For Assignment 5, we returned to the model assembled in assignment four in order to determine which of the initial conditions has the most influence on emerging patterns, i.e. to which parameters the model is most sensitive. We performed this sensitivity analysis by conducting a sweep of relevant model parameters. A parameter, as opposed to a variable, is a value that describes the conditions within which a simulation is conducted: it does not change within any given run, but is set prior. In this model, the most obvious parameters were the three primary inputs to the utility function: aesthetic quality preference (AQP), distance to service center preference (DSP), and nearest neighbor distance preference (NDP). (Ideal density was excluded due to difficulty of analysis: since the feedback examination from assignment four showed that the ideal density parameter could induce positive or negative feedbacks, its impact is ambiguous and would require a considerably more rigorous data analysis to determine. It was therefore held constant at .75 for this sensitivity analysis.) Based on model logic and informal observation from assignment four, the number of random candidate sites selected per home placement (n_test) also appears to have a strong effect on emerging patterns, and was also included in the sweep.

The parameter sweep thus included four parameters, each of which was run at three values:

	sweep values
AQP	0	0.5	1
DSP	0	0.5	1
NDP	0	0.5	1
n_test	16	32	64

Each unique combination of parameters was run ten times, for a total of 810 runs. In order to compare such a large number of runs, it is necessary to create a measurable emergent outcome. We created a measure of mean nearest neighbor distance (nnd): the average of the distance between each home and its nearest neighboring home.

Results

Descriptive statistics for the runs were generated using SPSS (http://www-01.ibm.com/software/analytics/spss/).

 

Figure 1. Histograms for AQP, DSP, NDP, and n_test.

A statistical aberration immediately jumps out: within each parameter, there is a group of runs at the “0” weighting with atypically high nnd values. Examining a visual output of these runs, the pattern becomes clear: these are runs where the weighting of all three of the utility function parameters were set at zero, leading to a distribution of homes with no expressed preference, i.e. random. Under these conditions, n_test becomes irrelevant as well.

 

Figure 2. Model runs with AQP = 0, DSP = 0, NDP = 0, n_test = 16, 32, 64.

In contrast to these essentially random runs, any preference at all quickly leads to the emergence of distinct patterns with considerably lower nnd values. Disregarding these outliers, more general observations about the sensitivity of the model to different parameters can be made. The histograms for AQP and DSP exhibit very little variation in nnd at 0, 0.5, and 1: while the interquartile range narrows at higher parameter values, the median and range remain quite similar. Insofar as analysis is limited to average nearest neighbor distance, these parameters appear to have very little impact on the model.

This stands in contrast to NDP and n_test parameters. The nnd histogram shows a marked change at different values for near-neighbor preference: the mean value for nnd goes from 1.63 at NDP = 0.0 to 1.32 at NDP = 0.5 to 1.28 at NDP = 1.0. As evidenced by the histogram. The impact of the n_test parameter is even more dramatic: of the four parameters analyzed, it exhibits the strongest influence on the model. The mean value for nnd goes from 1.63 at n_test = 16 to 1.37 at n_test = 32 to 1.22 at n_test = 64. Even more noticeable is the rapid narrowing of the range of values: at n_test = 64, the interquartile range is 0.089 (compared values between 0.44 and 0.30 for the other three parameters.)

Interpretation

The importance of nearest neighbor preference (NDP) to the measure of average nearest neighbor distance is rather self-evident: as it is the parameter that most directly encourages lower distances between neighbors, it is no surprise that it has a strong effect on it. In contrast, the other two parameters can only affect nnd indirectly, by drawing homes closer to attractions or service centers and thereby closer together. Indeed: the slight increase of nnd at high levels of AQP and DSP may indicate that they begin to select against nearest neighbor distance at extremes, though the difference is small enough to be statistical noise.

The even more significant impact of n_test is harder to account for, or to accept, within the confines of model logic. While the other three parameters incorporate some degree of real world validity (people do prefer certain levels of density, aesthetic quality, and service accessibility), n_test is relatively arbitrary in design: there must be some constraint on home siting, but whether 4 or 64 is more accurate is difficult to ascertain. That the model is more sensitive to n_test than to the “real world” parameters thus presents a challenge to the model’s validity which must be resolved. Further simulations could be used to calibrate n_test by finding the range within which it exhibits equal or lesser impact on outcomes than the other three.

n_test =                       16                                    32                                  64

NDP = 0.0   

NDP = 0.5   

NDP = 1.0   

 

Figure 3. A visual comparison of final model states from runs with varying NDP and n_test values. AQP and DSP = 0.5 in all. As is evident, high n_test values consistently result in tighter clusters and lower nnd values than high NDP values.

As discussed above, the choice of nearest neighbor distance as the measure of emergent model outcome naturally emphasizes the role of the NDP parameter. This parameter is relevant to real world applications because actual home-owners have been shown to have preferences for certain levels of neighborhood density: some prefer to have more neighbors while others prefer more secluded sites. Another measure that may be used to draw out other characteristics of the model would be average distance to initial service center. This measure (as opposed to say, average distance to nearest service center) would highlight the role of early changes in the model (i.e. the introduction of the first service center) on final outcomes: how much does the location of the first service center constrain future growth? This would allow the modeler to explore the role of path dependency in the model’s emergent outcomes.

Conclusion

This model describes how different factors in residential decision-making can lead to different patterns of growth. Some factors are static (location of attractions), and some evolve over the course of the model (service center locations and neighborhood density): those which can evolve exert greater influence over time, leading to path dependence. When emergent elements in the model are given greater weight, then outcomes become less predictable: for example, if aesthetic quality preference is the primary factor, then model outcomes will tend to have similar emergent patterns. When service center preference is given greater weight, emergent outcomes will become more diverse. Models such as this one can help researchers determine the degree to which these different elements play a role in residential decision making.

   

Figure 4. Runs with high AQP; low DSP & NDP. (Exact values in file name; mouse over to view.)