**C. Looking for the demonstration of the Eq. (21)**

$$Q\_{\rm et} = X \left(\mathbf{X}^T \mathbf{X}\right)^{-1} \mathbf{X}^T Q\_{\rm ref} \tag{21}$$

It must be analyzed

$$J = e^2 \tag{C1}$$

Moreover,

$$J = \left(Q\_{\rm ref} - Q\_{\rm est}\right)^T \left(Q\_{\rm ref} - Q\_{\rm est}\right) \tag{C2}$$

Therefore

*Optimal Analysis for the Enhancement in the Thermal Variables Measurement by Smart… DOI: http://dx.doi.org/10.5772/intechopen.112676*

$$J = \left(Q\_{\rm ref} - X\beta\right)^T \left(Q\_{\rm ref} - X\beta\right) \tag{C3}$$

Also

$$J = \left(Q\_{ref}^T - \beta^T X^T\right) \left(Q\_{ref} - X\beta\right) \tag{C4}$$

As well as

$$J = Q\_{r\!f}^T Q\_{r\!f} - \beta^T X^T Q\_{r\!f} - Q\_{r\!f}^T X \beta + \beta^T X^T X \beta \tag{C5}$$

It must be remembered

$$Q\_{\alpha t \, m \ge 1} = X\_{m \times n} \beta\_{n \ge 1} \tag{C6}$$

For which

$$X\_{m\ge n} = [Q\_{m\ge 1}(:, \, 1); Q\_{m\ge 1}(:, \, 2); \ldots; Q\_{m\ge 1}(:, \, n)]\_{m\ge n} \tag{C7}$$

Furthermore,

$$
\beta\_{1\text{xn}}^T X\_{n\text{xm}}^T Q\_{\text{ref}\_{m\text{x1}}} = number\_{1\text{x1}} \tag{C8}
$$

Looking for its Transpose

$$Q\_{ref\_{1\le n}}^T X\_{m\ge n} \beta\_{n\ge 1} = number\_{1\le 1} \tag{C9}$$

Therefore, comparing Eqs. (C8) and (C9).

$$
\boldsymbol{\beta}\_{1\text{x}m}^T \mathbf{X}\_{n\text{x}m}^T \mathbf{Q}\_{\text{ref}\_{m\text{x}1}} = \mathbf{Q}\_{\text{ref}\_{1\text{x}m}}^T \mathbf{X}\_{m\text{x}n} \boldsymbol{\beta}\_{n\text{x}1} \tag{\text{C10}}
$$

It means that in Eq. (C5)

$$J = Q\_{\rm rf}^T Q\_{\rm rf} - 2\beta^T \mathbf{X}^T Q\_{\rm rf} + \beta^T \mathbf{X}^T \mathbf{X} \beta \tag{\text{C11}}$$

It will be used the following properties (C12), (C13), and (C14)

$$\frac{\partial \left(\mathbf{A}^{\mathrm{T}} \mathbf{X}\right)}{\partial \mathbf{X}} = A \tag{C12}$$

$$\frac{\partial \mathbf{(A^T X)}}{\partial \mathbf{X^T}} = \mathbf{A^T} \tag{C13}$$

$$\frac{\partial \mathbf{(X^T A X)}}{\partial \mathbf{X}} = 2 \mathbf{A} \mathbf{X} \tag{\text{C14}}$$

Looking for the minimal value in the Eq. (C11)

$$\frac{\partial \mathcal{J}}{\partial \boldsymbol{\mathfrak{G}}} = \boldsymbol{0} - 2 \frac{\partial}{\partial \boldsymbol{\mathfrak{G}}} \left( \boldsymbol{\beta}^T \left( \mathbf{X}^T \mathbf{Q}\_{\text{ref}} \right) \right) + \frac{\partial}{\partial \boldsymbol{\mathfrak{G}}} \left( \boldsymbol{\beta}^T (\mathbf{X}^T \mathbf{X}) \boldsymbol{\beta} \right) \tag{C15}$$

By properties (C13) and (C14) in the Eq. (C15)

$$\frac{\partial \mathcal{J}}{\partial \boldsymbol{\beta}} = \mathbf{0} - 2 \frac{\partial \left( \left( \mathbf{X}^T \mathbf{Q}\_{\text{ref}} \right)^T \boldsymbol{\mathfrak{P}} \right)^T}{\left( \partial \boldsymbol{\mathfrak{P}}^T \right)^T} \ + 2 (\mathbf{X}^T \mathbf{X}) \boldsymbol{\beta} \tag{\text{C16}}$$

$$\mathbf{0} = -2\left(\frac{\partial \left(\left(X^T Q\_{\text{ref}}\right)^T \boldsymbol{\beta}\right)}{\partial \boldsymbol{\beta}^T}\right)^T + 2(X^T X)\boldsymbol{\beta} \tag{C17}$$

$$\left(\left(\mathbf{X}^T \mathbf{Q}\_{\text{ref}}\right)^T\right)^T = \left(\mathbf{X}^T \mathbf{X}\right)\beta\tag{\text{C18}}$$

$$\boldsymbol{\beta} = \left(\mathbf{X}^T\mathbf{X}\right)^{-1}\mathbf{X}^T\mathbf{Q}\_{r\!\!\!f\!\!f} \tag{\text{C19}}$$

$$\mathbf{Q\_{est}} = \mathbf{X(X^TX)}^{-1}\mathbf{X^TQ\_{ref}}\tag{C20}$$
