Trans RINA, Vol 156, Part B2, Intl J Small Craft Tech, Jul-Dec 2014


If f(λx) = λf(x), the function is linear, and if f(λx) ≠ λf(x), the function is nonlinear. We can see that, with a linear function, the real number  is extracted from the expression representing a function and, with a nonlinear function, the real number is a component part of the expression.


Now we put a linear function in expression (2) Yf(X) ε





and in the developed form 11 1 yf(x ) ε


yf(x ) ε 


 


yf(x ) ε    11  X) ε


 22 2


....................... nn n


We now put a nonlinear function in expression (2) Yf(


and in the developed form


yf(x ) ε 


yf(x ) ε  Thus, yf(x ) ε


 22 2


....................... nn n


our intention that the selected function, linear or nonlinear, describes


mathematical the “natural”


dependence between the measured quantities Y and X,


essentially means that the error i should be as small as possible.


4.1 REGRESSION ANALYSIS AND ARTIFICIAL NEURAL NETWORKS


To determine the coefficient  of a mathematic function from a condition of minimum error, regression analysis and the least squares method are used.


The regression analysis is a statistical procedure. Statistics is a science that deals with collecting, classifying and interpreting the numerical facts which it explains


by applying the mathematical theory of


probability. In this case, а dependence between measured quantities is seen as a statistical dependence.


Linear functions enable used the application for of direct


procedures by means of which the value of the coefficient is computed. Nonlinear functions force us to use iterative procedures. The first mathematical apparatus which is


both procedures for


computing the coefficient is the regression analysis. A more recent procedure for computing the coefficient  of nonlinear functions uses artificial neural networks.


(7) (4) These mathematic procedures both have a common


quality, they “work on error”, i.e. both methods of problem solving are founded on the existence of an error, see expressions (4) and (6), and an attempt that it be as small as possible.


When the coefficient  is found, the function is fully determined. It has the following properties:


 its graph passes through no given points, this means that it does not represent a “natural” law between variables Y and X, but an approximate one; of errors increases the


(5)


 the presence uncertainty; and


level of


 the function obtained in this way is defined on the whole coordinate axis X, i.e. xЄ(-∞,+∞), but it is determined on the basis of


the behaviour of the


function on a relatively small domain determined by the interval from x1 to xn. This function is depicted in Figure 1.


(6) 4.2 SPLINE FUNCTIONS


In the area of numerical mathematics there is a class of functions that are called spline functions. The term "spline function" is adopted for functions that most closely describe the behaviour of mechanical splines. There exist interpolating and approximating spline functions. In this paper only the interpolating ones are used.


A given set of n points subdivides a curve into m = n-1 segments and each segment is described by parametric third-order polynomials


x aa ti 0,i 1,i y bb ti 0,i 1,i i    2,i a i    2,i b ti   ti a ti   ti b


23 3,i


23 3,i


(8) (9)


where the index i denotes the i-th segment of the curve, and the parameter ti is the length of the chord of segment. All


m 2 cubic polynomials form the interpolating spline


function. A spline


function is fully determined when the


coefficients a0, a1, a2, a3 and b0, b1, b2, b3 are known for every parametric cubic polynomial. The coefficients are calculated separately for each of the variables x and y.


Given that one spline function has m segments, this means that there are


m 4 unknown coefficients. To


calculate them it is necessary to form the same number of equations. These equations are made from the conditions that a spline function must satisfy: the conditions of continuity and smoothness at the points of contact of the segments, and the conditions of the slope of the curve at final points.


©2014: The Royal Institution of Naval Architects


B-45


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