Why have so many measurement scales?

Transcription

1 Why have so many measurement scales? Blair Hall Measurement Standards Laboratory of New Zealand Industrial Research Ltd., PO Box , Lower Hutt, New Zealand.

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

3 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 different units, for example Reflection coefficient: Γ = b/a = Γ re + i Γ im Magnitude: ρ = Γ = Γ 2 re + Γ2 im Phase: φ = arg(γ) = tan 1 (Γ im /Γ re ) The raison d être may be one of convenience Return loss: RL = 20 log 10 ( Γ ) VSWR: r = (1 + ρ)/(1 ρ)

4 Notes As a newcomer to RF metrology I was struck by the variety of units used describe essentially the same quantity Sometimes, there may be a direct correspondence between a characteristic of the measured object and the unit, such as phase angle and the length of an airline. Sometimes, the raison d être may be one of convenience for computations (e.g., in the field). For instance, the additivity of logarithmic units is easier than multiplication. (So, if a component of known return loss is inserted in a transmission line and the incident power is known, the transmitted power is obtained by a simple difference calculation.) VSWR is the ratio of the maximum to minimum node heights for the standing wave on a mismatched transmission line. I am not aware that it is useful in computation these days. However, instruments are available to measure it directly. This made it a useful scale of measurement in the past, (Some reference texts would give tables of useful formulae in terms of the VSWR, e.g., Table 2.2 in Microwave Transmission Circuits, ed. G. L. Ragan, 1948)

5 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 in the terminology which it has fostered. For example, one still finds adaptors and attenuators specified in terms of the VSWR (in contrast to the more logical value of S 11 ) Legacy or headache: do we need all these scales?

6 Notes Tradition is hard to shrug off The convenient slide rule calculator suggests, however, that VSWR is not always the number wanted

7 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 vector quantity) N. Ridler, ANAlyse: 20 (uncertainty in the median) Rectangular-polar transformation, e.g.: J. P. Ide, ANAlyse: 6 N. Ridler, 23 rd ARMMS Conference Digest, 1995 G. J. French, ANAMET Report 023, April 1999

8 Notes A multitude of scales is not helpful in metrology, hence There have been quite a few ANAMET notes written to discuss the specifics of different cases. There is a risk that information about a measured quantity will be given in one scale, but then transformed to another. This may be done inadvertently, because one gets used to thinking about the simple 1-to-1 transformation of numbers on the scales and one forgets that other information may be transformed intact

9 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 the physical world and symbols (usually numbers) The information we obtain from measurements is not always a perfect match with the properties of numbers we use as symbols There are some transformations that we can apply to scales without losing information We can classify scales according to the types of transformations that preserve information Nominal Ordinal Interval Ratio

10 Notes In 1946 Stevens, a psychophysicist at Harvard, suggested that measurement scales could be classified according to the group of transformations that preserve information about measurements on the scale. (S S Stevens, 1946, Science 103(2684) pp ) Stevens regards the process of measurement as being the application of a mapping from one group of objects, or events, to another. (Usually, a group is mapped to numbers, 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)

11 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? S 11 of?nom?

12 Notes Classification: the same as set membership. Any one-to-one substitution is permitted if classes are re-labeled but the membership of each class is not changed We must know what we intend to measure and take steps to ensure that the entity that 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 need to establish a class of equivalent objects or events before measuring anything.

13 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 without loss of information Exam scores Performance measures (business world) In the field: too large, too small, need to get closer to x,

14 Notes Numbers or symbols (eg, letters of the alphabet) indicate relative order according to the measured property In practice, we sometimes work with data as if it were on an ordinal scale: it can be enough to have a few familiar values on a scale in mind and to compare actual results with these (thinking perhaps that s too small, or that needs to be increased etc).

15 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?

16 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 could be scaled off by noting equal volumes of expansion. So, when the temperature goes up by 3 degrees, say, the volume increase is the nominally same no matter what the starting temperature was. To define an interval scale two standards will be needed, because effectively and interval is being set. Since there is no absolute zero, one standard sets a point on the scale and another one is needed to define an interval (corresponding to a certain number of units). I can t think of any examples in RF

17 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) preserves information Voltage, current, impedance, Time interval, frequency Power, attenuation, Numerosity (numbers)

18 Notes The ratios of magnitudes can be compared, because zero is now well-defined Ratio scales place the greatest restriction on the transformations allowed: they cannot 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 (we can think of adding, multiplying, etc)

19 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

20 Notes There are perhaps conflicting interests in choosing the best scale. A field-engineer may find VSWR or RL very useful. A metrologist would probably prefer a scale in which it is natural to express the various sources of error that contribute to a measurement. Given the nature of measurement, such a scale is likely to be linear. However, if errors are multiplicative, a log scale may be desirable.

21 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 = Γ 2 re + Γ 2 im, tanφ = Γ im Γ re Rectangular-polar transformation is just a change of coordinates the same configuration of points in the complex plane can be represented in either set of coordinates however, problems arise when by treating the coordinates as independent 1-D quantities Was the transformation made to facilitate computations or to fit better with a conceptual model? it sometimes appears that the more natural conceptual domain is actually the complex plane, yet people still prefer to express data in polar coordinates

22 Notes There has not been an extension of Stevens ideas to 2-D quantities (that I know of) In RF there are problems when switching between polar and rectangular coordinates, but these arise because we treat the data as independent 1-D quantities after transformation. Clearly, the non-linearity of the coordinate transforms affects the statistics. (It is still possible to evaluate the complex mean of a set of data given in polar coordinates, but the result is not the same as the individual means of the r and φ data sets.) It is sometimes said that evaluating statistics in (r, φ) leads to errors. Actually, its just not the right way of treating the data. The question about whether to report measurement uncertainty of a complex quantity in polar or rectangular coordinates is important: what information are we trying to convey when reporting the uncertainty?

23 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. it is 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?

24 Notes Tradition is hard to shrug off The purpose for which data is being collected needs to be clear (why were ANAMET s splitter comparisons reporting VSWR, e.g.?) Transformation should be carried out with regard to the changes induced on the data If errors contribute to a measurement in a multiplicative way, then it might be sensible 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 reflection coefficient, rather than its rectangular coordinates. However, this not be the best way of dealing with the data in terms of measurement models. Perhaps we need to acknowledge the difference between useful summaries of multivariate data ( views, if you like) and the inherent information content of the actual data.

25 Thank You The Lower Hutt campus of Industrial Research Ltd

26 Notes (Notes added after a discussion during the 31 st ANAMET meeting on 2 April 2009) A quick poll was conducted asking how many participants would wish to retain use of 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 favour of the reflection coefficient. One participant spoke of the usefulness of the return-loss scale when tuning a filter to 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 the scale: the engineer is optimising a parameter by trying to make the return-loss as large as possible. I am not sure that other information is needed (ratios and intervals on the scale are irrelevant). One participant mentioned that in providing customer services, these scales are simply demanded and therefore measurements must be expressed in this way. My comment would be that, in such cases, it may be appropriate to work with other 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 with the client should be investigated (do they really need it, or are they just familiar with asking for it?.