lower bound parameters, defined originally as $\mathrm{UB}=\mathrm{max}(x)$ and $\mathrm{LB}=\mathrm{min}(x)$. The final Gramian matrix can be computed as follows: $${    \setlength{\eqhsize}{\minof{\widthof{\mbox{\ensuremath{               GAF = {X_{norm}}^{T} \cdot X_{norm} - \sqrt{I-{X_{norm}}^2}^{T}\cdot \sqrt{I-{X_{norm}}^2}}}}}{\linewidth}}
    \resizebox{\eqhsize}{!}{\ensuremath{               GAF = {X_{norm}}^{T} \cdot X_{norm} - \sqrt{I-{X_{norm}}^2}^{T}\cdot \sqrt{I-{X_{norm}}^2}}}}$$ $${    \setlength{\eqhsize}{\minof{\widthof{\mbox{\ensuremath{                  GAF =  \begin{pmatrix}
                          \cos(\phi_1+\phi_1) & \cos(\phi_1+\phi_2) & \cdots & \cos(\phi_1+\phi_N)\\
                          \cos(\phi_2+\phi_1) & \cos(\phi_2+\phi_2) & \cdots & \cos(\phi_2+\phi_N) \\
                          \vdots  & \vdots  & \ddots & \vdots  \\
                          \cos(\phi_N+\phi_1) & \cos(\phi_N+\phi_2) & \cdots & \cos(\phi_N+\phi_N)
                         \end{pmatrix}
                         }}}}{\linewidth}}
    \resizebox{\eqhsize}{!}{\ensuremath{                  GAF =  \begin{pmatrix}
                          \cos(\phi_1+\phi_1) & \cos(\phi_1+\phi_2) & \cdots & \cos(\phi_1+\phi_N)\\
                          \cos(\phi_2+\phi_1) & \cos(\phi_2+\phi_2) & \cdots & \cos(\phi_2+\phi_N) \\
                          \vdots  & \vdots  & \ddots & \vdots  \\
                          \cos(\phi_N+\phi_1) & \cos(\phi_N+\phi_2) & \cdots & \cos(\phi_N+\phi_N)
                         \end{pmatrix}
                         }}}$$ where $I$ refers to a unit row vector and the superscript $(\cdot)^T$ refers to the transpose operator. By analogy to Gram matrices $G_{ij} = \langle x_i,x_j\rangle$, where $\langle\cdot,\cdot\rangle$ is an inner product, the matrix is named as Gramian Angular Field. One can note, however, that $\cos(x_i+x_j)$ does not satisfy all the conditions of inner-products. In addition, it is noted that, the matrix does not depend on $r$. Therefore, it does not depend on time and not on the choice of $L$ in Eq \[eq:GAF-pol\]. As a consequence, the GAF transformation 