Approximate Bayesian Computation (ABC)"

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Approximate bayesian computation (ABC) with nlrx

Approximate bayesian computation (ABC) algorithms have been increasingly used for calibration of agent-based simulation models. The nlrx package provides different algorithms from the EasyABC package. These algorithms can be used by attaching the corresponding simdesigns (simdesign_ABCmcmc_Marjoram(), simdesign_ABCmcmc_Marjoram_original(), simdesign_ABCmcmc_Wegmann()). Example 1 shows the process of how to use ABC with nlrx. Additionally, Latin Hypercube Sampling output can be used to calculate parameter distributions based on rejection sampling and local linear regression. Example 2 shows, how the simdesign_lhs() can be used in combination with the abc package.

Example 1: Approximate bayesian computation with Monte-Carlo Markov-Chain

Here we present one example for the widely used Marjoram algorithm which combines ABC with a Markov-Chain Monte-Carlo parameter sampling scheme. However, the other two ABCmcmc simdesigns work in a very similar way except for the parameter definitions within the simdesigns (see respective documentation pages for help).

We use the Wolf Sheep Predation model from the models library to show a basic example of the calibration workflow.

Step 1: Create a nl object:

library(nlrx)
# Windows default NetLogo installation path (adjust to your needs!):
netlogopath <- file.path("C:/Program Files/NetLogo 6.0.3")
modelpath <- file.path(netlogopath, "app/models/Sample Models/Biology/Wolf Sheep Predation.nlogo")
outpath <- file.path("C:/out")
# Unix default NetLogo installation path (adjust to your needs!):
netlogopath <- file.path("/home/NetLogo 6.0.3")
modelpath <- file.path(netlogopath, "app/models/Sample Models/Biology/Wolf Sheep Predation.nlogo")
outpath <- file.path("/home/out")

nl <- nl(nlversion = "6.0.3",
         nlpath = netlogopath,
         modelpath = modelpath,
         jvmmem = 1024)

Step 2: Attach an experiment

Because we want to apply a calibration algorithm, we need to define proper variable ranges. The algorithm is allowed to change the values of these parameters within these ranges in order to reach our specified target values. Possible choices for the distribution type are qunif, qnorm, qlnorm and qexp. Each model run is evaluated for each specified metric within the metrics slot. When the simdesign is attached (see step 3) we will define target values for each metric. Thus, we should only enter metrics and reporters that should be used for calibrating the model.

For this simple example, we just want to check if we can find a parameterization that leads to a specific number of wolves and sheep after a runtime of 100 ticks. Thus, we define count sheep and count wolves as metrics, set the runtime to 100 and set tickmetrics to false (because we only want to measure the last simulation tick).

If more than one tick would be measured, the algorithm automatically calculates the mean value of the selected reporter over all measured ticks. If you wish to apply other functions to aggregate temporal information into one value, you can use a self-defined post-processing function when attaching the simdesign (see step 3).

nl@experiment <- experiment(expname="wolf-sheep",
                            outpath=outpath,
                            repetition=1,
                            tickmetrics="false",
                            idsetup="setup",
                            idgo="go",
                            runtime=100,
                            metrics=c("count sheep", "count wolves"),
                            variables = list("sheep-gain-from-food" = list(min=2, max=6, qfun="qunif"),
                                             "wolf-gain-from-food" = list(min=10, max=30, qfun="qunif")),
                            constants = list('initial-number-sheep' = 100,
                                             'initial-number-wolves' = 50,
                                             "grass-regrowth-time" = 30,
                                             "sheep-reproduce" = 4,
                                             "wolf-reproduce" = 5,
                                             "model-version" = "\"sheep-wolves-grass\"",
                                             "show-energy?" = "false"))

Step 3: Attach a simulation design

We use the simdesign_ABCmcmc_Marjoram() function to attach a calibration simdesign. The summary_stat_target represents the vector of our target values that we want to reach. These values corresponds to the defined metrics of the experiment and should have the same length and order. n_rec defines the number of samples and n_calibration defines the number of calibration runs. If this value is too low, a subscript out of bounds error message might appear. If use_seed is set to TRUE, the algorithm will automatically use a newly generated seed for each model run. If it is set to false, a user-specified seed (set in the run_nl_dyn() function) will be used instead. The progress_bar gives you expected runtime information during the execution of the algorithm. The nseeds command allows to generate a vector of random-seeds that may be used for setting model seeds.

nl@simdesign <- simdesign_ABCmcmc_Marjoram(nl=nl,
                                           summary_stat_target = c(100, 80),
                                           n_rec = 100, 
                                           n_calibration=200,
                                           use_seed = TRUE,
                                           progress_bar = TRUE,
                                           nseeds = 1)

