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RoyleNichols models for occupancyIn occupancy modelling we are interested in the probability that a species is present at a location, taking account of the possibility that it may be there but not be detected. So we need to jointly estimate probability of occupancy and probability of detection. In many studies we would expect probability of occupancy to be affected by the number of individuals at the location, so it would make sense to include that as a covariate in the analysis. But of course we don't know how many are present  if we did we would not be trying to estimate occupancy. The RoyleNichols model treats the number available for detection at a site as a latent variable which can be estimated from detection/nondetection data.In the simplest model, the number available for detection at site \( i \), \( N_i \), is modelled as being drawn from a Poisson distribution with parameter \( \lambda \). This is the biological model. The observation model assumes that detection of each individual is a Bernoulli trial, individuals being detected with probability \( r \). The data do not show which individuals are detected, just whether the species was detected, and that is recorded if at least one individual is observed.
More on overflow and underflowIn a recent post I described the limits on floating points numbers and the effect on probabilities close to 0 or 1. This can cause errors in manipulating probabilities, in particular in calculating likelihoods, which can become very small. We get around that by working with logs of probabilities. Multiplication is then easy (just add the logs), but addition and subtraction are more difficult. I proposed functions to add probabilities and to calculate 1  p avoiding underflow or overflow.In the course
of applying these ideas to maximum likelihood estimation in the
How long do animal signs remain visible?Great apes make nests to rest and ungulates produce clearlyvisible dung piles: both of these are used to estimate the density of animals. Signs are much easier to count than animals. There are more of them, and they don't run away or hide when you approach. To convert the number or density of signs to number of animals, we need two additional bits of information: how many signs each animal produces per day, and how many days the signs remain visible (the persistence time or decay time).If each animal produces p signs per day and signs remain visible for t days, then sign density will be: S = D x p x t where D is the density of animals. If we can estimate p and t, we can calculate D from S. In this post I want to provide code for estimating the persistence time of signs for retrospective studies of decay.
Probabilities and computer limitationsThere are limits to the size and precision of numbers which computers can handle, and they can cause problems when we try to calculate probabilities, and especially likelihoods. Numbers smaller than 10^{308} become zero and numbers greater than 0.9999999999999999 become 1. Although not important in the real world, these rounding errors mean that there are large ranges of parameter values that have likelihoods of 0 or 1, so that our algorithms to find maximum likelihoods or generate MCMC chains fail to work.The solution is to work with logarithms of probabilities instead of the actual values [log(p) instead of p], eg, we routinely work with loglikelihoods. Multiplying probabilities is then simply a matter of adding up the logs. But sometimes we need to add up probabilities or calculate the complement, 1  p, and we need to do that without falling in the 0 abyss or smashing into the 1 wall.
Arranging ArraysWe often have to deal with multidimensional data, which generally has to be squeezed into a 2D format for tables and spreadsheets and then later reconstituted. Whenever I have to do that, I need to rediscover how to do it. So here's a tutorial for my future self which might be useful for others.Here's a simple example: we have 3 sites, visited 4 times per year for 2 years. This is usually shoehorned into a table with 8 columns for the visits, like this:
These look like counts, but the data could be detection/nondetection (0/1) data, wind speed at each visit, or the name of the observer.
SECR with unpaired camerasWhen animals can be individually identified by their natural markings, as is the case with most cats, camera traps can be used to gather data for density estimation with SECR. Because markings differ on each side of the animal, cameras are usually set in pairs to simultaneously record both sides. I have just been looking at ways to analyse data from unpaired cameras, and was surprised to find that unpaired cameras can be preferable to paired cameras when the number of cameras is limited.This was based on simulations with just one set of parameter values, but it does suggest that projects with limited resources should consider using unpaired cameras, as this allows more locations to be sampled.
Correlated covariatesThere is usually some correlation between the covariates that we want to include in our model, a phenomenon know as "collinearity" or "multicollinearity". This is not a problem if the correlation is small, but you need to be careful if the absolute correlation between two covariates is > 0.7. You can avoid problems by discarding one of a pair of correlated covariates, but that can be a mistake if both have a real biological impact. We'll look at a toy example where both covariates should be included.
