Measurement as a interface between the language and tools ...

Measurement as a interface between the language and tools of the science :

Měření jako rozhraní jazyka a nástrojů vědy

Autor: Jiří Vaníček

Abstract:

One of the most important problem on the interfaces among mathematics, computer science and special scientific disciplines is the question of describing the empirical structures using numbers. The description of the selected part of real word using such formal objects as numbers is usually called measurement. On the narrow sense a measurement is a

mapping from the set E of empirical objects into the selected set R of, which is a homomorphism with respect to selected relation and operation on the set of empirical objects. Unfortunately using numbers as a aid for measurement is limited in case of many interesting empirical structures. The main two reasons for this limitation are:

1. The human empirical relation of priority is often not a week order. In many practical situation it is only a partial order (no comparable entities exist) or only a semiorder or interval order (the threshold preference with a nontransitive indifference). In this case the ordinal scale type measurement using numbers does not exist.

2. The concatenation operation in empirical structure should not be associative and therefore the additive representation using numbers does not exist.

On the other hand contemporary mathematics theory and contemporary computers is able to treat with more general formal objects that numbers only. Therefore more sophisticated formal structure can be used for the measurement in this situation.

The foundations of the advanced measurement theory, there is the theory of describing the real word by the more general formal structures as numbers shall be introduced in the contribution. The generalisation of the scale type concept for this generalisation shall be described. The relative good experience with the advanced measurement using formal objets such as vectors or subsets of a given finite set have been obtained by author for the problem of software complexity and software quality metrics on the imperative and object oriented environment.

Abstrakt:

Jedním z nejdůležitějších problémů rozhraní matematiky, informatiky a specializovaných vědních disciplin je otázka, do jaké míry lze empirické struktury popsat čísly. Takovému popisu vybrané části reálného světa pomocí formálních objektů, kterými jsou čísla, říkáme obvykle měření. V užším slova smyslu je měření zobrazením z množiny E empirických objektů do množiny R čísel, které je homomorfismem vzhledem k vybraným relacím a operacím na množině empirických objektů. Žel, užití čísel jako prostředku pro měření představuje v případě mnoha zajímavých empirických struktur značná omezení. Nejdůležitější příčiny těchto omezení jsou:

1. Lidské empirické relace upřednostňování jedné varianty před druhou často nejsou slabým uspořádáním. V mnoha praktických případech jde pouze o částečné uspořádání (existují navzájem neporovnatelné entity) nebo pouze semiuspořádání či intervalové uspořádání (podprahové rozdíly jsou nerozlišitelné s netransitivní relací nerozlišitelnosti). V takovém případě není možné měření ani ve stupnici ordinálního typu.

2. Operace slučování v empirické struktuře může být neasociativní a tedy aditivní representace pomocí čísel neexistuje.

Na druhé straně současná teoretická matematika a současné počítače mohou pracovat s obecnějšími objekty, než pouze s čísly. Pro měření lze tedy užít komplikovanějších struktur, pokud to povaha empirického světa vyžaduje.

V příspěvku jsou uvedeny základy tak zvaného zobecněného měření jako teorie popisu světa pomocí obecnějších matematických struktur, než jsou čísla. Je naznačeno, jak je možné pro tento případ zobecnit pojem typu měřicí stupnice (cca 20 rows)

Keywords:

science, mathematics and computer science interface, generalised measurement, metric scale type, metric scale signature, homomorphism, relation of preference, nonassociative concatenation

Klíčová slova:

rozhraní vědy, matematiky a informatiky, zobecněné měření, typ měřicí stupnice, signatura měřicí stupnice, homomorfismus, preferenční relace, neasociativní skládání,

Introduction:

A major difference between a ”well-developed” sciences as physics and chemistry an some of the less ”well-developed” sciences such as psychology and sociology is the degree to which the things are measured. The high-pitched intention of the author is do demonstrate, that this gap can be gradually contracted.

