extBatchMarking: Implementation of Extended Batch Marking Model

The primary objective of extBatchMarking is to facilitate the fitting of models developed by Cowen et al., 2017 for the ecologist. The marked models can be seamlessly integrated with unmarked models to estimate population size. The combined model harnesses the power of both the N-mixture model and the Viterbi algorithm for hidden markov model to provide accurate population size estimates.

The primary objective of extBatchMarking is to facilitate the fitting of models developed by Cowen et al., 2017 for the ecologist. The marked models can be seamlessly integrated with unmarked models to estimate population size. The combined model harnesses the power of both the N-mixture model and the Viterbi algorithm for hidden markov model to provide accurate population size estimates.

In ecological research, it’s often challenging to directly count every individual of a species due to various factors such as their elusive nature or inaccessible habitats. As a result, Cowen et al., 2017 employ two distinct modeling approaches: marked and unmarked models.

  1. Marked Models: These models focus on individuals that have been uniquely identified or ‘marked’ in some way, such as through tagging, banding, or other identification methods. These marked individuals are tracked over time, and their data is used to estimate parameters related to the population, such as survival rates, population growth, or movement patterns.

  2. Unmarked Models: Unmarked models, on the other hand, are designed to estimate population parameters without relying on individual identification.

The beauty of combining these two modeling approaches lies in their synergy. By leveraging both marked and unmarked data, ecologists can achieve more accurate and robust estimates of population abundance. Marked data provide insights into specific individuals, while unmarked data give a broader perspective on the entire population.

In practice, this combination is achieved through sophisticated statistical techniques, often utilizing concepts like the N-mixture model and algorithms like the Viterbi algorithm. These methods allow ecologists to integrate data from marked and unmarked individuals, resulting in more comprehensive and reliable population abundance estimates.

The models showcased in this example represent the foundational instances drawn from a set of four complex models available within the extBatchMarking package. These models serve as essential building blocks for understanding the advanced functionalities and capabilities offered by the package.

The extBatchMarking package is designed to empower researchers with a powerful tool set for the analysis of batch-marked data in ecological and population studies. It allows users to efficiently fit and assess batch-marked models, aiding in the estimation of critical population parameters such as survival and capture probabilities.

In particular, the models illustrated here provide a comprehensive introduction to the core concepts and methodologies underpinning the package’s functionality. They are intended to facilitate an initial grasp of how to work with batch-marked data, offering insights into the modeling techniques used in the field of population ecology.

It’s worth noting that the results obtained using the extBatchMarking package align with the findings presented in Cowen et al. (2017). This alignment demonstrates the package’s reliability and ability to replicate established research outcomes. The Cowen et al. (2017) results section serves as a benchmark against which the package’s performance can be validated, providing users with confidence in the accuracy of their analyses.

By starting with these basic examples, users can progressively delve into more intricate and tailored analyses within the extBatchMarking package, ultimately enabling them to make meaningful contributions to the understanding of population dynamics and ecology. The package’s versatility and fidelity to established research findings make it a valuable resource for both novice and experienced researchers in the field.

The example will guide you through the steps of how to employ this approach effectively, demonstrating its relevance and importance in ecological and wildlife studies. It showcases the power of merging marked and unmarked models to gain a deeper understanding of species populations and their dynamics within natural ecosystems.”

Installation

You can install the released version of extBatchMarking from CRAN with:

devtools::load_all(".")
#> ℹ Loading extBatchMarking
devtools::document()
#> ℹ Updating extBatchMarking documentation
#> ℹ Loading extBatchMarking
devtools::load_all()
#> ℹ Loading extBatchMarking

Example 1

This is a basic example which shows how to fit a Batch marking model with constant phi and p. Example 1 can also be found in the Cowen et al., 2017 results using the WeatherLoach data used in the same paper:

Load the data WeatherLoach from the extBatchMarking package: Here is the step-by-step guide on how to load data directly from extBatchMarking package. The default data discussed in Cowen et al. 2017.

data("WeatherLoach", package = "extBatchMarking")

First, we show with an example how to fit the batchMarkHmmLL and batchMarkUnmarkHmmLL functions. batchMarkHmmLL and batchMarkUnmarkHmmLL functions output the unoptimized log-likelihood values of marked only model and the combined models. These allow users know if the likelihood functions can be computed at the specified initial values. Otherwise, NAN or Inf will be returned. If so, the arguments of the functions should be revisited.

