribution); [Ancelet, Etienne, Benoît, et al., 2010](#bib-ancelet2010modelling); [Foster and Bravington, 2013](#bib-foster2013poisson)),
* Hidden layer : Spatially structured ( *Jean-Baptiste Lecomte's PhD* ) ([Lecomte, Benoît, Ancelet, et al., 2013](#bib-lecomte2013compound)),
* Integrate Commercial data where **sampling process** and **observations** are not independent (Commercial data) (*Baptiste Alglave's PhD*)
]

---
name:test
# A hierarchical spatial model to map the fish density
<img class="logopos_right" src="model.png" style="width:6%">
<img class="logopos_left" src="fish_model.png" style="height:7%">

## Estimation of Hierarchical model

* Integration over the hidden variables: a difficult task 
  * Use of Monte Carlo simulation
  * Approximation: Integrated Nested Laplace Approximation <a name=cite-rue2009approximate></a>([Rue, Martino, and Chopin, 2009](#bib-rue2009approximate)) combined with Gaussian Random Markov Field  for a sparse spatial model:

* Maximum likelihood estimation is also possible: the maximization  uses  Automatic differentiation  thanks to the TMB R package <a name=cite-kristensen2016automatic></a>([Kristensen, Nielsen, Berg, et al., 2016](#bib-kristensen2016automatic)), 
  
--

## Results 
.font80[<a name=cite-alglave2021integrated></a>[Alglave, Vermard, Etienne, et al. (2021)](#bib-alglave2021integrated) in review]

* Ignoring PS provides biased estimates
* Commercial data contain valuable information

---
template:test

<figure>
  <img src="sole_spatial_results.png" alt="Sole's results" style="width:75%" class = "centerimg" />
  .legend[Reconstruction of the spatial relative density for soles in 2018 in the Bay of Biscay during the Orhago scientific campaign]
</figure>

---
# A hierarchical spatial model to map the fish density
<img class="logopos_right" src="model.png" style="width:6%">
<img class="logopos_left" src="fish_model.png" style="height:7%">

## Around this model  (as part of Baptiste's Alglave PhD project)

* Adaptation for presence absence data (Florian Quemper Master's project) : a bit disappointing 
* Spatio temporal modelling (in progress)
* Account for aggregated data (initiated)

.pull-left[
<figure>
  <img src="aggregation.png" alt="Sole's results" style="width:100%" class = "centerimg" />
  .legend[Illustration of the reallocation process. Catch declaration = 50]
</figure>
]

.pull-right[
Most commercial vessels report one catch per day and administrative unit
  * Catch are then "reallocated"
  * Change the commercial observation process
  
`$$Y^{C}_s = \sum_{l \in \mathcal{F}_i} Y^{C}_{sl}$$`
]

---
# A statistical test to identify atypical genome region  
<img class="logopos_right" src="pvalue.png" style="width:6%">
<img class="logopos_left" src="dna.png" style="height:7%">

---
# A statistical test to identify atypical genome region  
<img class="logopos_right" src="pvalue.png" style="width:6%">
<img class="logopos_left" src="dna.png" style="height:7%">

## PhD thesis: a scoring scheme to measure atypicity within a sequence

see <a name=cite-Etienne03></a><a name=cite-Etienne04></a>[Daudin, Etienne, and Vallois (2003)](#bib-Etienne03); [Etienne and Vallois (2004)](#bib-Etienne04)

--
## Recurrent alteration in a cohort : CGH profiles

.pull-left[

<img src="recurrent_alt.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
]

.pull-right[

For a cohort of size `$N$`, a profile of length `$L$`, 
`$$\mathbb{P} ( \mbox{At least } m \mbox{ patients an alteration of length } l )$$`

*  <a name=cite-robin2009simultaneous></a>[Robin and Stefanov (2009)](#bib-robin2009simultaneous), considers  a profile `$(X^{i}_k)_{k=1,\ldots,n}$` as a 2 states discrete time Markov chain, the solution involved a Markov chain on a NL state space
 
]

---
# A statistical test to identify atypical genome region  
<img class="logopos_right" src="pvalue.png" style="width:6%">
<img class="logopos_left" src="dna.png" style="height:7%">

## Recurrent alteration in a cohort : CGH profiles

.pull-lefts[
* For large `$L$` and small `$N$`, <a name=cite-robin2015detection></a>[Robin and Stefanov (2015)](#bib-robin2015detection) considers  `$X^{i}$`  a 2 states continuous time Markov process
<figure>
  <img src="ind_profile.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
</figure>
 and define `$Y^{(N)}_{s}$` as the number of patients with alteration at position (time) `$s$`.

