Dear All, (01)
I just forgot about interval scale (thank you, Ed). Using the same notation
as before we have: (02)
interval scale: a scale where ratios can be defined for differences in Q and
S, such that: (03)
r.(f(q1) - f(q2)) = f(q3) - f(q4) if and only if r.(q1 - q2) = q3 - q4 (04)
The Celsius and Fahrenheit scales are straightforward examples of this type
of scale. (05)
Time is a difficult example. It fits with this definition, when time is
measured for an experiment. A clock is started at some arbitary instant
towards the beginning of the experiment and measured in seconds thereafter. (06)
Alternative terms for "interval scale" and "ratio scale" are "affine scale"
and "linear scale". These terms are used in the SysML Units of Measure
proposal. I suggest that we use them too because:
1) they have well established mathematical definitions; and
2) the definitions generalise to quantities with dimensions greater than 1. (07)
Best regards,
David (08)
At 18:33 11/08/2009 -0400, you wrote:
>David Leal wrote:
>> Dear Pat,
>>
>> I agree except for one thing - a scale is not a set of items/symbols in
>> itself, but a mapping from a set of "magnitudes of quantity" to a set of
>> items/symbols. Hence re-expressing the consensus in these terms we have:
>>
>> scale: a mapping f from Q (set of magnitudes of quantity) to S (set of
>> symbols - commonly numbers), such that:
>>
>> f(q1) = f(q2) if and only if q1 = q2
>>
>> ordinal scale: a scale where both Q and S are ordered, such that:
>>
>> f(q1) > f(q2) if and only if q1 > q2
>>
>> ratio scale: a scale where ratios can be defined for both Q and S, such that:
>>
>> r.f(q1) = f(q2) if and only if r.q1 = q2
>>
>I agree with all this. The one David left out is the "interval scale".
>I'm not quite sure what an "interval scale" is, because time and
>temperature are not similar in nature.
>
>The Celsius temperature scale assigns symbols to amounts/magnitudes of
>temperature. It is essentially just a ratio scale with 0 corresponding
>to a magnitude of temperature that isn't "none". So the scale is based
>on a ratio to the amount that is the difference in temperature between 1
>degrees and 0 degrees -- the degree Celsius -- offset by the magnitude
>at 0. It doesn't name "intervals" in any sense. It is just a "ratio
>scale with an offset" that maps symbols to (relative) magnitudes. And
>that is why it is easily converted to Kelvin.
>
>A "duration scale" is a ratio scale that assigns symbols to
>amounts/magnitudes of time. But a clock or a calendar is an "interval
>scale" that maps a set of symbols to a set of intervals on the
>(putative) Time Axis. It is not a map to "magnitudes of quantity" (my
>Q3) at all. In a similar way, the mileposts on a measured roadway
>constitute an "interval scale" of this kind. What characterizes it
>structurally is that there is a fixed reference point (in space time)
>that is the 0 point (without loss of generality) and each symbol denotes
>the interval that has size equal to the granularity of the scale and
>starts (or ends, depending on how you number them) at a point whose
>"difference" from the 0-point is equal to the magnitude that corresponds
>to the symbol on the corresponding ratio scale.
>
>The Celsius scale measures "magnitudes", whereas the clock/calendar kind
>of "interval scale" identifies intervals. So I don't know what an
>"interval scale" is.
>
>I raise this particular example, because it came up in the OMG date/time
>ontology effort and bogged down the process for months. My view of this
>is that "one size fits all" will waste everyone's time. We can write
>down the axioms we clearly want and see the differences in scales that
>don't satisfy certain axioms, and that may enable us to see other
>classes of scales and other relationships among them. And for OWL
>purposes, common axioms define classification hierarchies. So I what I
>am suggesting is that we formulate the applicable axioms and learn the
>classifications, rather than imposing them a priori. That is in essence
>what David is doing above.
>
>-Ed
>
>--
>Edward J. Barkmeyer Email: edbark@xxxxxxxx
>National Institute of Standards & Technology
>Manufacturing Systems Integration Division
>100 Bureau Drive, Stop 8263 Tel: +1 301-975-3528
>Gaithersburg, MD 20899-8263 FAX: +1 301-975-4694
>
>"The opinions expressed above do not reflect consensus of NIST,
> and have not been reviewed by any Government authority."
>
>
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> (09)
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