UA produces a distribution of model results Figure 1. It requires multiple model runs, where factor values are randomly chosen from their respective distributions. Because the results of quantitative UA-SA are computationally expensive, the selection of the sampling method used to perform UA is essential.
Following Saltelli et al. A sample is then used to execute the model, which produces an individual output value. In our case study, for example, the output is a measure of total area of land converted from agriculture to fallow. These results build a distribution of outputs that can be further analyzed using descriptive statistics. Two statistics are particularly useful: the mean that represents the central tendency of the stochastic process, and the variance that summarizes the variability of the results.
Commonly, SA involves modifying the value of one factor while keeping the other factors constant and observing the effects of this change on model results. This method, referred to as one-parameter-at-a-time OAT  ,  ,  , is most often used by socioecological modelers.
The prominent examples, closely related to socioecological ABMs, include the use of OAT in land use change cellular automata models to evaluate their sensitivity to map resolution and the size and configuration of neighborhoods  ,  and the use of OAT to identify the most sensitive factors in an epidemiological ABM of the spread of measles among humans . OAT popularity may be attributed to its simplicity, low computational cost, clear starting point in the form of a baseline parameter set, and the fact that the observed changes in outputs can be easily traced back to changes in specific factors .
The arbitrary choice of which factor to modify and by what amount is problematic when the magnitude of key system drivers is hard to determine . Also, OAT does not explore the variability of factors in combinations and, consequently, assumes a linear relationship between inputs and outputs. Finally, OAT is of limited use in exploratory modeling, because it does not test the full range of factor variability and therefore minimizes our ability to simulate extreme, but catastrophic, events.
As an alternative to OAT, we utilize a global SA approach, which is based upon simultaneous perturbations of the entire model factor space, examining the factors both individually and in combinations  , . Our global SA uses model output variance decomposition in which the variability of the area of fallow land resulting from farmer agent decision making is decomposed partitioned and distributed among model factors evaluated in various combinations  , .
Factor sensitivity is quantified using two measures referred to as a first order sensitivity index, S, and a total effects sensitivity index, ST  ,  , . Index S measures the independent, fractional contribution of each individual model factor to output variance. The ST index estimates the overall contribution of a given factor to output variance including its interactions with other factors. Assuming that model output Y has unconditional variance V, the indices of a given factor i are formalized as follows: Eq.
Therefore, the formula: Eq. This succinct measure of interactions can be further analyzed using the ST indices, which provide information about the total first and higher order influence of each factor on output variance. For more details on variance decomposition, the reader is referred to Saltelli et al. The S,ST pairs are quantified as ratios of the conditional output variances to the total variance and, thus, measure the relative contribution of each factor to output variance Figure 2A.
Factors with relatively high S ST values will have the greatest impact on the variability of model results. When these factors are refined or fixed to constant values, the result is a reduction in output variance. We use this property of the S,ST pairs to operationalize the explanatory power conception of modeling Figure 2B. The major premise of model explanatory power is that additional observations used for estimating the most influential factors get us closer to an accurate representation of the underlying system.
By better approximating values of the most influential factors, especially in cases where these factors dominate the output, we can unravel the interrelationships among other factors and expose model nonlinearities. Conversely, if we fix factors that have S ST values close to zero i. Instead, we derive a simplified model with quantitative exploratory power embodied in variance of a given output equal to this model's baseline implementation Figure 2C.
Applying variance decomposition to simplify a stochastic model A , and maintain its exploratory power embodied in outcome variability B or improving its explanatory power by reducing its outcome variability C. We use quantitative UA-SA for land use model simplification and factor prioritization. The goal is to build a simpler representation of an ABM with two distinct objectives: policy analysis that would benefit from exploratory modeling  , and advancing science through explanatory modeling . Our case study considers the participation of farmers in a land conservation program aimed at protecting ecologically valuable areas.
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Published research suggests that farmers' decision to participate in a land conservation program is driven by both financial and nonmonetary drivers  ,  ,  , . These findings are based on conventional statistical analyses of survey data. Few studies have explicitly modeled the decision processes and analyzed the resulting spatial configurations of conserved land .
Following this observation, we develop an ABM of agricultural land conservation decision making. The model simulates voluntary participation in the U. Figure 3. The area covers square km, with farmland parcels. Two types of decision makers are involved in this process Figure 4 :  farmers who decide whether or not to participate in CRP and  the Farm Service Agency FSA , which evaluates, selects, and accepts farmer enrollment offers.
