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Re: [ontolog-forum] "Ontology-based Standards" mini-series session-4 - T

To: ontolog-forum@xxxxxxxxxxxxxxxx
From: Tara Athan <taraathan@xxxxxxxxx>
Date: Sun, 13 Oct 2013 15:19:56 -0400
Message-id: <525AF25C.5000705@xxxxxxxxxx>
A classical reference regarding concepts in metrology is Krantz et al 
http://www.amazon.com/Foundations-Measurement-Volume-Representations-Mathematics/dp/0486453146    (01)

.    (02)

Since it is not downloadable, I'll attempt to give a summary of the 
points I think are relevant to the development of ontologies of measurement.    (03)

The foundational concepts employed in this reference are
* a "relational structure", consisting of a set of mathematical entities 
(such as positive real numbers) and one or more relations among them.
* A "numerical assignment" is a homomorphism that maps physical entities 
and measurement procedures into a relational structure in a way that the 
properties of the procedures are preserved.    (04)

 From these foundations, it can be shown that measurement theories 
corresponding to particular relational structures may be grouped into 
families according to the "permissible transformations" that may be 
applied to a numerical assignment that result in another numerical 
assignment. The familiar Stevens scale types correspond to certain families:    (05)

* Ratio scale: entities are 1-dimensional (e.g. real or positive real 
numbers), and permissible transformation are only multiplication by a 
positive scalar constant. Extensive measurements such as length fall 
into this family.    (06)

* Log-Interval scale: entities are again 1-dimensional, but typically 
only the positive real numbers, and permissible transformations include 
raising to a power in addition to multiplication by a positive scalar 
constant. Intensive measurements such as absolute temperature and 
density fall into this family. It's called log-interval because taking 
the log gives a theory in the "interval-scale" family, and Stevens 
recognized its existence in his original work, although it is usually 
not included in elementary treatments today.    (07)

and so on. For ordinal-scale measurements, permissible transformations 
include any strictly monotonic function.    (08)

However, there are plenty of measurement theories that do not fit into 
these families, notably, multi-dimensional theories.    (09)

Tara    (010)

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