«Urban Problems and sPatial methods VolUme 17, nUmber 1 • 2015 U.S. Department of Housing and Urban Development | Office of Policy Development and ...»
Intensity Ii is calculated based on the counts of endpoints and the area of each circular section. The series of intensities, then, is examined based on their relationship to the radius of the corresponding circular section di.
To detect the potential directional variation in distance decay, each circular range is separated into different directional sections. Exhibit 5 is showing an example in which each circular range is directionally partitioned into north, south, east, and west sections.
Then, the five specific forms of decay models are examined with respect to each directional section.
The similarity or difference between models for global decay and directionally partitioned decay can be evaluated by comparing the confidence intervals of regression parameters. Because the interest is decay tendency, the evaluation needs to focus on only the slope parameter b in a regression model. After a regression model for global data has been derived, y = α + b x + ε, a confidence interval with a specific significance level for the slope parameter b can be derived as well (b min, b max). Then, the slope parameter bi (i represents the four directions respectively) in the regression model for each directional partition is tested for whether bi ∈ (b min, b max). If the test derives a positive result, it indicates that the two regression models possess a significantly similar slope.
In practical terms, bi ∈ (b min, b max ) means that the data located within the i directionally partitioned data zone possess a similar decay tendency as the entire dataset. The global regression model for distance decay can be appropriately used to describe the decay tendency for movements along a specific direction.
Exhibit 5 Interface for Investigating Distance Decay Over Different Directions Application To test those functions in the framework, 2,363 residential changes, which were derived from 39,232 residential transactions of home sales and purchases in Franklin County, Ohio, from October 2004 to April 2006, are geocoded as matched pairs in a Geographic Information System, or GIS, database. The 2,363 records are regarded as 2,363 movement vectors, as shown in exhibit 6.
Exhibit 6 The 2,363 Residential Movements in Franklin County, Ohio According to a series of reports by the Mid-Ohio Regional Planning Commission, from 2000 to 2030 the northwest region of Franklin County is planning for more employment opportunities and increased economic growth. Thus, cities in the regions surrounding U.S. Highway 33, including Dublin, Upper Arlington, and townships in the northwest part of Franklin County, will likely attract more residents. Because the research data are derived from residential changes from 2004 to 2006, it is reasonable for the planners from the Mid-Ohio Regional Planning Commission who made the regional planning strategies for Franklin County to assume that the patterns of those residential movements will reflect aspects of planned behavior. To assess observed behavior, this application will therefore use the framework to thoroughly explore patterns in the 2,363 residential movements.
Exploratory Analysis of Distance and Direction To get more meaningful insights about spatial pattern for these residential movements, visual analytics are used. Following the procedures of vector standardization and the partition scheme introduced previously, a visual display for the distribution of the dual features is created. This scheme with standardized data offers an integrative version for the two spatial features, as illustrated in exhibit 7.
The counts of endpoints in each distance-direction section of the partition scheme reflect the number of movements for a corresponding distance and direction range. The counts of endpoints, or the frequencies of movements, can be visually illustrated by color gradation. By reading the color gradient tendency, the general pattern for the 2,363 movements becomes more evident. The eight sections of the innermost circle are much darker than those for any peripheral section, suggesting that distance decay is a dominant pattern in residential changes. For the distribution of directions, an imbalanced pattern is clear, which suggests a tendency for more residential changes to be oriented toward the northwest.
Directional Bias in Distance Decay Tendency To further explore the directional bias in distance decay for the 2,363 residential movements, the relationship between intensity of movements and moving distance is explored. Exhibit 8 illustrates the interface for such exploration.
Exhibit 8 Visual Analytics Exploring Movement Intensity (a) and Moving Distance (b)
Exhibit 8 also illustrates the global relationship between intensity of endpoints in each distance rings (without considering the directional partitions) and the radius of each distance circle.
Based on such an interface, the framework further estimates that, for all the movements, the most appropriate regression function is the exponential equation, because it has the highest R-square value, 0.9757. Thus, the function used for describing the global distance decay tendency is established as— I = exp [4.273 – (0.384 x d )]. (8) Such exploration is conducted for data that have been categorized into different directional zones as well. Exhibit 9 summarizes the quantitative descriptions of distance decay for the four directional zones. It is interesting to researchers to notice that square root exponential function is most appropriate for each of the four directions, but none of them follow the same trend as the global tendency.
The square root exponential function mathematically results in a more gradual rate of decay than the basic exponential function. This result actually indicates that the decay tendency is faster globally than it is for any of the directional zone subsets considered.
The framework further compares the global decay to the directionally partitioned decay statistically.
To implement this comparison, the global pattern must be modeled by the square root exponential function in accordance with the model for each partitioned subset. A square root exponential regression equation for the global pattern is— I = exp [7.503 – (2.377 x √d )], (9) with an R-square value of 0.964, which means this model can also appropriately summarize the global distance decay tendency. The test is to evaluate whether the value for the global slope,
-2.377, is significantly different from the slope for each of the directionally partitioned subsets.
