library(RPresence)
#> RPresence_2.13.47Program PRESENCE is a stand-alone program compiled to run under Windows(c) operating system. It can be run under MacOS or Linux using Wine. The program can be obtained from:
[https://www.mbr-pwrc.usgs.gov/software/bin/presence.zip]
RPresence is a package developed for R which allows users to build and run occupancy models from within R using simple function calls. The inner workings of PRESENCE are incorporated in RPresence, so there is no need to download PRESENCE to use RPresence. To install the RPresence package, type the following into the R console:
\(N\) sites are surveyed over time where the intent is to establish the presence or absence of a species. The sites may constitute a naturally occurring sampling unit such as a discrete pond or patch of vegetation; a monitoring station; or a quadrat chosen from a predefined area of interest. The occupancy state of sites may change over time, however during the study there are periods when it is reasonable to assume that, for all sites, no changes are occurring, (e.g, within a single breeding season for migratory birds). The study therefore comprises of \(K\) primary sampling periods (seasons), between which changes in the occupancy state of sites may occur. Within each season, investigators use an appropriate technique to detect the species at \(k_j\) surveys of each site.
The species may or may not be detected during a survey and is not falsely detected when absent (except false-positive model). The resulting detection history for each site may be expressed as \(N\) vectors of 1’s and 0’s, indicating detection and nondetection of the species respectively. We denote the detection history for site \(i\) at survey \(j\) as \(h_{i,j}\).
MacKenzie et al. (2002)1 present a model for estimating the site occupancy probability (or PAO) for a target species, in situations where the species is not guaranteed to be detected even when present at a site. Let \(\psi\) be the probability a site is occupied and \(p_j\) be the probability of detecting the species in the \(j^{th}\) survey, given it is present at the site. They use a probabilistic argument to describe the observed detection history for a site over a series of surveys. For example the probability of observing the history \(1001\) (denoting the species was detected in the first and fourth surveys of the site) is:
\[\psi \times p_1(1-p_2)(1-p_3)p_4\]
The probability of never detecting the species at a site (\(0000\)) would therefore be,
\[\psi \times (1-p_1)(1-p_2)(1-p_3)(1-p_4) + (1-\psi)\],
which represents the fact that either the species was there, but was never detected, or the species was genuinely absent from the site (\(1-\psi\)). By combining these probabilistic statements for all N sites, maximum likelihood estimates of the model parameters can be obtained.
The model framework of MacKenzie et al. (2002) is flexible enough to allow for missing observations: occasions when sites were not surveyed. Missing observations may result by design (it is not logistically possible to always sample all sites), or by accident (a technicians vehicle may breakdown enroute). In effect, a missing observation supplies no information about the detection or nondetection of the species, which is exactly how the model treats such values.
The model also enables parameters to be function of covariates. For example, occupancy probability may be a function of habitat, while detection probability is a function of environmental conditions such as air temperature. The model therefore allows relationships between occupancy state and site characteristics to be investigated. Covariates are entered into the model by way of the logistic model (or logit link).
A key assumption of the single season model is that all parameters are constant across sites. Failure of this creates heterogeneity. Unmodeled heterogeneity in detection probabilities will cause occupancy to be underestimated. If there is unmodelled heterogeneity in occupancy probabilities, then it is believed that the estimates will represent an average level of occupancy, provided detection probabilities are not directly related to the probability of occupancy.
Another major assumption of the MacKenzie et al. (2002) model (Single-season) is that the occupancy state of the sites does not change for the duration of the surveying. Situations where this may be violated, for instance, would be for species with large home ranges, where the species may temporarily be absent from the site during the surveying. If this process of temporary absence from the site may be viewed as a random process, (e.g., the species tosses a coin to decide whether it will be present at the site today), then this assumption may be relaxed. However, this will alter the interpretation of the model parameters (“occupancy” should be interpreted as “use” and “detection” as “in the site and detected”). More systematic mechanisms for temporary absences may be more problematic and create unknown biases. Although, users are reminded that the model assumes closure of the sites at the species level, not at the individual level, so there may be some movement of individuals to/from sites without overly affecting the model.