The principles of formal mathematical theory and computer technology utilisation for the extension of human knowledge can be described on the following schema:

Discussion:

Let us consider step by step all of this four transformations indicated on this picture in more detail:

1 The first step is usually called measurement. The in fact it is a representation of the real word carved area by means of formal objects, usually by means of numbers. Such formal objects have to keep the same roles as are valid in the empirical area. The properties of various kinds of numbers (natural numbers, integers, rationales or real numbers) are relatively convenient. The natural order of such a numbers is a simple order, addition and multiplication is associative and commutative. If the situation on the real word area is similar and the same rule are valid also for our priority relation and concatenation operation on the empirical word area, numbers can be considered as an appropriate mean for the measurement. On the antiquity time are only used for simple and positive rationales for the more complicated situation. On the Middle ages zero and negative integers and rationales had to be added. The differential and integral calculus and other mathematical analysis ideas leads to the extension of number domain to real numbers and in the specific situation also to more general domains of numbers.

On the 19th an 20th century the developing made in physics, chemistry, computer science, economy indicate the necessity of investigation more complicated empirical situation. For example structures with the binary relations which is not a simple order but only a partial order or semiorder, the structures with noncommutative or/and nonassociative binary operation and the structures with more complicated relations (operations). On the other hand modern mathematics is able to manipulate with more sophisticated formal structures which properties can be consider as appropriate for this new needs. Vectors are probably the most simple such a example, graphs the more complicated one. Many different mathematical objects can be select for the measurement instead of numbers. The important criterion for such formal structures only is the true representation of the empirical area important properties and rules.

Let us describe the situation more formally. The important (for us) area of the real word can be formalised as so called formal structure

A = (A, {Rj}jJ).

Where A is a nonempty set and Rj the nonempty finite sequence of relation (possible of a different ”-arity”). Binary relation in this sequence can represents for example the priority from some point of view, special type of ternary relations the concatenation operation of objects (parts) in the real word into the complex object. Similarly

B = (B, {Sj }jJ)

is the formal structure consisting from the formal objects and formal relations (operation).The mapping from A into B is called a metric and the value (x) the measure of xA, if and only if (iff) preserves all relations, there is iff it is a homomorphic mapping from A into B. Such a mapping of course can be not a bijection. Different objects from A can have the same measure. The triple

(A, B, )

in this situation is called measurement and the pair (A,B) the representation of the structure A using the structure B in this measurement. In the classical situation (measurement using numbers) the set of all real numbers or the set of all positive real numbers + is used instead of B. with the natural order ”>” or ”” and sometimes also with the operation of addition ”+” or multiplication ””.

Our first conclusion can be the following: Not only numbers can be selected as an appropriate tool for the representation of empirical structure using the formal one. The choice of more sophisticated object can be appropriate for the formal description of the selected real word area or the more bizarre relation then ”>” or ”” and more bizarre operation then ”+” or ”” may by useful to consider for numbers in formal structure for measurement.

The second step,represented by double-side arrow on the picture, is optional. It is a representation of data and algorithm in the form, which is convenient for the information technology tools. This turning branch is relatively new and its importance grows last fifteen years as a consequence of the computer boom in the second part on 20th century. This boom leads into spreading our wings in the possibility to realise various time consuming algorithms. But the following issues occur: Mathematical analysis and related disciplines describe the real word using real numbers and structures derived from real numbers. These structures forms a continuum and the continuity is essential for such fundamental properties as is for example the existence of the solution of the equation or the existence of extremes of the function or functional.

But the modern physics and chemistry used mainly a discrete paradigm for the real word description. The economical, psychological and social processes are also mainly discrete in fact. However the discrete representation in mathematics does not facilitate the application of the convenient mathematical theory.

On the other hand in many situation the computation complexity is so extensive, that the using of computer as a tool for data processing is indispensable. However all computers in the word are not so powerful to together one irrational number, or e, for example. The other discretisation by rounding is necessary to represent numbers and derived structures on the computer.

