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# The Point Process Framework for Integrated Modelling of Biodiversity Data

Kwaku Peprah Adjei  Department of Mathematical Sciences, Norwegian University
of Science and Technology, Norway  Center for Biodiversity Dynamics, Norwegian
University of Science and Technology, Norway  Corresponding author:
<EMAIL_ADDRESS>Philip Mostert  Department of Mathematical Sciences,
Norwegian University of Science and Technology, Norway  Center for
Biodiversity Dynamics, Norwegian University of Science and Technology, Norway
Jorge Sicacha Parada  Norwegian Computing Center, Oslo, Norway  Emma Skarstein
Department of Mathematical Sciences, Norwegian University of Science and
Technology, Norway  Robert B. O’Hara  Department of Mathematical Sciences,
Norwegian University of Science and Technology, Norway  Center for
Biodiversity Dynamics, Norwegian University of Science and Technology, Norway

###### Abstract

The quantity and types of biodiversity data being collected have increased in
recent years. If we are to model and monitor biodiversity effectively, we need
to respect how different data sets were collected, and effectively integrate
these data types together. The framework that has emerged to do this is based
on a point process formulation, with individuals as points and their
distribution as a realisation of a random field. We describe this formulation
and how the process model for the actual distribution is linked to the data
that is collected through observation models. The observation models describe
the data collection process, and its uncertainties and biases. We provide an
example of using these methods to model species of Norwegian freshwater fish,
which shows how integrated models can be adapted to the data we can collect.
We summarise the modelling issues that arise and the approaches that could be
taken to solve them.

Keywords: Bayesian models, data integration, observation errors, state-space
formulation, PointedSDMs

##  1 Introduction

In recent years, there has been an influx of biodiversity data collected and
hosted in openly accessible databases (Anderson, 2018; Farley et al., 2018;
Mandeville et al., 2021). These biodiversity data are collected from different
sources: monitoring programs that employ different sampling protocols, museum
collections and less structured citizen science collections. This biodiversity
data is vital for predicting species distribution and making inferences about
their ecology and conservation. The mapping process is, however, impeded by
inherent problems in the way the data have been collected, for example, due to
misidentification and errors in the data collection process (Kosmala et al.,
2016; Aceves-Bueno et al., 2017; Chambert et al., 2015; Royle and Link, 2006),
differences in the scale and resolution of the data (such as data collected at
different spatial resolutions Boakes et al., 2010; Donaldson et al., 2016;
Gonzalez et al., 2016) and availability of the data in various formats
(Bowker, 2000). Despite these challenges, integrating data from different
sources has been a growing field in ecology (Isaac et al., 2020).

Species distribution models (SDMs) are now widely used by ecologists and
conservation biologists to help us understand where in space and time species
presence can be expected (Higgins et al., 2012). SDMs, growing in their
popularity, have however come under some scrutiny from various perspectives
(for example, see A. Lee-Yaw et al. (2022) and Robinson et al. (2017) for a
review of some of these challenges). The problem of interest in this study is
the challenge of fitting SDMs without respecting the data type and data
collection protocol. For instance, most SDMs fitted with MaxEnt (Phillips et
al., 2009, 2006) and Biomod (Thuiller, 2003) assume that the data is generated
from a ‘presence-only’ process, i.e. it consists of reports of observations of
the species of interest and no information about possible absences, regardless
of the data collection process.

In practice, however, the available data are heterogeneous. For example,
presence-only observations, formal repeated surveys, atlas data and expert
range maps may all be available. Thus, we need methods to effectively combine
the different data types in a single analysis. Advances along these lines have
been made by Dorazio (2014) and Fithian et al. (2015), who both developed a
similar framework to combine presence-only data with occupancy data from
repeated surveys. Along similar lines, Pagel et al. (2014) use presence-
absence data to supplement data on abundance to improve the modelling of range
dynamics. Presence-only data and range maps have also been combined by Merow
et al. (2017).

