Introduction
This report will address the increase in foreclosures in Dane County, Wisconsin over the 2011-2012 time period. The results will confront the concern that more foreclosures will occur in 2013 by creating probabilities of the scenario.The reasons behind the increase cannot be determined with the data included, but this study will analyze the spatial patterns of the foreclosures to understand the likelihood that foreclosures will increase in 2013.
Methods
This study utilized z scores, which standardize the foreclosure data, to analyze the spatial differences between foreclosures in different census tracts. The z score takes takes an observation, subtracts the mean from it, and then divides that value by the standard deviation. This value gives readers an idea how far each observation is from the mean. These values can be placed on a normal distribution curve. Using a z score chart allows the user to find probabilities of any occurrence (Figure 1). The z-scores collected on foreclosures in 2011-2012 are used in this manner to analyze the probability that census tracts will exceed the mean. The data will also be used to view the probability that foreclosures in all of Dane County will be exceeded 70% of the time in 2013, and the probability that foreclosures in Dane County will be exceeded only 20% of the time.
| Figure 1: Complete table of z-score statistics and their probability of occurring. |
The census tract data is obtained through the US Census Bureau, and the foreclosure data has been collected and geocoded independently. The census tracts chosen to study within Dane County are 122.01, 31, and 114.01. Using ArcMap, the mean and stand deviations are collected to use in this study to analyze the results.
Table 1 shows the foreclosure counts for the chosen counties within this study.
Census Tract
|
2011 Count
|
2012 Count
|
122.01
|
6
|
6
|
31
|
24
|
18
|
114.01
|
32
|
39
|
Table 1: Foreclosure counts for the study area within Dane County.
2011 Mean: 11.39
2011 Standard Deviation: 8.776
2012 Mean: 12.299
2012 Standard Deviation: 9.906
Figure 2 shows the study area of Dane County within Wisconsin.
 |
| Figure 2: Image highlighting Dane County in Blue. |
Results
Table 2 displays the calculated z scores and probabilities for each census tract.
Census Tract
|
2011
Z Score
|
2011 Probability
|
2012 Z Score
|
2012 Probability
|
122.01
|
-.61
|
72.57
|
-.64
|
73.89
|
31
|
1.44
|
92.51
|
.58
|
71.9
|
114.01
|
2.35
|
99.06
|
2.7
|
99.65
|
Table 2: Z scores and probabilities calculated using Figure 1.
Figure 1 shows that for a 70% exceeded likelihood, the z-score would have to equal -.52.
-.52= (Xi-12.299)/9.906
(9.906*-.52)+12.999= Xi
Xi=7.85
The value of 7.85 means that there is a 70% chance that Dane County will have observations of 7.85 foreclosures or greater across each census tract. This seems to be the most realistic outlook
Figure 1 shows that for a 20% exceed likelihood, the z-score would have to equal .84.
.84= (Xi-12.299)/9.906
(.84*9.906)+12.299=Xi
Xi=20.62
The value of 20.62 means that there is a 20% chance that Dane County will have observations of 20.62 foreclosures or greater across each census tract.
Figure 3 illustrates the change in foreclosures from 2011-2012. The results were then converted to z scores to illustrate a diverging theme very clearly. Negative values indicate higher change in 2012. On can see that there are more extreme negative values (<-2.5) when looking at the map then extreme higher values (>2.5) throughout the country. Another observation that can be made is that it appears that there is more blue tract area than red tract throughout the country. Other data was needed to analyze this further.
 |
| Figure 3: Change map showing the difference in home foreclosures from 2011-2012. The data is normalized into z-scores to display a diverging scheme clearly. |
Figure 4 displays the statistics of the new field which calculates the difference between the 2011 and 2012 foreclosures. The key statistic in Figure 4 is the mean, -.906. Like Figure 3, a negative value means that there was growth in foreclosures in the county. Having a negative mean means that there is a trend upwards in the amount of foreclosures.
 |
| Figure 4: This image displays the statistics and graph describing the difference between 2011-2012 field. |