These are the most important simdesign parameters, but there are many more to fine control the behavior of the algorithm. Check out the simdesign help page for more information. As already mentioned before, it is also possible to define a custom post-processing function. To apply this function for model output, the function name needs to be entered as postpro_function within the simdesign. For example, we might want to use the maximum value of some measured metrics to calibrate our model. Or maybe we want to run some tests and use the test statistics as calibration criterion. In such cases, we can define a function that accepts the nl object (with simulation results attached) as function input and returns a vector of numerics. This vector needs to represent the values that we defined as sumary_stat_target and thus should have the same length and order. Below is an example of a custom post-processing function that calculates the maximum value of selected metrics over all simulation ticks.

post <- function(nl){
  res <- getsim(nl, "simoutput") %>% 
    dplyr::select(getexp(nl, "metrics")) %>% 
    dplyr::summarise_each(list(max=max))
  return(as.numeric(res))
}

nl@simdesign <- simdesign_ABCmcmc_Marjoram(nl=nl,
                                           postpro_function = post,
                                           summary_stat_target = c(100, 80),
                                           n_rec = 100, 
                                           n_calibration=200,
                                           use_seed = TRUE,
                                           progress_bar = TRUE,
                                           nseeds = 1)

Step 4: Run simulations

For calibration simdesigns, the run_nl_dyn() function lets you execute the simulations. There are some notable differences between run_nl_all() and run_nl_dyn(). First, because parameterizations depend of results from previous runs, run_nl_dyn() can not be parallelized. Second, the procedure does not automatically loop over created random seeds of the simdesign. If you want to repeat the same algorithm several times, just embed the run_nl_dyn() function in any kind of loop and iterate through the nl@simdesign@simseeds vector. We set the use_seed parameter of the simdesign to true, however we still need to define a random-seed in run_nl_dyn although it will be overwritten by the EasyABC functions.

results <- run_nl_dyn(nl, seed = nl@simdesign@simseeds[1])

Step 5: Investigate output

The output is reported as nested tibble which can be attached to the nl object. There are many possible ways to inspect the simulation output of the ABCmcmc functions. Below you find some guidance on how to summarize the output for calculating parameter statistics, sampling distributions, sampling density and exporting the best parameter combination.

nl@simdesign@simoutput <- results
saveRDS(nl, file.path(nl@experiment@outpath, "ABCmcmc.rds"))

## Calculate descriptive statistics
getsim(nl, "simoutput") %>% # get simulation results from nl object
  dplyr::select(param) %>% # select param column
  tidyr::unnest(cols=param) %>%  # unnest param column
  dplyr::summarise_each(list(min=min, max=max, mean=mean, median=median)) %>% # calculate statistics
  tidyr::gather(parameter, value) %>% # convert to long format
  tidyr::separate(parameter, into = c("parameter", "stat"), sep = "_") %>% # seperate parameter name and statistic
  tidyr::spread(stat, value) # convert back to wide format

## Plot histogram of parameter sampling distribution:
getsim(nl, "simoutput") %>% # get simulation results from nl object
  dplyr::select(param) %>% # select param column
  tidyr::unnest(cols=param) %>%  # unnest param column
  tidyr::gather(parameter, value) %>% # convert to long format
  ggplot2::ggplot() + # plot histogram with a facet for each parameter
  ggplot2::facet_wrap(~parameter, scales="free") +
  ggplot2::geom_histogram(ggplot2::aes(x=value), bins = 40)

## Plot density of parameter sampling distribution:
getsim(nl, "simoutput") %>% # get simulation results from nl object
  dplyr::select(param) %>% # select param column
  tidyr::unnest(cols=param) %>% # unnest param column
  tidyr::gather(parameter, value) %>% # convert to long format
  ggplot2::ggplot() + # plot density with a facet for each parameter
  ggplot2::facet_wrap(~parameter, scales="free") +
  ggplot2::geom_density(ggplot2::aes(x=value, fill=parameter))

## Get best parameter combinations and corresponding function values
getsim(nl, "simoutput") %>%  # get simulation results from nl object
  dplyr::select(dist,epsilon) %>%  # select dist and epsilon columns
  tidyr::unnest(cols=c(dist,epsilon)) %>%  # unnest dist and epsilon columns
  dplyr::mutate(runID=dplyr::row_number()) %>% # add row ID column
  dplyr::filter(dist == epsilon) %>% # only keep runs with dist=epsilon
  dplyr::left_join(getsim(nl, "simoutput") %>% # join parameter values of best runs
                     dplyr::select(param) %>%
                     tidyr::unnest(cols=param) %>% 
                     dplyr::mutate(runID=dplyr::row_number())) %>% 
  dplyr::left_join(getsim(nl, "simoutput") %>% # join output values best runs
                     dplyr::select(stats) %>%
                     tidyr::unnest(cols=stats) %>% 
                     dplyr::mutate(runID=dplyr::row_number())) %>% 
  dplyr::select(runID, dist, epsilon, dplyr::everything()) # update order of columns

## Analyse mcmc using coda summary and plot functions:
summary(coda::as.mcmc(getsim(nl, "simoutput") %>%
                        dplyr::select(param) %>%
                        tidyr::unnest(cols=param)), quantiles =c(0.05,0.95,0.5))

plot(coda::as.mcmc(getsim(nl, "simoutput") %>%
                        dplyr::select(param) %>%
                        tidyr::unnest(cols=param)))