Additionally, each of the available data types has their own quirks
(observation errors) that should be added to the integrated model
individually. Failure to properly account for these observation errors results
in biased inferences about the actual distribution of interest (Simmonds et
al., 2020). The sources of observation errors include uneven sampling effort
(Chakraborty et al., 2011; Sicacha-Parada et al., 2021; Ahmad Suhaimi et al.,
2021; Simmonds et al., 2020), imperfect detection (Kéry and Schmid, 2004;
MacKenzie et al., 2002a) and misclassification (Wright et al., 2020; Chambert
et al., 2015; Miller et al., 2011). By providing separate sub-models for each
available data in the integrated model, information is shared across the data
to produce identifiable model parameter estimates whilst accounting for the
observation errors in each data (Fletcher et al., 2016; Dorazio, 2014;
Guillera-Arroita et al., 2017; Isaac et al., 2020). For example, data
collected with preferential sampling (i.e. sampling more where the species is
present) and presence-only data can be integrated with presence-absence data,
camera trap and telemetry data to account for the sampling bias (Adde et al.,
2021; Koshkina et al., 2017; Fletcher et al., 2016; Simmonds et al., 2020).
Also, repeated counts from multiple visits to a site and double-observer or
site occupancy protocol data can be used to account for imperfect detection
(Dorazio, 2014; Kéry and Schmid, 2004; Case and Lawler, 2017). To account for
misclassification, data with information on the verification process, either
from Machine learning algorithms or experts can be used (Spiers et al., 2022;
Wright et al., 2020).

Fitting a distribution model with different data types requires a single model
for the actual distribution, which can then map onto the different observation
models for each data type. A common approach among those developing the models
is to use a point process model for the actual distribution (e.g. Aarts et
al., 2012; Fithian and Hastie, 2012; Renner and Warton, 2013; Dorazio, 2014;
Fithian et al., 2015), reviewed by Renner et al. (2015). This is a model in
continuous space, which has several advantages: (i) it removes the need to
discretise the disparate data into grid cells, with the accompanying loss of
spatial accuracy, (ii) a problem with presence-only models was the choice of
“pseudo-absences”, but with the point process formulation, their roles are
clarified as quadrature points in a numerical integration (Renner and Warton,
2013, see below).

The purpose of this paper is to present the statistical formulation of data
integration using the point process framework. We first present a brief review
of the data integration literature that use the point process framework and
collate the data types usually combined. Then, we present the process and
observation models using the point process framework and present an example of
an SDM fitted to data on freshwater fish in Norwegian lakes.

##  2 Brief review of literature

Before we describe the model framework, we briefly present a literature review
of integrated models that are fitted using the point process framework and the
various observation models used in these studies.

A conceptual workflow that integrates disparate datasets together to combine
the individual strengths of each in an SDM was communicated by Jetz et al.
(2012). They believed that doing so was beneficial to reduce geographic and
environmental biases inherent in single dataset types, as well as improving
quality control and cross-validation among the datasets. Since this call, a
multitude of techniques to integrate diverse data in a model-based framework
have been proposed through statistical and ecological journals. Fletcher Jr et
al. (2019) and Miller et al. (2019) provided introductions detailing some of
the methods to combine data: an informed prior approach (Marcantonio et al.,
2016; Talluto et al., 2016), an auxiliary data approach (Merow et al., 2016;
Regos et al., 2016; Huberman, 2020), an ensemble model approach (Douma et al.,
2012; Case and Lawler, 2017), a correlation model approach (Pacifici et al.,
2017) or through mere data pooling.

However, the method clearly predominating in literature is the one based on
the Poisson point process framework. The first paper to consider the point
process model approach for integrated distribution models was (Dorazio, 2014),
who used it to combine presence-only data with planned survey data to account
for sampling biases in the former. Since then, numerous other papers have
implemented similar methods using a range of different data types. Table 1
gives a list of some of these papers along with the data types integrated.

Observation models |  Citations
---|---
Presence absence |  Dorazio (2014), Fithian et al. (2015), Fletcher et al. (2016), Koshkina et al. (2017), Schank et al. (2017), Pacifici et al. (2017), Fletcher Jr et al. (2019), Miller et al. (2019), Gelfand and Shirota (2019), Peel et al. (2019), Duncan et al. (2020), Isaac et al. (2020), Simmonds et al. (2020), Chevalier et al. (2021), Adde et al. (2021), Bu et al. (2021), Watson et al. (2021), Gilbert et al. (2021), Ahmad Suhaimi et al. (2021), Fidino et al. (2022), Morera-Pujol et al. (2023), Grattarola et al. (2022), Mostert et al. (2022), Grabow et al. (2022)
Range maps |  Merow et al. (2017)
Abundance |  Giraud et al. (2016), Mäkinen and Vanhatalo (2018)
Distance sampling |  Martino et al. (2021), Farr et al. (2021), Pace et al. (2022)
Other |  Bowler et al. (2019) (Abundance and Presence absence), Zulian et al. (2021) (Presence absence only), Rufener et al. (2021) (Abundance only), Cunningham et al. (2021) (Counts and Density), Sultaire et al. (2022) (Presence absence only), Lauret et al. (2022) (Distance sampling and Presence absence), Cunningham et al. (2022) (Abundance only)
Table 1: Observation models used in conjunction with presence-only data found
across citations implementing the point-process framework for data
integration.