The basic spatial unit of CRP decision-making is a farmland parcel. During the model setup, a farmer agent FA is associated with various socio-demographic and economic factors land tenure , operator's retirement , and the value of production on a farm; described in later sections. The FA is then assigned to a parcel. As a first step, the FA calculates their willingness to enroll in the CRP based on decision criteria factor values including financial motives and nonmonetary drivers. To calculate the willingness to enroll, we apply a group of aggregation operations aka decision rules called Ordered Weighted Averaging - OWA  , .
OWA allows for simple representations of different conceptions of risk related to CRP enrollment, which, after acceptance, is mandatory for at least ten years. OWA decision rules range from the most risk-averse, where values of all decision criteria must be positive, to the most risk-taking where only one decision criterion needs to be nonzero.
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For example, if an agent makes a decision to participate in the program based on low value of production AND retirement, that decision is risk-averse, whereas if the decision is made based on low value of production OR retirement, the agent is risk-taking. Agent's willingness to enroll is extended with a simple group interaction mechanism, in which farmers adopt imitative behavior  based on the decisions made by other proximal farmers.
An FA incorporates into its decision mechanism the density of enrollment in its neighborhood, measured as the ratio of CRP-enrolled neighbors to the total number of neighbors within 0. If an FA's willingness to enroll exceeds an empirically derived threshold, the agent selects a fraction of its parcel for potential enrollment .
Eligible sites in the parcel pasture and cropland are rank ordered based on distance to water, distance to forest, and land slope, and the first fraction of sites is selected. Next, the FA builds an offer by calculating an expected annual payment based on soil rental rates . The FSA agent collects offers from the FAs and selects a subset n of them based on the environmental benefit index described in the following section , the signup budget, and the DAP.
Next, FSA announces the signup results leading to land use change from agriculture to fallow. In sum, the location and area of fallow land results both from the FAs' decisions to participate and the FSA's decision to accept their offers. Land use change maps constitute the output of the model. They are summarized into the total area of fallow land. This scalar is used in the UA and SA. The ABM uses a number of factors, some that are readily available and some that we derived from auxiliary resources, including land uses obtained from cropland raster layers  , freshwater ecosystems from a lakes and rivers geodatabase  , soil data  , and slope .
The major geoprocessing operations were mapping land eligible for CRP  , deriving spatial layers that influence FA's choice of area to enroll Figure 4 - distance to water, distance to forest, and land slope , and generating the soil rental rate SRR and the environmental benefits layers. The SRR layer Figure 5 was derived from a soil productivity index map for the State of Michigan  and county cash rental rates .
EBI is a composite index based on multiple rated criteria describing benefits for wildlife, water quality, soil erosion, long term maintenance of installed vegetation, and air quality . To optimize environmental benefits per dollar expended for rental payments, the EBI is adjusted by a cost and bid rating scale. Offers with lower total annual payments and higher bids voluntary reduction by the farmer of the offer value below the maximum payment receive highest priority. EBI can be quantified in many different ways, resulting in substantial uncertainty.
Consequently, we used alternative conceptions of the benefits Figure 6 , weighed by their respective point scores  in various combinations to generate six different benefit layers used interchangeably in the ABM. All N1, N2, and N3 layers are standardized based on their respective point scales . The remaining benefit criteria used in EBI calculation vegetation and air quality were not used due to their negligible role in the area of study.
Statistical and econometric studies of CRP enrollment point to five major categories of independent variables used to predict participation in land conservation programs: farm, household, and environmental characteristics; government assistance; and farmers' attitude and perception  ,  ,  ,  ,  , . We developed an a priori set of candidate multiple regression models to understand farmer participation in the CRP based on results of prior studies about farmer participation in land conservation programs.
Using a multimodel inference analytical approach based on the Akaike information criterion AIC , models with relatively low AIC values were considered the most parsimonious, balancing bias and variance of model predictions . We assigned relative strengths of evidence to each candidate model according to AIC weights and evaluated explanatory variables in terms of deviance explained.
Consequently, these three attributes became FA's decision criteria Figure 4. These functions epitomize the financial and nonmonetary drivers affecting the FA's land conservation decision.
