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Dynamic Population Models with Temporal Preferential
Sampling to Infer Phenology
Michael R. Schwob, Mevin B. Hooten, Travis McDevitt-Galles
Abstract
To study population dynamics, ecologists and wildlife biologists use relative abundance data,
which are often subject to temporal preferential sampling. Temporal preferential sampling
occurs when sampling effort varies across time. To account for preferential sampling, we
specify a Bayesian hierarchical abundance model that considers the dependence between
observation times and the ecological process of interest. The proposed model improves
abundance estimates during periods of infrequent observation and accounts for temporal
preferential sampling in discrete time. Additionally, our model facilitates posterior inference
for population growth rates and mechanistic phenometrics. We apply our model to analyze
both simulated data and mosquito count data collected by the National Ecological Observatory
Network. In the second case study, we characterize the population growth rate and abundance
of several mosquito species in the Aedes genus.
Key words: abundance, ignorability, mechanistic modeling, phenometrics, time series
1 Introduction
The timing of ecological events is a fundamental question in the field of phenology (Forrest
& Miller-Rushing 2010). Conventional approaches to studying phenology involve summary
1
arXiv:2212.05180v2 [stat.ME] 13 Dec 2022
statistics, or phenometrics, that capture the timing of cyclical biological events (Demidova
et al. 2021). Phenometrics for population dynamics may include the first or last date of
positive population growth (Anderson et al. 2013, Bewick et al. 2016). By contrast, ecologists
commonly use hierarchical temporal models to learn about how and why population size
changes over time (Potts & Elith 2006). We seek to make both phenological and ecological
inference on population dynamics using modern hierarchical Bayesian models that allow for
embedded mechanistic and observation processes. We specify the model in such a way to
extract relevant phenometrics and associated uncertainties as derived quantities based on our
defined mechanistic process (i.e., the latent population growth rate).
Ecologists working in temperate environments tend to design field studies that align with
the growing season (Pau et al. 2011). The resulting data sets are collected using preferential
sampling designs, which may induce nonignorable missingness (Zachmann et al. 2022). In the
presence of temporal preferential sampling, inference during periods of infrequent observation
will be biased (Monteiro et al. 2019). We adapt preferential sampling approaches for the
discrete temporal domain to improve posterior inference on the abundance process during
periods of infrequent observation.
Remedies for spatial preferential sampling have been more popular than those for temporal
preferential sampling in ecology (Watson 2021). However, phenological studies may be subject
to temporal preferential sampling, including frequent and short gaps (Zhang et al. 2009),
fewer and longer gaps (Roleˇcek et al. 2007), and asynchronous and inconsistent gaps in data
collection (Cole et al. 2012). Records of population dynamics have infrequent and long gaps
in observation because the data are collected seasonally. However, the existing remedies to
temporal preferential sampling focus solely on frequent and short gaps in observations (Karcher
et al. 2016, Monteiro et al. 2019, 2020). Furthermore, existing temporal preferential sampling
remedies are only applicable in the continuous-time domain, whereas most abundance studies
collect counts for discrete-time population dynamics. In studies that use hidden Markov
models, temporal preferential sampling is referred to as state-dependent missingness. Several
2
recent algorithms accommodate state-dependent missingness in observations of homogeneous
Markov chains (Speekenbrink & Visser 2021, Hoskovec et al. 2022); however, the existence of
seasonal variation in population dynamics precludes the use of these algorithms. Therefore,
existing solutions to temporal preferential sampling and state-dependent missingness are not
applicable to phenological or abundance data sets. We account for discrete-time temporal
preferential sampling in the form of a probit regression that is suitable for many phenological
and abundance data sets that exhibit seasonal variation.
We propose a hierarchical Bayesian abundance model in section 2 that characterizes
population dynamics using relative abundance data from irregular surveys. We specify
the process model to characterize the population growth rate and infer phenometrics, and
we modify the data model to account for temporal preferential sampling. In section 3, we
demonstrate the proposed model along with its non-preferential variant on simulated data and
a mosquito case study with data collected by the National Ecological Observatory Network
(NEON). The data collection associated with the NEON study involved temporal preferential
sampling. We seek to determine if posterior inference resulting from a preferential and a
non-preferential model differs under the NEON sampling protocol. We also show that the
bias varies when interpolating population dynamics during infrequently observed periods. In
section 4, we conclude with a discussion of mechanistic modeling and temporal preferential
sampling for phenology and population dynamics.
