91904, Fax: 02-6520797, E-mail: kisets@netvision.net.il

Abstract: The paper discusses the linear, concave and convex diagrams of weights as mathematical models of relative significances in a system of independent quality factors, representing in terms of probability law the complete group of events. On the basis of proposed criteria the paper compares the diagrams and proves that the linear diagram may successfully serve as unified diagram of weights that enables the extensive application of the principle of information cyclicity as the instrument of optimization.

The purpose of this paper is to fill a gap in the problem of the practical use of the recently discovered principle of information cyclicity [1]. The principle is based on qualimetry [2] and information theory [3]. For the complete group of n independent factors with weights K₁ ³ K₂ ³ . . . ³ K_n (relative significances expressed as probabilities) the principle establishes the following proportion:

(n - j )/n = Kj / K₁ = r _o = 1/2p , (1)

where: j = exp (-) is the optimum number of so-called informative

factors, unlike the (n - j ) redundant ones;

p = 3.14159 is the fundamental mathematical constant.

Severely the proportion (1) is true for a linear diagram of weights (D K_j = K_j – K_j+1 = constant > 0). Any deviation from the linearity creates an error in determining the j informative factors.

The equivalency between the informative number of factors (j ) and the lower limit of their weights (Kj ) depends on the configuration of weights diagram represented as step-functions K(j). The conventional illustration of concave, convex and mixed diagrams is given in the Appendix 2.

The mixed diagram may involve the parts of K(j) curve which are characterized by equal weights and therefore can cause the determination uncertainty when selecting the factors with j . In these cases the equivalency between two possible ways of selection (with j and Kj ) is broken, and the selection with Kj becomes the only solution, because the selection reliability increases due to allowing for as informative all equally weighted factors of the curve part where j is located.

The two different criteria, which being taken separately reflect the informative and qualimetry approaches appear to be true for solving the problem of unified weights diagram. There are 1) the selection error (L_q), which is the relative systematic error of quality estimation dependent on the form of diagram and on the weights forming the part of the subsystem of redundant quality factors, and 2) the maximum averaged information sensitivity (L _m) of the function K(j) represented by a weights diagram

The selection error can be expressed in the form of relative quality loss (L_q) as follows:

L_q = k_f (Kj + Kj ₊₁ + . . . + K_n) = (1/nK₁)(Kj + Kj ₊₁ + . . . +K_n), (2)

where: k_f = 1/nK₁ is the form-factor of weights diagram.

The form-factor is determined as the function of n and K₁ as the specific parameters of a diagram which satisfy the condition: 0 £ k_f £ 1 (where: k_f = 0 when K₁ = 1; n = 1), and k_f = 1 (when K₁ = 1/n). The following three ways that give the same result are possible for the determination of form-factor:

1. As an average of the effective number of factors n_e = 1/K₁ as regards one step (j = 1) of weights diagram, i.e. k_f = n_e/n = 1/nK₁.
2. As an average of the functional insensitivity of the diagram as follows:

k_f = (1/n)= (1/nK₁) = 1/nK₁, (4)

where: D j = 1 = constant;

= 1 (by the definition of weights diagram).

When choosing the unified model of weights diagram, the selection error of all diagrams in the table shall (a) not exceed the optimum accuracy coefficient (r _o = 1/2p ), and (b) be less for the linear diagram then that determined for the maximum concave diagram. The graphical presentation of the errors as functions of n (Fig. 1) illustrates the conformity with these requirements to the full. Besides, the linear diagram represents the approximately intermediate form between the limiting concave and convex diagrams.

Fig. 1 Selection errors (L_q) as functions of number (n) of quality factors for the

Each step (j) of weights diagram may be characterized by the information sensitivity calculated in the form of entropy (H_j) as follows:

H_j = -{[(D K_j/D j)] / []}* ln{[(D K_j/D j)]/[]} (5)

Taking into account that D j = 1, and = K₁, the following expression for the averaged information sensitivity (L ) of a weights diagram is true:

• = (1/n) = - (1/n)

The averaged information sensitivity must as far as possible be maximum in order to provide minimum theoretical uncertainty when determining Kj .

In accordance with the nature of entropy the maximum value of the averaged information sensitivity (L _m) is achieved when D K_j = K₁/n = constant, which is a characteristic of a linear diagram of weights, and in this case L _m = (1 / n) ln n. In terms of Table 1 L _m is equal to L ₃.

As for the limiting diagrams of weights considered in the previous paragraph, one can be easily convinced that according to the formula (6) they are characterized by the same sensitivity coefficient L _{1, 2}, which is expressed as shown in Table 1. For the comparison both L ₃ and L _1,2 are presented as functions of n (Fig. 2).

Fig. 2 Information sensitivity coefficients (L ) as functions of number n of quality

K_j = (h_j + ^-1, (7)

where: h_j = the component of weight determined without allowance for group interactions;

h_jr = the component of weight determined for r (combination of group interactions);

l = (2^{n – 1}) = the number of combinations of each component with all others.

The physical and geometrical characteristics of the current transformer and the results of their variance analysis [4] serve here as an initial data (quality factors) to exemplify the practical use of formula (7).The following parameters were considered as influencing the transformer ratio (output quality characteristic): the performance load (A), the ratio between the direct and alternative rms current (B), and the width of the air gap, via which the transformer is linearized (C). The relative variances as the components of weights (h) of the parameters and their interactions (AxB, AxC, BxC, and AxBxC) are presented in Table 2.

The weights (K) were calculated in accordance with formula (6) as follows:

K_A = (h_A + h_AB + h_AC + h_ABC) / (h_A + h_B + h_C + h_AB + h_AC + h_ABC) = 0.165;

K_B = (h_B + h_AB + h_BC + h_ABC) / (h_A + h_B + h_C + h_AB + h_AC + h_ABC) =0.021;

K_C = (h_C + h_AC + h_BC + h_ABC) / (h_A + h_B + h_C + h_AB + h_AC + h_ABC) = 0.841.

The parameters A and C, which weights exceed Kj = K_C /2p = 0.129, can be recognized as informative ones and subject to measurement control.