On Modeling Assumptions and Artificial Neural Networks Stephan Rudolph and Bernd-Helmut Kr¨oplin Institute for Statics and Dynamics of Aerospace Structures University of Stuttgart, Pfaffenwaldring 27, D-70569 Stuttgart http://www.isd.uni-stuttgart.de/rudolph
[email protected] Workshop Proceedings Selbstorganisation von adaptivem Verhalten (SOAVE’97) University of Ilmenau, Germany, 23.-24. September, 1997. Published as: VDI Fortschrittsberichte, VDI Verlag, Reihe 8, Nr. 663, Seite 222-231.
Abstract. As a consequence of the epistemological principle of dimensional homogeneity follows the existence proof for the so-called Pi-Theorem, which is valid for all dimensionally homogeneous function equations. Any real-valued functional model from biology, chemistry, physics and engineering can therefore be subjected to a so-called similarity transform and be checked with the Pi-Theorem. This is possible because it is generally agreed on that any dimensionally not homogeneous model cannot be correct. As an interdisciplinary example, the construction of artificial neural networks (ANN) models currently used in many applications for function approximation based on a set of input-output patterns is subjected to this principle of dimensional homogeneity. Several important restrictions, properties and conclusions about the ANN generalization are then proved by the Pi-Theorem and justified by the principle of dimensional homogeneity. Keywords: Epistemological principles, Pi-Theorem, similarity transforms, similarity functions, dimensional homogeneity, artificial neural networks, network generalization.
1 Modeling In the absence of any other well established source of knowledge, humans often rely on abstract ideas and/or general principles as decision guidelines for their actions. In the natural sciences, it is the act of so-called epistemological reasoning which serves as a guidance in the process of investigation, understanding and model building of unknown natural phenomena or objects. The individual procedural steps to follow, the possible findings as well as the current state of knowledge about this epistemological reasoning process itself represents in its last consequence nothing less than the history of science of mankind and is essentially the summation of knowledge gained about model building by human reasoning, imagination and thoughtful experimental observation. The article points out several far reaching interdisciplinary consequences from the existence of the epistemological principle of so-called homogeneity emphasized by the greek philosopher T HEON and introduced to the modern natural sciences by N EWTON. This principle of homogeneity has later 1
been refined among others by F OURIER to the so-called concept of dimensional homogeneity. It has become since then one of the basic assumptions of classical group theoretic methods in mathematical physics developed by L IE and others. Due to the philosophical nature of the principle of homogeneity, it is generally agreed on in all natural sciences that any possibly correct functional model equation complies to this epistemological principle. The establishment of such epistemological principles is however qualitatively quiet distinct from the purely theoretical postulation of mathematical axioms often freely combined to form logical statement systems. In the building process of a theoretical geometry model for example, it is agreed on that the fifth axiom can be exchanged by an alternative axiom based on a different thought concept and will thus lead to a different type of geometry (Euclidean geometry versus Riemann geometry or Lobatschevsky geometry [4]). The selection of epistemological principles (such as heat has only been observed to flow from the ” hotter to the colder object“ as one of the fundamental assumptions in classical thermodynamics [8]) however is based on and justified by experimental observation and lacks therefore to a certain degree such above conceptual mathematical arbitrariness. More strictly speaking it is often very likely to be even impossible to vary or change these epistemological principles, because a change in these epistemological principles would on the one hand affect many aspects of our presently already agreed on scientific thought constructs and models and could even prohibit or deny the existence of already well established pieces of knowledge in the natural sciences such as in parts of engineering, chemistry, biology or physics 1. On the other hand such variations would as a further consequence eliminate the epistemological justification of the future logical statement system and thus no further link would be available to point from new statements derived inside the logical statement system back into the original space of the natural phenomena or object. This would result in the the real object world in semantically no longer interpretable, in an engineering sense meaningless“ logical statements. ”
1.1 Representations In order to be able to build models of reality or of objects or phenomena contained therein, it is necessary to find an appropriate representation for the model characteristics of interest. There exists an old discussion in artificial intelligence (AI) about whether this representation has to be of symbolic or sub-symbolic form to allow for the modeling of certain reasoning processes commonly considered as characteristic for intelligent“ behavior. ” While many open questions in this central discussion to AI have not been answered up to present and are therefore left for more or less open speculations, it is undoubtedly a feature of any representation form candidate that it has to model reality (or objects or phenomena contained therein) adequately. From a mathematical viewpoint, any representation possesses (or is at least intended to possess) therefore a bijective mapping property from the problem domain of interest onto the chosen representation form. Otherwise the representation could be shown not to model the reality adequately. Consequently, the problem of finding an appropriate representation form might not have one single unique solution, since any other bijective mapping applied to a bijective representation form would immediately lead to the construction of an alternative, equivalent bijective representation. Just the notational aspects of the two representations would vary, mathematically they would be equivalent. However, due to the notational differences, some aspects could be more apparent and be more clearly visible in one representation form than in the other. 1
A permissible way to consistently extend an already existing formal statement system is to embed the existing piece of theory into a more general one [8]. This can be done by replacing the former epistemological foundation by a new, more general one.