2 Methods
We let
T
denote the number of days throughout the study period and
J
denote the number of
species in the study. We also let
y
(
t
)
≡
(
y1
(
t
)
, ..., yJ
(
t
))
0
and
λ
(
t
)
≡
(
λ1
(
t
)
, ..., λJ
(
t
))
0
denote
the observed and true abundance of all species on day
t∈ {
1
, ..., T }
and aim to recover
the latent abundance process
Λ≡
(
λ
(1)
, ..., λ
(
T
)) using temporal state-space modeling.
The specification of the observation distribution [
y
(
t
)
|λ
(
t
)] and the process distribution
3
[
λ
(
t
)
|λ
(
t−
1)] are fundamental to obtaining interpretable inference in temporal state-space
models. We denote the data model as
y
(
t
) =
H
(
λ
(
t
);
Θd
;
(
t
)), where
H
is an observation
function that maps the latent process
λ
(
t
) to the data with noise process
(
t
). Similarly, we
can write the latent process evolution model as
λ
(
t
) =
M
(
λ
(
t−
1);
Θe
;
η
(
t
)), where
M
is the
evolution operator and
η
(
t
) is a noise process (Wikle & Hooten 2010). In general, the noise
processes may be Gaussian or non-Gaussian. We assume a first-order Markov process for the
evolution model, which is common in ecological studies of population dynamics (Usher 1979).
2.1 The Process Model
We model abundance dynamically and focus on specifications involving mechanistic population
models. A common mechanistic approach to modeling population dynamics is to specify the
latent process evolution as
M(λ(t−1); Θe;η(t)) = exp (Alog λ(t−1) + Bx(t) + η(t)) ,(1)
where
Θe≡ {A,B}
,
η
(
t
)
∼N
(
0, σ2I
), and
x
(
t
)
≡
(
x1
(
t
)
, ..., xp
(
t
))
0
are species-independent
covariates of interest at time
t
(Ross et al. 2015). In the multivariate setting, we let
exp
(
·
)
and
log
(
·
) be element-wise operations over vectors. We let
A≡diag
(
α1, ..., αJ
) and
I
be
the
J×J
identity matrix. Species
j
experiences density dependence when
αj<
1, positive
association with population density when
αj>
1, and density independence when
αj
= 1
(Dennis et al. 2006). We let
B≡
(
β1, ..., βJ
)
0
, where
βj≡
(
βj,1, ..., βj,p
) are the time-invariant
coefficients of covariates xfor species j. The process model in (1) can be written as
log λ(t)∼NAlog λ(t−1) + Bx(t), σ2I,(2)
where the species-level parameters βjconvey how the abundance of species jresponds to a
change in environment and resources.
Most phenology studies focus on species with population dynamics that experience a high
4
degree of seasonality. We modify the process model in (2) for species with population dynamics
that are intrinsically tied to the environment and, therefore, seasonal. The more seasonality
influences population dynamics, the more the dynamic is centered on environmental trends.
We account for the seasonality of population dynamics by anchoring the log-abundance using
the environmental trend Bx(t) such that
log λ(t)−Bx(t)∼NA(log λ(t−1) −Bx(t−1)), σ2I,(3)
where
|αj|<
1 because the population is assumed to be density dependent (Royama 2012);
this provides a trend-stationary and autoregressive stochastic model for population dynamics
that are heavily influenced by the environment (Shumway & Stoffer 2000). Thus, our process
model for log-abundance can be expressed as
log λ(t)∼NBx(t)−ABx(t−1) + Alog λ(t−1), σ2I,(4)
for
t
= 2
, ..., T
. When
t
= 1, we assume
log λ
(1)
∼N
(
µ1, σ2
1I
). Although trend-stationary
stochastic processes are commonly found in econometric studies, they are underutilized in
population dynamic studies for species with seasonal abundance processes. When applicable,
accounting for trend stationarity improves the stability and performance of MCMC algorithms
(McCulloch & Tsay 1994).