Currently, there are many types of models can be fit to detection/nondetection data within Program PRESENCE.
In all models, estimated parameters (\(\psi\), p, \(\gamma\), \(\epsilon\), …) may be modelled as functions of site-specific, or site-survey-specific covariates.
Program PRESENCE allows you to build models where the model parameters ( occupancy, detection, colonization, extinction,…) depend on various covariate information collected in conjunction with the detection data. To allow flexibility, models are defined by using a “design-matrix”. Although the mention of “design-matrices” may cause anxiety in non-statistician biologists, they can be thought of as a list of simple mathematical equations relating quantities to estimate (beta parameters) to real parameters of interest (real parameters, \(\psi, p, \gamma, \epsilon\)).
RPresence allows you to build models by specifying a formula for each parameter rather than design matrices. These formula are specified like formula for general linear models (GLM) in R. Although this greatly simplifies model specification, it is a good idea to know how the formulae are translated into design matrices and equations.
In the design-matrix, the beta-parameters are represented by the columns, and the real parameters are represented by the rows. As an example, let’s look at building a simple single-season model where occupancy is constant across all sites and detection is constant across all sites and surveys. For occupancy, we need to create an equation relating the real parameter, \(psi\), with one or more beta parameters. For constant occupancy, we’ll use the equation,
\[psi = beta_1\]
Then, we re-write the equation using all beta’s associated with psi with a zero or one coefficient. OK, this is nonsense in this case, but we get:
\[psi = 1 \times beta_1\]
Next, we create a matrix with only the coefficients of the \(beta\)’s:
\[ \left(\begin{array}{cc} 1 \end{array}\right) \]
and that’s our design matrix for psi - a \(1 \times 1\) matrix! The formula notation used in RPresence would simply be:
For detection, we have \(T\) detection parameters (one for each of \(T\) surveys) and we’ll use the equations,
\[p1 = beta_2 \\ p2 = beta_3 \\ p3 = beta_4 \\ p4 = beta_5\]
Since PRESENCE estimates the beta parameters, we’ll have different estimates of \(p1\) through \(p4\) since each beta will be different. To convert these equations into a design matrix, re-write each equation using all beta’s for p:
\[p1 = 1 \times beta_2 + 0 \times beta_3 + 0 \times beta_4 + 0 \times beta_5 \\ p2 = 0 \times beta_2 + 1 \times beta_3 + 0 \times beta_4 + 0 \times beta_5 \\ p3 = 0 \times beta_2 + 0 \times beta_3 + 1 \times beta_4 + 0 \times beta_5 \\ p4 = 0 \times beta_2 + 0 \times beta_3 + 0 \times beta_4 + 1 \times beta_5\]
Then, create a matrix containing the coeffecients of the betas:
\[ \left(\begin{array}{cc} 1 & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\\ \end{array}\right) \]
The formula notation used in RPresence for detection (p) would simply be:
where SURVEY is a categorical covariate indicating the survey number. (More on covariates later.)
Why is there a “-1” in the formula? There are many ways to specify how the beta parameters relate to the “real” parameters (\(\psi, p\)). In this case, we used a separate beta for each of the \(p\)’s. If you’re used to doing GLM’s, you probably are used to seeing one of the \(beta\)’s as an intercept, and the other 3 \(beta\)’s as “effects” (difference between the intercept and each of the other \(p\)’s). Using an intercept, you would specify the equation,
and get the design matrix:
\[ \left(\begin{array}{cc} 1 & 0 & 0 & 0\\ 1 & 1 & 0 & 0\\ 1 & 0 & 1 & 0\\ 1 & 0 & 0 & 1\\ \end{array}\right) \]
The resulting real parameters will be the same, but the \(beta\)’s would be different since \(p_2, p_3\) and \(p_4\) would be computed using two \(beta\)’s (intercept \(beta\) and effect \(beta\)) instead of one. So, the “-1” means “don’t use an intercept”.