It is evident through this short review that the data types mostly integrated
together are presence-only, presence-absence, range maps, abundance and
distance sampling occurrence records. The combination of the disparate
datasets are either with different data types (for example, abundance and
presence-only data) or the same data type but from different sampling
protocols. Therefore, we proceed to describe the point process framework for
the process model and the various observation process models for the data
types available.

##  3 Point Process Framework

Because we have to deal with both the ecological and observation processes, it
is natural to use a state-space formulation to separate the model into process
and observation models. We define the unobserved distribution as a spatial
field, $\lambda(\mathbf{s},t)$ in a space $\mathbf{s}\in\mathbb{R}^{2}$ and
time $t$ (here we will consider time to be discrete).

For each of the $M$ datasets we want to use in our model, we need a data-
generating model that links the observations to the underlying state. That is,
if dataset $d$ is $Y_{d}$, we want
$Pr(Y_{d}|\lambda(\mathbf{s},t),\theta_{d})$, where $\theta_{d}$ are
parameters of the data generating model. The full likelihood for the model is
then:

$L(Y_{d}|\theta_{d})=p(\lambda(\mathbf{s},t))\prod_{d=1}^{M}Pr(Y_{d}|\lambda(\mathbf{s},t),\theta_{d}).$
(1)

The process model is $p(\lambda(\mathbf{s},t))$: clearly there can only be one
model at this level. This formulation is clearly general enough to cover a
much wider range of models, so here, we will focus on the types of data
typical for species distributions, using a point process framework.

###  3.1 Process Model

Our process model follows Aarts et al. (2012); Dorazio (2014) by using a point
process model. We assume that the locations of individuals are points,
possibly with marks (such as which species the individual is), and we can
model the distribution of points as a field with an intensity,
$\lambda(\mathbf{s})$. For convenience in presenting the process model, we
describe this intensity with only spatial variation. We model this field as a
log-Gaussian Cox process with intensity
$\lambda(\mathbf{s})=e^{\eta(\mathbf{s})}$ (Møller and Waagepetersen, 2003)
with:

$\eta(\mathbf{s})=\sum_{i=1}^{P}\beta_{i}X_{i}(\mathbf{s})+u(\mathbf{s}),$ (2)

where $X_{i}(\mathbf{s})$ is the field for the $i^{th}$ covariate (or, if more
complex responses are needed, a feature sensu Elith et al., 2011).

Residual spatial effects are modeled through $u(\mathbf{s})$. This is a random
field which is set up so that there is a covariance between points that depend
on the distance between them (i.e. there is a spatial autocovariance).
Although there are several alternatives to modelling this spatial effect, in
practice, a Gaussian Markov Random Field (Lindgren et al., 2011) is a common
and computationally efficient choice. Technically, this means assuming
$u(\mathbf{s})$ is a Gaussian field with a Matèrn correlation function:

$Cor(u(s_{i}),u(s_{j}))=\frac{2^{1-\nu}}{\Gamma(\nu)}(\kappa\|s_{i}-s_{j}\|)^{\nu}K_{\nu}(\kappa\|s_{i}-s_{j}\|).$
(3)

with covariance function
$Cov(u(s_{i}),u(s_{j}))=\sigma_{u}Cor(u(s_{i}),u(s_{j}))$, where
$\sigma_{u}^{2}$ is the marginal variance. In the analyses below we fix
$\nu=1$, and parameterise the covariance as
$\theta=\\{log(\tau),log(\kappa)\\}$, where $\tau$ is a local variance
parameter, so that

$\sigma_{u}^{2}=\frac{1}{4\pi\tau^{2}\kappa^{2}},$ (4)

and thus $\log(\tau)=-\log(4\pi\sigma_{u}^{2}\kappa^{2})/2$. Finally,
$u(\mathbf{s})$ is approximated through basis functions with local support and
defined over a triangulation of the space. These basis functions are weighted
by Gaussian random variables. The local nature of the basis functions
guarantees the sparseness of the representation (Blangiardo and Cameletti,
2015).