The Gompertz form of density dependence has performed well in various population
dynamic studies (Dennis et al. 2006, Knape & de Valpine 2012). Our process model implies
stochastic Gompertz population growth, which can be written as the heterogeneous Malthusian
growth function
λ
(
t
) = (
I
+
diag
(
g
(
t
)))
λ
(
t−
1)
,
where
g
(
t
)
≡
(
g1
(
t
)
, ..., gJ
(
t
))
0
. The species-
level per capita growth rate on day tcan be written as
gj(t) = exp βj(x(t)−αjx(t−1))λj(t−1)αj−1ej,t −1,(5)
5
where
j,t ∼N
(0
, σ2
). We consider the posterior median of the log-normal distributed random
variable
ej,t
, which assumes an absolute loss function; we assume this loss function because
it is less sensitive to skewness resulting from exponentiating the log-abundance process.
Additionally, the posterior median is more efficient and robust than the posterior mean for
log-normally distributed random variables with
σ >
0
.
546, which is true in our case studies
(Zellner 1971, Rao & D’Cunha 2016). Thus, our derived phenometric is
ψ≡min t∈ {1, ..., T }: exp βj(x(t)−αjx(t−1))λj(t−1)αj−1−1>0,(6)
which is the first day that the posterior median of
gj
(
t
) is positive. We use posterior densities
to quantify uncertainty for the derived phenometric
ψ
, which is not possible using conventional
phenological approaches. We present the derivation of gj(t) and its median in Appendix A.
2.2 The Data Model
For the observation function H, we specify
yj(t)∼Pois(λj(t)·ω(t)),(7)
where
ω
(
t
) captures heterogeneity in sampling effort. The conditional Poisson data model is
a common choice in temporal abundance modeling (Hooten & Hefley 2019). For invertebrate
trapping studies, we specify
ω
(
t
) =
w
(
t
)
·h
(
t
), where
w
(
t
) is the proportion of the trap
sampled and
h
(
t
) is the sampling duration of the trap standardized to a single day length in
hours. If sampling effort was constant across traps, we set ω(t) = 1,∀t.
To account for temporal preferential sampling, we introduce new notation and indices for
time. We let
N
denote the number of days within the study period and
n
denote the number
of observed days. Further, we let
ti
denote the
i
th day of the study and
˜
tk
denote the
k
th
6
observed day of the study. We define Tas the set of observed days, such that
T ≡ {˜
t1,˜
t2, ..., ˜
tn}⊆{t1, t2, ..., tN}.(8)
In the event that every day is observed, temporal preferential sampling is not present and
T={t1, t2, ..., tN}. We let τ≡(τ1, τ2, ..., τN)0be a binary vector of length Nsuch that
τi=
1, ti∈ T
0, ti6∈ T
.(9)
Finally, we let
λ
(
ti
) denote the abundance processes for all species at time
ti
. The abundance
processes exist throughout the entire study regardless of which days were observed.
We express the joint distribution of the underlying process
Λ
, observation indicators
τ
,
and the observations of the underlying process Y≡(y(1), ..., y(n)) as
[Λ,τ,Y]=[Y|τ,Λ][τ|Λ][Λ],(10)
where “[
·|·
]” represents the conditional probability density or distribution function (Gelfand
& Smith 1990). We condition
τ
on
Λ
because the observation of days depends on the process
of interest. The observation time model is usually specified as a discretized continuous-time
inhomogeneous Poisson process, which is ideal for continuous time series with many short
gaps of missing data (Karcher et al. 2016, Monteiro et al. 2019). However, it is more common
that there are fewer, longer gaps of missing data in discrete-time population dynamic studies.