Replication of surveys may be temporal or spatial in nature. That is, the same sites can be sampled at different points in time, or a site may be broken into multiple locations and sampled once. A common example of spatial replication is transect sampling, where observers walk along a trail and attempt to detect sign of the species (eg., bird singing, mammal scat, visual spotting) at specific points along the transect. A common problem with spatial replication is that an individual of a wide-ranging species (eg., tigers) may roam among several spatial replicate stops. So, if one point along a transect has an individual nearby, detection probability will be higher than points which are not near an individual. This leads to a form of detection heterogeneity where there is autocorrelation in sample detection probabilities. The same sort of autocorrelation can occur in aural surveys where a 10-minute detection survey is broken into intervals of 3,3 and 4 minute detection intervals. The “Single-season correlated detections” model addresses this problem. Surveys are conducted along trails such that when a species is found at one sample, nearby samples have a much higher (or lower) probability of the species being present than those farther away. This can be accounted for by adding two new parameters, \(\theta\), \(\theta\)‘. \(\theta\) is the probability that the species is present locally, given the species was not present in the previous sample, but globally present at some point along the transect. \(\theta\)’ is the probability that a species is present locally, given it was locally present at the previous sample and globally present along the transect. An example detection history might be:
01011
Here, the species was detected at the 2nd, 4th and last samples (segments of transect line), but not detected at the 1st and 3rd samples. The probability of this history would be represented by:
\[ \psi \left\{ \\ (1-\pi) [(1-\theta_1)\theta_2+\theta_1(1-p_1)\theta'_2] + \\ \pi [(1-\theta'_1)\theta_2+\theta'_1(1-p_1)\theta'_2] \right\} \\ p_2[(1-\theta'_3)\theta_4+ \theta'_3(1-p_3)\theta'_4] p_4\theta'_5p_5 \]
In this model, we’ve added a new parameter, \(\pi\), which is the proportion of sites which are locally occupied before the first survey. If all transects begin at a boundary, such that it is impossible for the species to be locally present before the first survey (eg., start at a lake, or road), then this parameter may be fixed to zero. Another option for the \(\pi\) parameter is to assume that it is equal to some sort of average of \(\theta\) and \(\theta\)’. This can be achieved in PRESENCE by fixing \(\pi\) to “EQ”, which is an abbreviation for “equalibrium”, meaning the value of local occupancy obtained if a random survey is chosen from all surveys.
When this model is chosen, the \(\theta\) parameters will appear in the design matrix window in the same tab as the occupancy (\(\psi\)) parameter. This model is described in Hines (2010).
Note: this model allows the theta’s to be segment-specific. This may be possible if the detection parameters (p) are constant among segments (or a function of a survey-specific covariate). This model is not identifiable (no single solution for parameters) if both \(\theta\)’s and detection probabilities \(p\)’s are allowed to be segment-specific.
This model is a special case of new multi-season Spatial Dependence model where there is only 1 season. So, if you have single-season data (number of surveys per season = total number of surveys), just run the multi-season correlated detections model.
The False-positive detection model (Miller et. al. 2011) extends the single season model by relaxing the assumption that detections do not occur when the species is not present. This type of model might be used when one suspects that it is likely that detections are not being recorded perfectly. For example, a detection for species A might actually be a similar species, but recorded for species A. A common example of the need for this model is the case where some observers are untrained or inexperienced and others are “experts”. A requirement for this model is that some observations must “known”, so the probability of a false detection can be estimated. For example, in aural surveys, where some detections are based on auditory information (where there is a possibility of mistaking the call of another species for the target species) and some detections are based on visual information (where the observer is sure the species is present). Both types of data are required in order to obtain estimates for this model.