`$$\mathbb{P}(\exists t, \forall 0 \leq s < l, Y^{(N)}_{t+s} \geq m )$$`
]

.pull-rightb[
<figure>
  <img src="cum_profile1.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
</figure>
]

---
count: false
# A statistical test to identify atypical genome region  
<img class="logopos_right" src="pvalue.png" style="width:6%">
<img class="logopos_left" src="dna.png" style="height:7%">

## Recurrent alteration in a cohort : CGH profiles

.pull-lefts[
* For large `$L$` and small `$N$`, [Robin and Stefanov (2015)](#bib-robin2015detection) considers  `$X^{i}$`  a 2 states continuous time Markov process
<figure>
  <img src="ind_profile.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
</figure>
 and define `$Y^{(N)}_{s}$` as the number of patients with alteration at position (time) `$s$`.

`$$\mathbb{P}(\exists t, \forall 0 \leq s < l, Y^{(N)}_{t+s} \geq m )$$`
]

.pull-rightb[
<figure>
  <img src="cum_profile2.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
</figure>
]

---
count: false
# A statistical test to identify atypical genome region  
<img class="logopos_right" src="pvalue.png" style="width:6%">
<img class="logopos_left" src="dna.png" style="height:7%">

## Recurrent alteration in a cohort : CGH profiles

`$$\mathbb{P}(\exists t, \forall 0 \leq s < l, Y^{(N)}_{t+s} \geq m )$$`
]

.pull-rightb[
<figure>
  <img src="cum_profile3.png" alt="at the beginning is" style="width:100%" class = "centerimg" />
</figure>
]

<br>
But the computational complexity drastically increases with `$N$`.

.question[Could we identify some limit behavior when N increases ?]

---
count:false
# A statistical test to identify atypical genome region  
<img class="logopos_right" src="pvalue.png" style="width:6%">
<img class="logopos_left" src="dna.png" style="height:7%">

## Recurrent alteration in a cohort : CGH profiles

Let `$S^{N}= \sum_{i=1}^N \frac{X^{i}-E(X^i)}{\sqrt{N/4}}$` a normalized version of `$Y^{(N)}$`.

`$$\mathbb{P}( \exists t, \forall 0 \leq s < l, S^{(N)}_{t+s} \geq m^\prime )=  \mathbb{P}( E^{*,m^\prime}_{S^N} > l ),$$`
 with  `$E^{*,m^\prime}_{S^N}$` the length of the longest excursion of the cumulative  profile `$S^{N}$` above `$m^\prime$`

.addspace[$$ $$]

.question[ `$$\mathbb{P}( E^{*,m^\prime}_{S^N} > l )\underset{N\to\infty}{\longrightarrow} ??$$`]

---
# A statistical test to identify atypical genome region  
<img class="logopos_right" src="pvalue.png" style="width:6%">
<img class="logopos_left" src="dna.png" style="height:7%">

## Recurrent alteration in a cohort : CGH profiles

<a name=cite-Decreusefond19a></a>[Decreusefond, Etienne, Lang, et al. (2021)](#bib-Decreusefond19a) prove
<div class= "theorem">
Let Z be a stationary standard Ornstein Uhlenbeck process,

`$$\left \vert  \mathbb{P}( E^{*,m}_{S^N} > l)  - \mathbb{P}(E^{*,m}_Z > l))\right\vert \leq \frac{C\ln{N}}{N^{1/8}}.$$`
</div>
Key points of the proof:

--
* Convergence  of `$S^{N}$`  to `$Z$` : a Gaussian process with covariance function `$c(s) = e^{-\tau s}.$`

--
* Rate of convergence: combined a Brownian representation of the Ornstein Uhlenbeck process and a result from <a name=cite-kubilius1994rate></a>[Kubilius (1994)](#bib-kubilius1994rate) on the rate of convergence of the martingale increments to the Brownian motion.