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In engineering, there is no strong separation in symbolic and sub-symbolic approaches by tradition. In contrary, it is expected from the equivalence between symbolic and sub-symbolic computational procedures that in principle numerical (i.e. sub-symbolic) and analytical (i.e. symbolic) algorithms yield identical results within reasonable error bounds. If this expectation couldn’t be justified by scientific principles, the attempt of computing numerical solutions for differential equations for which analytical solutions are not known or are difficult to obtain would be a lost effort a priori. It is therefore clear that to every numerical algorithm (i.e. a sub-symbolic representation) on a computer an equivalent analytical formula (i.e. a symbolic representation) exists. Once a representation is specified, it is also necessary to specify a set of permissible operations on the representation objects. As an instructive but simple example, the following two typical representations taken from computer science and physics are displayed and compared. In computer science, as shown in figure 1, expressions formed in computer languages are used to represent data objects and the algorithms for their manipulation. The specification of grammars by means of production rules is used to create parsers and syntax checkers for the syntactical correctness of a computer program. It is needless to say and generally agreed on that any syntactical incorrect program can never be semantically correct. grammar (example) list list list digit
?! ?! ?! ?!
list + digit list ? digit digit 0j1j2j3j4j5j6j7j8j9
Figure 1: Syntax Restrictions in Computer Science In physics, a typical representation form is a formula like the one shown in figure 2. These representations not only have the property that they are numerically correct functional relationships, but that also the physical dimensions in form of SI-units attributed to each language object (i.e. the variables) have to be subjected to the functional relationship as well. This means that due to the principle of dimensional homogeneity, physical dimensions cannot be created from or vanish in the void. The so-called dimensions check is a well-known example of this syntax check for functional equations. It is therefore generally agreed on that any dimensionally not homogeneous function equation can never be semantically correct. formula (example)
F =ma
%
quantitative
R R
qualitative
[kg ][ sm2 ] 7?! [N ] [ kgs2m ]
&
7?! R
Figure 2: Syntax Restrictions in Physics As is probably not widely known, the property of dimensional homogeneity has a proven record of usefulness in the natural sciences. As can be easily seen from the shown example equation F = m a (or written as F ? m a = 0), and ignoring the specific form of the functional relationship, one could just guess“ that a functional relationship f (x1 ; x2 ; x3 ) = 0 of three variables exists (here x1 = m; x2”= a and x3 = F ). In the special case shown above, the (syntactical) knowledge of the physical dimensions (i.e. the measurement unit scales of x1 in [kg ], x2 in [m=s2 ] and x3 in [N ]) of each variable is even so restrictive, that the only possible equation as a special solution of the most general 3
P Q formula x3 = i ( i j x1 ji x2 ji ) with i ; ji and ji as free constants is x3 = x1 x2 , e.g. F = m a. This strong (semantical)2 result can in this special case be achieved by a simple (syntactical) dimensions check3 . The general proof for this statement follows from the so-called Pi-Theorem, which is quoted from the literature in the appendix. 1.2 Biological Modeling Within the broad field of biology, a part of biology is concerned with the modeling of biological systems. These biological systems commonly possess sensors, which measure“ (local distributions ” of) force, temperature, pressure, amount of substance or light (list uncomplete). How these signals are processed in later stages inside the biological system is not yet known and the topic of current research in connectionist models of neural networks. Output“ of the biological information processing chain ” is used and needed to walk, think or speak (list uncomplete). A central point of human modeling techniques of real systems now needs to be discussed. As stated in the introduction, humans use principles used and/or model assumptions to come from a real object to a representation. By no means it is assumed that the object itself (i.e. an apple which falls from a tree as used in N EWTON’s thought experiment and which is said to have lead to f = ma) is required to be conscious“ about this or needs to have some knowledge about things like a force ” concept F which is the product of a mass m and an acceleration a. The strong syntactical consequence of these concepts however is valid and has been emphasized in previous paragraphs. It is therefore a central question to any modeling effort, whether the data acquired are a measured forces, temperatures, pressures, amount of substances or light and do therefore possess physical dimensions in form of SI-units (see appendix for definition of terms). While it is evident that all measured quantities possess physical dimensions as the word measuring already implies, it is important to realize that also dimensionless distribution functions do implicitly possess physical dimensions which can be formally constructed by a integration process [7]. All so-called higher moments of a (formerly dimensionless) function possess physical dimensions (An example for this rather abstract statement is the pressure p of a gas, whose mathematical model in classical thermodynamics is a second order moment [8]). The unexpected consequences of the idealized existence of the philosophical concept of physical dimensions in form of the SI-units are described in the next section. There the central question will be investigated what consequences follow from the epistemological concept of dimensional homogeneity if the input and output patterns of artificial neural networks explicitly possess physical dimensions.