Parameters:
Data input for this model is the same as the standard single-season model, except “assured” detections are coded with “2” and non-assured detections are coded as “1”. For example, the detection history, “12001”, would indicate:
Instead of having surveys with combined uncertain and certain detections, the sampling scheme may involve two methods of detection, where one method yields certain detections (eg., DNA sample, visual detection), and one method yields uncertain detections (eg., auditory detection). The False-positive model can still be used for these data. The input data would have to be entered such that each survey would occupy two columns in the input file: one column for the uncertain detections and another for the certain detections. Uncertain detections are coded as ‘1’ and certain detections are coded as ‘2’ (as above). The ‘trick’ needed to make this model work is to fix all of the ‘b’ parameters (since the probability of a detection being ‘certain’ is known for all detections). For the columns which correspond to uncertain surveys, the \(b\) parameter should be fixed to 0, and the \(b\) parameter for columns which correspond to certain detections should be fixed to 1. The probability of false detections, \(p^{10}\), should be fixed to 0 for columns which correspond to ‘certain’ detection surveys. For example, if there are 3 surveys with ‘certain’ and ‘uncertain’ detections in each survey, the input detection-history data would contain 6 columns. Columns 1,3,5 would contain uncertain detections in each of the 3 surveys (0=not detected, 1=detected). Columns 2,4,6 would contain certain detections (0=not detected, 2=detected). As far as PRESENCE is concerned, there are 6 surveys, but you will know that there are really only 3 surveys, with two methods per survey.
This model relaxes closure assumption such that a site may locally colonize and go locally extinct once during the surveys (ie., delayed arrival and/or early departure). It is descirbed by Kendall et. al 2013. Input data is of the same format as the single-season model.
Parameters:
The Royle/Nichols heterogeneity model (Royle and Nichols, 2003) estimates population size from temporally replicated presence/absence data (from point-counts or other types of surveys) at a number of sample sites. This model assumes that heterogeneity in detection probability among sites is due to heterogeneity in abundance (more individuals lead to higher probability of detecting the species at the site). Input data for this model are the presence/absence (1/0) of the species at each survey at each sample site. Parameters estimated under the assumption of a Poisson distribution:
The species-level probability of detection can be computed using \(r\) and \(\lambda\) as:
\(p_{i,j} = 1 - (1-r_{i,j})\lambda_i\)
The two-species model (MacKenzie et al., 2004) 16 extends the single season model in another way by allowing the computation of occupancy parameters of two species along with conditional probabilities of occupancy when the other species is present or detected. Parameters:
Input data for this model is in the same form as the single-species, single-season model except that the first half of the detection history records are assumed to be species A, and the second half of the records are assumed to be species B. So, if there are 60 sites, the input would consist of 120 detection history records. Records 1-60 would be the site-detection history records for sites 1-60, species A, and records 61-120 would be the site-detection history records for sites 1-60, species B.
Alternatively, data could be coded without doubling the number of sites. In this case, detections would consist of the following codes:
Since two of the parameters in the default parameterization are not probabilities bounded by the interval (0 - 1), numerical problems can arise. For example, if \(\psi^A\) is zero, \(\phi\) would be undefined since it’s computation involves dividing by \(psi^A\). An alternate parameterization was developed, using conditional probabilities as parameters, which is more numerically stable. The parameters are:
Using this parameterization, quantities from the other parametrization can be computed. (eg.,
\(\psi^B = \psi^A\psi^{BA}+(1-\psi^A)\psi^{Ba}\) \(\phi = \frac{\psi^A\psi^{BA}}{\psi^A\psi^B}\)
The Single-season,two-species model (MacKenzie et al., 2004) is extended to allow mis-identification of the species (see Chambert et. al. 2018). Additional parameters:
Additional data: In order to obtain estimates for this model, additional data are required, in addition to the standard two-species detection data. This additional data consists of a code for each site and survey, indicating which species were confirmed at the site-survey. So, these data should be in the same format as the detection data. For example:
| survey1 | survey2 | survey3 | survey4 | survey5 | survey6 | |
|---|---|---|---|---|---|---|
| site1 | 2 | 2 | 0 | 2 | 0 | 2 |
| site2 | 0 | 0 | 0 | 3 | 0 | 2 |
| site3 | 2 | 0 | 0 | 2 | 0 | 0 |
| site4 | 3 | 2 | 2 | 2 | 0 | 2 |
| survey1 | survey2 | survey3 | survey4 | survey5 | survey6 | |
|---|---|---|---|---|---|---|
| site1 | 0 | 2 | 0 | 2 | 0 | 2 |
| site2 | 0 | 0 | 0 | 0 | 0 | 0 |
| site3 | 1 | 0 | 0 | 0 | 0 | 0 |
| site4 | 3 | 0 | 2 | 0 | 0 | 0 |
Explanation:
Input to PRESENCE: The additional confirmation data is entered into PRESENCE as the first “survey covariate”, and should be named, “conf”.