--
* The quantity of interest is expressed through a continuous functional to preserve the convergence

---
# A statistical test to identify atypical genome region  
<img class="logopos_right" src="pvalue.png" style="width:6%">
<img class="logopos_left" src="dna.png" style="height:7%">

## Perspective : Practical implementation

The distribution of the length of the OU excursions is unknown in general

* Conditionally on the first hitting time `$\sigma_m$`,  express the change of measure from OU to a Wiener process with a quantity which only depends on  the signs and the lengths of  Brownian excursions.

* Simulation of the signs and lengths of Brownian excursion using the size biased algorithm porposed by <a name=cite-devroye2010exact></a>[Devroye (2010)](#bib-devroye2010exact).

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

--
<figure>
  <img src="overview5_app.png" alt="at the beginning is" style="width:90%" class = "centerimg" />
</figure>

---

# Efficient estimation methods in movement ecology
<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) ]
]

.pull-right[

* Motion capacities
* Internal state
* Environment
]
  
--

.question[Movement informs on internal states and habitat preferences]

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## Movement data

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

---
count: false

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## Movement data

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

---
count: false

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## Movement data

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

.pull-right[
A continuous process sampled at some discrete potentially irregular times.

Time series with values in `$\mathbb{R}^2$`.

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

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## Heterogeneity in movement pattern interpretated as different internal states

.pull-left[
<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>
]

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## 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>
]

.pull-right[
## Accounting for internal states

Classically addressed with Hidden Markov Model

### Exploring the change point detection approach. <a name=cite-lavielle2005using></a><a name=cite-picard2007segmentation></a>([Lavielle, 2005](#bib-lavielle2005using); [Picard, Robin, Lebarbier, et al., 2007](#bib-picard2007segmentation))
]

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## Signal processing approach for movement ecology ([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 <a name=cite-patin2019identifying></a>([Patin, Etienne, Lebarbier, et al., 2019](#bib-patin2019identifying))]
</figure>

Let `$\boldsymbol{\tau}={\tau_1,...,\tau_{K-1}}$` ( `$\tau_0=-1$` ) be a partition in K segments of `$\{1,\ldots n\}$`
`$$\boldsymbol{X}_{i}\overset{i.i.d}{\sim}\mathcal{L}(\theta_k),\quad \forall i \in \{ \tau_{k-1}+1:\tau_k \}$$`
The .care[Dynamic Programming algorithm] allows to explore efficiently all possible segmentation and to estimate `$\boldsymbol{\hat{\tau}}$`
]

--
.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>

Let  `$Z_k$` stand for the class of segment `$k,$` `$\forall i \in \{ \tau_{k-1}+1:\tau_k \}$` 
`$$Z_k \overset{i.i.d}{\sim} \mathcal{M}(\pi), \quad\boldsymbol{X}_{i}\vert Z_k=l \overset{i.i.d}{\sim}\mathcal{L}(\theta_l)$$`
The Dynamic Programming coupled with EM algorithm allows to explore efficiently all possible segmentation and to estimate `$\boldsymbol{\hat{\tau}}$`
]

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

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## Signal processing approach for movement ecology ([Lavielle, 2005](#bib-lavielle2005using); [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>
]

.question[Movement path is more than time series, importance of considering the space.]

.center[.care[Proposing ecologically meaningful movement models]]

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

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## 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.

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## 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.
]

---

# Efficient estimation methods in movement ecology
<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)).

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## Flexible movement model

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## Flexible movement model

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

---

# Efficient estimation methods in movement ecology
<img class="logopos_right" src="compute.png" style="height:6%">
<img class="logopos_left" src="paw.png" style="height:7%">

## Linking covariates and stationary distribution

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).
    	
---
# Perspectives

### 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>
* 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

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

.pull-right[
* 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
]
---
# Many thanks to

---
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