2 Artificial Neural Networks Artificial neural networks (ANNs) are simple models of much more complex, still not fully understood information processing structures found in biological systems. The computational power commonly attributed to these networks is mainly based on the existence of adjustable parameters (e.g. the 2
N EWTON at the time came to his conclusion by interpretation of observations, for which he needed semantical knowledge about physics. He united the at the time distinct phenomena of inertia and gravity into one single concept of force. Modern physicists therefore sometimes joke that the true great achievement of N EWTON was the identification of the measurement unit scale for force F in [N ] = [ kgs2m ], which is considered in the context of this work as syntactical knowledge (see also next footnote). 3 The terms syntactical and semantical are used in this work in a more general fashion as defined in computer science. By syntax or syntactical considerations it is meant what conclusions can be drawn from the knowledge about the restrictions valid for the representation form. By semantic or semantical considerations it is meant what conclusions can be drawn from the interpretation of the domain knowledge or parts thereof.
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weights) and the use of non-linear, so-called transfer functions used to model individual neuron characteristics (e.g. sigmoidal functions). While the limitations of older linear neural network models are now mathematically well understood [12, 13, 15], it is mainly the nesting of the non-linear transfer functions inside the ANN which makes any deeper mathematical analysis of the network properties and performance difficult [11, 16]. Neural network research has therefore mainly been concentrated on the establishment of certain classes of non-linear multi-layered feed-forward neural networks, for which theorems and theoretical bounds for the approximation properties can be stated [1, 2, 6, 10, 17]. For time-independent process models feed-forward ANNs are used, while for time-dependent processes recurrent neural networks are applied. But until today no general theory has been presented for the a priori topology design, the explanation of the generalization properties and the a posteriori interpretation of non-linear multilayered neural networks.
2.1 Feed-Forward ANNs Figure 3 shows one typical application example of an feed-forward ANN for function approximation. There, so-called training patterns of p input and output values are presented to the ANN during a training period. It is hoped that by an appropriate adjustment of the inner weights of the
0x 1 BB x12 CC BB .. CC @ . A xn?1 p
approximation function
f
- xn p
Figure 3: Classical Feed-Forward Neural Network f feed-forward ANN, the feed-forward ANN will be able to approximate the functional relationship f (x1 ; : : : ; xn) = 0 encoded in the set of p training patterns fx1 ; : : : ; xngp. The training patterns are usually available in a data table, similar to the one displayed in Figure 4, in which the numerical data represent measurements of a beam under deflection (here now n = 5 and p = 1; : : : ; 5). The beam data 1 2 3 4 5
x1 = P x2 = l 3000 2875 2750 2625 2500
x3 = E x4 = I
0.50 125000.000 0.75 96643.525 1.00 68287.050 1.25 39930.575 1.50 11574.100
2 42 82 122 162
x5 = u 0.5000e-3 0.0996e-3 0.1637e-3 0.3508e-3 1.5000e-3
Figure 4: Original Pattern Data is of length l, with a material of Young’s modulus E , a cross sectional moment I and exhibits a deflection u under a load P . These measurements of the variables (x1 ; : : : ; x5 )p represent the available knowledge about the unknown process f (x1 ; : : : ; x5 ) = 0 of bending“. ” Since the underlying mathematical model in form of a differential equation can be solved, a closed analytical solution in the form of f (l; E; P; I; u) = 0 exists and is well known in the literature [18] to be equal to 1 1 P l3 = P 1 l3 E ?1 I ?1 (1) 3EI 3 From the data in figure 4 it is quite clear that the approximation problem during the training phases of the neural network consists in the identification of the correct n-dimensional hyper-surface of physical
u
=
5
variables f (x1 ; : : : ; xn ) = 0 with n = 5. A suitable topology for the computation sequence inside such an ANN with five adjustable weights w0 ; w1 ; w2 ; w3 and w4 is shown in figure 5. This ANN sums the weighted logarithms of its inputs in s and propagates it through the exponential h(s) = es as output function. According to equation (1), the correct solution for the ANN after training should be w1 = 1; w2 = 3; w3 = ?1; w4 = ?1 and w0 = 1=3 to generalize correctly.
