Chapter 2 Feature Representation

2.1 Vector vs. Raster Data Models

To work in a GIS environment, real world observations (objects or events that can be recorded in 2D or 3D space) need to be reduced to spatial representations. These spatial representations can be presented in a GIS as a vector data model or a raster data model.

Figure 2.1: Vector and raster representations of a river feature.

Figure 2.2: Vector and raster data model example.

2.1.1 Vector

Vector features can be decomposed into three different geometric primitives: points, lines/polylines and polygons.

2.1.1.1 Points

Figure 2.3: Three point objects defined by their X and Y coordinate values.

A point is composed of one coordinate pair representing a specific location in a coordinate system. Points are the most basic geometric primitives having no length or area. By definition a point can’t be “seen” since it has no area; but this is not practical if such primitives are to be mapped. So points on a map are represented using symbols that have both area and shape (e.g. circle, square, plus signs).

We seem capable of interpreting such symbols as points, but there may be instances when such interpretation may be ambiguous (e.g. is a round symbol delineating the area of a round feature on the ground such as a large oil storage tank or is it representing the point location of that tank?).

2.1.1.2 Polylines

Figure 2.4: A simple polyline object defined by connected vertices.

A polyline is composed of a sequence of two or more coordinate pairs called vertices. A vertex is defined by coordinate pairs, just like a point, but what differentiates a vertex from a point is its explicitly defined relationship with neighboring vertices. A vertex is connected to at least one other vertex.

Like a point, a true line can’t be seen since it has no area. And like a point, a line is symbolized using shapes that have a color, width and style (e.g. solid, dashed, dotted, etc…). Roads and rivers are commonly stored as polylines in a GIS.

2.1.1.3 Polygons

Figure 2.5: A simple polygon object defined by an area enclosed by connected vertices.

A polygon is composed of three or more line segments whose starting and ending coordinate pairs are the same. Sometimes you will see the words lattice or area used in lieu of ‘polygon’. Polygons represent both length (i.e. the perimeter of the area) and area. They also embody the idea of an inside and an outside; in fact, the area that a polygon encloses is explicitly defined in a GIS environment. If it isn’t, then you are working with a polyline feature. If this does not seem intuitive, think of three connected lines defining a triangle: they can represent three connected road segments (thus polyline features), or they can represent the grassy strip enclosed by the connected roads (in which case an ‘inside’ is implied thus defining a polygon).

2.1.2 Raster

Figure 2.6: A simple raster object defined by a 10x10 array of cells or pixels.

A raster data model uses an array of cells, or pixels, to represent real-world objects. Raster data sets are commonly used for representing and managing imagery, surface temperatures, digital elevation models, and numerous other entities.

A raster can be thought of as a special case of an area object where the area is divided into a regular grid of cells. But a regularly spaced array of marked points may be a better analogy since rasters are stored as an array of values where each cell is defined by a single coordinate pair inside of most GIS environments.

Implicit in a raster data model is a value associated with each cell or pixel. This is in contrast to a vector model that may or may not have a value associated with the geometric primitive.

Also note that a raster data structure is square or rectangular. So, if the features in a raster do not cover the full square or rectangular extent, their pixel values will be set to no data values (e.g. NULL or NoData).

2.2 Object vs. Field

The traditional vector/raster perspective of our world is one that has been driven by software and data storage environments. But this perspective is not particularly helpful if one is interested in analyzing the pattern. In fact, it can mask some important properties of the entity being studied. An object vs. field view of the world proves to be more insightful even though it may seem more abstract.

2.2.1 Object view

An object view of the world treats entities as discrete objects; they need not occur at every location within a study area. Point locations of cities would be an example of an object. So would be polygonal representations of urban areas which may be non-contiguous.

2.2.2 Field view

A field view of the world treats entities as a scalar field. This is a mathematical concept in which a scalar is a quantity having a magnitude. It is measurable at every location within the study region. Two popular examples of a scalar field are surface elevation and surface temperature. Each represents a property that can be measured at any location.

Another example of a scalar field is the presence and absence of a building. This is a binary scalar where a value of 0 is assigned to a location devoid of buildings and a value of 1 is assigned to locations having one or more buildings. A field representation of buildings may not seem intuitive, in fact, given the definition of an object view of the world in the last section, it would seem only fitting to view buildings as objects. In fact, buildings can be viewed as both field or objects. The context of the analysis is ultimately what will dictate which view to adopt. If we’re interested in studying the distribution of buildings over a study area, then an object view of the features makes sense. If, on the other hand, we are interested in identifying all locations where buildings don’t exist, then a binary field view of these entities would make sense.

