Consistency in the spatial structure of surfaces Yukio SADAHIRO Department of Urban Engineering...
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![Page 1: Consistency in the spatial structure of surfaces Yukio SADAHIRO Department of Urban Engineering University of Tokyo Analysis of similarity among surfaces.](https://reader035.fdocuments.us/reader035/viewer/2022070415/5697bf7d1a28abf838c84cef/html5/thumbnails/1.jpg)
Consistency in the spatial structure of surfaces
Yukio SADAHIRO
Department of Urban Engineering
University of Tokyo
Analysis of similarity among surfacesdefined in the identical region
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Surfaces (scalar fields) in a region
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Objective of the study
To propose
1) a method for measuring the similarity of spatial structure among surfaces
and
2) a method for detecting the spatial structure shared by surfaces.
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Outline of the method
“To what extent are surfaces similar ?”
1. Evaluation of the spatial similarity among surfaces■ Representation of surface slope as a binary function■ Evaluation of similarity among surfaces as a function■ Evaluation of total similarity among surfaces by a quantitative m
easure
“How are surfaces similar?”
2. Detection of the spatial structure shared by surfaces■ -peak regions: regions in which many surfaces have peaks■ -pit regions: regions in which many surfaces have pits■ -monotonic lines: lines on which many surfaces have the same m
onotonic trend
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1. Slope function
Slope function represents the trend of a slope, whether it increases or decreases, given a location and a direction.
Original surfaces
1
0
Slope functions
Similarity of surface structure
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Surfaces having the same slope function
Original surfaces
Slope function 1
0
Similarity of surface structure
Slope function regards surface functions of the same spatial trend as equivalent.
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Slope function in a two-dimensional space
A binary function of location and direction
Slope functions of different directionsOriginal surface
Similarity of surface structure
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2. Similarity function
The maximum ratio of surfaces of the same slope function value
1.0
0.0
0.2
0.4
0.6
0.8
Similarity of surface structure
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3. Similarity index
Total similarity among surfaces in a given region
1.0
0.0
0.2
0.4
0.6
0.8
Similarity of surface structure
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1. -peak regions and -pit regions
Regions in which many surfaces have peaks and pits
Detection of common spatial structure
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Various -peak regions
Detection of common spatial structure
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-peak regions useful for analysis
1. Contain peaks of many surfaces
2. Small
3. Simple shape
4. Not overlap with each other
1. Contain peaks of a certain proportion () of surfaces
2. Smaller than a certain size
3. Circular
4. Not overlap with each other
Detection of common spatial structure
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Extraction of useful -peak regions
1. Put a lattice on points.
2. Draw small circles centered at lattice points.
3. Expand the circles until peaks of a certain proportion () of surfaces are contained.
4. Shrink the circles as small as possible, keeping the peaks contained
5. Extract circles from smaller ones in turn without spatial overlap.
Detection of common spatial structure
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2. -monotonic lines
-monotonic lines are line segments on which many surfaces have the same trend, either increase or decrease, in the same direction.
Detection of common spatial structure
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Extraction of -monotonic lines useful for analysis
1. A part of radial lines extending from centers of -peak regions.
2. Lines on which a certain proportion () of surfaces have the same trend.
-peak region
-monotonic lines
Detection of common spatial structure
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Empirical study: market area analysis
Shop:
A ‘gourmet’ supermarket in the neighborhood of Tokyo, Japan
Source data:
Purchase history of customers from May 13 to June 9, 2002
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Average distribution of customers
RailwaysSupermarkets
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Customer distribution from May 13 (Mon) to June 9 (Sat), 2002
Mon FriTue ThuWed SatSun
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Similarity function of customer distribution
RailwaysSupermarkets
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-peak regions, -pit regions, and -monotonic lines
RailwaysSupermarkets
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Conclusions
New methods for evaluating the similarity of spatial structure among surfaces and detecting the spatial structure shared by surfaces was proposed.
The method was applied to market area analysis of customer purchase data of a supermarket in the neighborhood of Tokyo, Japan.
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For details,
Sadahiro, Y. and Masui, M. (2002): “Analysis of similarity among surfaces defined in the identical region.” Discussion Paper, 93, Department of Urban Engineering, University of Tokyo.
http://ua.t.u-tokyo.ac.jp/okabelab/lab/due-dp/93.pdf
Thank you for your attention.