P
w1
l
w2 w3
E
s
w0
u
w4
I
Figure 5: Neural Network Topology for f
2.2 Dimensional Homogeneity In this context it is important to note that in classical ANN function approximation, the weights w0; w1; w2; w3; w4 are initialized by random before the training and are iteratively updated according to the employed learning rule. To understand the purely syntactical consequence, the condition of dimensional homogeneity of the weights in the ANN in figure 5 is written as [u]1 = [P ]w1 [l]w2 [E ]w3 [I ]w4
(2)
since the dimensional representation of u is [u] = [F ]0 [L]1 and [u]; [P ]; : : : denote the dimensional representation of the variables u; P; : : : . From the general equation (2) one obtains two equations in the two dimensions [F ]orce and [L]ength used to represent the variables in the dimensional matrix. This leads to a linear equation system of the weights w1 ; : : : ; w4 written in [F ] : in [L] :
w1 + w3 w2 ? 2w3 + 4w4
= 0 = 1
(3) (4)
This means that all numerical values of the weights w1 ; : : : ; w4 in figure 5 which do not satisfy equations (3) and (4) do not represent dimensionally homogeneous equations. This however means that the initial neural network state as well as most of the intermediate neural network states do encode dimensionally not homogeneous functions which are known to be syntactically (and thus also semantically) incorrect by definition (see introduction).
2.3 Similarity Networks Using the a priori knowledge of the physical dimensions of the variables l; E; P; I; u, the following dimensional matrix can be established in a ([M ]ass, [L]ength, [T ]ime)-System. By adding multiples of the matrix columns to each other in the left hand side of figure 6, the modified dimensional matrix in the right hand side of figure 6 is obtained. Concerning the right hand side of figure 6 the third
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Variable SI-units
[kg m=s2]
P l E I u
[m] [kg=m s2 ] [m4] [m]
[M ] [L] [T ]
[F ] [L]
1 1 -2 0 1 0 1 -1 -2 0 4 0 0 1 0
1 0 0 1 1 -2 0 4 0 1
=)
Figure 6: Dimensional Matrix Computations column (i.e. [T ]ime) has been omitted, since it contains only zeros. Since the rank of the dimensional matrix is r = 2, m = n ? r = 3 dimensionless products from n = 5 physical variables are obtained.
1 = E P ?1 l2 2 = 3 =
I l?4 u l?1
=
E
(5)
P 1 l?2
I u = 1 l
= 4 l
(6) (7)
This means that every data point in x1 ; : : : ; xn corresponds to a data point in 1 ; : : : ; m . In figure 7 the numerical values of this mapping for the data points in figure 4 are shown. From this transformed data 1 2 3 4 5
1
2
3
10.4167 18.9085 24.8317 23.7682 10.4167
32.0000 132.7407 82.0000 49.9712 32.0000
0.100e-2 0.133e-3 0.164e-3 0.281e-3 0.100e-2
Figure 7: Transformed Pattern Data data table, as well as from the knowledge of the exact dimensionless solution in form of equation (9), it is clear that the approximation problem during the training phases of the neural network has now been transformed into the problem of the identification of the correct m-dimensional hyper-surface F (1; : : : ; m) = 0 with only m = 3. This is shown in figure 8.