2.3 Scale

Scale has two separate meanings in geography:  

1. Map Scale or Cartographic Scale: the measurement on a map (ratio of space on map to space on the globe); and 

2. Geographic Scale: the hierarchy of spaces or places

2.3.1 Map Scale

How one chooses to represent a real-world entity will be in large part dictated by the scale of the analysis. In a GIS, scale has a specific meaning: it’s the ratio of distance on the map to that in the real world. So a large scale map implies a relatively large ratio and thus a small extent. This is counter to the common interpretation of large scale which focuses on the scope or extent of a study; so a large scale analysis would imply one that covers a large area.

A measure of the size at which features in a map are represented. The scale is expressed as a fraction, or ratio, of the size of features represented by the map to the size of the actual objects on the ground. A large-scale map is one in which the ratio is large (i.e., the denominator is small). Thus, a 1:24,000 scale map has a larger scale than a 1:100,000 scale map.

The following two maps represent the same entity: the Boston region. At a small scale (e.g. 1:10,000,000), Boston and other cities may be best represented as points. At a large scale (e.g. 1:34,000), Boston may be best represented as a polygon. Note that at this large scale, roads may also be represented as polygon features instead of polylines.

Figure 2.7: Map of the Boston area at a 1:10,000,000 scale. Note that in geography, this is considered small scale whereas in layperson terms, this extent is often referred to as a large scale (i.e. covering a large area).

Figure 2.8: Map of the Boston area at a 1:34,000 scale. Note that in geography, this is considered large scale whereas in layperson terms, this extent is often referred to as a small scale (i.e. covering a small area).

2.3.2 Geographic scale

Geographic conceptual hierarchy of spaces, from small to large that reflects actual levels of organization in the real world.

Example from small to large:  Neighborhood → Urban Area → Metropolitan Area → Region → Nation → World

Geographic scales used in research:

• Human Settlements: community, home, body

• Nation States: subdivided into de jure (“concerning law”) regions or functional regions

• World Regions: major clusters of humankind with broadly similar cultural attributes

• World Economy: subdivided into core, semipheriphery, and periphery

Example: Unemployment and other socio-economic characteristics, Murders in the U.S.

2.4 Attribute Tables

Non-spatial information associated with a spatial feature is referred to as an attribute. A feature on a GIS map is linked to its record in the attribute table by a unique numerical identifier (ID). Every feature in a layer has an identifier. It is important to understand the one-to-one or many-to-one relationship between feature, and attribute record. Because features on the map are linked to their records in the table, many GIS software will allow you to click on a map feature and see its related attributes in the table.

Raster data can also have attributes only if pixels are represented using a small set of unique integer values. Raster data sets that contain attribute tables typically have cell values that represent or define a class, group, category, or membership. NOTE: not all GIS raster data formats can store attribute information; in fact many raster data sets you will work with in this course will not have attribute tables.

2.4.1 Data measurement types

A variable’s measurement level describes what its values mean and which comparisons they support. Nominal and ordinal variables are categorical. Interval and ratio variables are numerical (quantitative).

Measurement level	Meaning	Park example
Nominal	Labels or categories with no inherent order	Park ID; public or private ownership
Ordinal	Ordered categories, without established equal differences between categories	Condition: poor, fair, good
Interval	Equal differences are meaningful; zero is a reference point rather than an absence of the quantity	Temperature in degrees Celsius or Fahrenheit
Ratio	Equal differences and ratios are meaningful; zero represents an absence of the quantity	Number of trees; park area in square meters

Interval and ratio: differences and multiples

On an interval scale, a change from 10°C to 20°C is the same temperature increase as a change from 20°C to 30°C. Both differences are 10°C. Equal intervals describe the measurement scale: observed temperatures such as 12.3°C, 19.8°C, and 31.1°C do not have to be evenly spaced.

Zero Celsius is a reference temperature. It does not mean an absence of temperature. Although 40 is twice 20, 40°C is not twice as hot as 20°C. Those same temperatures are 104°F and 68°F, which do not have a ratio of two. Differences and averages of comparable temperature observations are meaningful; dividing the Celsius readings does not establish a meaningful temperature multiple.

Ratio data support those multiplicative comparisons. A park with 40 trees has twice as many trees as a park with 20 trees, and zero trees means none were counted. A park covering 2,000 m² has twice the area of one covering 1,000 m². Converting both areas to the same alternative unit preserves that ratio.

The meaning of zero comes from the variable’s definition. A dataset does not need to contain an observed zero to use a ratio scale. Missing or unavailable observations are different from zero: an unknown tree count does not establish that a park has no trees.

Categories and category counts

Ownership is a nominal variable. The number of parks in each ownership category is a separate, numerical count. Counting observations within categories does not turn the category labels themselves into quantities.