# Initial parameter
theta <- c(0, -1)

res1 <- batchMarkHmmLL(par           = theta,
                       data          = WeatherLoach,
                       choiceModel   = "model4",
                       covariate_phi = NULL,
                       covariate_p   = NULL)

res1
#> [1] 132.3349

[1] 132.3349

Example 2

thet <- c(0.1, 0.1, 7, -1.5)

res3 <- batchMarkUnmarkHmmLL(par           = thet,
                             data          = WeatherLoach,
                             choiceModel   = "model4",
                             Umax          = 1800,
                             nBins         = 600,
                             covariate_phi = NULL,
                             covariate_p   = NULL)

res3
#> [1] 870.0261

[1] 870.0261

Example 3

#> initial  value 132.334856 
#> final  value 124.984186 
#> converged

initial value 132.334856 final value 124.984186 converged

Print the results with the print.batchMarkOptim() function: This prints to the console the log-likelihood, AIC, recapture probability and survival probability with their corresponding standard errors.

print(res)
#> [[1]]
#> 
#> 
#> | log.likelihood|      AIC|
#> |--------------:|--------:|
#> |       124.9842| 253.9684|
#> 
#> [[2]]
#> 
#> 
#> |      p| p_S.Error|    phi| phi_S.Error|
#> |------:|---------:|------:|-----------:|
#> | 0.1964|    0.0206| 0.6042|      0.0261|

[[1]]

log.likelihood AIC
124.9842 253.9684

[[2]]

p p_S.Error phi phi_S.Error
0.1964 0.0206 0.6042 0.0261

Plot the results with the plot.batchMarkOptim() function: This plots the goodness-of-fit plot to perform a goodness-of-fit test for the model fit.

plot(res)

Example 4

This example serves as a fundamental illustration of the process of combining both marked and unmarked models to estimate the population abundance of a species. It demonstrates a key approach used in ecological and wildlife studies to gain insights into the size of a specific species population within a given habitat.

theta <- c(0.1, 0.1, 7, -1.5)

res2 <- batchMarkUnmarkOptim(par=theta,
                            data=WeatherLoach,
                            Umax=1800,
                            nBins=600,
                            choiceModel="model4",
                            popSize = "Horvitz_Thompson",
                            method="BFGS",
                            control=list(trace = 1),
                            covariate_phi = NULL,
                            covariate_p   = NULL)
#> initial  value 870.026082 
#> iter  10 value 205.041347
#> final  value 205.031738 
#> converged

initial value 870.026082 iter 10 value 205.041347 final value 205.031738 converged

Print the results with the print.batchMarkUnmarkOptim() function: This prints to the console the log-likelihood, AIC, recapture probability and survival probability with their corresponding standard errors. The species abundances are also provided which include number of unmarked individuals, number of marked individuals, and Total abundance.

print(res2)
#> [[1]]
#> 
#> 
#> | log.likelihood|      AIC|
#> |--------------:|--------:|
#> |       205.0317| 418.0635|
#> 
#> [[2]]
#> 
#> 
#> |      p| p_S.Error|   phi| phi_S.Error|
#> |------:|---------:|-----:|-----------:|
#> | 0.1866|    0.0048| 0.614|      0.0169|
#> 
#> [[3]]
#> 
#> 
#> | No.of.Unmarked.U.| No.of.Marked.M.| Abundance.N.|
#> |-----------------:|---------------:|------------:|
#> |              1500|               0|         1500|
#> |               900|             171|         1071|
#> |               900|             268|         1168|
#> |               900|             322|         1222|
#> |               300|             118|          418|
#> |               300|             107|          407|
#> |               300|              54|          354|
#> |               300|              86|          386|
#> |               300|             107|          407|
#> |               300|             139|          439|
#> |               300|              64|          364|
#> 
#> [[4]]
#> 
#> 
#> |   Lambda| Lambda_S.Error|    Gam| Gam_S.Error|
#> |--------:|--------------:|------:|-----------:|
#> | 1346.844|       1903.361| 0.0599|       0.022|

[[1]]

log.likelihood AIC
205.0317 418.0635

[[2]]

p p_S.Error phi phi_S.Error
0.1866 0.0048 0.614 0.0169

[[3]]

No.of.Unmarked.U. No.of.Marked.M. Abundance.N.
1500 0 1500
900 171 1071
900 268 1168
900 322 1222
300 118 418
300 107 407
300 54 354
300 86 386
300 107 407
300 139 439
300 64 364

[[4]]

Lambda Lambda_S.Error Gam Gam_S.Error
1346.844 1903.361 0.0599 0.022

Plot the results with the plot.batchMarkOptim() function: This plots the googdness-of-fit plot to perform a goodness-of-fit test for the model fit.

plot(res2)

Model 2 for marked model with covariate for phi

This provides some level of flexibility to the marked model by adding a placeholder for time-dependent covariate to better estimate parameter phi. For example, we might want to understand how temperature affects the survival rate of a species.