Empirical data for other factors were only partially available. Consequently, we used a uniform PDF for the other model factors . The density of enrollment in the FA's neighborhood depends on how the neighborhood is defined. In our model, we delineated neighborhood based on distance from FA's parcel, which varied from to m. For OWA, we assumed various magnitudes of attitude to risk, where each level had an equal probability of selection.
Our simulations include three computational experiments. In experiment one EXP1, model runs , our base scenario Figure 2 , we run Monte Carlo simulations using all nine factors. In experiment 2 EXP2, runs , the simplified exploratory scenario, we include only those factors that most influence the variability of the total area of fallow land AREA , calculated at the end of model execution.
The simplified explanatory scenario with variance reduction is implemented in experiment 3 EXP3, runs , where we fix the value of the most influential factor from EXP1, leaving the remaining factors unchanged. All simulations were run using high performance computing at Michigan State University.
Factor samples were produced using the quasi-random Sobol' experimental design  , which is the most optimal method to approximate the values of the S and ST indices  , . Statistical regressions were completed utilizing R, version 3. The results of our ABM simulations are land use maps with one additional category, fallow land , when compared to our input land use maps. Because FAs make decisions on a site-by-site basis, most of the parcels enrolled in CRP at the end of model execution have only a portion of their land enrolled in CRP.
Figure 7B illustrates the frequency of site enrollment number of times a site is enrolled for all ABM executions. Note the considerable spatial variability in site enrollment. We hypothesize that this dispersed enrollment is caused by the complex interactions between the nine factors. For clarity, only the southeast portion of the study area is shown. We also calculated the mean and variance of AREA per experiment. Since our experimental design uses a more uniform quasi-random sampling compared to the typical ABM Monte Carlo simulations that are based on simple random sampling, we can infer that the calculated mean is indeed the true accurate measure of central tendency in AREA distribution.
Consequently, we can use this value to validate the model against an independent dataset. The U. Fallow land area is reported in map units equivalent of 30 m. We used the variance to evaluate the degree of AREA variability. Consequently, the simplified model used in EXP2 can be used in exploratory analysis without the loss of variability necessary when evaluating the CRP policies. Conversely, EXP3 in which data on the most sensitive factor was refined produces a distribution much more centered around the mean when compared to the baseline.
Consequently, the simplified and refined ABM used in EXP3 can be used in explanatory analysis of the social, economic, and ecological processes associated with CRP participation. The following section explains the details on how we arrived at these two ABM simplifications. UA alone does not provide any information about the influence of individual factors on AREA variability. Figure 8B shows pie charts of the S and ST indices for all three experiments. Because factors with relatively high values of S have the most effect on the of total fallow land area, we look for factors that, if fixed singly, would most reduce the variance of AREA.
Trivially then, the extent of farmland conservation is first and foremost driven by the FSA signup choices. Given that CRP is competitive among farmers, the ABM confirms the observation that program participation depends on the federal budget allocated to annual payments. Due to their influence, these five factors were included in a simplified version of the model in EXP2 central box plot and pie charts of Figure 8A and 8B.
Because we only excluded factors that had negligible influence on the distribution of AREA which were set to constant values - either their mean or median , the resulting and baseline distributions are nearly identical, including their means and variances. More importantly, variance decomposition generated S and ST indices consistent with the original model formulation.
This simplified model is more efficient computationally - an indispensable feature for models used in policy analysis . At the same time it maintains result variability, which can be of use when identifying the less probable but highly consequential policy scenarios. This scenario imitates a situation in which we obtain more accurate data on the most sensitive factor of the model.
EXP3 is also characterized by a more complex behavior than the first two experiments. We hypothesize that a portion of these interactions can be attributed to the functional relationships between factors. For example, if the fraction of land to convert in a particular parcel has a relatively high value while the OWA rule is conjunctive AND-only  , a large portion of land has the potential to become fallow.
In summary, while we reduced the range of the distribution for AREA in EXP3, we also exposed more complex dependency among the remaining factors than initially observed. We can therefore postulate that this simplified ABM carries more explanatory power than the original model. It leads to equivalent but simpler representations of a given socioecological system. Output uncertainty can be greatly reduced if more effort is put into improving the quality of data on the most influential factors factor prioritization through additional field studies, surveys, or auxiliary databases.
However, the UA-SA framework also has limitations due to two design aspects: factor distributions and the type of output variables used i. A different output variable e. For example, the use of spatial metrics applied to output land use change maps  may lead to alternative explanations of model uncertainty . Similarly, the type and characteristics of the probability distributions used for each factor e. The ABM presented here is of limited use for natural resource management practice.