Therefore, we specify [
τ|Λ
] for discrete-time abundance counts that experience long, seasonal
gaps in data collection. We consider the following distribution of observed days conditioned
on the process of interest:
[τi|θ,λ(ti),˜
λ]≡Bern(p(ti)) (11)
7
with the probit link function
p(ti)=Φθ0+θ1
1
{λ(ti)01≥˜
λ},(12)
where Φ(
µ
) is the standard normal cumulative distribution function evaluated at
µ
and
θ≡
(
θ0, θ1
)
0
are probit regression coefficients. The indicator function
1
{·}
evaluates to 1 when
the number of observed mosquitoes exceeds the threshold
˜
λ
, and 0 otherwise. We specify a
prior for the threshold as ˜
λ∼Ga(αλ, βλ).
Probit regression is a common regression technique for analyzing binary ecological data
(Trexler & Travis 1993, Hooten & Hefley 2019). We consider probit regression because we can
induce dependence of the binary vector
τ
on the abundance process using data augmentation
(Albert & Chib 1993). Thus, we introduce the latent variables
z
(
ti
) and model [
τi|θ,λ
(
ti
)
,˜
λ
]
using
τi=
1, z(ti)>0
0, z(ti)≤0
,(13)
z(ti)∼Nθ0+θ1
1
{λ(ti)01≥˜
λ},1,(14)
which implies the same model as (11)-(12); however, the specification in (13)-(14) results in
conjugate updates for both the latent variables
z
(
ti
) and
θ
assuming a multivariate normal
prior on
θ
. The observations
Y
are no longer assumed to be conditionally independent
because we explicitly define the data model to account for the dependence of
τ
on
Λ
. To
complete the data model, we specify
yj
(
˜
tk
)
∼Pois
(
λj
(
˜
tk
)
·ω
(
˜
tk
)) based on the observations
in set T.
Tests for temporal preferential sampling are underdeveloped (Watson 2021). However,
θ1
in (12) provides a convenient model-based quantity that can be used to assess discrete-
time temporal preferential sampling. When sampling mechanisms favor periods of higher
8
abundance, we expect
θ1>
0. Thus, we may test for the presence of temporal preferential
sampling using posterior inference on θ1.
The full Bayesian hierarchical model is provided in Appendix B. We implemented the
model in Julia to reduce computation time (Bezanson et al. 2017); alternatively, it could be
fit using standard MCMC software. We used conventional, conjugate priors for
θ
,
µβ
,
Σβ
,
α
,
σ2, and ˜
λto facilitate computation.
3 Application
We analyzed two abundance data sets subject to temporal preferential sampling: A simulated
data set and a terrestrial site observed by the National Ecological Observatory Network. To
understand the effect of temporal preferential sampling on posterior inference, we compared
the proposed preferential model to its non-preferential variant, which does not contain the
preferential component in (13)-(14). The discrepancy between the resulting inference indicates
the effects of neglecting the presence of temporal preferential sampling. We implemented
both case studies using MCMC algorithms and 500,000 iterations with a burn-in of 200,000
iterations.
3.1 Simulated Abundance
We simulated a log-abundance process for a single species using the process model in (4). With
J= 1, we let A= 0.98 be a scalar, B= (β1, β2) = (0.1,0.3) be a row vector, and σ2= 0.03.
For the environmental covariates
x
, we used an intercept and growing degree day (GDD)
estimates at Harvard Forest in Massachusetts, USA, from 2014-2016. We quantified GDD
using estimated temperature from the Oregon State PRISM Climate data with a baseline
temperature of 10
◦
C and a cutoff temperature of 30
◦
C (Neteler et al. 2011, PRISM Climate
Group 2019). We simulated the sequence of fully observed counts
y∗≡
(
y
(
t1
)
, ..., y
(
tN
))
0
using a Poisson distribution with abundance as the intensity; we let
ω
(
˜
tk
) = 1 because we
9
assume homogeneous sampling effort across traps.
We considered three common sampling mechanisms found in abundance studies to de-
termine which counts in
y∗
were retained for analysis: Random sampling, where each day
in the study was observed with equiprobability (0.3); Preferential switch sampling, where
each day that abundance exceeded a threshold (15) was observed with equiprobability (0.3)
and each day that abundance did not exceed that threshold was unobserved; and logistic
sampling, where the probability of observing day
ti
was
logit−1
(
−
10 + 0
.