Example 2: Using Latin Hypercube Sampling for Approximate bayesian computation

Step 1: Create a nl object:

library(nlrx)
# Windows default NetLogo installation path (adjust to your needs!):
netlogopath <- file.path("C:/Program Files/NetLogo 6.0.3")
modelpath <- file.path(netlogopath, "app/models/Sample Models/Biology/Wolf Sheep Predation.nlogo")
outpath <- file.path("C:/out")
# Unix default NetLogo installation path (adjust to your needs!):
netlogopath <- file.path("/home/NetLogo 6.0.3")
modelpath <- file.path(netlogopath, "app/models/Sample Models/Biology/Wolf Sheep Predation.nlogo")
outpath <- file.path("/home/out")

nl <- nl(nlversion = "6.0.3",
         nlpath = netlogopath,
         modelpath = modelpath,
         jvmmem = 1024)

Step 2: Attach an experiment

We want to run a Latin Hypercube sampling, thus we need to define proper variable ranges. We also need to define our output metrics. These metrics are also used for the rejection sampling later on.

nl@experiment <- experiment(expname="wolf-sheep",
                            outpath=outpath,
                            repetition=1,
                            tickmetrics="false",
                            idsetup="setup",
                            idgo="go",
                            runtime=100,
                            metrics=c("count sheep", "count wolves"),
                            variables = list("sheep-gain-from-food" = list(min=2, max=6, qfun="qunif"),
                                             "wolf-gain-from-food" = list(min=10, max=30, qfun="qunif")),
                            constants = list('initial-number-sheep' = 100,
                                             'initial-number-wolves' = 50,
                                             "grass-regrowth-time" = 30,
                                             "sheep-reproduce" = 4,
                                             "wolf-reproduce" = 5,
                                             "model-version" = "\"sheep-wolves-grass\"",
                                             "show-energy?" = "false"))

Step 3: Attach a simulation design

We use the simdesign_lhs() helper function to generate a Latin Hypercube Sampling with 500 samples

nl@simdesign <- simdesign_lhs(nl, 
                              samples=500, 
                              nseeds=1, 
                              precision=3)

Step 4: Run simulations

We can simply use run_nl_all() to execute our simulations.

results <- run_nl_all(nl)

Step 5: Investigate output

We first attach the output results to our nl object and store a copy of the nl object on disk.

nl@simdesign@simoutput <- results
saveRDS(nl, file.path(nl@experiment@outpath, "ABClhs.rds"))

For post-processing, we need a tibble with input parameter distributions. This can be easily extracted from the siminput slot of the simdesign by selecting only the columns with variable (non-constant) parameters. Next, we need corresponding outputs for these parameters. In this example we just take the measured output tibble (simoutput) and only select the columns with our metrics. Of course, you can also perform additional post-processing of these outputs if desired. Third, we need to define expected values for our outputs. In our case, we just assume that count sheep should have a value of 100, whereas count wolves should have a value of 80.

input <- getsim(nl, "siminput") %>% 
  dplyr::select(names(getexp(nl, "variables")))
output <- getsim(nl, "simoutput") %>% 
  dplyr::select(getexp(nl, "metrics"))
target <- c("count sheep"=100, "count wolves"=80)

We use the abc function of the abc package to perform the rejection sampling. For this example, we perform both algorithms provided by this function ("rejection" and "loclinear").

results.abc.reject <- abc::abc(target=target, 
                        param=input,
                        sumstat=output,
                        tol=0.3, 
                        method="rejection")

results.abc.loclin <- abc::abc(target=target, 
                               param=input,
                               sumstat=output,
                               tol=0.3, 
                               method="loclinear")

Finally, we might want to compare the accepted parameter distributions of both algorithms and the initial distribution of the Latin Hypercube sampling. Thus, we reformat the results to a tidy data format and attach the initial parameter distributions. This dataset can now be used for displaying the parameter distributions with ggplot.

results.abc.all <- tibble::as_tibble(results.abc.reject$unadj.values) %>% # results from rejection method
  tidyr::gather(parameter, value) %>% 
  dplyr::mutate(method="rejection") %>% 
  dplyr::bind_rows(tibble::as_tibble(results.abc.loclin$adj.values) %>% # results from local linear regression method
                     tidyr::gather(parameter, value) %>% 
                     dplyr::mutate(method="loclinear")) %>% 
  dplyr::bind_rows(input %>%                # initial parameter distribution (lhs)
                     tidyr::gather(parameter, value) %>% 
                     dplyr::mutate(method="lhs"))

ggplot2::ggplot(results.abc.all) +
  ggplot2::facet_grid(method~parameter, scales="free") +
  ggplot2::geom_histogram(ggplot2::aes(x=value))

ggplot2::ggplot(results.abc.all) +
  ggplot2::facet_wrap(~parameter, scales="free") +
  ggplot2::geom_density(ggplot2::aes(x=value, fill=method), alpha=0.1)


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nlrx documentation built on Nov. 14, 2020, 1:08 a.m.