Data for most of the factors are either simulated or come from secondary sources and some of the mechanisms are poorly defined. Future model improvements will require surveys of and interviews with farmers and government officials. The recent decline in CRP enrollment suggests that increasing crop prices and government subsidies may play a significant role in the extent of land conservation  , indicating the importance of such research. Finally, more insight into the spatial configuration of fallow land connectivity, clustering, or dispersion of fallow land may be necessary to better evaluate the ecological benefits of land conservation resulting in the prioritization of protected areas.
ABMs have distinct advantages over other modeling approaches due to their abilities to couple human and natural systems, to incorporate micro-level behaviors among interacting agents, and to understand emergent phenomena due to these interactions. Their use thus far has been primarily by researchers for descriptive and predictive purposes . This fact may explain their limited use in policy-making; ABMs' abilities to make accurate predictions have been questioned  , .
We have addressed this perceived limitation using our quantitative UA-SA approach by identifying and fixing the values of the most influential factors, thereby reducing the variance of model results. Doing so allows researchers to gain a greater understanding of the individual and interactive effects of different model factors. Further, by controlling the factors that explain the most variation in the output, researchers can expose the smallest number of factors that influence the steady state of a system. In our CRP example, we fixed the number of offers accepted by the FSA in our exploratory model EXP 3 , thereby reducing the number of factors by one as compared to the baseline model.
Although the mean of our output variable, fallow land area, was essentially the same as that of the baseline, the variance decreased dramatically.
Thus, this explanatory model revealed complex and important interactions among the remaining factors. Lempert  argued that ABM policy relevance might be improved if utilized for exploratory rather than predictive purposes, reflecting the fact that there is often great uncertainty and little agreement among stakeholders regarding complex, dynamic processes and corresponding decisions. Whereas his suggestion was to exercise large numbers of model runs and use various criteria including robustness, resilience, and stability to evaluate different policies, we have offered a more tractable approach.
By identifying the most influential factors and ignoring others, we developed an ABM model for exploratory purposes; a simplified model with no loss in output that allows for the exploration of various policy scenarios, including rare but potentially catastrophic events. In our example, our exploratory model EXP 2 used only five factors as compared to nine in the baseline model.
Yet, the mean and variance of our output variable, fallow land in conservation, changed little from the baseline. Thus, by reducing model factors, we are able to efficiently explore different, policy-relevant scenarios. Interest in the study of complex socioecological systems or coupled human and natural systems has risen concomitantly with the recognition of profound challenges in the Anthropocene including climate change, biodiversity loss, land use change, alteration of nitrogen and phosphorus cycles, and the depletion of freshwater .
Our ability to address these challenges depends greatly on how well we can make decisions despite great uncertainty. Although utilizing a variety of approaches is certainly of value  , ABMs will likely play an important role in these efforts. Our intent, in utilizing a quantitative UA-SA approach, was to expand ABMs explanatory and exploratory potentials, contributing both to scientific efforts to increase our knowledge and predictive abilities and to policy requirements of making good decisions without complete knowledge.
Factors are presented using uppercase bold fonts. Conceived and designed the experiments: ALZ.
Performed the experiments: ALZ. Browse Subject Areas? Click through the PLOS taxonomy to find articles in your field. Abstract Agent-based models ABMs have been widely used to study socioecological systems. Introduction Socioecological systems are perpetually dynamic and nonlinear  ,  ,  ,  ,  ,  , . Download: PPT. Figure 1. Uncertainty and sensitivity analyses of model output. Sensitivity analysis using output variance decomposition Commonly, SA involves modifying the value of one factor while keeping the other factors constant and observing the effects of this change on model results.
Figure 2. A framework for uncertainty and sensitivity analysis of ABMs of socioecological systems. Figure 4. Agent-based model of enrollment in Conservation Reserve Program. Model input data The ABM uses a number of factors, some that are readily available and some that we derived from auxiliary resources, including land uses obtained from cropland raster layers  , freshwater ecosystems from a lakes and rivers geodatabase  , soil data  , and slope .
Figure 5. This chapter provides an overview of the programming language and concepts that are used within NetLogo. NetLogo basics, such as how to create a simple environment, commands and procedures, are presented with step by step instructions for creating a simple model. Following this basic model, more advanced features are introduced. The overall aim of this chapter is to provide an understanding of the main components that make a NetLogo program. Subsequent chapters build upon the basics presented here. Click here to find accompanying resources for Chapter 4.