4
·λ
(
ti
)). We fit the
preferential and non-preferential models to the retained observed counts to determine the
effect of ignoring temporal preferential sampling on posterior inference. Observations and
posterior estimates for abundance under the three scenarios are shown in Figure 1.
To quantify the error for each model in each scenario, we considered root-mean-squared
error (RMSE) for the abundance process
{λ
(
ti
)
}
, which is conventional in ecological model
comparisons for species richness (Mouillot & Lepretre 1999). Table 2 contains the RMSE for
the preferential and non-preferential models for the three sampling scenarios.
With random sampling, we did not expect the competing models to provide significantly
different results because the sampling mechanism did not engage in temporal preferential
sampling. Empirically, the addition of the preferential data model in (13)-(14) did not
improve posterior inference because the missingness in the data was ignorable. The posterior
estimates for the parameters governing the log-abundance process were recovered by both
models, as depicted in Figure 3. The 95% posterior credible interval for
θ1
was (-0.53,0.46),
which implies that temporal preferential sampling is not detected under the preferential
model.
Under preferential switch sampling, abundance was overestimated using the non-preferential
model during periods of infrequent observation, whereas the preferential model resulted in
more accurate estimates during such periods. Additionally, the RMSE for the preferential
model was lower than that for the non-preferential model. Both models adequately recovered
σ2
. The posterior median for
α
under the preferential model was closer to the true value
10
than under the non-preferential model. Finally, the 95% posterior credible intervals for
β1
and β2contained the true value under only the preferential model.
For the logistic sampling scenario, the preferential model outperformed the non-preferential
model because the observed days were highly concentrated during summers when abundance
was notably high. During periods of infrequent observation, the estimated abundance from the
preferential model was much closer to the truth than the estimates from the non-preferential
model. Both models accurately recovered
σ2
. Additionally, the posterior medians for the
remaining parameters were close to the true values under the preferential model. The true
values for
α
and
β2
were recovered under the non-preferential model; however, the 95%
credible interval for β1did not contain the true value.
In the presence of temporal preferential sampling, the preferential model resulted in
more precise estimates for the abundance process throughout the entire study period, as
evidenced by the narrower 95% credible intervals in Figure 1. The abundance process was
recovered accurately and precisely under the preferential model during periods of infrequent
observation, whereas the non-preferential model tended to overestimate abundance during
these periods. Additionally, the true value for each parameter in the log-abundance process
was captured under the preferential model, whereas
β1
and
β2
were poorly estimated under
the non-preferential model for both preferential sampling scenarios. Thus, we obtained
inadequate posterior inference via the non-preferential model in the presence of temporal
preferential sampling.
3.2 Mosquito Abundance
We analyzed mosquito count data collected by NEON. The NEON mosquito monitoring
program followed a standardized sampling protocol conducted across a broad geographical
range and aimed to collect adult mosquito activity data for phenological modeling (Hoekman
et al. 2016, National Ecological Observatory Network 2021). NEON collected data throughout
each year with two distinct sampling approaches based on levels of mosquito activity: An
11
“off-season” with infrequent sampling during periods of low abundance, and a “field season,”
where sampling intensity increased concurrently with abundance. Because the decision to
start and stop consistent data collection was motivated by the abundance process, NEON
engaged in temporal preferential sampling. We sought to determine if posterior inference
resulting from the preferential and non-preferential model differed under the NEON sampling
protocol.
We expressed the environmental covariates as a convolution to account for latency in the
reproductive and larval stages of the mosquito life cycle:
x(t) = ZG(φ, t, ˜τ)˜
x(˜τ)d˜τ , (15)
where
˜
x
(
˜τ
) represents the original continuous-time environmental variables,
x
(
t
) is a smoothed
version of
˜
x
(
˜τ
) resulting from the convolution, and
G
is a diagonal matrix of basis function
gl
(
φ, t, ˜τ
) for
l
= 1
, ..., p
associated with each covariate. The convolution in (15) is generalized
to account for temporal lag in population dynamics (Aukema et al. 2008, Chi & Zhu 2008).