Burn... or adapt?Most runs of MCMC chains for Bayesian estimation begin with an adapt phase, when the samplers used to produce the chains are tuned for best performance. I recently did a long run for a complex model, and the output was flagged with "convergence failure". In fact the problem was caused by cutting short the tuning or adapt phase. The default adapt phase for the R package
jagsUI is only 100 iterations, which was woefully
inadequate for my model. We should be specifying big numbers for
n.adapt and small
numbers, or even zero, for n.burnin .
Viewing previous graphs in R for WindowsYou have produced several plots in R, and now you want to go back and take a look at one of those old plots. On a Mac or RStudio, you can do that with the navigation keys. R for Windows has the same functionality but it is not enabled by default. Here's how to enable it.
The wiqid package is now on CRANThe wiqid package has been accepted on CRAN, with the first version being 0.1.0. This means that it can now be installed  together with its dependencies  with install.packages("wiqid"), and update.packages() will also get you the newest version.
Making a habitat mask for SECR in JAGSI have just put together an R package to automate the conversion of habitat masks produced with Murray Efford'ssecr
package to the format needed to run in JAGS or WinBUGS/OpenBUGS.
Updated 28 May 2017: You can install it from Github using the
githubinstall package by opening R and running
Installing R and JAGS on WindowsI've previously posted on compatibility issues encountered when installing R and JAGS on Macs or Ubuntu, but Windows seemed immune to these problems ... until now!The issue arises because current R versions, 3.3.0 or later, are not compatible with the current default installer for JAGS. Note that the problem is with the Windows installer, not JAGS source code, and doesn't affect other platforms. A compatible installer is available, but it's not the default. See Martyn Plummer's post for more details.
Installing R and JAGS on Apple MacI don't have an Apple computer, but I have picked up some hints about installing R and JAGS on a Mac from trying to troubleshoot friend's installations. 30 Oct 2017, changes in red.People seem to run into problems with different versions of the Mac OS, R, JAGS and the rjags package. The only way to stay sane is to use recent versions of all four. Check your Mac OS version!From the Apple menu, choose About This Mac; the version number appears below the name. Note whether you have v. 10.11 (El Capitan) or later. If you have an earlier version, upgrade your OS before going further.
Installing R and JAGS on Ubuntu OSI recently tried installing R and JAGS on my machine running Ubuntu. I wanted to test myBEST and wiqid packages with the new
version of JAGS on Ubuntu. It took me a while, but I finally
found a simple way to do this which might be of interest to
others.I already had R and JAGS 3 installed, together with
the
Bayesian estimation with a random walk samplerIn the last post we looked at a way to use conjugate distributions for several parameters via a Gibbs sampler. The output from this was an MCMC sample of random draws from the posterior distribution. We can produce similar MCMC samples without using conjugate distributions with a method often called "MetropoliswithinGibbs".The idea for the sampler was developed by Nicholas Metropolis and colleagues in a paper in 1953. This was before the Gibbs sampler was proposed, but it uses the same idea of updating the parameters one by one. A better name would be "componentwise random walk Metropolis sampler". The rules for the random walk ensure that a large number of samples will be a good description of the posterior distribution.
Bayesian estimation with a simple Gibbs samplerAs discussed in the last post, conjugate distributions provide an easy way to calculate posterior distributions for a single parameter, such as detection of a species during a single visit to a site where it is present. If we have more than one unknown parameter in our model  as with a simple occupancy model, where we have detection and occupancy  we may still be able to use conjugacy via a Gibbs sampler.Gibbs sampling works if we can describe the posterior for each parameter if we know all the other parameters in the model.
Bayesian estimation with conjugate priorsConjugate distributions provide useful tricks for combining informative priors with likelihoods to produce posterior distributions. In the days before powerful computers and clever algorithms, they were often the only way. They only work for a single variable. Nevertheless, Gibbs sampling, which the wiqid package uses when possible, builds on the idea of conjugate distributions.As our example, we'll use estimation of detection probability from data for repeat visits to a site which is known to be occupied by our target species. First, we'll describe the beta distribution, then see how that can be combined with our data. A discussion of priors will follow, and we'll finish with brief descriptions of conjugate priors for other types of data.
Likelihood and maximum likelihood estimationI'm planning a series of posts looking at what happens under the hood when we analyse a data set using some of the estimation functions in thewiqid package. I'll focus mainly on Bayesian
methods, but this first post will look at the likelihood, which
is used for both Bayesian analysis and maximum likelihood
estimation.We'll use a simple occupancy model. It has just two parameters and both must between 0 and 1. That means that we can plot all possible combinations of the two parameters in a simple twodimensional graph. As we'll see we need to add a third dimension, but three is still manageable.