The multi-method model (Nichols et al. 2008) extends the single season model by allowing detection probabilities to be different for different methods of observation. This allows the computation of an additional parameter, \(\theta\) which is the probability that individuals are available for detection at the site, given that they are present.
Parameters:
Data input for this model is as follows:
| s1m1 | s1m2 | s2m1 | s2m2 | s3m1 | s3m2 | |
|---|---|---|---|---|---|---|
| site1 | 0 | 0 | 0 | 0 | 1 | 1 |
| site2 | 0 | 0 | 0 | 0 | 1 | 0 |
| site3 | 1 | 1 | 1 | 0 | 1 | 0 |
where s1m1 is survey1, method1, s1m2 is survey1, method2, s2m1 is survey2, method1,…etc.
In this input dataset, there were detections of the species in both methods for the 3rd survey for site1. For site2, only survey3, method1 had a detection, and for site3, there were detections for both methods of survey1, only method1 for surveys 2 and 3.
In the multi-state model (MacKenzie et al., 2009), two kinds of detections are recorded. Detections where only adults observed are recorded as ‘1’ in the data, and detections of known breeding adults (adults seen with young) are recorded as ‘2’ in the data. This allows the computation of an additional parameter, \(R\) which is the probability that adults breed, given that they are present. Parameters:
Input data for this model is in the same form as the single-species, single-season model except that breeding status (‘1’=adults only, or ‘2’=adults and young) is recorded instead of presence (‘1’).
A more general multi-state model allows for more than just two occupied states. In this case, input consists of:
Parameters under this parameterization:
In order for the multi-state model to be identifiable, constraints must be made on the parameters.
This model is a special case of the multi-season-multi-state model and can be run in PRESENCE as a multi-season-multi-state model with only one season.
The Royle N-Mixture model (Royle, 2004) estimates population size from temporally replicated point-count data at a number of sample sites. The variation in these point-counts provides information about the distribution of site-specific population size (N). Input data for this model are the counts of the number of individuals observed at each survey (instead of the usual ‘1’ or ‘0’) at each sample site. Parameters estimated under the assumption of a Poisson distribution:
\(\lambda\) - population density (per site), \(r\) - probability of detection (per individual of the species) per survey
The multiple season model (MacKenzie et al., 2003) extends the single season model by introducing two additional parameters, \(\epsilon_t\) and \(\gamma_t\). These parameters are, respectively, the probability a species becomes locally extinct or colonizes a site between seasons \(t\) and \(t+1\).
Parameters:
For example, if the detection history 101 000 was observed at a site (denoting the species was detected in the first and third survey of the site in the first season; not detected otherwise), the probability of this occurring could be expressed as;
\(\psi \text{ } p_{1,1}(1-p_{1,2})p_{1,3} [ (1-\epsilon_1) (1-p_{2,1}) (1-p_{2,2}) (1-p_{2,3}) + \epsilon_1 ]\).
This represents the fact that after the first season, the species may have not gone locally extinct \((1-\epsilon_1)\), but was undetected in the 3 surveys in season 2 \(((1-_{p2,1})(1-p_{2,2})(1-p_{2,3}))\) or the species did go locally extinct (\(\epsilon_1\)) between the first and second seasons.
The model may also be reparameterized in terms of \(\psi_t\) and \(\epsilon_t\); or \(\psi_t\) and \(\gamma_t\), as in some situations this may be a more meaningful parameterization (in terms of overall occupancy) than in terms of the underlying processes. As in the single season model, parameters may be functions of covariates using the logit link.