Exhibit 10 summarizes the confidence interval (with 95-percent significance level) of the global slope as well as the slope of each partitioned subset.
Exhibit 10 clearly shows that the slope values for the north, east, and west subsets are not within the 95-percent confidence interval (-2.621, -2.134) of the global slope. This result indicates that the decay tendency for the north, east, or west partition is significantly (with 95-percent significance level) different from the globally estimated decay tendency. While the globally estimated tendency indicates a sharper decay slope, the decay tendency for the north, east, and west partitions is significantly smoother. So, if the globally based model is used to describe the pattern of distance decay for the entire set of movements, such a directional effect cannot be appropriately captured.
Pattern Comparison Between Global and Local Exploration of whether the pattern for a subset of the data is consistent with the global pattern can help reveal whether residential changes lead to social differentiation in a city or region. As an example for such exploration, the 20 richest census tracts are considered locally interesting. Then, the residential movements with their destinations located inside these 20 census tracts are selected to form a sample subset. A research question is: “Is the distribution of distances and directions for those movements with destinations in the richest neighborhoods significantly similar to the respective distributions for all movements in Franklin County?” In exhibit 11a, highlighted polygons are the 20 richest census tracts in Franklin County. In exhibit 11b, the movements whose destinations fall within the 20 census tracts are highlighted. This sample contains 538 movements. Exhibit 11c is similar to exhibit 7 but the vectors are standardized by destinations. Those highlighted vectors correspond to the selections in exhibit 11b.
Researchers in this study noticed that most of the highlighted vectors are oriented toward the north, because 18 of the 20 selected tracts are in the northern portion of Franklin County.
Further, based on the comparison metrics introduced previously, the framework tests the significant similarity or difference between the local pattern of the 538 selected movements and the global pattern of all the data by linear regression model. Exhibit 12 summarizes the counts of endpoints with respect to the entire (global) and subset (local) movements in each partition section.
Exhibit 11 Selected Movements With Destinations in the 20 Richest Census Tracts
with a coefficient of determination (R-square value) of 0.87. This R-square value indicates that the two patterns, in general, are statistically similar. In short, the local pattern, which is for the subset of movements with their destinations in the richest regions of the study area, is similar to the global pattern for the complete set of 2,363 residential changes.
Summary Through a series of analyses, the framework facilitates pattern exploration for the 2,363 residential movements in Franklin County, Ohio. At the global level, distance decay has been confirmed as a significant tendency, but the framework also quantitatively detects that a single equation derived from the entire dataset is not able to fully describe characteristics in the distance decay tendency, because the directional bias is also established as a significant effect. For these northwest-oriented movements, the slope for their distance decay tendency is not as steep as the general trend.
The framework also facilitates investigation of local characteristics associated with the identified movement patterns as well. The research examines movements into the richest 20 census tracts in Franklin County. Their movement pattern is established and compared with the global pattern. The framework confirms that the local pattern is actually similar to the global. This finding indicates that these “rich” residential movements did not exhibit a significant difference in spatial behavior from the complete set of movements in this research.
Conclusion This research introduces a framework consisting of a series of functions and methods for exploratory and confirmatory examination of spatial patterns from mass residential movements at a micro scale. The spatial pattern of movement in this research is treated as the arrangement of distances and directions over space. Then, to efficiently detect patterns from mass movements, the framework provides ways to visually standardize movements. Based on standardized movements, a partition scheme is introduced. The distribution of movement distance and direction can then be qualitatively, as well as quantitatively, investigated. By using such an analytical process, movement patterns for 2,363 residential changes within Franklin County, Ohio, have been evaluated. Further, local deviations associated with subsets of movements can be identified from global movements.
This technology has been used to examine patterns for 538 movements from the complete dataset.
The successful application has confirmed the effectiveness of the proposed framework.
Some future developments can be made for improvement. First, some functions of the framework should be more objective for achieving statistically robust insights; for example, the size of the distance-directional interval in the partition scheme. Second, temporal dimensions should be included for analysis as well, because the distribution of residential movements over time can also suggest insights into how humans and the environment interact. Such developments indicate that this framework can exhibit additional strength for effectively and efficiently detecting information from mass residential movements.
Acknowledgments The authors thank Sergio Rey and Elizabeth Wentz at Arizona State University for their review of and suggestions on the context of this paper. They also thank Ron Wilson and the journal editors for their editorial advice.
Authors Yin Liu is an assistant professor in the Key Laboratory of Land Resources Evaluation and Monitoring in Southwest China at the School of Geographical and Resource Science, Sichuan Normal University, Chengdu, China.
Alan T. Murray is a professor in the Center for Spatial Analytics and GeoComputation at the College of Computing & Informatics and in the School of Public Health, Drexel University, Philadelphia, Pennsylvania, United States.
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