The following paradoxical situation occur. We have to make an forced artificial completition of the problem to reach the possibility to use mathematical theory. this theory leads to some algorithm, which can be realised only via other discretisation of the problem. This artificial unnatural discretisation is of course completely different from the discrete constitution of the nature. Numerical unstability of the algorithms is only one of the bunch of problems, which can be evoked by this two incompatible transformation of information and data. The consistent description and managing of this two completition/discretisation and discretisation/completition processes are from my point of view the great challenge for the current science and research.

Our second conclusion can be the following: There is a essential gap between the nature and mathematical model of nature and another substantial gap between the mathematical theory and its computer realisation. We have to be very careful jumping across these gaps.

The third step on the picture coheres with the proper formal process, which is usually called the computation. This part of the process have been investigated in huge number of researches. It will be not a realistic idea to try to advert all, which is eminent or significant and we have no intention to make it. Let us mention very briefly only two particular points, related to the second arrow on our picture:

· First of these selected problems is computational complexity. The huge accretion of the speed and memory capacity of modern computers is able to evoke the wrongful idea, that everything is possible to settle, only is our computer will be sufficiently powerful. It is a total misunderstanding and illusion. The theory of computational complexity showed, that time (or the space), which is necessary to realise some algorithm can depends from the volume of input data like a exponential or factorial function and for many important problems we dont know an algorithm, which is substantially better. If we are able to solve the problem of the volume 100, but no larger one and buy another one with the 100-times better processing rate, we are able to solve in case of exponential time complexity only the problem of the volume 109 and in the case of factorial complexity only the 101 volume problem. The roll of hardware parameters upgrowth is therefore often overrated.

· Second problems is the numerical stability of algorithms. This problem has a connection with the rounding, necessary for the representation of numbers on the computer memory. Various algorithms handle with this errors differently. The accumulation of these errors can be minimise, can be acceptable, but can be also abnormal, that leads to the debasing of the computation. In all elementary textbooks for the integral calculus the problem to compute using ”per-partes” method and derived the reccurent formula for

The attempt to realise this algorithm for n = 10 is the best occasion to embrace into the unveiled problem.

Our third conclusion can be the following: Computers bring a significant gain for the knowledge extension process. However there are not all-powerful tool for everything.

The last fourth step on the picture is called interpretation. It has a close connection with the firs one. The metrics in measurement are in general not determine uniquely. If we use a metric and another metric for the representation (A,B), both and can be a homomorphism between structures A and B and therefore both (A,B, ) and (A,B, ) can be considered as a equal competent measurement. Measures obtained by and are however different and the results of computation can be different also. Such concept, which is not invariant to a particular choice of metrics is of course not meaningful in the empirical structure. Let us describe the situation more precisely.

Representation (A,B) is called to be regular iff for each two measurements (A,B,) and (A,B,) there exists a function f from (A) B into B, such that (a) = f((a)) for each aA. It is reasonable to confine to regular representations only. If (A,B,) is a measurement the function f from (A) B into B is called to be an admissible transformation iff (A,B, ), where is defined by the equation (a) = f((a)) for each aA, is also a measurement, there is iff is also a homomorphism from A to B. The representation with the appropriate set of admissible transformations is called usually a metric scale type of measurement. It is clear, that only such results, obtained using the calculation in the formal structure, can be meaningfully interpreted as a rule for the empirical structure, which are invariant to any admissible transformation of measures concordant with the metric scale type.

It is clear, that extensive set of admissible transformation results in a narrow class of meaningful interpretable statements and vice versa, limited set of admissible transformation leads to a broad class of interpretable results from formal structure into empirical one.

For measurement using numbers the following Stevens scale type classification is known more then years ago:

· Absolute scale type, where the set of admissible transformation contain only identity mapping and all results are interpretable. The example of such measurement where any measure which is define as a ratio of some value and the fixed reference value.

