Why have so many measurement scales?

Blair Hall

Measurement Standards Laboratoryof New Zealand

Industrial Research Ltd.,PO Box 31-310, Lower Hutt,New Zealand.

b.hall@irl.cri.nz

Notes

Presented (via internet) to the 31th ANAMET meeting, 2nd April, 2009.

So many units

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

■ RF and microwave measurements are represented in many differentunits, for example

Reflection coefficient: Γ = b/a = Γre + i Γim

Magnitude: ρ = |Γ| =√

Γ2re

+ Γ2

im

Phase: φ = arg(Γ) = tan−1(Γim/Γre)

■ The raison d’etre may be one of convenience

Return loss: RL = −20 log10(|Γ|)

VSWR: r = (1 + ρ)/(1 − ρ)

Notes

■ As a newcomer to RF metrology I was struck by the variety of units used describeessentially the same quantity

■ Sometimes, there may be a direct correspondence between a characteristic of themeasured object and the unit, such as phase angle and the length of an airline.

■ Sometimes, the raison d’etre may be one of convenience for computations (e.g., inthe field). For instance, the additivity of logarithmic units is easier than multipli-cation. (So, if a component of known return loss is inserted in a transmission lineand the incident power is known, the transmitted power is obtained by a simpledifference calculation.)

■ VSWR is the ratio of the maximum to minimum node heights for the standing waveon a mismatched transmission line. I am not aware that it is useful in computationthese days. However, instruments are available to measure it directly. This made ita useful scale of measurement in the past, (Some reference texts would give tablesof useful formulae in terms of the VSWR, e.g., Table 2.2 in Microwave TransmissionCircuits, ed. G. L. Ragan, 1948)

There is probably an historical reason

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

Glen Engen 1992:

Perhaps [VSWR]’s major legacy to the current art is inthe terminology which it has fostered. For example, onestill finds adaptors and attenuators specified in terms of the‘VSWR’ (in contrast to the more logical value of |S11|) · · ·

■ Legacy or headache: do we need all these scales?

Notes

■ Tradition is hard to shrug off

■ The convenient slide rule calculator suggests, however, that VSWR is not alwaysthe number wanted

ANAMET has had much to say about all this

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

■ Transformations, e.g.:

◆ N. Ridler, ANATips: 4 and 5 (log and linear formats)

◆ N. Ridler, ANATips: 8 and 9 (VSWR and VRC formats)

◆ J. P. Ide, ANAlyse: 22 (uncertainty in magnitude and phase)

■ Ordinal statistics, e.g.:

◆ N. Ridler and J. Medley, ANAlyse: 15 and 16 (estimating a vectorquantity)

◆ N. Ridler, ANAlyse: 20 (uncertainty in the median)

■ Rectangular-polar transformation, e.g.:

◆ J. P. Ide, ANAlyse: 6

◆ N. Ridler, 23rd ARMMS Conference Digest, 1995

◆ G. J. French, ANAMET Report 023, April 1999

Notes

■ A multitude of scales is not helpful in metrology, hence · · ·

■ There have been quite a few ANAMET notes written to discuss the specifics ofdifferent cases.

■ There is a risk that information about a measured quantity will be given in onescale, but then transformed to another.

■ This may be done inadvertently, because one gets used to thinking about the simple1-to-1 transformation of numbers on the scales and one forgets that other informa-tion may be transformed intact

Back to basics · · ·

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

■ Measurement involves a mapping between a phenomenon in thephysical world and symbols (usually numbers)

■ The information we obtain from measurements is not always aperfect match with the properties of numbers we use as symbols

■ There are some transformations that we can apply to scales with-out losing information

■ We can classify scales according to the types of transformationsthat preserve information

◆ Nominal

◆ Ordinal

◆ Interval

◆ Ratio

Notes

■ In 1946 Stevens, a psychophysicist at Harvard, suggested that measurement scalescould be classified according to the group of transformations that preserve informa-tion about measurements on the scale. (S S Stevens, 1946, Science 103(2684) pp677–680.)

