chapter 6 visual analysis - 東京大学ua.t.u-tokyo.ac.jp/okabelab/sada/docs/pdf_class/ch06.pdf2 6....

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Chapter 6Visual Analysis

6.　Visual Analysis

Visual analysis is the first step of spatial analysis.

It greatly helps us finding interesting or important patterns in the distribution of spatial objects, scalar and vector fields, relationship between spatial distributions, etc..

Figure: Dr. John Snow’s Map of London (cases of cholera, 1854)http://www.ph.ucla.edu/epi/snow.html


You can visualize spatial objects in various ways. One method is to visualize them in their original form.

However, it often helps your analysis to convert spatial objects into a different form, which is the main subject of this chapter.


• References - visualization

1. Monmonier, M. (1993): Mapping It Out : Expository Cartography for the Humanities and Social Sciences , University of Chicago Press.

2. MacEachren, A. M. (1994): Some Truth with Maps: A Primer on Symbolization and Design, Association of American Geographers.

3. MacEachren, A. M. and Taylor, D. R. F. (1994): Visualization in Modern Cartography, Pergamon.

4. MacEachren, A. M. (1995): How Maps Work, Guildford Press.


5. Robinson, A. H., Morrison, A. L., Muehrcke, P. C., Kimerling, A. J., and Guptill, S. C. (1995): Elements of Cartography, John Wiley.

6. Kraak, M.- J. and Ormeling, F. (1996): Cartography: Visualization of Spatial Data, Addison-Wesley.

7. Slocum, T. A. (1998): Thematic Cartography and Visualization, Prentice-Hall.

8. Campbell, J. (2000): Map Use & Analysis, McGraw Hill.

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6.1 Visualization of point distributions

Point is the most fundamental spatial object in GIS. There are many ways of visualizing the distribution of points.

Figure: Point distribution


6.1.1 Quadrat method

Quadrat method converts point data into raster data by

1. overlaying a square lattice, and2. count the number of points in each cell.

Quadrat method is thus a kind of data aggregation.

Figure: Quadrat method


Limitations of quadrat method

Though quite simple and easy to perform, quadratmethod has two limitations.

1. The distribution represented by raster data is not smooth, so it does not look nice.2. The result depends on cell size. Users may have to try various cell sizes to obtain a good result.


6.1.2 Kernel method

Kernel method overcomes the first limitation of quadratmethod. Kernel method generates smooth surfaces by

1. putings small bumps called kernels on the points, and2. visualizing the accumulated small bumps.

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Kernel method was originally developed for estimating the probability density function from a set of observed data (sample data).


• References - kernel method

1. Silverman, B. W. (1986): Density Estimation, Chapman & Hall.

2. Scott, D. W. (1992): Multivariate Density Estimation, John Wiley.


zi: Location of point ik(x, t): Kernel function at x


Kernel functions

1. Gaussian (normal) kernel

( ) 22 2

1 1, exp2 2

kπσ σ

= − − x t x t

Figure: Gaussian kernel


2. Epanechnikov kernel (a part of paraboroid)

( ) ( )22 23 1 if 1,

0 otherwisek π

− − − ≤=

x t x tx t

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Figure: Epanechnikov kernel


3. Triangle kernel (quadrangular pyramid)

Though this kernel function has a simple shape, its mathematical representation is rather complicated.

Figure: Triangle kernel


A point distribution is converted into a surface function by

( )( )

( )

,

, d

ii

ii

kf

k=∑∑∫

x zx

t z t

Figure: Kernel method


The result depends on the shape of the kernel function, especially the steepness of its slope.

If we use a gentle kernel, we obtain a smooth surface, which is suitable for understanding the global structure of point distribution.

A steep kernel yields a rough surface, which is suitable for looking at local variation of point distribution.

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This property is quite similar to the effect of cell size in quadrat method.

If we use large cells, we have a smooth, though not continuous, surface. Large cells corresponds to gentle kernel. Similarly, small cells yields rough surface as steep kernel does.

Figure: Rough surface obtained by a steep kernel

Figure: Smooth surface obtained by a gentle kernel Figure: Smoother surface obtained by a gentler kernel

Figure: Customer distribution of a shopping street Figure: Expected customer distribution of a new supermarket

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6.1.3 Visualization of attributes of point objects

Quadrat and kernel methods are used for visualizing ‘spatial information’ about point objects.

Point objects, on the other hand, also have ‘aspatialinformation’, that is, attribute data.


To visualize attribute data of point objects, we use point symbols. Properties of point symbols are

1. shape,2. size,3. color, and4. orientation.

Difference in attribute data is represented by that in properties of point symbols.