We constructed the convolution in the form of a backward moving average to account for the
lagged effect of environmental conditions on the mosquito life cycle. If the basis functions are
specified as
gl(φ, t, ˜τ) =
1
φ, t −φ < ˜τ < t
0,otherwise
,(16)
then the population growth rate is a function of the average environmental conditions over the
past
φ
days. The parameter
φ
can be specified using expert knowledge of mosquito biology
or estimated in the model when enough data exist to identify it in addition to other model
parameters. We used a 14-day backward moving average of GDD at each site throughout the
study period because GDD has been reported to influence mosquito dynamics (Field et al.
2019); the 14-day time frame was chosen to account for the stages in the mosquito life cycle
preceding sexual maturity (Crans 2004).
12
We restricted analysis to the Aedes genus, in which several species are vectors for dengue
fever, yellow fever, and the Zika virus (Suwanmanee & Luplertlop 2017). Of the NEON
terrestrial sites where the Aedes genus was present, we analyzed counts at the University
of Notre Dame Environmental Research Center (UNDE) because the counts were obtained
daily during the field season. UNDE observed counts for three species in the Aedes genus:
Aedes canadensis,Aedes excrucians, and Aedes punctor. We analyzed counts from 2016 to
2019 in this study.
We considered an absolute loss function when providing inference because it is robust
to skewness that may result when exponentiating the log-abundance process. Therefore, we
reported posterior medians for the abundance process of Aedes punctor in Figure 4; the
analogous plots for Aedes canadensis and Aedes excrucians can be found in Supplemental
Figures 1 and 2. We also reported posterior interquartile ranges (IQR) to accompany the
posterior point estimates. As expected, some observed counts exceed the posterior IQR due
to the inflated counts resulting from extended trapping duration. Similarly, we computed
population growth rates by considering the posterior median of
g
(
t
) in (5), which depends
on
α
and
B
. The estimated growth rates are also provided with each species’ abundance
process.
We inferred the first day in which a species has positive population growth by computing
the derived phenometric
ψ
. The posterior estimates of
ψ
are depicted in Supplemental
Figure 3 for 2017-2019 alongside the same phenometric derived using conventional summary
statistics (Jones & Daehler 2018). Many phenological approaches cannot provide uncertainty
quantification, whereas our mechanistic hierarchical model formally allowed us to quantify
uncertainty with ψvia posterior densities.
The posterior inference provided by the preferential and non-preferential models matched
throughout the entire study period for each species. Neither model overestimated the relatively
low mosquito abundance during the winter, a behavior evident in the non-preferential model
in the simulated case study. The resulting inference from both models was similar because
13
NEON consistently obtained enough zero counts preceding the annual rise and following
the annual decline in mosquito populations; fitting these zero-counts to the non-preferential
model resulted in adequate estimates for
α
and
B
. Although NEON explicitly engaged in
temporal preferential sampling, their sampling mechanism mitigated the effect that temporal
preferential sampling had on posterior inference by sampling infrequently during the off-season
until a mosquito is detected.
We also analyzed a subset of the UNDE data, where observations were removed for
days in which zero mosquitoes were detected; we refer to this subset as the second scenario,
which resembles many abundance studies that only experience positive counts (Turchin 2013).
The posterior inference under the competing models significantly differed under the second
scenario, as seen in Figure 5. The first observed counts of Aedes punctor were positive in
2016, 2017, and 2019. Consequently, abundance was overestimated via the non-preferential
model during the springs of 2016, 2017, and 2019. However, the resulting inference from the
preferential model was similar to that provided in Figure 4 and better coincides with the
removed data and scientific knowledge of the abundance process of mosquitoes; this indicates
that more robust inference is obtained under the preferential model for population dynamics
at the rise and decline of abundance.
In the second scenario, the posterior inference for the phenometric
ψ
substantially changed
for the non-preferential model. As depicted in Figure 6, the non-preferential model more
heavily weighted the beginning of the year, whereas the preferential model provided similar
inference as in the first scenario (see Supplemental Figure 3). The inference provided by the
preferential model aligns with scientific knowledge of mosquito population dynamics under
both scenarios. Posterior estimates for
α
and
B
were inadequate under the non-preferential
model in the second scenario, which significantly affected the ability to obtain robust estimates
for the phenometric.
