This section discusses the visualization of bibliometric networks, Hankel's matrix, window length, and the orthogonality theorem.
VOSviewer is a software that allows its users to check bibliometric network maps for free (Yatscoff dan Hayter, 1983). VOSviewer has an advantage over other analysis programs because it uses text mining methods to find related phrase mapping combinations as well as creating grouping methods for data analysis VOSviewer has an advantage over other analysis programs because it uses text mining methods to find related phrase mapping combinations as well as creating grouping methods for data analysis (Saputro et al., 2023).
Based on Figure 1, several clusters are formed which are marked with 8 different colors. The red cluster describes the majority of the research on methodology, the purple cluster describes the majority of the research on analysis, the blue cluster describes the majority of the research on data, the orange cluster describes the majority of the research on the matrix, and the brown cluster describes the majority of the research on the matrix. component. Thus, there are still a few clusters that examine Hankel and SSA matrices.
Hankel Matrix
The Hankel matrix is a matrix of size L × K that has a special characteristic where each element on its anti-diagonal is of the same value. The Hankel matrix has a special structure in which each element on its anti-diagonal is of equal value, which plays an important role in the time series transformation of the SSA embedding process (Golyandina dan Zhigljavsky, 2013). The main advantage of the Hankel matrix in time series analysis (SSA) lies in its ability to maintain temporal relationships between elements. The concept of anti-diagonal invariants explains this, as it allows mapping of time-series elements into a more orderly structure. From a mathematical point of view, this provides the basis for a more stable approach to decomposition, particularly in separating the deterministic and stochastic components in a time series. Advanced matrix analysis techniques that provide a new perspective on the conditions of parability in SSA (Rodriguez and González, 2022) . The nature of the Hankel matrix has direct implications for the effectiveness of the SSA algorithm. A comprehensive mathematical framework for the analysis of parallelability based on the structure of the Hankel matrix, with a special focus on the orthogonality conditions between components. The Hankel matrix (Li et al., 2022)(Rodriguez dan Martinez, 2021)H = (hij) has the basic property hij = huv for i + j = u + v, which provides consistency of temporal information along the anti-diagonal. This structure is essential for the transformation of univariate time series into decomposable multivariate representations.
In the context of SSA, a Hankel matrix is formed during an embedding process that converts the univariate time series x = (x1,...,xn) into a trajectory matrix. For window length L, the Hankel H matrix can be written as
𝑋1 𝑋2 ⋯ 𝑋𝑘 𝐻 = ( 𝑋2 𝑋3 𝑋𝑘+1) ⋮ ⋮ ⋱ ⋮ 𝑋𝑙 𝑋𝑙+1 ⋯ 𝑋𝑛 |
(1) |
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where L is the selected window length, K = N - L + 1 is the number of columns, N is the length of the original time series, and hij = xi𝗁j₋₁ for 1 ≤ i ≤ L and 1 ≤ j ≤ K.
The numerical stability in SSA is greatly influenced by the structure of the Hankel matrix. In numerical analysis, the existence of extreme eigenvalues, both very small and very large, can lead to high sensitivity in the calculation of singular values, which in turn affects the decomposition results. Therefore, the application of a regularization approach or optimal parameter selection is very important to ensure stability in the use of SSA. The anti-diagonal structure of the Hankel matrix plays a crucial role in maintaining the numerical stability of the
SSA. The uniqueness of this structure not only affects the representation of data in higher-dimensional vector spaces, but also contributes to retaining temporal information contained in time series. In the context of linear algebra, Hankel matrices are often analyzed in conjunction with spectral decomposition to understand how the characteristics of singular values contribute to the separation of components. Further, numerical stability in Hankel matrix processing is highly dependent on the selection of the right embedding parameters. A small error in parameter selection can result in a mixing effect that can blur the boundary between deterministic and stochastic components. Therefore, the development of methods to determine optimal parameters is crucial in the implementation of a more accurate and stable SSA.
The structure of the Hankel H matrix in SSA has a special characteristic whereby each element of the ƒ anti-diagonal k fulfills i + j = k + 1. The anti- diagonal structure of the Hankel matrix has a significant role in maintaining numerical stability in Singular Spectrum Analysis (SSA). The uniqueness of this structure not only affects the way data is represented in higher vector spaces, but also contributes to the preservation of temporal information contained in time series. In the study of linear algebra, Hankel matrices are often analyzed in relation to spectral decomposition to understand the contribution of singular value properties in component separation. One of the main advantages of the Hankel matrix in SSA is its ability to maintain temporal relationships between elements. This can be understood through the concept of anti-diagonal invariants that allow the mapping of elements in a time series into a more orderly structure. Mathematically, this provides the basis for a more stable decomposition approach, especially in separating the deterministic component from the stochastic component in a time series.
The selection of window length (L) is a crucial aspect that affects the effectiveness of SSA decomposition. According to (James, 2019), window The optimal length must meet two main criteria: large enough to capture patterns in the data and not large enough to avoid mixing effects. A theoretical approach based on matrix properties for the selection of optimal window lengths, taking into account aspects of dimensionality and preservation of temporal information (Kim dan Park, 2021) . A more comprehensive theoretical framework for the analysis of SSA based on matrix properties (Moskvina dan Zhigljavsky, 2021). Matrix-based optimization approach for optimal window length selection(Park dan Kim , 2021). The parallelibility of the time series components depends on the orthogonality conditions affected by the structure of the Hankel matrix and the selection of window lengths.
