## 2 Preliminaries
### The H-Rank as a Measure of Algebraic Complexity of a Time Series  
The Hankel transform of a sequence of real numbers \((x_{k})_{k=1}^{+\infty}\) yields a sequence \((h_{k})_{k=1}^{+\infty}\), where \(h_{k}=det(H_{k})\) and \(H_{k}\) is a \(k\)th -order Hankel matrix Equation (1):  
\[H_{k}= \begin{bmatrix}x_{1}&x_{2}&\dots&x_{k}\\ x_{2}&x_{3}&\dots&x_{(k+1)}\\ \vdots&\vdots&\ddots&\vdots\\ x_{k}&x_{(k+1)}&\dots&x_{(2k-1)}\end{bmatrix} . \tag{1}\]  
If there exists such \(m>1\) that \(h_{m}\
eq 0\), but \(h_{j}=0\) for all \(j>m\), then the sequence \((x_{k})_{k=1}^{+\infty}\) is an \(m\)th-order linear recurrence sequence (LRS) [23]. A straightforward computation of a sequence of determinants is evidently an impractical strategy for determining the order of a linear recurrence sequence. Moreover, an additive would compromise such an approach. For example, the detectable order of a linear recurrence sequence with additive noise is infinite (it is not possible to reconstruct a mathematical model for random noise) [24]. In other words, an alternative strategy should be employed to determine the orders of real-world time series that are inevitably contaminated by external noise. It is well known that the singular value decomposition (SVD) of the Hankel matrix can efficiently determine the rank of the linear model that governs the evolution of a time series [25]. The number of singular values of the Hankel matrix of a linear recurrence sequence greater than zero coincides with the number of non-zero roots of the characteristic polynomial of the linear recurrence. It appears that the SVD algorithm can also be used to efficiently determine the orders of linear recurrences contaminated by noise [24]. The number of squared singular values of the Hankel matrix \(H_{d}\) greater than \(\varepsilon\) is defined as the H-rank of a sequence (even if the sequence is not the LRS) [24]. The H-rank algorithm can be tuned by choosing two essential parameters (the order of the Hankel matrix \(d\) and the threshold \(\varepsilon\)). If the sequence is the LRS, \(d>m\), and \(\varepsilon\) is set to the machine epsilon, the H-rank is equal to the order of the linear recurrence [24]. However, a proper selection of \(d\) and \(\varepsilon\) helps to assess the algebraic complexity of any time series (not necessarily a linear recurrence) [24]. The SVD decomposition of the matrix \(H_{k}\) yields \(k\) squared real singular values \(\sigma_{l}^{2}\); \(l=1,2,\dots,k\), sorted in descending order. A proper selection of \(\varepsilon\) helps to determine the \(\varepsilon\)-H- pseudospectrum of \(H_{k}\)[24]; the calculation of the number of singular values greater than \(\varepsilon\) yields the H-rank of the sequence [24]; see Equation (2):  
\[\text{H-rank}(H_{k})=\sum_{l=1}^{k}I(\sigma_{l}^{2}\geq\varepsilon) \tag{2}\]  
where \(I(S)\) is the indicator function (a Boolean to integer conversion function), which is equal to one if the statement \(S\) is true, and equal to zero otherwise. **Example 1**. : _Let us consider a period-3 sequence \((x_{k})_{k=1}^{+\infty}=1,4,5,1,4,5,\dots\). The Hankel transform reads as Equation (3):_  
\[h_{1}=x_{1}=1;h_{2}= \begin{vmatrix}1&4\\ 4&5\end{vmatrix} =-11;h_{3}= \begin{vmatrix}1&4&5\\ 4&5&1\\ 5&1&4\end{vmatrix} =-130;h_{4}= \begin{vmatrix}1&4&5&1\\ 4&5&1&4\\ 5&1&4&5\\ 1&4&5&1\end{vmatrix} =0. \tag{3}\]  
_All higher-order determinants are equal to zero: \(h_{k}=0,k=4,5,\dots\). Therefore, the order of this period-3 sequence is 3._  
_The same result can be determined using the SVD algorithm. Set \(d=5\). Then, the squared singular values of \(H_{5}\) are \((\sigma_{k}^{2})_{k=1}^{5}=(17.0254,6.063,5.0376,0,0)\). Thus, the H-rank of the period-3 sequence is also equal to 3.__Now, let us consider that this period-\(3\) sequence is perturbed by the additive noise (choose a Gaussian distribution with zero mean and a standard deviation equal to one).