Shape Size Color Orientation

Figure: Four elements of point symbols


Size and color are used for visualizing numerical variables. Larger symbols usually correspond to (relatively) larger values of an attribute variable.

Shape is used for categorical variables. Difference in shape is associated with qualitative difference of spatial objects.

Orientation is used for wind direction, flows, etc..


Two relationships of symbol size

In principle, point symbols get larger with an increase of attribute value.

There are two quantitative relationships between symbol size and attribute value that are frequently used in GIS.


1. Proportional method

The size (area) of point symbols is proportional to the attribute value.

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-10,000

-25,000

-50,000

-100,000

Figure: Proportional method


2. Perceptual method

Symbol size is determined so as to fit the human perception of point symbols.

Perceptual method exaggerates the difference in symbol size than proportional method. For instance, symbol size is proportional to the square of attribute value.

-10,000-25,000

-50,000

-100,000

Figure: Perceptual method


In commercial GIS, perceptual method is used more frequently than proportional method.


6.2 Visualization of spatial tessellation

Spatial tessellation is a set of spatial units that extensively cover a bounded region. In this sense it is a special case of a set of polygons.

Typical examples include census tracts, administrative units, and school districts.


Spatial tessellation is a very important spatial structure found in GIS and spatial analysis, because a lot of detailed data are aggregated by spatial units to ensure confidentiality when they are opened to the public.

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‘Spatial information’ about spatial tessellation is visualized by indicating boundaries of spatial units.

Attribute data may include numerical variables, say, population density and average income, or categorical variables such as land use category, or both numerical and categorical variables.

Figure: Tessellations with numerical and categorical variables


Choropleth map

To visualize attribute data of spatial tessellations, a map called ‘choropleth map’ is used. Choropleth map is a map showing attribute data of a spatial tessellation by colors and textures.

Categorical variable is directly visualized by colors and textures: different colors indicate different categories. Numerical variable is first categorized into several classes, and then visualized by the progression of colors and textures.


To make a choropleth map of a numerical variable, we determine

1. classification scheme,2. class number,3. class boundaries, and4. colors and textures.


6.2.1 Classification schemes

There are four schemes for categorizing numerical variables.

1. Equal interval2. Quantile3. Nonuniform4. Irregular interval


Equal interval

Equal interval scheme categorize a numerical variable by an equal interval value.

If we specify interval value, GIS calculates boundary values, classifies spatial units into categories, and visualize the categories by different colors or textures.

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84 149 74 162 91

0 20 40 60 80 100

Figure: Equal Interval


Quantile

Quantile scheme categorize a numerical variable so that each category has the same number of spatial units.

In this scheme we give the number of categories instead of interval value. GIS then calculates boundary values, classifies spatial units into categories, and visualize the categories by different colors or textures.

100 100 100 100 100

0 22 32 58 78 100

Figure: Quantile


Nonequal interval

When we are interested in a certain range of attribute value, we want to use finer categories in the range. In this case we use nonequal interval scheme.

Typical examples include monotonically increasing (decreasing) interval scheme, in which the interval monotonically increases (decreases) with attribute value.

0 33.3 60..0 80.0 93.3100

33.3 26.0 20.0 13.3 6.7

03.2 9.7 22.6 48.4 100

3.2 6.5 12.9 25.8 51.6

0 51.6 77.4 90.393.3100

51.6 25.8 12.9 6.53.2

0 10.7 26.7 47.5 72.5 100

10.7 16.0 20.8 25.0 27.5

0 9.3 20.4 34.9 56.6 100

9.3 11.1 14.5 21.7 43.4

Arithmeticprogression

Geometricprogression

Irregularlyincreasing

progression

Figure: Nonequal interval


Irregular interval

Frequency distribution of attribute value often shows ‘breakpoints’, where the frequency suddenly drops.

In such a case, we can obtain a natural classification of attribute variable if we take the breakpoints as boundaries of intervals.

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0 34 56 78 100

Breakpoints

Figure: Irregular interval

Breakpoints

0 34 56 78 100944416

Figure: Irregular interval


6.2.2 Class number

In theory, we can use any number of categories in classification of numerical variables. You may think that the more categories you use, the better map you obtain.

In practice, however, we can discriminate only a limited number of colors used for visualizing categories. It is not always useful to increase class number.


We generally use from five to seven classes in choropleth maps when using the single hue progression.

Figure: Effect of class number


If we want a finer classification of the variable, we should use two or more hues, or we should combine several progressions together.

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Bi-polar progression

Partial spectralhue progression

Blendedhue progression

Full spectralprogression

Value progression

Two-variablecolor progression

Figure: Color progressions


6.2.3 Class boundary

Class boundary greatly affects the appearance of choropleth maps that represent numerical variables.

