**8. Multivariate analysis**

The Multivariate Analysis represents a set of statistical method in which most of variables of a data set are comprise information for decision-making (Rajalahti; Kvalheim, 2011), such as 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 process quality that involves a large number of characteristics (Paiva, 2006).

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 (Hopke, 2003).

Paiva, Ferreira and Balestrassi (2007) combined DOE with Multivariate Analysis when optimizing multiple correlated responses in a manufacturing process.

#### **8.1. Multiple linear regression**

68 Multivariate Analysis in Management, Engineering and the Sciences

time by reducing the number of experimental conditions.

**7. Taguchi orthogonal array** 

according to equations 1, 2, 3 and 4:


SS= ∑ (yi - y� � <sup>n</sup> <sup>2</sup>

df=N-1

MQ= SS


Where N means number of level for each factor


Scarminio, 1995).


**8. Multivariate analysis** 

2009).

According to Franceschini and Macchietto (2008), DOE is a statistical tool used to maximize the value of variable responses obtained on each experiment and also to minimize cost and

Interactions between variables are considered in the experimental design and can be used for optimizing the operating parameters in multivariable systems (Ay; Catalkaya; Kargi,

According to Salazar (2009), the experimental has been studied as an important mathematical tool in the area of Advanced Oxidation Processes (Heterogeneous Photocatalysis). In this study, fractionated schedules for the degradation of organic matter

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;

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

> � = �������� �������

The Multivariate Analysis represents a set of statistical method in which most of variables of a data set are comprise information for decision-making (Rajalahti; Kvalheim, 2011), such as

i=1 (1)

df (3)

(4)

and COD percentage of dairy effluent were used to obtain 93.70% of the treatment.

The Multiple linear regression is a determining method of combinations of variables to achieve an optimal process or product. (Beebe; Pell; Seasholtz, 1998).

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 al., 2008)

$$\mathbf{y}\_{i} \mathbf{=} \boldsymbol{\beta}\_{0} + \boldsymbol{\Sigma}\_{\mathsf{T}^{\mathsf{T}}}^{\mathsf{k}} \boldsymbol{\beta}\_{i} \mathbf{X}\_{i\mathsf{i}} + \mathbf{e}\_{i} \tag{5}$$

According to Montgomery (2001), the method of Ordinary Least Squares (OLS - Ordinary Least Squares) to determine βi, minimize the sum of squared errors:

$$\mathbf{L} = \begin{bmatrix} \sum\_{i=0}^{n} \varepsilon\_{i}^{2} \\\\ \sum\_{i=1}^{n} \left( \mathbf{y}\_{i} - \mathbf{\beta}\_{0} - \sum\_{j=1}^{k} \beta\_{i} \mathbf{x}\_{j} \right)^{2} \end{bmatrix}$$

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

$$\frac{\partial L}{\partial \beta\_l} = -2\,\Sigma\_l^n = \,\_1\{\mathcal{Y}\_l - \beta\_0 - \Sigma\_j^k = \,\_1\beta\_l \ge\_{fl} \}$$

Simply stated, there is equality:

$$\hat{\beta}\_0 \Sigma\_{l=1}^n \mathbf{x}\_{lk} + \hat{\beta}\_1 \Sigma\_{l=1}^n \mathbf{x}\_{lk} \mathbf{x}\_{l1} + \hat{\beta}\_2 \Sigma\_{l=1}^n \mathbf{x}\_{lk} \mathbf{x}\_{l2} + \dots + \hat{\beta}\_k \Sigma\_{l=1}^n \mathbf{x}\_{lk} \mathbf{z}^2 = \Sigma\_{l=1}^n \mathbf{x}\_{lk} \mathbf{y}\_{lk}$$

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

$$\mathbf{y} = \begin{bmatrix} \mathbf{y}\_1 \\ \mathbf{y}\_2 \\ \vdots \\ \mathbf{y}\_n \end{bmatrix}, \mathbf{X} = \begin{bmatrix} 1 & \mathbf{x}\_{11} & \mathbf{x}\_{11} & \cdots & \mathbf{x}\_{1k} \\ 1 & \mathbf{x}\_{21} & \mathbf{x}\_{22} & \cdots & \mathbf{x}\_{2k} \\ \vdots & \vdots & \vdots & \vdots & \vdots \\ 1 & \mathbf{x}\_{n1} & \mathbf{x}\_{n2} & \cdots & \mathbf{x}\_{nk} \end{bmatrix}, \boldsymbol{\beta} = \begin{bmatrix} \boldsymbol{\beta}\_1 \\ \boldsymbol{\beta}\_2 \\ \vdots \\ \boldsymbol{\beta}\_k \end{bmatrix} \mathbf{e} \ \boldsymbol{\varepsilon} = \begin{bmatrix} \boldsymbol{\varepsilon}\_1 \\ \boldsymbol{\varepsilon}\_2 \\ \vdots \\ \boldsymbol{\varepsilon}\_n \end{bmatrix}$$

