BULETINUL INSTITUTULUI POLITEHNIC DIN IAŞI

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44 Petronela Paraschiv case of biomechanical models lead to the necessity `correcting` the models achieved through permanent comparison with the experimental registered values (Fisher et al., 2007). Therefore, if through experiment can be achieved numerical values, respectively, a matrix of real values, then, applying the method of correlation coefficient between the matrix of real values and the matrix of theoretical values, it is determined the correlation degree between values or, other way stated, in which measure the variation of a variable from the analytical model has as effect a proportional variation of the values achieved through experimental determinations (Azar & Calandruccio, 2008). Moreover, it can be applied the method of perturbation functions over the analytical model in such a way that the values theoretically determined to coincide or to be very closed to those experimentally determined. However, if through the experiment cannot be achieved quantitative information, but only qualitative, then it would be possible the analysis of all variation charts of experimental sizes, respectively the determination of interpolation functions that will approximate the best the analysed curves, for each portion of curve, and then the comparison of these functions with the ones corresponding to the analytical model and the effectuation of the afferent changes, if it is imposed, in the theoretical model (Marciuk, 1983). The comparison of these curves can reveal different speeds of variation, through curves gradient, of the analyzed size, so that one can intervene in the theoretical model for approaching it to the experimental one 2. Theoretical Approach on Correction Methods There are discussed three correction methods of biomechanical model in order to approach the experimental model to the theoretical one. 2.1. The Correlation Coefficient Method Correlation is a descriptive static method that shows the existence or not of a link between two analyzed variables, but does not highlight the causality relation. Therefore, between the forces of the tendon achieved through experimental and analytical path can be established the type of link between them, respectively the linear or non-linear link, proportional or converse proportional, of weak, medium or strong or lack of strength, i.e. the independence of the two variables (Antonescu et al., 1986). The determination of the correlation starts from calculating the note or the standard score, that shows the deviation of a given value of the analysed variable from the arithmetic average of all values, this deviation being expressed in standard deviations. The standard note has the expression xi − x Zi = , (1) σ
Bul. Inst. Polit. Iaşi, t. LVIII (LXII), f. 3, 2012 45 where x i is a value of the analyzed variable, x is the arithmetic average of all values and σ is the standard deviation, calculated with 1 n i n i = 1 ∑ ( ) 2 , (2) σ = x −x n being the total number of values of the variable. In this way, there can be calculated the standard notes for the values of the first and second variable. These notes can be expressed with Eq. (3) Z 1 i 1 1 xi − x = , (3) 1 σ for the first variable and 2 2 2 xi − x Zi = , (4) 2 σ for the second variable. The correlation coefficient can be calculated using the Person correlation formula such as r = n ∑ i= 1 Z Z 1 2 i i n . (5) The correlation coefficient can have values between –1 şi +1, with the following meanings: r = –1, the link between variables is perfect and converse proportional; r = 0, between variables there is no link, respectively variables are independent; r = +1, the link between variables is perfect and direct proportional; r ∈ (–1, 1 )\ 0, ithout value 0, the link is not perfect between the two variables, meaning that for the variation in a variable will correspond a smaller variation within the other, converse or direct proportional; in this situation, between the two variables there is a regressive variation, being also possible to establish the corresponding equation of regression. For being able to calculate the correlation coefficient there are necessary, therefore, the values of the two variables that are analysed. These values are, actually, matrixes with a single dimension, with n values, to which apply the usual calculations for statistic analysis, for checking the two arrays of data, as well as achieving the statistical values. 2.2. The Method of Perturbation Functions In the case of some “converse problems” there can be applied methods of the perturbations theory with the help of which one can generate algorithms of
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