Note this model does not allow for a so-called “rescue effect”, where the local extinction of a colony is “rescued” by the re-colonization of the site before the unoccupied site can be observed (i.e., the site becomes unoccupied then re-occupied all between a single season). Such an effect is sometimes included in metapopulation models, however while a rescue effect is biologically plausible, it can not be estimated (without some potentially unrealistic strict assumptions) from the type of data we are considering here, nor from the type of data often collected in metapopulation studies. The main argument for not including a rescue effect is: why should the rescue of the colony be limited to an arbitrary single event, when possibly there may be a number of opportunities between two seasons for the rescue to occur? To reduce the possibility of having unobserved changes in the occupancy state of sites, the sampling scheme should be designed to reflect the appropriate time scale of the system under study.
The initial parameterizaton uses a single initial occupancy paramter, \(k-1\) extinction parameters (assuming \(k\) seasons), \(k-1\) colonization parameters, and \(T\) detection parameters (assuming \(T\) surveys). Once these parameters are estimated, other quantities of interest can be computed. Occupancy in other seasons can be computed as:
Sometimes, it is desirable to model seasonal occupancy as a function of some covariates. Since seasonal occupancy is computed from \(\epsilon_i\) and \(\gamma_i\), this cannot be done with these parameters. An alternate parameterization in PRESENCE uses \(k\) occupancy parameters, \(k-1\) extinction parameters, and \(T\) detection parameters. The \(k-1\) colonization parameters are then computed from the seasonal \(\psi\)’s and \(\epsilon\)’s by solving the above equations for \(\gamma_i\).
\(\gamma_1 = \frac{\psi_2 - \psi_{initial}(1-\epsilon_1)}{1-\psi_{initial}}\)
By selecting this parameterization, it’s now possible to build a model where seasonal occupancy (\(\psi_i\)) is a function of a seasonal covariate. Similarly, we could have estimated the colonization parameters and computed the extinction parameters. This parameterization is sometimes useful if the above parameterization fails to converge on reasonable estimates.
Finally, PRESENCE can model extinction and colonization in such a way that the proportion that go locally extinct is the same as the proportion that don’t colonize (\(\epsilon=1-\gamma\)).
This model relaxes closure assumption such that a site may locally colonize and go locally extinct once during the surveys (ie., delayed arrival and/or early departure). It is descirbed by Kendall et. al 2013 and is a simple extension of the single-season-staggered-entry model. Input data is of the same format as the multi-season model.
Parameters:
The False-positive detection model (Miller et. al. (2013)) extends the single season-false-positive model to allow change in occupancy between seasons. Input format is similar to the single-season-false-positive model.
Parameters:
This model is an extension of the Single-season Spatial Dependence model. The parameters for the single-season Spatial Dependence model are replicated for each season, with additional parameters for colonization (\(\gamma\)) and extinction (\(\epsilon\)) for each interval between seasons.
With the multi-season model, it may be of interest to model extinction or colonization between seasons \(t-1\) and \(t\) as a function of site occupancy for neighboring sites in season \(t-1\). If you’re willing to assume that all sites are ‘neighbors’ of each other (i.e., if the individual organisms can move anywhere within the study area), then this model can be run by simply using “psi1” as a covariate name in the design matrix. When PRESENCE goes to compute colonization or extinction, it will compute the average “conditional” occupancy of neighboring sites in season \(t-1\) and use that value as a covariate in computing colonization/extinction in season \(t\). By specifying “upsi” instead of “psi1”, PRESENCE will compute average “unconditional” occupancy of neighboring sites instead of “conditional” occupancy.
Note: “Conditional occupancy” in this case, means conditional on the detection history for the season, not on the entire detection history.