P l E I
?1 +2
1
+1 v1
?4 +1 2
v2
v0
+1
+1
3
+1
u
Figure 8: Network Topology for f via F According to equation (9) the correct solution after training is v1 = ?1; v2 = ?1 and v0 = 1=3 for the neural network to generalize correctly. In contrary to the previous classical neural network function approximation of f it is important to note in this context that the neural network can now permanently represent only dimensionally 7
homogeneous functions f , regardless of the random initialization and the iterative update of v0 ; v1 ; v2 during the training. Taking now a look back to the exact solution in form of equation (1), an algebraic manipulation (multiplication of the right hand side of equation (1) by unity in form of the factor (l2 =l2 )) leads with elementary calculus to
1 P l4 ( )( ) (8) 3 El2 I Substituting now equations (5), (6) and (7) into equation (8) yields the dimensionless equation 3 = 13 1?1 2?1 (9) Equation (9) is a practical example of the fact that every complete and dimensionally homogeneous equation of n physical variables can be written in the form of a dimensionless equation of its m = n ? r dimensionless groups (i.e. the dimensionless products). The dimensionless products can thus be interpreted as the necessary and sufficient “building blocks” of the correct solution. This is stated by the Pi-Theorem in the form of equations (10), (11) and (12). The function F can thus be understood as being nested inside of f , enclosed by the appropriate similarity mappings. This mapping scheme based on the Pi-Theorem can thus be generalized to the new neural network similarity topology design and interpretation scheme as shown in figure 9. This
u l
=
0x 1 BB x12 CC BB .. CC - 1 @ . A xn?1
function
F
?1
- xn
Figure 9: Similarity Network for f via F means that the first and last network layer represent the (here predetermined and fixed) for- and backtransform 1 and ?1 into and from dimensionless space, while the learning during the training is done through adjustment of the weights of the sought after similarity function F only. This in direct comparison to figure 3 advantageous exploitation of the principle of dimensional homogeneity in the design of feed-forward neural networks guarantees always dimensionally homogeneous functions regardless of random initialization or iterative weight update during the training. As previously stated, this restriction reduces the weight search space to the class of all possibly syntactical correct solutions without loss of generality.
2.4 Generalization Proof Based on this neural network topology design scheme as shown in figure 9, a generally valid proof of the two necessary and sufficient conditions for the correct generalization in neural networks can now be established in form of the two following consecutive steps. The formerly unresolved generalization capability of non-linear multi-layered feed-forward neural networks can be now proven to be
pointwise correct, if and only if a training pattern p can be learned and recalled error-free by the new similarity neural network topology F . This proof is due to the fact that well distinct data points in ‘x-space’ may fall onto the very same point in ‘ -space’. (An example of this is data point 1 and 5 in tables 4 and 6). This behaviour is called complete similarity. totally correct, if and only if the neural network approximates after the training the correct similarity function F (1 ; : : : ; m ) = 0 which is associated with f (x1 ; : : : ; xn ) = 0. The correct similarity function F is approximated if and only if the correct pointwise generalization property is fulfilled for each point in the whole domain of definition of F . 8
3 Summary The two mathematically necessary and sufficient conditions for the correct generalization capability of artificial neural networks (ANN) have been shown. It should however be clear that because of the necessary and sufficient condition for the totally correct generalization in form of the identification of the correct similarity function F , there is made no explicit or implicit claim that this correct result is in fact achieved after the training of the neural network. It is evident that the underlying basic problem of function approximation based on sparse data samples remains. It is however claimed that every correct result can always be written in this form and that otherwise the search space after random initialization or in intermediate stages of the sought after approximation of f without the use of the dimensionless groups is very likely to violate the property of dimensional homogeneity [14].