The selection of window length affects the dimensions of the formed Hankel matrix. The selection of window length L has theoretical implications for the structure of the Hankel matrix formed. The N/4 < L < N/2 boundaries are based on the inferior boundaries (N/4 < L) which provide minimal dimensions to capture temporal patterns, provide ample space for spectral analysis, and the formation of a representative Hankel matrix. Superior limit (L < N/2) that
optimizes decomposition without information redundancy, prevents information redundancy in the Hangel matrix, and minimizes mixing effects between components (Sanei dan Hassani, 2020) . If the L is too small, the information obtained from the embedding becomes limited, thus not allowing effective separation of components. Conversely, if the L is too large, excessive information redundancy occurs, which can lead to inaccuracies in the separation of the main signal and noise. Parameter selection also affects numerical stability in spectral decomposition. If the selected value is too small, the singular value of the Hankel matrix may not reflect the existing patterns in the data well, which may result in errors in the signal reconstruction. On the other hand, if the selected value is too large, the resulting singular value can contain a considerable amount of noise, which risks obscuring important information in the time series. Based on the matrix theory approach, the selection of L must also consider the singular value distribution of the Hankel matrix. The orthogonality properties of singular vectors in SVD play a key role in determining the extent to which a component can be separated. Thus, the selection of L must take into account the spectral characteristics of the Hankel matrix in order to maintain the decomposition quality. The spectral structure of the Hankel matrix can be analyzed through the decomposition of singular values written as
𝐻 = UΣV𝑇 |
(2) |
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where
To measure the quality of the parapherability of the time series components F⁽¹⁾ and F⁽²⁾, the concept of w-orthogonality is used. The two components of the time series F⁽¹⁾ and F⁽²⁾ are said to be w-orthogonal if the inner product of the column of the matrix of its trajectory is close to zero which is expressed as
〈𝑤1, 𝑤2〉 = 0 |
(3) |
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where w₁ and w₂ are the columns of the trajectory matrix H⁽¹⁾ and H⁽²⁾. An inner product value that is close to zero indicates that the two components can be well separated.
The strength of the parability (ν) can be measured by stating as
‖𝑈𝑇𝑉‖ 𝑣 = √(𝑡𝑟(𝑈𝑇𝑈)𝑡𝑟(𝑉𝑇𝑉)) |
(4) |
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where U and V are the trajectory matrices of different components, tr indicates the trace operator is the sum of the main diagonal elements of the matrix to provide a quantitative measure of how well the two components can be separated in the space of the Hankel matrix, and a value of V close to zero indicates good compatibility.
Hankel's matrix analysis in SSA has an important theorem basis. introduces the orthogonality theorem which states that the two components F⁽¹⁾ and F⁽²⁾ of the time series F are said to be separated if the corresponding trajectory matrix has an inner product of zero. In the context of SSA, the additive decomposition (Hassani dan Mahmoudvand, 2013) F = F⁽¹⁾ + F⁽²⁾ induces the decomposition of the Hankel matrix H = H⁽¹⁾ + H⁽²⁾, where H⁽k⁾ inherits the Hankel properties of the F⁽k⁾ component. The threshold ε represents the theoretical upper limit for weak coherbility and the theoretical lower limit for strong coherability in Hankel's matrix space. The weak and strong separability conditions are interrelated through the spectral structure of the Hankel matrix. The threshold ε of 0.1 was established based on a theoretical analysis of the balance between the sensitivity of component separation and the numerical stability of decomposition. This value is the theoretical upper limit for weak parability, where the two components are considered quite separate if the normalized inner product is less than this threshold. W-correlation ρij provides a quantitative measure of orthogonality that is directly related to the separation capacity of components in Hankel's matrix space.
The orthogonality theorem states that the two components F⁽¹⁾ and F⁽²⁾ are said to be weak parallelibility if they meet the stated as
‖𝑋(1)𝑋(2)𝑇‖ ‖𝑋(1)‖‖𝑋(2)‖ < 𝜀 |
(5) |
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where ε is the threshold value that determines the degree of separability, and
X⁽i⁾ is the trajectory matrix of F⁽i⁾.
For strong parallelivity, additional conditions are required at singular values (Chen dan Zhang, 2019) which is stated as
|𝜎𝑖(1) − 𝜎𝑗(2)| > 𝛿 |
(6) |
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where all
〈𝑓̃𝜄, 𝑓̃𝑦〉 ρ𝑖𝑗 = ‖𝑓̃‖‖𝑓̃ ‖ 𝜄 𝑦 |
(7) |
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with
The SSA not only serves as a tool for analyzing time series, but it also has relevance in the field of pure mathematics, especially in the context of spectral analysis and matrix decomposition. The Hankel matrix decomposition applied in SSA is often associated with eigen-value theory, which describes how the spectral information of a system can be broken down into smaller, more understandable components.
The relationship between SSA and other spectral methods illustrates the utilization of Hankel matrix decomposition in a variety of fields, including numerical optimization and multivariate data analysis. One of the interesting aspects is the application of SSA in dimension reduction, where the decomposition of singular values allows for a deeper understanding of the underlying data structure. In particular, SSA representations utilizing singular value decomposition (SVD) can be attributed to spectral transformations in orthogonal function analysis. This approach suggests that SSA is essentially a development of classical methods such as Karhunen-Loève transformations, which are widely applied in signal processing and data compression.