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process studied over a small range. The experiment is normally planned around a
working point [ref. 170].
Factorial design is now being used routinely in welding applications. It is
mainly applied to evaluate how tolerant a procedure is to changing welding
parameters [ref. 171].
If the objective is to develop a model spanning a process operating range,
factorial design might be restrictive as the physical combination of some welding
parameters might lead to defects and instability. Hence, it is not always possible to use
factorial experimental design. Its structure can however be used for initial
experimental plan and then adapted to avoid (ie. change for better) unsatisfactory
parameter combinations [ref. 51].
2.6.4.2 Regression analysis
Regression analysis is a statistical way to derive a quantitative relationship
between variables [ref. 172]. It is frequently used to model complex multifactor
processes, in which a theoretical approach is not yet fully developed. The models
allow for the main quantitative relationships and can be obtained with a comparatively
small amount of experimental studies [ref. 166]. However, it cannot prove cause and
effect, since these can only be inferred from physical or chemical principles or direct
observation [ref. 172].
Regression methods are frequently used to analyse data from unplanned
experiments, but can also be used for designed experiments [refs. 167,172]. The
regression models that are normally applied to fit a set of experimental points can
have different mathematical structures (e. g. polinomial, multiplicative, exponential,
trigonometric) [refs. 51,167], the most commonly applied being the polinomial ones,
which can be interpreted as an expansion of the relationship investigated into a Taylor
series [ref. 166].
The modelling process consists of two stages, the development of a model
structure and the estimation of the model parameters [ref. 166]. The model structure
normally presents the generallised form of equation (2.33).
Y= f(X,, X29**,, xk)
(2.33)
where xi (i=1,2,
..., k) are independent or regressor variables, y is a dependent
variable and R. ) is the regression equation, which can be the true functional
relationship between the dependent and independent variables, if known, or an
appropriate function which approximates the true functional relationship within the
range of the investigated variables [ref. 167].
The most common regression method is multiple linear regression. Linear in
the sense that the response variable is linear in the unknown parameters. Multiple
linear regression uses the linear model of equation (2.34) to fit a set of experimental
data points [ref. 167].
y =10 + ß1x1 + 32X2+... +ß&Xh +C
46
(2.34)