· Ratio scale type, where the set of admissible transformation contains all function of the type f(x) = ax, where a>0 is a given number. Example is the measurement of length, size, weight, … . The set of meaningful statements is broad and contains for example geometrical and arithmetical mean, percentage etc.

· Interval scale type, where the set of admissible transformation contains all function of the type f(x) = ax + b, where a>0 and b are a given numbers. Example is measurement of temperature (Celsius, Fahrenheit, …). Arithmetic mean is meaningful, but geometric mean and for example the statement :”two time more” is meaningless.

· Ordinal scale type, where the set of admissible transformation contains all function of the type f(x), where f is any strictly increasing function. Only such results which depends only on the order of numbers, not on the operation are meaningful. Not only geometric, but also an arithmetic mean is meaningless. Meaningful is only median and extrema.

· Nominal scale type, where the set of admissible transformation contains all function of the type f(x), where f is any one to one function. The example is using numbers for the classification of categories. Only a statements concerning the category classification are meaningful.

This classification of metric scale types for numbers is of course not exhausting. In some application the log-interval scale type measurement with the set of admissible functions of the form f(x) = axb , or a difference scale type measurement with the set of admissible functions of the form f(x) = x + a, are appropriate. The careful analysis points that the prevalent notion that ratio scale type is a special case of interval scale measurement is not correct. The ratio scale type is a special type of log-interval scale type (with b = 0) . The special type of interval scale is the difference scale type measurement.

It is also important to point out, that for any measurement on the scale type which is more bountiful that the ordinal one some concatenation operation on the empirical structure have to be consider. The absence of such a operation leads to completely meaningless statements and considerations. Let us give two forbidding examples:

1. ”The temperature growth rapidly. It was 20 degrees of Celsius yesterday, today it is 300C. 50% in one day is to much.”. The defect is easy to find. Measurement of temperature is of the interval scale type. Not a ratio scale type.

2. ”The dog has the flair five times better then the man” What is wrong now? The question if it is a true or false statement is not meaningful till the concatenation operation between fliers is not specified. Let us try to clarify this statement. The attempt ”If the dog smell something it 5 men have to come to smell it” is probably not convenient. The second attempt may be ”:If the dog smell one deer, it have to come 5 dears so that the men smell this dears also”. Is it correct now or it is necessary to try explain the statement again for example using the distance? If we are fair, we have to avouch that the initial statement does not have any correct meaning from the very beginning.

What is important for the measurement with the correct interpretation using the metric scale type theory is to decide, what scale type measurement exists for a given empirical structure. It is to late make the analysis after the measurement. without this consideration before. For the described measurement scale type for numbers such conditions are known and can be found in the excellent monographs from Roberts [ROB] or Krants, Luce, Suppers and Tversky [KLST]

For the measurement using more sophisticated formal objects then only numbers the problem of interpretability of results from the formal structure into empirical one is also important, but more complicated. First difficulty consists in the non-existence of a integral theory of metric scale types for a general case (there isfor the case where measures are not simple numbers).

Let us sketch some results of an attempt to create such a theory for structures with several relations (operations), such that one of these relations are binary relation of our preferences, which is a week order. Such a structure can be called a ordered relational structure. The study of the scale type is in fact the study of the subsets H of the group G of all automorphisms of such structures. For the characterisation of such subgroups two numbers can be useful:

· H is said to be M-point homogeneous iff for each a1 .a2, … aM and b1 b2 … bM from A there exists a automorphism H such that ( aj,) = bj for j = 1, 2, … ,M.

· H is said to be N-point unique iff for each a1 .a2, … aN the following is valid: if there is (ai,) = (ai,) for i = 1, 2, … ,N, than (a) = (a) for all aA.

The representation is said to be M-point homogeneous or N-point unique, depending on the related group H.