This chapter presents the main concepts and terminology that students are required to understand geographical information systems. The main data types are presented, along with a discussion of relevant issues such as accuracy and precision. A brief overview of the development of GIS is given along with a flavour of the main software available. Where appropriate, we highlight the main issues that need to be considered when using a GIS and agent-based modelling. Click here to find accompanying resources for Chapter 5.
Building on previous chapters outlining the fundamentals of GIS and agent-based modelling, what are the benefits to linking these approaches? How is this undertaken? This chapter will explain loose and tight coupling, critiquing the relative advantages and disadvantages of both. We present an overview of open source toolkits that can be used for the creation of geographically explicit agent-based models, before providing a critical look at where and how GIS and ABM should be combined, offering practical advice on best practice.
Click here to find accompanying resources for Chapter 6. This chapter explores the most common approaches by which researchers incorporate human behaviour into agent-based models. We explain why it can be necessary to model human behaviour and the main considerations that the researcher needs to be aware of when developing an agent-based model. From this, we present an overview of the two main broad approaches, mathematical and conceptual cognitive models when it comes to modelling human behaviour in agent-based models.
We supplement this discussion with two case-studies that provide examples of how these approaches can be implemented, both examples have the model code available that can be downloaded and experimented with. The chapter finishes with a discussion of some of the thorny issues that researchers need to be aware of when attempting to simulate behaviour within agent-based models.
Click here to find accompanying resources for Chapter 7. Networks play a critical role in our lives in terms of physical networks we use to navigate upon, our social networks and more recently how we communicate via cyber networks e. This chapter provides a brief introduction to such networks and shows how they can be integrated into agent-based models. Importantly, a model is also introduced that demonstrates how to navigate agents along a physical road network this is a common requirement for spatially-explicit agent-based models. Click here to find accompanying resources for Chapter 8.
This chapter presents a range of statistics and algorithms that can be used to compare two spatial data sets. These are important for modelling because, at some point, it will be necessary to compare a model outcome to some real-world data in order to assess how reliable the model is. This chapter examines the statistics themselves, before Chapter 10 elaborates on how to evaluate the success of a model more broadly, part of which includes making use of the methods discussed here.
Click here to find accompanying resources for Chapter 9. Model evaluation is one of the central challenges associated with agent-based models.
While there are no universally accepted methods for evaluating agent-based models, researchers often adopt the same three stage process of verification, calibration and validation. This chapter presents an overview of the methods that are commonly used within each of these stages. The overarching aim of this chapter is to provide the reader with the knowledge to design their own approach to evaluating agent-based models. Click here to find accompanying resources for Chapter Agent-based modelling is one of the most popular approaches used in social and spatial simulation.
This chapter presents an overview of these other approaches giving simple examples on how they can be used and summarising the main differences between them. To compare these models, they are applied to the same issue, the spread of a disease using a Susceptible-Infected-Recovered SIR epidemic model. This shows that while the same general patterns emerge, the reasons for this are very different.
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This chapter reflects on the current state of the art of agent-based models and factors that may shape the future of this discipline. Specifically we discuss the key challenges for developing robust agent-based models of geographical systems as well as potential solutions. We argue that we need to make progress on these challenges if these models are to be used to offer insight into key societal challenges, for example climate change, urban growth and migration.
While the entirety of this book has given a readers a selection of models mainly in NetLogo with respect to agent-based modelling more generally and linking such models to real world geographical information. We thought it would be useful for readers to have exposure to agent-based models developed not only with NetLogo but also other toolkits. The criteria for inclusion here is that the source code and data of the model is available and it is based on real world geographical information.
In each case we provide a screen shot of the graphical user interface of the model, a short description of the model, its full citation where possible and where the model and data can be downloaded from. Our hope is that the models provide readers with exposure to the possibilities of agent-based models and its potential for analysing a wide array of geographical systems and also share their own models and data. Click here to find accompanying resources for Appendix A. Home Endorsements Authors Get The Book Book Outline Agent-Based Modelling and Geographical Information Systems A Practical Primer Agent-based models - computational models that simulate complex social interactions - have become a well established simulation tool in the social sciences, but until recently their potential within the spatial sciences has been limited.