We should be careful when classification scheme involves subjective choice of class boundaries, as seen in nonequaland irregular interval schemes.

Figure: Effect of class boundary


In choropleth map, map readers tend to pay attention to large polygons while they often overlook small polygons.

Attribute data of large polygons are more influential than those of small polygons in the perception of map readers. Large polygons are emphasized as a result of visualization. This often leads to misunderstanding of the spatial distribution of attribute values.

6.2.4 Effect of the size of spatial units

Figure: Effect of the size of spatial units


To avoid this perceptual illusion, we often perform areal interpolation. We convert spatial data reported by irregular spatial units into those based on regular lattice, where all the spatial units are identical.

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Figure: Spatial data based on a square lattice


Kernel smoothing is also useful to avoid the perceptual illusion. The result is visualized by isopleth (isarithm) map or bird’s eye view.

20

6060

80100

8040

Figure: Isopleth map Figure: Isopleth map with color

Figure: Bird’s eye view Figure: Bird’s eye view

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6.3 Cartogram

Cartogram is a cartographic technique that transforms a map to be ‘uniform’ with respect to the distribution of spatial objects. Spatial units containing many objects are represented by large polygons, while those of fewer objects are small polygons.


A typical example is found in the analysis of the distribution of disease cases. If we find a cluster of cases, we think that the disease is epidemic. However, it may not be true because we can find clusters in regions where many people live.

Cartogram rescales the map of case distribution according to the population count to make the population distribution uniform. Cartogram helps map readers’understanding of a spatial distribution considered to be closely related to another non-uniform spatial distribution.

Figure: Census tracts of San Francisco City/County (1980) Figure: Original map

Figure: Map of uniform population distribution Figure: Hypothetical cases of a disease

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Figure: The cases under the uniform population distribution


Cartogram is useful

1. when we analyze the distribution of spatial objects that seems affected by other non-uniform distributions, say, population distribution, terrain elevation, etc., and

2. when we determine the location of urban facilities that should be uniformly located with respect to other nonuniform distributions: schools, post offices, etc..


6.4 Visualization of higher-dimensional objects

Visualization of two-dimensional spatial objects is relatively straightforward because computer displays and papers are both two-dimensional spaces.

On the other hand, visualization of two-dimensional spatial objects is far more difficult and is a recent topic in GIS research.

1. Visualization of three-dimensional spatial objects2. Visualization of four-dimentional (spatiotemporal) objects


• References - modern visualization

1. Cromley, R. G. (1992): Digital Cartography, Prentice-Hall.

2. MacEachren, A. M. and Taylor, D. R. F. (eds) (1994): Visualization in Modern Cartography, Elsevier.

3. Slocum, T. A. (1998): Thematic Cartography and Visualization, Prentice-Hall.

Figure: Three-dimensional point distribution Figure: Three-dimensional scalar field

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Figure: Spatiotemporal scalar field Figure: Spatiotemporal scalar field


6.5 GeoVISTA

GeoVISTA Center at the Pennsylvania State University

Its goals are to support research in GIS, with an emphasis on geographic visualization (geovisualization), and to facilitate application of advances in GIS and related technologies within science more broadly as well as within business, government, and education.

http://www.geovista.psu.edu


6.6 Visualization and sound

Human beings have five senses -- sight, smell, hearing,taste, and touch. This implies that we can use other than sight in order to communicate spatial information.

J. B. Krygier (1994): “Sound and Geographic Visualization”, in MacEachren, A. and Taylor, D.R.F. (eds.), Visualization in Modern Cartography, Pergamon, pp. 149-166.

http://www.owu.edu/~jbkrygie/krygier_html/krysound.html

Figure: Communication of spatial information by sound


Homework Q.6.1 (20 pts)

Suppose that 10, 000 points are distributed randomly in a bounded region. We put a lattice of 100 (10 by 10) cells on the region, and count the number of points in each cell. We visualize the cell count by quadrat method in which the cells are classified into l categories by quantile scheme.

Evaluate the expected values of class boundaries when

1) l=2, 2) l=3, 3) l=5.

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Homework Q.6.2 (40 pts)

Suppose that m points are distributed randomly in a bounded region. We put a lattice of n cells on the region, and visualize the point distribution by quadrat method.

Given a point distribution, we can calculate the frequency distribution of the point count in the cells. When a point distribution follows the random distribution (the binomial distribution), evaluate the variance of point count as a function of m and n, and discuss how cell size affects the appearance of lattice data representing point distributions.

chapter 6 visual analysis - 東京大学ua.t.u-tokyo.ac.jp/okabelab/sada/docs/pdf_class/ch06.pdf2 6....

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