Then:

$$\mathbf{L} = \begin{array}{c} \Sigma\_{l=0}^{n} \ \varepsilon\_{l}^{2} \ \ = \ \varepsilon^{T} \ \varepsilon \ = \ \{\mathbf{y} - \ \mathbf{X}\boldsymbol{\beta}\}^{T} \ \{\mathbf{y} - \ \mathbf{X}\boldsymbol{\beta}\} \\\\ \mathbf{L} = \mathbf{y}^{\mathrm{T}} \mathbf{y} - \boldsymbol{\beta}^{\mathrm{T}} \mathbf{X}^{\mathrm{T}} \mathbf{y} \ \cdot \ \mathbf{y}^{\mathrm{T}} \mathbf{X}\boldsymbol{\beta} + \boldsymbol{\beta}^{\mathrm{T}} \mathbf{X}^{\mathrm{T}} \mathbf{X}\boldsymbol{\beta} \\\\ \mathbf{L} = \mathbf{y}^{\mathrm{T}} \mathbf{y} - 2\boldsymbol{\beta}^{\mathrm{T}} \mathbf{X}^{\mathrm{T}} \mathbf{y} + \boldsymbol{\beta}^{\mathrm{T}} \mathbf{X}^{\mathrm{T}} \mathbf{X}\boldsymbol{\beta} \end{array}$$

Minimizing the function:

$$\frac{\partial \mathbf{L}}{\partial \boldsymbol{\beta}} = \mathbf{-2X^{T}y + 2X^{T}X\widehat{\boldsymbol{\beta}} = 0}$$

Hence, is obtained:

XTXβ� <sup>=</sup>XTy

and consequently, the coefficients are determined by Equation 6:

$$
\widehat{\boldsymbol{\beta}} = \left(\mathbf{X}^{\mathrm{T}}\mathbf{X}\right)^{\cdot 1} \mathbf{X}^{\mathrm{T}} \mathbf{y} \tag{6}
$$

Multivariate Analysis in Advanced Oxidation Process 71

Experimental design Taguchi L9 with advanced oxidation process and heterogeneous photocatalysis were used for 1.0 liters of fresh effluent and 2 liters of distilled water, previously homogenized and conditioned at room temperature. Semiconductor titanium dioxide (TiO2), and the amount of H2O2 (30% w / v) were added during the initial 50 minutes of 1-hour total reaction, using burettes of 25 and 50 ml. The temperature of the medium reaction during the whole period of the photocatalytic process was controlled at 25 °C by using an Ophterm DC1 thermostatic bath. pH reaction was performed using a combined glass electrode adapted to the shell. This was connected to the potentiostat digital Digimed. A centrifugal pump was used for conducting the effluent from the tubular reactor to the storage tank. Ultraviolet lamps of 15 and 21 watts were used. Figure 4 presents the

detailed scheme for treatment with AOP showing the experimental procedure.

**Figure 4.** Layout of the process of photochemical treatment

### **9. Materials and methods**

This work was performed at the Laboratory of the Environmental Engineering Department of the Chemical Engineering of School of Lorena EEL-USP. Polyester resin effluent was supplied by Valspar industry, located in São Bernardo do Campo, State of São Paulo. Statistical analyses were performed by Statistica version 2.0, available at the College.

Samples were stored cold chamber at EEL-USP at 4ºC. The effluent from the oxidation reaction was conducted into a Germetec tubular reactor, model FPG-463/1, with a nominal volume of approximately 1 L, receiving irradiation from a GPH-463T5L mercury lamp of low pressure, which emits a UV radiation at 254 nm with a power of 15 W and 21 W and protected by a quartz tube. The manufactured reactor model is shown in Figure 3.

**Figure 3.** Tubular reactor used for photochemical treatment

Experimental design Taguchi L9 with advanced oxidation process and heterogeneous photocatalysis were used for 1.0 liters of fresh effluent and 2 liters of distilled water, previously homogenized and conditioned at room temperature. Semiconductor titanium dioxide (TiO2), and the amount of H2O2 (30% w / v) were added during the initial 50 minutes of 1-hour total reaction, using burettes of 25 and 50 ml. The temperature of the medium reaction during the whole period of the photocatalytic process was controlled at 25 °C by using an Ophterm DC1 thermostatic bath. pH reaction was performed using a combined glass electrode adapted to the shell. This was connected to the potentiostat digital Digimed. A centrifugal pump was used for conducting the effluent from the tubular reactor to the storage tank. Ultraviolet lamps of 15 and 21 watts were used. Figure 4 presents the detailed scheme for treatment with AOP showing the experimental procedure.

70 Multivariate Analysis in Management, Engineering and the Sciences

� = ∑ ��

∂L

and consequently, the coefficients are determined by Equation 6:

� = ��� = �� � ��� � � � � � �� � ���

L = yTy – βTXTy - yTXβ + βTXTXβ

L = yTy – 2βTXTy + βTXTXβ

∂β = -2XTy + 2XTXβ�= 0

XTXβ� <sup>=</sup>XTy

This work was performed at the Laboratory of the Environmental Engineering Department of the Chemical Engineering of School of Lorena EEL-USP. Polyester resin effluent was supplied by Valspar industry, located in São Bernardo do Campo, State of São Paulo.

Samples were stored cold chamber at EEL-USP at 4ºC. The effluent from the oxidation reaction was conducted into a Germetec tubular reactor, model FPG-463/1, with a nominal volume of approximately 1 L, receiving irradiation from a GPH-463T5L mercury lamp of low pressure, which emits a UV radiation at 254 nm with a power of 15 W and 21 W and


XTy (6)

β� = (XTX)

Statistical analyses were performed by Statistica version 2.0, available at the College.

protected by a quartz tube. The manufactured reactor model is shown in Figure 3.

**Figure 3.** Tubular reactor used for photochemical treatment

Then:

Minimizing the function:

**9. Materials and methods** 

Hence, is obtained:

**Figure 4.** Layout of the process of photochemical treatment
