Spatial Modeling – Julie StringhamThis page explores path dependence and feedbacks, two interrelated concepts, in a model proposed by Brown et al.
Part 1. Why is it important to test each of your sub models independently?
Explain how you tested the sub-components of the main model using images and text to illustrate your methods and explain how your diagnostics conformed (or not) to your expectations.
When constructing a model that replicates a complex system, we find that testing the model’s subcomponents benefits us in the two following ways: (1) it allows us to systematically test and debug our model, and (2) in doing so, we familiarize ourselves with the components of the model – how they complement and interact with one another – to better understand the organization of our system and its emergent patterns.
Three sub-models are presented in this project (arguably four, if you include development). Two of the sub-models show how far away a given patch is away from either a service center, or area of aesthetic quality. Another component calculates the utility of an area.
Since our sub-models are deterministic in nature, we are able to hypothesize the effects a parameter change will have on each of the models.
Our first two models are very basic in that they simply calculate the distance a given area is to either an area of aesthetic quality, or service center. Below, we can see that there is no change in the pattern even with different parameters.
Utility on the other hand is affected by parameter changes. In the model proposed by Brown et al., agents are given a set of patches, and choose the patch that appeals to them the most through a simple calculation that pays respect to agent preference parameters.
We find an expected outcome when calculating utility with different combinations of preference parameters. When agents are given a high preference of aesthetic quality, and no preference of distance to service centers, we find that areas of high utility center around our predefined points of high aesthetic quality. A similar reaction when flipping the parameters, and giving agents a high preference to service centers. Lastly, when we give agents an equal preference of both parameters, we see a blend between the two patterns.
Part 2. The methods for calculating the components of utility are deterministic in this model. How is stochasticity implemented? What parameter influences the degree of stochasticity?
The “development” component in this model incorporates stochasticity in a few different ways. The most influential implementation of stochasticity is seen in housing settlement, which is dictated by the variable n_test. In each time step, an agent is presented with a number of randomly selected patches and chooses the patch with the highest calculated utility to settle in. The number of patches in which the agent is able to choose from is defined as the user-defined variable n-test.
Theoretically, the higher the number of patches an agent is able to choose from, the higher the likelihood the agent will be able to choose an ideal patch. When the variable is low, a house has fewer options to settle, and thus may not have the opportunity to settle near its preferred location. These two patterns are expressed in the videos below:
Part 3. How is the initial environment heterogeneity implemented in this model and what parameter(s) determine its importance? Do you think that greater environmental heterogeneity corresponds to greater variability among replicate model runs, and why? Support your argument with evidence from replicate simulations with and without initial environment heterogeneity.
In this model, environmental heterogeneity is implemented by determining two spots in our environment that we define as points of high aesthetic quality. The parameter aesthetic_quality_preference determines how much aesthetic quality influences the utility of a patch, and thus influences where clusters of development will form. For instance, if aesthetic quality is highly preferred, agents will cluster by points of aesthetic quality, or near them depending on whether or not we choose to have an initial service center.
The utility visualization gives us clear clues into why the spatial clusters appear where they do. In the previous videos, we note that clusters appear in areas of high utility. However, in the case where there is no distinction between areas’ utility, we still find that agents create spatial clusters.
Without reference to aesthetic quality, we see patterns that are driven by path dependence. This is because no attribute of our environment affects the settlement patterns of agents. However, we e can clearly see clusters of agents forming in the space independently from the areas of high aesthetic quality. Since agents still prefer being near service centers, and since service centers are placed near houses, we see clusters forming both spatially and temporally.
This is replicated in the video above. Note that since the cluster is not controlled by environmental attributes, the cluster forms in different spaces when the model is run multiple times with the same parameter. Without environmental heterogeneity, we see the location of spatial clusters to be highly variable between model runs.
Part 4. Explain the roles of feedbacks and path dependence in this model. How are the two concepts related? How do the three parameters for preference relate to feedbacks and path dependence? What parameter settings would you use to eliminate any effects of feedbacks or path dependence?
Feedback (namely, positive feedback) and path dependence are two interrelated concepts which both appear in this model and deeply impact the spatial patterns we observe. In our model, these two concepts are integral to the formation of urban clusters.
For instance, an initial, random cluster of agents eventually influences where service centers are placed, and will thus attract more houses to appear in the area. This is both an example of path dependence, and a positive feedback, as it is an instance where the initial placement of agents influences areas where future agents will be placed, as it is a self-reinforcing, spatial and temporal pattern.
Here we display the spatial patterns that are the result of an extremely simple parameter sweep, where all other parameters were kept at their average value (either 0.5 or 0.75 for ideal density):
We’ve already recognized that environmental heterogeneity hinders the effects of path dependence, and that the aesthetic_quality_preference parameter disrupts path dependence when it is high, and emphasizes it when it is low. With regard to the other preference parameters, we find that the overall pattern is not phased by dramatic changes in neighborhood_density_preference, but is influenced by the preferred distance to service centers.
Since the location of services is directly related to the location of homes, we will see a prominent feedback loop when houses are attracted to service centers. Changing the aesthetic_quality_preference will not on its own influence feedback, but will focus urban clusters to either end of the environment. Lastly the neighborhood_density attribute does not affect the feedback or path dependence boldly.
To eliminate all feedbacks and path dependence, we would eliminate the initial_center, and aesthetic_quality_preference (to create a homogenous environment). Creating a homogenous environment, and then lowering the distance_service_preference attribute will create an entirely stochastic pattern in both residential and commercial settlement. This results in the following spatial pattern:
Part 5. Describe at least two of the model assumption or simplification and how they could influence your interpretation of model results. Despite these assumptions and limitations, what can we learn from this model?
In the proposal of this model, Brown et. al mentions that in a real urban landscape, we would take into account road networks and transportation to calculate euclidean distance to service centers. In this simplified model, we do not do this. This could have a large impact on house distribution (although perhaps you could argue in a more generalized sense that a service center could be a transportation hub). Brown et al. also mention that our method for determining residential housing space is a brash generalization, and does not take into account areas of reserved land-use such as parks, public spaces, and industrial complexes.
Despite the generalization of this model, we recognize interesting feedbacks in the model, and begin to gain an understanding of how commercial land use both influences and is influenced by residential units and areas of aesthetic quality.