The signature of the measurement is the ordered pair (M,N), where M is the largest degree of homogeneity and M the least degree of uniqueness. In case the structure is M-point homogeneous for all M, we set M = . In case the structure is not unique for any finite N, we set N = . See [VAN1] for more details.

Various important theorems concerning this signature can be proved. It seems that this concept can be considered as a hopeful for the generalisation of the notion of the metric scale type. Anyway the problem for the structures without any binary relation, which is week order is still open.

Another problem is to bring under control such empirical structures, which does not allow the measurement using numbers because of the nonassociative concatenation operation. In this situation the associative operation of adding numbers are not appropriate. For this case some conditions for the empirical structure can be derived, which guaranties the metrics into the subset of real numbers with some unusual operation of adding numbers, which, of course, have to be also nonassociative. Some analogy between this conditions and the condition for the existence of ratio scale type measurement in the case of associatively can be found. See [VAN2] for more details.

Important question also is to find the conditions, which are necessary for the application of mathematical analysis for numbers obtained as measures. After necessary simplification this question can be reduced to a problem under which condition is the range (A) of measures an interval in the set of all real numbers. The condition for the empirical structures with the binary priority relation which is a week order is natural:

1. The structure (A, ) must not have gaps (gap is the pair ab, such that does not exist any c, such that a c b).

2. The structure (A, ) must not have a holes (hole is a division of A into two subsets U and L, such that A = LU, if l L and u U, there is ul and the set L has not a maximum and the set U has not a minimum). There is A is Dodekind complete.

3. A contains a countable subset, order-dense with respect to .

The conditions for the completition of the set of empirical object A and the method how to do it can be also derived. See [VAN3] for more details.

Our fourth conclusion can be following: The problem of interpretation of results obtained by formal method is important and can be dangerous. Any incautiousness can raise a shortening in the prestige and credit of mathematics and computer science. For measurement using numbers there exists a consistent theory. For general measurement only few results are obtained. This area is open for the complex and detail investigation.

Reference:

[KLST] Krantz, D.H.; Luce, R.D.; Suppers, P and Tversky, A.:Foundation of Measurement,Vol. I. Additive and Polynomial Representation. Accademic Press Inc., San Diego, New York, Boston, London, Sydney, Tokyo, Toronto, 1971, 384p., ISBN 0-12-425403-9

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Luce, R.D.; Krantz, D.H.; Suppers, P and Tversky, A.:Foundation of Measurement, Vol. III. Representation, Axiomatization and Invariance. Academic Press Inc., San Diego, New York, Boston, London, Sydney, Tokyo, Toronto, 1971, 341p., ISBN 0-12-425403-9

[ROB] Roberts, F.S.:Measurement Theory with Applications to Decisionmaking, Utility, and the Social Sciences, Encyclopaedia of Mathematics and its Applications, Vol. 7, Addison/Wesely Publ. Comp., London, Amsterdam, Don Mills .Ontario, Sydney, Tokyo, 1979, 420 p., ISBN 0-201-13506-X,

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[VAN1] Vaníček, J.:O možném zobecnění pojmu měřicí stupnice - On the Possible Generalisation of the Measure Scale Type Concept. In: Sborník mezinárodní konference Agrární perspektivy 98, ČZU Praha, 1998, s. 781 - 786, ISBN 80-213-0420-0

[VAN2] Vaníček, J.: Continuous Measurable Concatenation Structures - Spojité měřitelné struktury s operací skládání. In: Sborník mezinárodní konference Agrární perspektivy 99, ČZU Praha, 1999, s. 728 - 732, ISBN 80-213-05630-0,

[VAN3] Vaníček, J.: Nonadditive Representation of Measurement Results.Sborník příspěvků z odborné konference k aktuálním otázkám české ekonomiky a univerzitního ekonomického vzdělávání. Provozně ekonomická fakulta Mendelovy zemědělské a lesnické univerzity v Brně, Brno 1999. 2. díl, ISBN 80-86515-87-8