Given this structure, the number of individuals in an area $B$ follows a
Poisson distribution with mean:

$\mu(B)=\int_{B}\lambda(\mathbf{s})ds=\int_{B}e^{\sum_{i=1}^{P}\beta_{i}X_{i}(\mathbf{s})+u(\mathbf{s})}d\mathbf{s}.$
(5)

This then means that the probability that area $B$ is occupied is:

$Pr(N_{B}>0)=1-Pr(N_{B}=0)=1-e^{-\mu(B)}.$ (6)

The integral in equation (5) is hard to handle analytically, so it is
approximated by numerical integration in practice. We use the approach of
Simpson et al. (2016), which makes use of the discretised version of the study
region to approximate $\mu(B)$ in Equation (5) as

$\mu(B)\approx\sum_{s=1}^{n_{B}}A(s)e^{\eta(B(s))},$ (7)

where $n_{B}$ is the number of integration points in $B$, each located at
$B(s)$, and $A(s)$ is the area of the polygon around $s$, defined through a
dual mesh. We thus only need to estimate the value of the intensity at the
integration points. If we want to estimate it for any other point, we do it as
an interpolation between the three points that form the corners of the
triangle that contains the point.

###  3.2 Observation Process

With the definition of the model for the actual abundance in any area, we can
add a variety of observation process models, appropriate to the data at hand,
fit the model to the data and estimate this abundance layer.

####  3.2.1 Point Counts

Data is often collected in the form of counts of individuals of a species, for
example, the Breeding Bird Survey in North America (Pardieck et al., 2015). If
we assume that observers observe at a site $\mathbf{s}$ for a time $t$, and
the probability that they observe each individual in the site over the period
is $p$, then the number of individuals counted follows a Poisson distribution.
That is;

$Pr(N(\mathbf{s},t)=r)=\frac{\eta(\mathbf{s},t)^{r}e^{-\eta(\mathbf{s},t)}}{r!},$
(8)

where

$\eta(\mathbf{s},t)=pt\int_{\mathbf{s}}\lambda(\mathbf{s})d\mathbf{s}.$ (9)

If we assume that the site is small, then we can treat the intensity and
covariates as constant over the whole site, so:

$\eta(\mathbf{s},t)\approx ptA(\mathbf{s})\bar{\lambda}(\mathbf{s}).$ (10)

where $A(\mathbf{s})$ is the area of $\mathbf{s}$. In practice, we are
unlikely to know $A(\mathbf{s})$ or $p$. But both (as well as $t$) are part of
the observation model, and can be seen as different elements of the
observation effort, so we collapse these into a single parameter,
$E(\mathbf{s})=ptA(\mathbf{s})$. Hence, we only have a single parameter to be
estimated. With repeated counts at a site, we can estimate how these vary over
time, introducing extra variance as an overdispersion parameter, i.e.

$\theta(\mathbf{s},t)\approx
ptA(\mathbf{s})\bar{\lambda}(\mathbf{s})\epsilon(\mathbf{s})$ (11)

where $\varepsilon(\mathbf{s})$ follows some distribution, e.g. a Gamma
distribution (or equivalently, a $\chi^{2}$ distribution) leads to a negative
binomial.

In practice, equation (11) is modeled on the log scale:

$\log{\theta(\mathbf{s},t)}\approx\log(E)+\bar{\eta}(\mathbf{s})+\varepsilon(\mathbf{s}),$
(12)

where $\varepsilon(\mathbf{s})$ could be modeled as a Gaussian noise term. If
$E$ varies between sites in a known way, this can be added to the model. For
example, if the observation time, $t$, varies, this can be included as an
offset, i.e. $\log(E)=\alpha+\log(t)$.

####  3.2.2 Occupancy Models

Data from surveys of small sites may not be of the form of a count, but simply
of presence/absence. With a single visit to a site, the likelihood is simply,
from equation (8),
$Pr(N(\mathbf{s},t)>0)=1-Pr(N(\mathbf{s},t)=0)=1-e^{\eta(\mathbf{s},t)}$.
Working on the log scale for $\eta(\mathbf{s},t)$, this becomes:

$Pr(N(\mathbf{s},t)>0)=p(\mathbf{s},t)=1-e^{-e^{\eta}(\mathbf{s},t))},$ (13)

which is the inverse of the cloglog link function (e.g. McCullagh and Nelder,
1989, §1.2.4), i.e.

