Consistency in the spatial structure of surfaces Yukio SADAHIRO Department of Urban Engineering University of Tokyo Analysis of similarity among surfaces.

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

Surfaces (scalar fields) in a region

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.

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

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

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.

Slope function in a two-dimensional space

A binary function of location and direction

Slope functions of different directionsOriginal surface

Similarity of surface structure

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

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

1. -peak regions and -pit regions

Regions in which many surfaces have peaks and pits

Detection of common spatial structure

Various -peak regions

Detection of common spatial structure

-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

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

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

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

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

Average distribution of customers

RailwaysSupermarkets

Customer distribution from May 13 (Mon) to June 9 (Sat), 2002

Mon FriTue ThuWed SatSun

Similarity function of customer distribution

RailwaysSupermarkets

-peak regions, -pit regions, and -monotonic lines

RailwaysSupermarkets

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.

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.