Introducing the wiqid packageThe wiqid package for R statistical software provides Quick and Dirty functions for the analysis of Wildlife data.Currently it has functions for estimating occupancy, abundance from closed captures, density from spatial capturerecaptures, and survival from markrecapture data, plus a slew of functions for species richness and alpha and beta diversity. It is intended to be used for (1) simulations and bootstraps, (2) teaching, and (3) introducing Bayesian methods. And it should work on all platforms: Windows, Linux, and Mac.
The jackknife estimatorJackknife estimators are used in ecology in two situations:
SECR in BUGS/JAGS with patchy habitatAnalysis of spatially explicit capturerecapture (SECR) data can be done in a maximum likelihood (ML) or a Bayesian framework. Program DENSITY and thesecr package take care of the former. Bayesian
analysis with the usual workhorses, WinBUGS, OpenBUGS and JAGS,
is straightforward if the traps are laid out in a large area of
homogenous habitat.Faced with patches of suitable habitat surrounded by inhospitable terrain, or a large extent of habitat punctuated with patches of nonhabitat, we had the choice of ML methods or one of the packages designed specifically for Bayesian SECR analysis, such as SPACECAP or SCRbayes. But then we are limited to the range of models provided by package authors: we don't have the flexibility to specify our own models that comes with WinBUGS, OpenBUGS or JAGS. Here I present a way to incorporate patchy habitat into a BUGS/JAGS model specification.
Probability densities and spinnersIn our basic data analysis workshops, we use an idea from John Kruschke's Doing Bayesian Data Analysis: we use spinners to generate random values for continuous variables and introduce the concept of probability density.We start off with simple spinners representing a uniform distribution over a range from, say, 0 to 0.5. We discuss the problems of attaching a probability to an exact value, which leads to probability of a range of values and hence probability density.
Cameratrap layout for SECRI have recently been looking at the design of cameratrap studies to estimate the population density of tigers when populations are very sparse. The intention is to use recentlydeveloped spatially explicit capturerecapture (SECR) methods to analyse the data. The optimal cameratrap layout for SECR may well differ from the design used for older methods.Before the advent of SECR methods, putting all your traps into a single cluster with minimal perimeter length made sense, as you needed to estimate the area trapped animals came from to get a density. SECR estimates density directly, without needing to estimate area, so a single, large cluster may no longer be advantageous.
SECR and circular home rangesDo the models used for SECR (spatially explicit capture recapture) assume that animals' home ranges are approximately circular?I've seen this asserted a couple of times, in particular in Tobler and Powell (2013, p.110), and I've myself drawn circular home ranges when discussing the interpretation of the capture parameters, but I don't think it is a necessary assumption.
SECR : spatially explicit capture recaptureI've a couple of ideas for blog posts on SECR (spatially explicit capturerecapture), and this post sets out the basic concepts of SECR which I will need to refer to in later posts.Capturerecapture methods (also know as markrecapture or capturemarkrecapture) have been used to estimate the size of animal populations for many years: the first software package for analysis of this kind of data, CAPTURE (Otis et al 1978), is now 35 years old. Early methods did not use the spatial component in the data, the capture locations, and spatially explicit capturerecapture models (SECR, or just spatial capturerecapture, SCR) first appeared in 2004 (Efford 2004).
BEST  Bayesian Estimation Supersedes the tTestJohn Kruschke's BEST code for R is a nice introduction to Bayesian thinking for folks used to ttests. I've referred to it, linked to it, and used it in workshops before now.The idea is to provide an R function which is as easy to use as t.test but which gives not a mere pvalue but the kind of output Bayesians are used to  posterior probability distributions. John's BESTmcmc function uses JAGS, but handles all the preliminaries automatically and produces a result in a simple format.
