Linear Regressions in Matrix Forms

The regression analysis, when the relationship between the process variables is given by a matrix, is frequently used to solve the problems of identification and confidence of the coefficients as well as the problem of a model confidence. The matrix expression is used frequently in processes with more than two independent variables which present simultaneous interactive effects with a dependent variable. In this case, the formulation of the problem is similar to the formulation described in the previous section. Thus, we will use the statistical data from Table 5.11 again in order to identify the coefficients with the following relation  

The first step in this discussion concerns the presentation of the matrix of the independent variables (X), the experimental observation vector of the dependent variable (Y) and the column matrix of the coefficients (B) as well as the transposed matrix of the independent variables (X. All these terms are introduced by relation (5.84). A fictive variable Xq, which takes the permanent value of 1, has been considered in the matrix of the independent variables  

The particularization of the system of the normal equations (5.9) into an equivalent form of the relationship between the process variables (5.83), results in the system of equations (5.85). In matrix forms, the system can be represented by relation (5.86), and the matrix of the coefficients is given by relation (5.87). According to the inversion formula for a matrix, we obtain the elements for the inverse matrix of the matrix multiplication (XX ), where (X ) is the transpose matrix of the matrix of independent variables. 

Relation (5.89) gives the value of each element of matrix (XX ), where the symbol dj]j represents a current element, as shown in relation (5.88), 

In order to obtain the value of the residual variance, we first define the matrix of the expected observations Y = XB and then we observe that the quadratic displacement betiveen the measured and computed output values of the variables can be ivritten as  

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