**7. Taguchi orthogonal array**

Taguchi method uses orthogonal arrays to study various factors with a small number of experiments (Sharma et al., 2005). Furthermore, the method provides other benefits, such as the reduction of process variability, low operating costs and expected results. (Barros; Bruns; Scarminio, 1995).

Rosa et al. (2009) define the Analysis of Variance (ANOVA) in the application of statistical analysis of Taguchi method in order to evaluate the significance of the parameters used in the process. A ANOVA table determines the most relevant parameters for the process according to equations 1, 2, 3 and 4:


$$\mathbf{SS} \triangleq \sum\_{l=1}^{n} (\mathbf{y}\_l \cdot \overline{\mathbf{y}})^2 \tag{1}$$

Multivariate Analysis in Advanced Oxidation Process 69

Multiple Linear Regression (MLR), Principal Component Analysis (PCA), Principal Component Regression (PCR) and Partial Least Squares Regression (PLS) (Otto, 2007).

The manufacturing processes can generally have correlated variables depending on the

The field of chemometrics is a multivariate analysis defined as the application of designs and mathematical and statistical methods to solve chemical problems. It is utilized to improve data collection and to allow extraction of useful and obtained data information

Paiva, Ferreira and Balestrassi (2007) combined DOE with Multivariate Analysis when

The Multiple linear regression is a determining method of combinations of variables to

MLR property of interest relates to a linear combination of independent measurements. The modeling MRL can be represented by equation 5, where a set with n samples, *i* = 1 to *n*, *Y* is the response variable, *X* is the independent variable and *i* is the error estimation (Steiner et

+ ∑ β<sup>i</sup>

According to Montgomery (2001), the method of Ordinary Least Squares (OLS - Ordinary

 L= ∑ ε<sup>i</sup> 2n i=0

= ∑ ��� – β� – ∑ β�x��

 = – 2 ∑ ��� – �� – ∑ ��

� � �

1 x11 x11 ⋯ x1k 1 x21 x22 ⋯ x2k ⋮⋮ ⋮ ⋮ ⋮ 1 xn1 xn2 ⋯ xnk

k

Xij+ ei

� � � � �

� � � � � �

� � � �

�, β =

� � � � � β1 β2 ⋮ βk� � � � �

� � � = ∑ ��� �

����

� ∑ ��� � �

e ε = �

�� �� ⋮ �� � � � � ��

� �

� ∑ ������ + � + ��

j=1 (5)

=β<sup>0</sup>

process quality that involves a large number of characteristics (Paiva, 2006).

optimizing multiple correlated responses in a manufacturing process.

achieve an optimal process or product. (Beebe; Pell; Seasholtz, 1998).

Least Squares) to determine βi, minimize the sum of squared errors:

Function L must be minimized in terms of β0, β1, ..., βk.

�� ���

� ∑ ������ + ��

�, X =�

In matrix notation, there is Y = βX + ε, where:

y1 y2 ⋮ yn

y = �

� � � �

� � �

(Hopke, 2003).

al., 2008)

��

**8.1. Multiple linear regression** 

yi

Simply stated, there is equality:

� ∑ ��� + � � � � ��


df=N-1

Where N means number of level for each factor


$$\mathbf{M}\mathbf{Q} \stackrel{\text{SS}}{=}\tag{3}$$


$$F = \frac{\text{MS}\_{effect}}{\text{MS}\_{error}}\tag{4}$$