$\log(-\log(1-Pr(N(\mathbf{s},t)>0)))=\eta(\mathbf{s},t).$ (14)

If we have multiple visits to the site, we can extend this from a Bernoulli to
Binomial model, i.e. likelihood that the species is observed $n$ times in $N$
visits is:

$Pr(n=r|N,p(\mathbf{s},t))\propto p(\mathbf{s},t)^{n}(1-p^{N-n}),$ (15)

where $p=Pr(N(\mathbf{s},t)>0)$, from equation (6). Note that this differs
from a classical occupancy model (MacKenzie et al., 2002b), as we assume that
occupancy in the area can change (although a full occupancy model could also
be developed).

####  3.2.3 Point observations and presence-only models

Data such as from eBird (Sullivan et al., 2014), where presences alone are
recorded, can be treated as a point process (e.g. Warton and Shepherd, 2010).
Models for point process data can be reduced to a generalized linear model
(Møller and Waagepetersen, 2003), and in the context of distribution models
have been shown to be equivalent to the model assumed by MaxEnt (Renner and
Warton, 2013; Fithian and Hastie, 2012). If all individuals are observed, then
the observation model is trivial: the model is just the process model. But
this is seldom the case, and instead, we use a thinned point process model: if
each individual is observed with probability $q(\mathbf{s})$, the intensity of
observation is $\phi(\mathbf{s})=q(\mathbf{s})\lambda(s)$. If we observe $M$
points, at locations $\mathbf{s}_{1},...,\mathbf{s}_{M}$, the log-likelihood
is:

$l(\beta|{\bf
s})\propto\sum_{i=1}^{M}\phi(\mathbf{s})-\int_{A}\phi(a)da-\ln(M!).$ (16)

(Møller and Waagepetersen, 2003).

The integral cannot usually be calculated, so instead, we resort to a
numerical approximation, summing the intensity over discrete plates. These
become weighted sums of quadrature points (the corners of the planes). If the
quadrature points are placed at points $r\in R$, the likelihood becomes:

$l(\beta|{\bf
s})\approx\sum_{i=1}^{M+R}w_{i}[z_{w,i}\ln{\phi(\mathbf{s})}-\phi(\mathbf{s})]$
(17)

where $w_{i}$ is a quadrature weight, and $z_{w,i}=w_{i}^{-1}$ if the point is
a data point or 0 otherwise. This is, by inspection, a Poisson likelihood,
albeit with a non-integer response. We can thus use a standard GLM
formulation, with a log link, for the model (Renner et al., 2015). On the log
scale, the model for the intensity of the observations is
$\eta(\mathbf{s})+\log(q(\mathbf{s}))$, so observation bias can be added as
additive terms (e.g. Fithian and Hastie, 2012), and made a function of
covariates (including, potentially, a residual spatial term).

####  3.2.4 Regional lists

We can also combine equations (9) and (6) to model list of species from larger
regions (e.g. nature reserves or states). It would be reasonable to assume
that the data is certain, so the likelihood is simply equation (6).
Unfortunately, this does not easily fit within the model fitting approach used
here. If the area is small enough, it could be approximated by a constant
surface. If the area is larger, an alternative approach is needed, e.g.
section 3.2.5.

####  3.2.5 Expert Range Maps

In principle, we can adapt the model used in section 3.2.4 for expert range
maps, but the problem of variation within the area cannot be ignored. We thus
take another approach, including the range map as a covariate in the process
model (equation 2). The simplest way to do this is to use it as a binary
variable: in/out. But the range map may be wrong at a local scale, for
example, if the scale of the range map is too coarse, or if the species has
expanded beyond its range since the map was drawn. Thus, we use a more
flexible model:

$\rho(s)=-\gamma\min(0,||s-r||),$ (18)

and we can assume $\gamma>0$. This can be done by placing informative priors
on the parameters. With sufficient data, we would expect to overcome the
priors, but for more data-poor species (limited data on species), they should
influence the predicted distribution. This approach is similar to that taken
by Merow et al. (2017), except that they assume a more complex functional
form, but also fix the effect of distance from range edge, rather than
estimating it. Thus, the cost of using a less flexible model is offset
(hopefully) by the advantage of allowing the range maps to be less accurate.
Of course, informative prior distributions can be used to increase the
importance of the range map.