■ Stevens regards the process of measurement as being the application of a mappingfrom one group of objects, or events, to another. (Usually, a group is mapped tonumbers, but this need not be so.)

■ Permissible transformations ‘keep intact the empirical information depicted by the

scale’

■ ‘Information’ here can be thought of as statistics (e.g., mean, median, etc)

Nominal scales

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

■ A nominal scale represents a classification scheme

■ It identifies members of discrete sets

■ One-to-one substitution of class labels is permitted

■ It says nothing about order

◆ Library books

◆ Product catalogs

◆ Testing (pass-fail)

◆ Measurable properties

■ temperature of ?nom?

■ mass of ?nom?

■ S11 of ?nom?

■ · · ·

Notes

■ Classification: the same as set membership.

■ Any one-to-one substitution is permitted if classes are re-labeled but the membershipof each class is not changed

■ We must know what we intend to measure and take steps to ensure that the entitythat is measured belongs to a category of objects that posses the required property.We might ask ”Does an object x have a temperature; does it have a mass, etc?”.One cannot measure the mass of an electromagnetic wave, for example. We needto establish a class of equivalent objects or events before measuring anything.

Ordinal scales

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

■ Numbers or symbols can only represent relative order

■ Order statistics can be used, e.g., median

■ Intervals on the scale cannot be compared

■ Any ’order-preserving’ function can transform ordinal data withoutloss of information

◆ Exam scores

◆ Performance measures (business world)

◆ In the field: too large, too small,need to get closer to x, · · ·

Notes

■ Numbers or symbols (eg, letters of the alphabet) indicate relative order accordingto the measured property

■ In practice, we sometimes work with data as if it were on an ordinal scale: it can beenough to have a few familiar values on a scale in mind and to compare actual resultswith these (thinking perhaps ‘that’s too small’, or ’that needs to be increased’ etc).

Interval scales

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

■ Zero is conventional (need two standards to define the scale)

■ Intervals on the scale may be compared

■ Familiar statistics can be used (mean, SD, correlation)

■ Any linear transformation (x′ = ax + b)preserves information

◆ Temperature (◦F, ◦C)

◆ Date

◆ Location

◆ Potential energy?

Notes

■ The interval between pairs of points can be compared.

■ The ratio of scale points is not meaningful: 30 ◦C is not twice as hot as 15 ◦C

■ Ratios are meaningless

■ Early thermometers used the empirical fact that equal temperature increments couldbe scaled off by noting equal volumes of expansion. So, when the temperature goesup by 3 degrees, say, the volume increase is the nominally same no matter what thestarting temperature was.

■ To define an interval scale two standards will be needed, because effectively andinterval is being set. Since there is no ‘absolute’ zero, one standard sets a pointon the scale and another one is needed to define an interval (corresponding to acertain number of units).

■ I can’t think of any examples in RF

Ratio scales

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

■ Zero is defined (not arbitrary - only one standard required)

■ Ratios of points on the scale can be compared

■ Most scales in physics are ratio scales

■ Transformation by an arbitrary scale factor (x′ = ax) preservesinformation

◆ Voltage, current, impedance, · · ·

◆ Time interval, frequency

◆ Power, attenuation, · · ·

◆ Numerosity (numbers)

Notes

■ The ratios of magnitudes can be compared, because zero is now well-defined

■ Ratio scales place the greatest restriction on the transformations allowed: theycannot be ‘shifted’ by adding a constant offset.

■ Transform by scaling only, e.g., the SI prefixes, e.g. nano-, milli-, etc

■ The properties of numbers closely represent the associated physical quantity (wecan think of adding, multiplying, etc)

In RF there are many different ratio scales

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

E.g., these three scales are all giving information about the same quantity

ρ = |Γ|

RL = −20 log ρ

r = (1 + ρ)/(1 − ρ)

■ Only the ordinal information is preserved between the scales

◆ and even then a big RL implies small ρ!

■ The amount of information lost depends on the data

◆ hence, e.g., GUM methods may apply

Notes

■ There are perhaps conflicting interests in choosing the best scale. A field-engineermay find VSWR or RL very useful.