14
4 Discussion
We augmented a hierarchical abundance model to account for temporal preferential sampling
and to obtain phenometrics. Temporal preferential sampling is common in abundance studies,
and our extension of the model improved abundance estimates during periods of infrequent
observation using the readily available data
τ
. Consequently, the derived phenometrics are
greatly improved when accounting for preferential sampling. We derived population growth
rates explicitly and inferred phenometrics by relating the latent abundance process to the
stochastic Gompertz population growth function in a way that accounts for preferential
sampling. The phenometrics can be compared to existing phenological studies to confirm
or present new results. In addition, our model formally allowed us to quantify uncertainty
associated with ψ, whereas most phenological approaches lack uncertainty quantification in
the relevant phenometrics.
Our first case study revealed the advantages of accounting for temporal preferential
sampling in the abundance model under three sampling scenarios. Under preferential switch
sampling and logistic sampling, fitting the preferential model resulted in more precision when
predicting unobserved abundances. In data sets with fewer, longer gaps of missing data, the
proposed preferential model will outperform its non-preferential variant.
In the mosquito case study, we found that enough zero-abundance counts were recorded
in the UNDE data for the non-preferential model to estimate all model parameters and,
consequently, abundance. Thus, the NEON sampling mechanism did not affect posterior
inference on abundance. However, the non-preferential model overestimated the growth rate
during periods of infrequent observation when analyzing the subset of NEON data that only
included positive counts. The second scenario is realistic because many abundance studies
observe strictly positive counts. For the second scenario, the posterior inference under the
non-preferential model was imprecise during the mosquito off-season, indicating that temporal
preferential sampling should be accounted for when working with abundance data that are
mostly positive.
15
During abundance study design, ecologists and wildlife biologists may apply the proposed
temporal preferential sampling methods to obtain more accurate inference with less data.
Researchers may allocate less resources to collect data at the start and end of the off-season
and more resources during the field season when studying species with population dynamics
that depend on the environment. Additionally, smaller ecological surveys may find the
preferential model useful if they lack the resources to collect data consistently during an
off-season.
Temporal preferential sampling has been accounted for exclusively in the continuous-time
domain. Applications include monitoring air pollution (Shaddick & Zidek 2014), clinical
trials (Monteiro et al. 2019), and neuroimaging (Poeppel 2003). By contrast, we account
for preferential sampling in discrete time to align with data from many abundance studies.
When explicitly accounting for the dependence of observations on the process of interest, we
obtain more robust inference during periods of infrequent observation.
The field of temporal preferential sampling is rapidly developing, and there is much work
to be done regarding its theory and methodology. Additionally, the implementation of models
that account for preferential sampling may be improved. For example, the development of
algorithms that induce conjugacy for the abundance process may increase the efficiency of
MCMC implementations (e.g., Bradley et al. 2018). Recursive and distributed computing
approaches would increase the computational efficiency of temporal preferential sampling
models (e.g., Hooten et al. 2021). If we seek to fit a hidden Markov model while accounting for
preferential sampling, then the use of particle Gibbs filters or forward-backward algorithms
may significantly improve computational efficiency (Beal et al. 2001, Tripuraneni et al. 2015).
Finally, interspecies interactions can be accounted in a multivariate version of our model to
improve species conservation, network analysis, and community ecology.
16
Acknowledgements
Data were collected by the National Ecological Observatory Network and the Oregon State
PRISM Climate Group. The National Ecological Observatory Network is a program sponsored
by the National Science Foundation (NSF) and operated under cooperative agreement by
Battelle. This material is based in part upon work supported by the NSF through the NEON
Program. Any use of trade, firm, or product names is for descriptive purposes only and does
not imply endorsement by the U.S. Government.
Funding
This research was supported by the NSF Graduate Research Fellowship Program.
Disclosure Statement
The authors report that there are no competing interests to declare.
Data Availability
The datasets analyzed during this study are publicly available from the National Ecological Ob-
servatory Network (DOI: dp1.10043.001) and PRISM climate group (https://prism.oregonstate.edu/).
Code Availability
We will provide a GitHub link containing the code for publication.