###  3.3 Putting it all together: Using the framework to model biodiversity
observation error

To provide a conceptual link of the point process model framework in data
integration, we describe an integrated model that accounts for biodiversity
observation errors. We assume that biodiversity data contains observation
errors caused by imperfect detection, uneven sampling effort and reporting
bias. These sources of observation errors are assumed to thin the actual
intensity $\lambda(\mathbf{s})$ (Dorazio, 2014; Sicacha-Parada et al., 2021;
Adjei et al., 2023) in a hierarchical way (Adjei et al., 2023). An observer
first samples a given location to collect data (often at the most accessible
locations) and collects data at these sampled locations. However, it is
possible that the observer may be unable to detect all the species of
interest, and in reporting the detected species, choose to ignore some of the
detected species.

Let $b(\textbf{s})$ be the sampling probability, $\psi(\textbf{s})$ be the
detection probability and $\gamma(\textbf{s})$ be the reporting probability.
Following the data generating model defined in equation (1), the likelihood of
the integrated model in this illustration is defined as:

$\begin{split}L(Y_{d}|b(\textbf{s}),\psi(\textbf{s}),\gamma(\textbf{s}))=p(\lambda(\textbf{s}))\prod_{d=1}^{M}Pr(Y_{d}|\lambda(\textbf{s}),b(\textbf{s}),\psi(\textbf{s}),\gamma(\textbf{s})).\end{split}$
(19)

None of the observation models described in section 3.2.1 to 3.2.5 \- fitted
with one dataset - can provide unique estimates of the model parameters in the
equation (19). However, it is possible to integrate data from multiple types
and sampling protocols to ensure that the model parameters are identifiable.

If the sources of biases are assumed to be conditionally independent on the
datasets available, then the model likelihood in equation (19) now becomes:

$\begin{split}L(Y_{d}|b(\textbf{s}),\psi(\textbf{s}),\gamma(\textbf{s}))&=p(\lambda(\textbf{s}))\prod_{d_{1}=1}^{M_{1}}Pr(Y_{d_{1}}|\lambda(\textbf{s}),b(\textbf{s}))\\\
&\prod_{d_{2}=1}^{M_{2}}Pr(Y_{d_{2}}|\lambda(\textbf{s}),\psi(\textbf{s}))\prod_{d_{3}=1}^{M_{3}}Pr(Y_{d_{3}}|\lambda(\textbf{s}),\gamma(\textbf{s})),\end{split}$
(20)

where $M_{1}$ is the number of datasets that inform the model on the
estimation of the sampling probability, $M_{2}$ is the number of datasets that
inform the model on the estimation of the detection probability and $M_{3}$ is
the number of datasets that inform the model on the estimation of the
reporting probability.

For illustration, we assume that we have available occupancy ($Y_{d_{1}}$),
count data ($Y_{d_{2}}$) and presence-only ($Y_{d_{3}}$) to integrate
together. We further, assume that the presence-only data provide information
on the sampling process of observers, the occupancy data provides information
on the detection process and count data provides information on the reporting
process. Following the definition of the observation models in equations (8),
(13) and (16), the log-likelihood becomes:

$\begin{split}l(b(\textbf{s}),\psi(\textbf{s}),\gamma(\textbf{s})|Y_{d_{1}},Y_{d_{2}},Y_{d_{3}})&\propto\ln(p(\lambda(\textbf{s})))+\sum_{i=1}^{n_{d_{1}}}\ln\bigg{(}1-e^{\int_{\textbf{s}}\psi(\textbf{s})\lambda(\textbf{s})d\textbf{s}}\bigg{)}\\\
&+\sum_{i=1}^{n_{d_{2}}}\ln\bigg{(}1-e^{\int_{\textbf{s}}\gamma(\textbf{s})\lambda(\textbf{s})d\textbf{s}}\bigg{)}-\sum_{i=1}^{n_{d_{2}}}e^{\int_{\textbf{s}}\gamma(\textbf{s})\lambda(s)ds}-n_{d_{2}}\ln(r!)\\\
&+\sum_{i=1}^{n_{d_{3}}}b(\textbf{s})\lambda(\textbf{s})-\int_{A}b(a)\lambda(a)da-\ln(n_{d_{3}}!),\end{split}$
(21)

where $n_{d_{1}}$ is the number of occurrence data points, $n_{d_{2}}$ is the
number of count data points and $n_{d_{3}}$ is the number of location points
in the presence-only data. This log-likelihood can either be maximised in the
frequentist approach or simulated from in the Bayesian framework. It must be
stated again that the integral in equation (21) needs to be approximated with
numerical methods. Moreover, the information used to estimate the underlying
state $\lambda(\textbf{s})$ comes from the three datasets integrated together,
and this sharing of information produces precise estimates of
$\lambda(\textbf{s})$. This has been noted by Dorazio (2014); Fletcher et al.
(2016); Koshkina et al. (2017) as the advantage of data integration.