■ A metrologist would probably prefer a scale in which it is natural to express thevarious sources of error that contribute to a measurement.

■ Given the nature of measurement, such a scale is likely to be linear. However, iferrors are multiplicative, a log scale may be desirable.

Transformation of coordinates

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

Some problems have been associated with changes in 2-D coordinates

ρ =√

Γ2re

+ Γ2

im, tanφ =

Γim

Γre

■ Rectangular-polar transformation is just a change of coordinates

◆ the same configuration of points in the complex plane can berepresented in either set of coordinates

◆ however, problems arise when by treating the coordinates as inde-pendent 1-D quantities

■ Was the transformation made to facilitate computations or to fitbetter with a conceptual model?

◆ it sometimes appears that the more natural conceptual domain isactually the complex plane, yet people still prefer to express datain polar coordinates

Notes

■ There has not been an extension of Stevens ideas to 2-D quantities (that I knowof)

■ In RF there are problems when switching between polar and rectangular coordi-nates, but these arise because we treat the data as independent 1-D quantities aftertransformation. Clearly, the non-linearity of the coordinate transforms affects thestatistics. (It is still possible to evaluate the complex mean of a set of data givenin polar coordinates, but the result is not the same as the individual means of ther and φ data sets.)

■ It is sometimes said that evaluating statistics in (r, φ) leads to errors. Actually, itsjust not the right way of treating the data.

■ The question about whether to report measurement uncertainty of a complex quan-tity in polar or rectangular coordinates is important: what information are we tryingto convey when reporting the uncertainty?

Discussion

Introduction

A legacy?

ANAMET Interest

Steven’s Scales

- nominal

- ordinal

- interval

- ratio

Scales in RF

Two dimensions

Discussion

■ How useful are all the different scales?

■ What is the role of the different scales in metrology?

■ Which are the best scales to represent measurement errors?

■ Are there RF measurements that clearly require a log scale (i.e. itis the natural measurement scale)?

■ Do some RF measurements clearly require polar coordinates?

■ Are some measurements best retained in complex format?

■ Should we make use of summary statistics?

Notes

■ Tradition is hard to shrug off

■ The purpose for which data is being collected needs to be clear (why wereANAMET’s splitter comparisons reporting VSWR, e.g.?)

■ Transformation should be carried out with regard to the changes induced on thedata

■ If errors contribute to a measurement in a multiplicative way, then it might besensible to work in logarithmic units. Is that why we have them?

■ Our minds find it much easier to manipulate information on one number scale,hence it may be easier to contemplate the magnitude and phase of a reflectioncoefficient, rather than its rectangular coordinates. However, this not be the bestway of dealing with the data in terms of measurement models. Perhaps we need toacknowledge the difference between useful summaries of multivariate data (‘views’,if you like) and the inherent information content of the actual data.

Thank You

The Lower Hutt campus of Industrial Research Ltd

Notes

(Notes added after a discussion during the 31st ANAMET meeting on 2 April 2009)

■ A quick poll was conducted asking how many participants would wish to retain useof the scales: return-loss, VSWR and (complex) reflection coefficient. Roughly 50%voted in favour of return-loss, about 25% in favour of VSWR and 100% in favourof the reflection coefficient.

■ One participant spoke of the usefulness of the return-loss scale when tuning a filterto obtain a good match. The scale becomes very sensitive as the desired result(match) is approached.

◆ My own comment would be that this is using the ordinal information of thescale: the engineer is optimising a parameter by trying to make the return-lossas large as possible. I am not sure that other information is needed (ratiosand intervals on the scale are irrelevant).

■ One participant mentioned that in providing customer services, these scales aresimply demanded and therefore measurements must be expressed in this way.

◆ My comment would be that, in such cases, it may be appropriate to work withother scales while performing the measurement and assessing the accuracy,leaving the transformation until the final stage of reporting to the client.

◆ I would also say that the opportunity to discuss the best choice of scale withthe client should be investigated (do they really need it, or are they justfamiliar with asking for it?.