In the case where it is desired to define neighbors specifically for each site, PRESENCE can read a file which defines the neighbors of each site. This file should be a text file containing 1’s and 0’s where 1 denotes that site \(s\) is a neighbor of site \(r\). The format of the file is rows of contigous 1’s and 0’s, one row for each focal site. Each row should contain a string of \(k\) characters (where \(k\) = number of sites in the study area). For example if there were 10 sites, the neighbor file might look like this:
| Neighbors | |
|---|---|
| 1 | 0100000000 |
| 2 | 1000000000 |
| 3 | 0001100000 |
| 4 | 0010100000 |
| 5 | 0011000000 |
| 6 | 0000001111 |
| 7 | 0000010111 |
| 8 | 0000011011 |
| 9 | 0000011101 |
| 10 | 0000011110 |
In this example, sites 1 and 2 are neighbors of each other (row 1, col 2=1 and row 2, col 1=1), sites 3,4,5 are neighbors, and sites 6-10 are neighbors. So, if colonization between seasons \(t-1\) and \(t\) is modeled as a function of average neighborhood occupancy at time \(t-1\), colonization for site 2 in season \(t\) will be a function of the average occupancy for sites 1 and 2 in season \(t-1\). In some cases, sites may not all be of the same size or habitat quality. In these cases, it would be preferable to weight the average occupancy of neighboring sites by a value indicating the size/quality of each neighboring site. For example, if all sites except site 5 are nearly the same size, but site 5 is twice as large as the other sites, we would want the occupancy for site 5 to have more influence on colonization/extinction of other sites than the occupancy of other sites. So, the neighbor file would be:
| site | Neighbors | Weight |
|---|---|---|
| 1 | 0100000000 | 1 |
| 2 | 1000000000 | 1 |
| 3 | 0001100000 | 1 |
| 4 | 0010100000 | 1 |
| 5 | 0011000000 | 2 |
| 6 | 0000001111 | 1 |
| 7 | 0000010111 | 1 |
| 8 | 0000011011 | 1 |
| 9 | 0000011101 | 1 |
| 10 | 0000011110 | 1 |
In this example, sites 3,4,5 are neighbors, but the average occupancy used in the computation of colonization for those sites will be computed as:
\(logit(\gamma_{site4,t}) = \beta_0 + \beta_1 \times X_{site4,t-1}\)
where \(X\) is the auto-logistic covariate and is computed as:
\(X_{site4,t-1} = \frac{W_{site3} \psi_{site3,t-1} + W_{site5} \psi_{site5,t-1}}{Wsite3 + Wsite5}\)
Using the weights in the example:
\(X_{site4,t-1} = \frac{1.0 \times \psi_{site3,t-1} + 2.0 \times \psi_{site5,t-1}} {1.0 + 2.0}\)
(note: \(X\) = auto-logistic covariate and \(W\) = weight.)
If PRESENCE finds this auto-logistic covariate name, ‘psi1’, in the design matrix, it will ask for a neighbor text file. If you don’t specifiy a file, PRESENCE will assume all sites are neighbors of each other and all have equal weight (as described initially). If a file is specified, the filename will be saved in the results file (*.pa3), and will be used in all future auto-logistic models. If the file is specified and does not contain a column of weights, all sites will be assumed to have equal weight.
This is an extension of the Single-season multi-state model. After the first season, occupancy status can change between each season. Parameters:
A more general way of parameterizing this model is as follows:
Like the single-season-multi-state model, the parameter \(p^{21}_i\) would be zero since it would be impossible to observe breeding (state=2) if the species is in state=1 (non-breeding).
So, the first parameterization assumes that there are only 3 occupancy states:
The other parameterization can also be used and the relationship between the parameters is:
This parameterization is more general than the first parameterization, but needs constraints like the ones mentioned above in order to be identifiable. The advantage of this parameterization is that it is possible to have more than 3 occupancy states.
This model generates estimates of changes in occupancy state in relation to changes in habitat state.
Input data consists of the following codes:
Parameters:
Note: X = habitat state can be A or B. All parameters except \(\pi\) and \(\psi\) can be indexed by survey (subscript \(i,j\) in multi-season model framework).
This is an extension of the two-species model (MacKenzie et al., 2004) allowing the computation of occupancy, colonization, extinction and detection parameters of two species along with conditional probabilities when the other species is present or detected.
Parameters:
Input data is the same as in the Single-season Two-Species Model.
Instead of repeating each site for each species, the following codes can be used for this model:
If there are any 2’s or 3’s in the data, PRESENCE will assume this form of input.
Below is a list of options available to specify what to compute
and/or print from program PRESENCE. These options can be passed
to the RPresence occMod function as
arguments (see help('occMod')).