A revised approach to measurement

for geographic information

Charlotte, North Carolina; March 1995

by Nicholas Chrisman

University of Washington

- Background: Measurement Theory
- Classical versus Representationalist
- Applied to cartography

- Steven's Scales of Measurement
- What is missing
- Measurement Frameworks for Geographic Information

abbreviated intellectual history

developed in physics (and related sciences) by late 19th century

numerical
relationship between standard& object measured; inherent in
object

Example: US
Office of Weights and Measures | standard kilogram | Legal
Weights and Measures in the State of Indiana*[still a tacit part of
geographic databases?]*

Bertand Russell; Campbell; Bridgeman; Stevens ...

properties not inherent
in object; numbers derive from measurement operations

20th century dominated
by representationalists.

Measurement theorists concentrated on physics

'extensive' measures (eg.
length) emphasized strict limits; special role for addition

'Operationism'
enshrines the procedure;

result: no room for social sciences

Stevens:
a psychophysicist

research on perceived loudness of sound

measurements
rejected by operationists

General movement for quantitative approach to
social sciences

Published 1946 in *Science* (without any references)

Adopts
'nominalist' form of representationalism:

Defines measurement as 'assignment
of numbers to objects according to a rule'

Measurement scale defined by
properties of numbers (not properties of the rules or objects)

Scales
organized by transformations under which the meaning remains invariant.

1946:
Nominal, Ordinal, Interval, Ratio

(log-interval added in 1959)

Links
scales to permissible statistics

Scale Empirical Invariance Permissible Operations Group Statistics NOMINAL equality any 1:1 substitution Mode Number of cases ORDINAL , > any monotonic Median function Percentiles INTERVAL equality of x' = ax + b Mean, std. dev. intervals Correlations RATIO equality of x' = ax Coefficient of ratios variation

Objects classified by shared attributes.

All members belong
equally.*NOTE: Examples based on a footrace. "Objects" in this case are
people.*

[Critique of
traditional categories]

Order of arrival of contestants in footrace

Women's race Men's race First Jane Tom Second Melissa Dick Third Leila Harry ...

Ordinal measures may be complete orderings, partial orderings or ranked
categories

(High, Medium, Low)

(slight, medium, severe, very severe
limitations)

Examples: Grading Sweet
Potato crops in Indiana | Plant Hardiness in
Texas

Interval measures require a fixed distance, but the zero point is
arbitrary.

Differences between two interval measures are ratio.

Interval
measures are often raw results, with some additional relationships they become
ratio.

A frequent example of interval measurement is temperature in degrees F, or
°C.

The 'interval' equation does apply:

degrees F = ( 5/9 · degrees C ) +
32**BUT** degrees K is not the 'true' ratio scale (extensive).

To
combine regular extensive measures: **ADD**

To combine
temperatures:

Two objects brought in contact reach a weighted equilibrium
temperature.

Current Local Weather
data: (Use BACK to return)

Ratio measures have:

a fixed zero
that means 'no quantity';

a constant interval (distance on the
scale).

May be rescaled because '1' is arbitrary.

Examples of
ratio scales: SI
units | Physical
Reference Data (CODATA 86) | Bushels
in Indiana

Stevens conflated two types of ratio measures.

Extensive: based on
addition rule for combination

examples: distance, mass, time

Derived:
created by division of extensive measures;

velocity = distance / time (speed of light meters
per second) , Newtonian
constant of gravitation, etc.

Different axioms apply

(recognized
in thematic cartography practice)

Extensive: proportional symbols

Derived: choropleth [See Figure 1-8 and
1-9 in Exploring GIS]

In 1959, Stevens recognizes that the invariance rules could allow another
scale on the same 'level' as interval.

The invariance function would have
the form: x' = a times x to the b power

This 'log-interval' scale is
mostly theoretical, though earthquake intensities on Richter and Mercali scales
use this form.

None of the social science statistics textbooks take
notice; cartographers still use the basic 4.

Ratio is not the 'highest' level of measurement, '1' need not be
arbitrary.

Using Stevens' invariance scheme, an **'absolute'** scale
could not be rescaled by any multipier.

Both zero AND one are
fixed.

Probability is an absolute scale.

Bayes' Law and other axioms
depend on absolute scaling.

Just as a difference of intervals is ratio, some
divisions of ratios produce absolute measures.

Some measures are obtained by tabulating the total of some objects inside a larger collecting unit.

- Counts are discrete, since only integers can result.
- There is a real zero, and 2x does mean twice the quantity.
- The unit is not arbitrary; 'one' cannot be reset.

Are counts higher than ratio?

Are counts 'discrete' like categorical
measures?

One answer: Counts are counts;

Extend the system to handle all
the cases.

Stevens scales only accomodate open-ended linear scales.

Some measures are
cyclical, with a fixed range.

Example: Angles

- Fixed zero (no turn)
- Arbitrary unit (degrees, grads, radians)
- Upper limit (360°, 100 grad, 2 pi radians)

Additional axioms are required to handle the fact that 359° is the same distance from 0° as 1° is.

Multidimensional scales have extra relationships.

In Stevens' scheme,
an unbounded linear axis could not be fit into a fixed interval without losing
information.**yet**, two unbounded Cartesian axes can be transformed
into radial coordinates with only one unbounded axis, and one angle.

Physics Devices at Cornell <Dead? too bad>

'Nominal' measures depend upon Aristotle's view:

categories imply shared
values of attributes.

Hence, test for equivalence only.

Two
alternative approaches to categories:

- Graded membership : some scale of partial connection (eg. probability or fuzzy set)
- Distance from
**prototype**(not normalized); [object fits with closest exemplar]; Common in photointerpretation, land use, etc. (eg. "dominant" species)

Both provide a **metric** for membership beyond the simple Yes/No of
traditional categories.

Matthews Datasets | Fish and Wildlife Service "Cowardin" manual for wetland classification (and its official hierarchy); National list of plants associated with wetlands (1996) with 5 categories of association <now dead> | FGDC Vegetation Standard <relocated?>|

The four levels are not adequate.

New Requirements:

- Nominal must be expanded
- Extensive and derived must be distinguished
- Higher levels recognized
- Multidimensional relationships handled

Measurement involves more axioms than just those involved in the number line.

Stevens never questions the nature of the 'object'.

[in psychology, the
'case' is not a problem]

Standard model: case 'has' attributes

(not
relationships that can be measured)

Geographical Matrix: places 'have'
attributes

'Case' permits measurement as a form of control.

Geographic
control is not completely routine and accepted.

Scheme based on Sinton:
fixed, controlled, measured

based on relationships between geometry & attribute

[Lecture on Measurement Frameworks]

Object Control Isolated objects: 'features'; isolines Connected objects: network; coverage Spatial Control Point-based: center point Area-based: (many rules) Relationship Control Triangular Irregular Network (TIN) Composite Frameworks (eg. Choropleth)

*basic rules: attribute serves as control, positions measured to
suit*

- Isolated Objects: each category taken as Yes/No
- 'Spatial Object' (or cartographic feature view): each entity (point,line,area) surrounded by the void
- Isoline: continuous attribute sliced, disjoint

- Connected Objects: multinomial (more than one) categories
- Network: object connectivity relationships
- Categorical Coverage: exhaustive classification divides space with contiguous boundaries

Index from Here: |