##  4 Case study

Now, we provide an application of the point process framework to fit data on
freshwater fish in Norwegian lakes, to illustrate how the model can be
tailored to fit the data sources we have available, and the ecological
questions we wish to answer.

In this case study, we consider citizen science observations of four fish
species (pike, Esox lucius; perch, Perca fluviatilis; brown trout, Salmo
trutta, and Actic char, Salvelinus alpinus) available through Global
Biodiversity Information Facility (GBIF; https://www.gbif.org/). The citizen
science observations are presence-only occurrence records. Additionally, we
also have access to a presence/absence data on freshwater fish in Norway from
the Fish Status Survey of Nordic Lakes (Tammi and Finstad, 2019).

The two data sources each have their advantages and disadvantages. The citizen
science observations are opportunistic, and are likely to be spatially biased,
since no sampling plan has been followed. However, we have a larger number of
observations, particularly more recent observations in this dataset. The
survey data, on the other hand, is from 1996, and so does not reflect the
current distribution of these species. It does, however, present a less
spatially biased sample than the citizen science observations. The presence
points from both data sets can be seen in Figure 1. To take advantage of the
strengths of both these data sets, we intend to integrate both in order to
make predictions on the abundance of these freshwater fish in Norway.

Figure 1: The observed presences of the different fish species in each of the
two data sets. Locations are slightly jittered to avoid overplotting.

For the presence/absence survey data, we use a Bernoulli distribution (as
described in section 3.2.2), where the presence probability depends on some
covariates $x(s)$, along with a spatial field $\xi_{j}(s)$ for species
$j\in\\{\textit{Salmo trutta, Perca fluviatilis, Esox lucius, Salvelinus
alpinus}\\}$,

$\displaystyle Y_{PA,j}(s_{i})$
$\displaystyle\sim\text{Bernoulli}(p_{PA,j}(s_{i}))$ (22)
$\displaystyle\text{cloglog}(p_{PA,j}(s_{i}))$
$\displaystyle=\alpha_{PA,j}+x(s_{i})^{T}\beta_{j}+\xi_{j}(s_{i}).$ (23)

The presence-only data is fitted with a Poisson point process model (as
described in section 3.2.3), where the intensity depends on the same
covariates $x(s)$ and the same spatial field $\xi_{j}(s)$, plus an additional
spatial field $\xi_{\text{bias}}(s)$ that is unique to the citizen science
data, but shared across all fish species:

$\displaystyle Y_{PO,j}(s_{i})$
$\displaystyle\sim\text{Poisson}(e^{\eta_{PO,j}(s_{i})})$ (24)
$\displaystyle\eta_{PO,j}(s_{i})$
$\displaystyle=\alpha_{PO,j}+x(s)^{T}\beta_{j}+\xi_{j}(s_{i})+\xi_{\text{bias}}(s_{i}).$
(25)

In this integrated distribution model, $\xi_{j}(s)$ describes the spatial
autocorrelation shared between the datasets. $\xi_{\text{bias}}(s)$ is a bias
field that captures the extra spatial variation in the presence-only records,
and therefore captures the variation that is likely due to human observation
biases rather than the distribution of the fish. The model was fitted using
the PointedSDMs (Mostert and O’Hara, 2023a) R package. Code and data for
fitting the model, along with a more detailed description, can be found in the
Supporting Information at
https://github.com/emmaSkarstein/IntegratedLakefish/.

Comparing the observations in the two data sets shown in figure 1, we see that
the citizen science data shows certain geographical presences that the survey
data does not. That is to be expected, since the survey data is from 1996, and
there is reason to believe that the citizen science data shows a more updated
view of the species distributions. We present the predictions of the mean log
intensity fields for each species from the model in Figure 2.