17
Appendix A
We solve for the species-level per capita growth rate
gj
(
t
) in (5) and the posterior median of
gj(t) in (6). From (4), we can write the abundance of species jon day tas
λj(t) = exp βj(x(t)−αjx(t−1)) + αjlog λj(t−1) + j,t(17)
= exp βj(x(t)−αjx(t−1))λj(t−1)αjej,t ,(18)
where
j,t ∼N
(0
, σ2
). If we consider the heterogeneous Malthusian growth function
λj
(
t
) =
(1 + gj(t))λj(t−1), we can solve for gj(t) to obtain
gj(t) = λj(t)
λj(t−1) −1 (19)
=exp βj(x(t)−αjx(t−1))λj(t−1)αjej,t
λj(t−1) −1 (20)
= exp βj(x(t)−αjx(t−1))λj(t−1)αj−1ej,t −1.(21)
Finally,
exp
(
j,t
) is log-normally distributed with parameters
µ
= 0 and
σ2
. Thus,
M
(
j,t
) =
exp(µ) = 1 and
M(gj(t)) = exp βj(x(t)−αjx(t−1))λj(t−1)αj−1−1,(22)
which is the expression in (6). We let M(·) denote the median of the random variable.
18
Appendix B
The full Bayesian hierarchical model that we used in our case studies is
yj(˜
tk)∼Pois(λj(˜
tk)·ω(˜
tk)),(23)
τi=
1, z(ti)>0
0, z(ti)≤0
,(24)
z(ti)∼Nθ0+θ1
1
{λ(ti)01≥˜
λ},1,(25)
log λ(ti)∼NBx(ti)−ABx(ti−1) + Alog λ(ti−1), σ2I, i = 2, ..., N, (26)
log λ(t1)∼N(µ1, σ2
1I),(27)
β0
j∼N(µβ,Σβ),(28)
µβ∼N(µ0,Σ0),(29)
Σβ∼IW(Ψ, ν),(30)
α∼N(µα, σ2
αI),(31)
θ∼N(µθ,Σθ),(32)
σ2∼IG(q, r),(33)
˜
λ∼Ga(αλ, βλ),(34)
for
k
= 1
, ..., n
,
j
= 1
, ..., J
, and
i
= 1
, ..., N
with the exception of (26). We let
A≡diag
(
α
),
B≡(β1, ..., βJ)0, and βj≡(βj,1, ..., βj,p).
19
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Figure 1: Abundance estimates under three sampling mechanisms. In (a)-(c), the true
abundance and the observed abundance are depicted as black lines and dots, respectively.
The red lines and polygons depict the posterior means and 95% credible intervals under the
non-preferential model. The blue lines and polygons depict the posterior means and 95%
credible intervals under the preferential model.
Table 2: Root-mean-squared errors for the two competing models under three sampling
scenarios.
Sampling Mechanism Non-preferential Model Preferential Model
Random 4.393 4.332
Preferential Switch 12.127 8.016
Logistic 8.711 2.578
26
Figure 3: Posterior estimates under the preferential and non-preferential model for random,
preferential switch, and logistic sampling. True values are indicated by horizontal black lines.
(a) α(b) σ2
(c) β1(d) β2
27
Figure 4: Top: Posterior IQR and median for the abundance process of Aedes punctor at
the UNDE site from 2016-2019. Black dots denote observed abundances, lines denote the
posterior medians, and polygons denote the posterior IQRs. Bottom: Derived posterior
means for the growth rate of Aedes punctor at the UNDE site from 2016-2019. The blue
lines and polygons correspond to inference provided by the preferential model, whereas the
red corresponds to the non-preferential model.
28
Figure 5: Top: Posterior IQR and median for the abundance process of Aedes punctor at
the UNDE site when removing zero-count observations. Full black dots denote observed
abundances, black circles denote removed zero-count observations, lines denote the posterior
medians, and polygons denote the posterior IQRs. Bottom: Derived posterior means for the
growth rate of Aedes punctor at the UNDE site when removing zero-count observations.
29
Figure 6: Posterior distributions for the first day in which each species experiences population
growth,
ψ
, for 2017-2019 when removing zero-count observations. Black lines depict the
phenometric derived via summary statistics.
30