References [1] A. Barron, Universal approximation bounds for superpositions of a sigmoidal function, IEEE Transactions on Information Theory, 39 (3) (1993), 930–945. [2] E. Blum and L. Li, Approximation theory and neural networks, Neural Networks, 4 (4) (1991), 511–515. [3] G. Bluman and S. Kumei, Symmetries and Differential Equations. (Springer, New York, 1989). [4] I. Bronstein and K. Semendjajew, Taschenbuch der Mathematik, 23. Auflage, Verlag Harri Deutsch, 1987. [5] E. Buckingham, The principle of similitude, Nature 96 (1915), 396-397. [6] G. Cybenko, Approximations by superpositions of a sigmoidal function, Math. Control, Signals, Systems, 2 (1989), 303–314. [7] G. Emrich. Bilderkennung mit neuen, multi-skaleninvarianten Zentralmomenten, in: Kr¨oplin, B.: Internationales Workshop Neuronale Netze in Ingenieuranwendungen, Institut f¨ur Statik und Dynamik der Luftund Raumfahrtkonstruktionen, Universit¨at Stuttgart, 15.-16. Februar 1996, 111-121. [8] C. Gerthsen, H. Kneser and H. Vogel, Physik, 12. Auflage, Springer, 1974. [9] H. G¨ortler, Dimensionsanalyse (Springer, Berlin, 1975). [10] K. Hornik, M. Stinchcombe, and H. White, Multilayer feedforward networks are universal approximators, Neural Networks, 2 (5) (1989), 359–366. [11] T. Masters, Practical Neural Network Recipes in C++ (chapter 6: Multilayer Feedforward Networks, 85–90, Academic Press, Boston, 1993). [12] M. Minsky and S. Papert, Perceptron: An Introduction to Computational Geometry (MIT Press, Cambridge, MA, 1969). [13] Y.-H. Pao, Adaptive Pattern Recognition and Neural Networks (Addison-Wesley, Reading MA, 1989). [14] RUDOLPH , S.: On topology, size and generalization of non-linear feed-forward neural networks. To appear in Neurocomputing, 1997. [15] D. Rumelhart and J. McClelland, Parallel Distributed Processing. Volume I and II (MIT Press, Cambridge, MA, 1986). [16] E. Sanchez-Sinencio and C. Lau (eds), Artificial Neural Networks (IEEE Press, New York, 1992). [17] K.-Y. Siu, V. Roychowdhury, and T. Kailath, Rational approximation techniques for analysis of neural networks, IEEE Transactions on Information Theory, 40 (2) (1994), 455–466. [18] S. Timoshenko and J. Goodier, Theory of Elasticity, McGraw-Hill, London, 1987.
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Appendix The term dimensional homogeneity means that in any physical function equation, the functional relationship of the function parameters xi ; xj applies to the physical dimensions of the parameters (usually expressed in SI-units) just like it does to their numerical values. The principle of dimensional homogeneity guarantees that every correct equation fulfills the dimensions check. The term dimensionless product stands for a special class of monomial expressions in the form Q xj ni=1 x?i ji which have no physical dimensions (e.g. are dimensionless) and are formed out of a (sub)set of physical variables xi ; xj of the very same function equation f (x1 ; : : : ; xn ) = 0. For all dimensionally homogeneous function equations, the Pi-Theorem of Buckingham [5, 9] holds: Theorem 1 (Pi-Theorem [5, 9]) From the existence of a complete and dimensionally homogeneous function f of n physical quantities xi 2 R + follows the existence of a dimensionless function F of only m n dimensionless quantities j 2 R + (10) f (x1; :::; xn) = 0 F (1; : : : ; m) = 0 (11) where m = n ? r is reduced by the rank r of the dimensional matrix formed by the n dimensional
quantities. The dimensionless quantities (also dimensionless products or dimensionless groups) have the form Yr j = xj x?i ji (12) i=1 for j = 1; : : :
; m 2 N + and with the ji 2 R as constants.
The so-called dimensional matrix associated with the relevance list of variables x1 ; : : : ; xn is shown in the left hand side of figure 10. This dimensional matrix has n rows for the variables xi and up to k columns for the representation of the dimensional exponents eij of the variables xi in the k base dimensions sk of the employed unit system.
x1 x2: :: :: :: :: :: x: n
s1 s2 : : : : : sk eij
66 xx1
r n
2:
:: :: :
?6 xx1r :: m : ?? xm
s1 s2 : : : : : sr 1
1
::
::
ji
::
1
9 > > > =variables > xi “ > > ; ” ) variables x“ ” j
Figure 10: Definition of Dimensional Matrix [14] To calculate the dimensionless products j in equation (12), the dimensional matrix of the relevance list of variables x1 ; : : : ; xn as shown in the left hand side of figure 10 needs to be created. By rank preserving operations the upper diagonal form of the dimensional matrix as shown in the right hand side of figure 10 is obtained. The unknown exponents ? ji of the dimensionless products in equation (12) are then automatically determined by negation of the values of the resulting matrix elements ji in the hatched part of the matrix on the lower right hand side of figure 10.
10