Figure 2: The mean log intensity of the estimated spatial fields for each of
the fish species.

##  5 Discussion

We present a statistical formulation of data integration using a point process
framework, including a review of the current literature on data integration,
an overview of the data types that are commonly integrated together, and a
detailed description of the process model using the point process formulation
and various observation models. As an illustration, we have applied the
framework to predict the distributions of four fish species in Norwegian
lakes, illustrating how this framework allows us to smoothly incorporate
information about the observation processes for each of the data sources in a
way that respects the data collection, and also makes sense ecologically. In
this paper, we focus on the point-process methodology estimated via a joint-
likelihood approach, though other methods to construct ISDMs are also
possible. Methods such as data pooling and combining independent models may
fall within the definition of data integration, but are not recommended for
most case scenarios (Fletcher Jr et al., 2019). While other popular methods
may be justified, including using an informed prior in a Bayesian setup, using
one dataset as a covariate for the other, or by connecting datasets together
through a shared covariance matrix, designed to capture patterns present
throughout the datasets (Pacifici et al., 2017). Simulation studies comparing
the strengths and weaknesses of each of these methods have been completed
(Pacifici et al., 2017; Ahmad Suhaimi et al., 2021); they found that the
joint-likelihood method worked well when spatial bias in the presence only
data was low or statistical components to account the bias were included in
the model.

Integrated distribution models fitted with the point process framework has
emerged as an ideal approach to utilising the strengths of various data types
to estimate state variables in a single analysis better (Dorazio, 2014;
Fletcher et al., 2016; Koshkina et al., 2017). Another interesting feature of
data integration is the adjustment of the biodiversity biases by using other
data types. Data from well designed surveys would be an ideal complement to
unstructured data (Koshkina et al., 2017; Fletcher et al., 2016; Pacifici et
al., 2017; Giraud et al., 2016). Although data from structured surveys are
less numerous, a good design will not be spatially biased, and this can be
used to estimate and correct for the bias in the unstructured data (Simmonds
et al., 2020). Thus, even in this model-based approach to modelling
distributions, a design-based approach will be important if we are to utilise
the data that is being collected fully.

Any state variable of interest can be seen as derived quantities of point
patterns (Kéry and Royle, 2015). With this, various data types can be
integrated together through their basic derived quantity (point patterns)
through the point process framework. In contrast to other papers summarised in
Table 1 that focus on making inferences and predictions with integrated
models, we focused on the statistical properties of the data in the integrated
model. Here, we have described the data in terms of its statistical
properties, but a more practically useful typology would describe them in
terms of the methods used to collect it. Although we may see line transect
data and eDNA data both as counts, there are of course large differences in
how they are collected, and thus how we should treat them. For example, the
correlation between read number (i.e. count) of a sequence in eDNA data and
abundance of the species is an area of active research (e.g. Skelton et al.,
). A lot of work will therefore be needed to develop specific models for
different data sources.

Our framework will only be helpful if it can be implemented in software that
can be used by analysts, who may often be ecologists, potentially without
formal training in statistics. The approach we have taken is closely linked to
models that can be fitted with R-INLA (Rue et al., 2009a) or inlabru Bachl et
al. (2019), but for the case study we have used the R-package PointedSDMs
(Mostert and O’Hara, 2023a), which builds on inlabru but makes the model
fitting a bit more convenient and user friendly. Additionally, the full
potential of ISDMs will only be met if we can build a large-scale workflow
that designs pipelines to move data from species occurrence databases (notably
GBIF) to the modelling framework. An early implementation of such a workflow
has been discussed and implemented through the R-package intSDM (Mostert et
al., 2022), which has been built to obtain species’ occurrence and
environmental data from popular repositories, standardise the data into a
coherent framework, and obtain estimates from the model.

We have outlined a framework for integrated distribution models and shown that
it can be used for real problems. Within this framework, there is a lot of
flexibility, and thus many developments to be explored. One of the key issues
is how different datasets may support each other: surveys (for example) will
still be important, because they provide reliable data that citizen science
data can be calibrated to. But will this mean we should re-evaluate how
surveys should be designed? This is the sort of issue that we will have to
grapple with if we want to better understand and monitor biodiversity in the
future.

## Data and code availability

Instructions for accessing the data and code used in the case study are
available in the Supporting Information at
https://github.com/emmaSkarstein/IntegratedLakefish/. All data is publicly
available through GBIF.

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