#-------------------------------------------------
# Model 2: 10 phis and 1 prob
#-------------------------------------------------

theta <- c(-1, rep(0, 10))

cv <- matrix(seq(2, 10, length = 10), ncol = 1)
batchMarkHmmLL(theta, WeatherLoach, "model2", covariate_phi = cv, covariate_p = NULL)
#> [1] 132.3349
res <- batchMarkOptim(par=theta, data=WeatherLoach, covariate_phi = cv, covariate_p = NULL, 
                      choiceModel = "model2", method="BFGS", control = list(trace = 1))
#> initial  value 132.334856 
#> iter  10 value 97.474508
#> iter  20 value 95.823647
#> iter  30 value 95.787362
#> iter  40 value 95.784462
#> iter  50 value 95.783391
#> iter  60 value 95.783072
#> final  value 95.783042 
#> converged

initial value 132.334856 iter 10 value 97.474508 iter 20 value 95.823647 iter 30 value 95.787362 iter 40 value 95.784462 iter 50 value 95.783391 iter 60 value 95.783072 final value 95.783042 converged

#-------------------------------------------------
# Model 2: 10 phis and 1 prob
#-------------------------------------------------

print(res)
#> [[1]]
#> 
#> 
#> | log.likelihood|      AIC|
#> |--------------:|--------:|
#> |         95.783| 213.5661|
#> 
#> [[2]]
#> 
#> 
#> |     p| p_S.Error|    phi| phi_S.Error|
#> |-----:|---------:|------:|-----------:|
#> | 0.181|    0.0316| 0.3268|      0.0000|
#> | 0.181|    0.0316| 0.0000|      0.0000|
#> | 0.181|    0.0316| 1.0000|      0.8321|
#> | 0.181|    0.0316| 0.1082|      0.1306|
#> | 0.181|    0.0316| 0.5680|      0.1388|
#> | 0.181|    0.0316| 0.4722|      0.0009|
#> | 0.181|    0.0316| 0.9973|      8.2224|
#> | 0.181|    0.0316| 0.6458|      0.0985|
#> | 0.181|    0.0316| 0.6619|      0.1316|
#> | 0.181|    0.0316| 0.4397|      0.0285|

[[1]]

log.likelihood AIC
95.783 213.5661

[[2]]

p p_S.Error phi phi_S.Error
0.181 0.0316 0.3268 0.0000
0.181 0.0316 0.0000 0.0000
0.181 0.0316 1.0000 0.8321
0.181 0.0316 0.1082 0.1306
0.181 0.0316 0.5680 0.1388
0.181 0.0316 0.4722 0.0009
0.181 0.0316 0.9973 8.2224
0.181 0.0316 0.6458 0.0985
0.181 0.0316 0.6619 0.1316
0.181 0.0316 0.4397 0.0285 
plot(res)

Model 3 for marked model with covariate for p

This provides some level of flexibility to the marked model by adding a placeholder for time-dependent covariate to better estimate parameter p. For example, we might want to understand how temperature affects the survival rate of a species.

#-------------------------------------------------
# Model 3: 1 phis and 10 prob
#-------------------------------------------------

theta <- c(-1, rep(0, 10))

cv <- matrix(seq(2, 10, length = 10), ncol = 1)
batchMarkHmmLL(theta, WeatherLoach, "model3", covariate_phi = NULL, covariate_p = cv)
#> [1] 200.0016
res <- batchMarkOptim(par=theta, data=WeatherLoach, covariate_phi = NULL, covariate_p = cv, 
                      choiceModel = "model3", method="BFGS", control = list(trace = 1))
#> initial  value 200.001592 
#> iter  10 value 114.293180
#> final  value 97.288967 
#> converged

initial value 200.001592 iter 10 value 114.293180 final value 97.288967 converged

#-------------------------------------------------
# Model 2: 1 phis and 10 prob
#-------------------------------------------------

print(res)
#> [[1]]
#> 
#> 
#> | log.likelihood|      AIC|
#> |--------------:|--------:|
#> |         97.289| 216.5779|
#> 
#> [[2]]
#> 
#> 
#> |      p| p_S.Error|   phi| phi_S.Error|
#> |------:|---------:|-----:|-----------:|
#> | 0.9559|    0.0182| 0.627|      0.0297|
#> | 0.9991|    0.0011| 0.627|      0.0297|
#> | 0.0084|    0.0099| 0.627|      0.0297|
#> | 0.0156|    0.0078| 0.627|      0.0297|
#> | 0.0793|    0.0241| 0.627|      0.0297|
#> | 0.0933|    0.0260| 0.627|      0.0297|
#> | 0.2411|    0.0410| 0.627|      0.0297|
#> | 0.3234|    0.0388| 0.627|      0.0297|
#> | 0.3903|    0.0352| 0.627|      0.0297|
#> | 0.3164|    0.0311| 0.627|      0.0297|
plot(res)

[[1]]

log.likelihood AIC
97.289 216.5779

[[2]]

p p_S.Error phi phi_S.Error
0.9559 0.0182 0.627 0.0297
0.9991 0.0011 0.627 0.0297
0.0084 0.0099 0.627 0.0297
0.0156 0.0078 0.627 0.0297
0.0793 0.0241 0.627 0.0297
0.0933 0.0260 0.627 0.0297
0.2411 0.0410 0.627 0.0297
0.3234 0.0388 0.627 0.0297
0.3903 0.0352 0.627 0.0297
0.3164 0.0311 0.627 0.0297