Animal activity patterns and overlapA new R package called overlap to estimate the overlapping of animal activity patterns from data derived from camera traps has now arrived on R's central depository, CRAN.As soon as cameras with "data backs" came along in the early 90s, biologists realised that they could harvest data on the activity patterns of rare, secretive forest animals. Were they diurnal, nocturnal, crepuscular, or maybe cathemeral (active all around the clock)? More recently, people have tried to get clues about how species interact  competition or preypredator interactions  from activity patterns, by examining the extent of overlap. In our corner of the biological world, Martin Ridout and Matt Linkie published a paper (2009) on the activity patterns of tigers, clouded leopards and golden cats in Sumatra, with a lot of technical detail on how overlap could be quantified and confidence intervals estimated. They followed up (2011) with a paper on tigers and their prey, also in Sumatra. ...
Comparing confidence intervalsOften we are interested in the difference between the means of two populations and whether we can infer from samples from the populations that the means are different.This is often a silly question: the means of real populations are almost always different, even if the difference is microscopic. More useful would be to estimate the difference and the probability that it is big enough to be of practical importance. See the BEST software for a way to do this in R. Sometimes we are presented with confidence intervals for each of the means. This happens in particular with the standard packages we use for wildlife data analysis, where the output includes confidence intervals for each coefficient or real value. Can we infer evidence of a difference from confidence intervals in the same way as for a pvalue from a test of significance?
What if my data aren't normal?Sometimes people I talk to are worried because their data aren't normally distributed, and they believe that they can't use the usual techniques such as ttests or ANOVA without first transforming the data to be normal, or they must resort to nonparametric methods. There are many good reasons for transforming data or NOT using t tests or F tests, but nonnormal data is not usually one of them!
Installing JAGS with AVG antivirus softwareA couple of people on a recent workshop had trouble with their AVG antivirus software when installing JAGS 3.3.0. This appears to be due to AVG's paranoia: see
Martyn Plummer's comment. No malware is detected by McAfee
AntiVirus Plus or Trend Micro Office Scan. See also information
on false positives at the
AVG forum.
Displaying rasters in QGISIn ecology and wildlife studies, a lot of our spatial data takes the form of rasters rather than vector files. When you first add a raster in QGIS, you usually get a plain grey rectangle, or maybe just a grey outline on a white background, as most raster file formats have no styling information. To make sense of a raster, you need to change the style. Here I'll give some hints for "quickanddirty" styling to display the contents of a raster. For a more detailed tutorial, see here.
Creating a GIS layer for "Distance from..." in QGISIn a recent post, I showed how to deal with "distance from..." data in GIS layers using the R packages for handling spatial information. The example I used there involved
Here we will see how to do the same thing in QGIS.
Importing data into R for home range analysisAt our recent workshop on Geographical Information Systems (GIS) using Quantum GIS we had a number of people interested in working with radio telemetry or GPS data to model animal home ranges. The home range plugin for QGIS doesn't work with current versions, at least with Windows. It is designed to pass data to R and get the adehabitat package to do the home range estimation and pass the result back to QGIS. QGIS uses Python code, and to get it to talk to R requires a bit of software called "RPy2". This was always difficult to set up on Windows, but Python has been upgraded and RPy2 no longer works. In any case, the adehabitat package has been replaced by new packages with a wider ranger of options. So now it's better to prepare spatial data in QGIS, read the files into R, process with adehabitatHR, write the results to new files, and load into QGIS.
Creating a GIS layer for "Distance from..." in RWe recently ran a workshop on Geographical Information Systems (GIS) using Quantum GIS for ecologists and wildlife researchers. For many species, distance from water, a road, forest edge, or a settlement may be an important habitat variable. For example, we may be using automatic cameras to investigate occupancy of sites by leopards. Probability of occupancy may depend on distance from the nearest road. Given vector layers with roads and camera locations, we want to do two things:
Mathematical formulae with MathJaxI sometimes need to put formulae into my web pages, and I've been exploring the use of MathJax. In the past I've inserted the formula into MS Word with MS Equation 3.0, doing a screen capture, cropping the image to the formula I want, saving as a .GIF file, and then displaying it on the web page as an image. So I get something like this for the Poisson distribution:
That's not ideal. If I want to change anything, I have to start all over again from Word. It's also messy if I want to put something like into the text; for a start it doesn't line up properly. MathJax allows me to type the formula in LaTeX style directly into the HTML code for my web page.
Format for data filesI have a collection of data sets for use during workshops or just to play with when trying out new statistical techniques or computer code. A big question is what format to use, and I've changed my mind on this several times already! After looking at this blog post by John Mount I've decided to try using tabseparated files with a .tsv extension.
 