Ordinal values indicate an order without specifying equal gaps. For example, poor, fair, and good describe ordered park conditions. Coding them as 1, 2, and 3 does not establish that each step represents the same improvement in condition.

The Home Owners’ Loan Corporation redlining grades also illustrate ordered categories. Assigning numbers to the grades would not establish equal differences in the underlying assessments or justify treating one grade as twice another.

Figure 2.9: Redlining map of Chicago, as created by the Home Owners’ Loan Corporation.

Discrete and continuous numerical variables

Discrete versus continuous describes the possible numerical values of a variable. This distinction is separate from its measurement level.

• Discrete numerical variables have separate, countable possible values. Tree counts can be 0, 1, 2, and so on. A count of individual trees cannot be 2.5.
• Continuous numerical variables can, in principle, take any value within a range. Temperature, distance, and area are examples. Instruments and recording choices limit the precision of the values we actually store.

Tree counts are discrete and ratio. Celsius temperature is continuous and interval. Park area is continuous and ratio. Recording a temperature or area to the nearest whole unit reduces its precision without changing its measurement level.

In GIS, discrete geographic objects is another use of the word discrete you will encounter q . A park represented as a polygon can have a discrete tree count and a continuous area measurement in the same attribute table. Vector and raster describe spatial data structures: a raster may store land-cover categories, counts, or continuous measurements, and a vector layer may also carry each kind of attribute.

Applying the distinctions to an attribute table

Attribute	Measurement level	Kind of values	Suitable storage
Park ID, such as 001	Nominal	Identifier	Text, preserving leading zeros
Ownership	Nominal	Categories	Text or category codes
Condition	Ordinal	Ordered categories	Text or category codes
Number of trees	Ratio	Discrete numerical	Integer
Temperature in °C	Interval	Continuous numerical	Float or double
Park area in m²	Ratio	Continuous numerical	Float or double

The next section explains storage types. Storing a category as a number does not change its measurement level.

Map design may group numerical values into ranges and assign a color or symbol to each group. This is classification. An equal-interval classification uses equal-width ranges; it can be applied to either interval or ratio measurements. Classification is a design choice: unclassed colors and proportional symbols can also represent numerical values without grouping them into classes.

For additional examples, see OpenStax’s explanations of measurement levels and discrete and continuous data.

2.4.2 Data type

The storage type determines which values a field can hold and with what precision. Choose it to preserve the variable’s meaning and required detail. ArcGIS supports integer, float, double, text, and other field types.

Type	Values stored	Storage per numeric value
Short integer	Whole numbers from -32,768 to 32,767	16 bits (2 bytes)
Long integer	Whole numbers from -2,147,483,648 to 2,147,483,647	32 bits (4 bytes)
Float	Approximate numerical values, including fractions; single precision	32 bits (4 bytes)
Double	Approximate numerical values, including fractions; greater precision than float	64 bits (8 bytes)
Text	Characters, including labels and identifiers such as 001	Length and storage depend on the field and data format

This table covers common choices. Supported types and limits vary by data format; consult ArcGIS field data types for the full list.

The decimal point is not a test of measurement level. The values 2 and 2.0 can represent the same tree count, while an integer code such as 1 can represent an ownership category. Use text for identifiers when leading zeros are part of the identifier.

Float and double fields can store whole numbers as well as fractions, but their precision is finite. Some large integers cannot be represented exactly in a float. Choose an integer field with an adequate range for exact counts. A long integer and a float each use 4 bytes per value, so changing an integer field to a float does not always increase storage.

Converting fractional measurements to integers loses information. If we explicitly round 0.2, 0.01, 0.34, 0.1, and 0.876 to the nearest whole number, the results are 0, 0, 0, 0, and 1. A conversion tool may instead truncate or reject fractional values; check its behavior. The maps below demonstrate explicit rounding with R’s round() function.

For a missing observation, use the missing-data value supported by the data format. Replacing an unknown value with zero changes its meaning and can distort summaries and maps.

Figure 2.10: Map of data represented as decimal (float) values.

Figure 2.11: Map of same data represented as integers instead of float.

2.5 Resolution

Resolution refers to the sampling interval at which data are acquired. Resolution may be spatial (what is represented by one pixel or one spatial unit like census tracts or counties), thematic, or temporal (Decennial Census data vs. American Community Survey estimates, ACS).

Figure 2.12: Map resolution example

High resolution entails a higher frequency of measurement or counting and smaller areal units. while low resolution entails a lower frequency of measurement or counting. In the raster data model, a higher resolution spatial representation of data entails pixels that represent a smaller area on the earth. 1 meter resolution is higher resolution than 1 mile. In the vector data model, census tracts represent a higher spatial resolution than counties.

Figure 2.13: Map resolution example: from vector to raster

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