In TSA, hidden patterns and correlations can be revealed by transforming a univariate sequence into a multidimensional structure. A crucial step in this process is the construction of a trajectory matrix, specifically using the HM structure. In this study, a lag of 1 is chosen to construct the HM, as temporal dependencies in weakly stationary processes, particularly in AR(1) and AR(2) models, can be efectively captured. This choice is aligned with PACF interpretation and is utilized to facilitate SVD-based dimensionality reduction. By structuring the HM in this way, a balance between information preservation and computational eficiency is ensured in time series modeling.

Given a time series generated from a weakly stationary process, the objective is to transform this series into an HM of dimensions , constructed by defining a window length L and setting Within the HM, there exists an index such that . Each data point is then mapped to a lagged vector of length represented as:

The resulting trajectory matrix has an HM structure as follows [1], [2], [3]:

The use of trajectory matrices with H aligns with the Singular Spectrum Analysis (SSA) method, introduced by Broomhead and King in 1986. SSA has been extensively studied, covering theoretical and methodological aspects [4] , [5] , [6] software implementation [7], [8], and practical applications in research and industry [9], [10], [11]. SSA, known for its flexibility, comprises four stages: embedding, SVD, grouping, and reconstruction. This study focuses on embedding and SVD for dimensionality reduction methods to assess their efectiveness in capturing temporal patterns in time series data.

SVD involves the factorization of H into three elementary matrices: two orthogonal matrices and and one diagonal matrix containing scale factors, termed singular values , arranged in descending order, from the largest to the smallest value, denoted as 0 \end{document} ]]> . [12], [13], [14]. The columns of U and V, referred to as left and right singular vectors. This decomposition can be expressed as follows:

SSA is a model-free method that disregards weak stationarity and autocorrelation assumptions. Given that TSA is inherently stochastic with autocorrelation behavior, integrating SSA with classical TSA methods would be insightful. Among these, the Box-Jenkins method remains prominent, influencing diverse fields such as economics [15], social sciences [16], healthcare [17], tourism [18], industries [19], and agriculture [20]. Its ,broad and lasting impact is well-documented [21], highlighting its continued relevance.

The Box-Jenkins methodology consists of three main components: Autoregressive (AR), Moving Average (MA), and Integrated (I) models, which together form advanced models like ARIMA ( p , d , q ) [22], [23], [24], [25], [26], [27]. In TSA, AR models highlight data dependency, where the order is identified using the autocorrelation function, particularly PACF [28], [29], [30]. A PACF plot helps determine the appropriate lag order p; a sharp drop after lag p suggests an process, while a gradual decline may indicate an ARMA process. Understanding these functions is essential for selecting an optimal forecasting model.

Inspired by this method, this study explores data matrix fields through trajectory matrices with HM structures, providing a novel analytical approach. Investigating SVD on HM is particularly intriguing, as singular values may ofer crucial insights into variance within AR models. Therefore, adhering to the principle of parsimony, the study initiates with the simplest mean-centred AR(1) and AR(2) model structure for , with a general form of

This model provides a foundational framework for understanding dynamics and variance captured by the HM structure and SVD. Implementing SVD for dimensionality reduction presents challenges, from optimizing matrix size to analyzing singular value sequences in AR models. Addressing these challenges, this study examines the properties and conditions of H in SVD to enhance its implementation. Insights from this exploration are applied to simulated AR model data, enabling a comprehensive evaluation of SVD’s efectiveness.

Since this method applies to all types of observational data, the notation X is used for generality. A dataset with N observations represents onedimensional data and is transformed into a matrix H with elements . From this formulation, several properties of H can be deduced. Literature on matrix analysis [3], [14], [31] provides the basis for constructing lemmas and theorems in this study. These results are broadly applicable to observational data, including time series.

Definition 2.1. MatrixHhas the following properties:

The structure of HM for time series data, as defined in Definition 2.1, progresses either downwards or rightwards along its columns or rows as time moves forward [32]. This structural property aligns the columns of the HM as delayed versions of the time series, efectively capturing a lag-one relationship within a weak stationary process. Given the unique characteristics of HM in time series, it is crucial to understand their general properties, as outlined in the following lemma. This lemma will lead to the properties and conditions for suficient SVD in HM which is a benchmark for processing time series data

Lemma 2.2.Based onDefinition 2.1point (ii) , hence

Proof. Let be an eigenvalue of H and y be the corresponding eigenvector. Then:

Thus,

Consequently, it follows that , and hence

Lemma 2.3.According toDefinition 2.1point (iii)are self adjoint and thus share the same λ, including multiplicities.

Proof. Given , it is concluded that is self-adjoint. Similarly, = H^∗H is obtained, hence HH^∗ is also self-adjoint. Let λ be an eigenvalue of HH^∗ with eigenspace ; for , it is found that:

Multiply both sides by H:

Thus, Hv is an eigenvector of HH^∗ with eigenvalue λ, because:

If , then each eigenvector v from HH^∗ maps to an eigenvector Hv of one-toone. This is because if with Hv = Hw:

which implies , thus . Similarly, every eigenvalue of maps to an eigenvalue of with eigenvector , mapping from the eigenspace of , one to one, for . For each of and , the associated eigenspaces have the same dimension. Because and are self-adjoint and have the same dimension, the associated eigenvalues for also have the same dimension. Consequently, the multiplicities are also the same. □

The matrix H in Eq. (1) , which consists of reflects its properties through in the multiplication with its transpose, namely and resulting in a matrix that resembles the autocovariance matrix (C). The of (C) represents the covariance between at diferent lags defined as:

Since the process is weakly stationary, the only depend on the lag . The multiplication of the HH^∗ results in an matrix:

Meanwhile, the multiplication H^∗H produces a matrix:

When normalized by the length L or , these forms approximate as follows:

Although and have diferent dimensions, they exhibit similar spectra of eigenvalues and share the same eigenvalues as stated in Lemma2.3. Since they are symmetric and positive semi-definite, are guaranteed to be nonnegative real numbers, as described in Lemma2.2. Moreover, the singular values of H are directly related to these eigenvalues, as

This implies that the choice of L or W does not afect the fundamental spectral properties of the matrices. Both matrices capture essential statistical properties of . For instance, in autocorrelation behavior, the autocorrelation matrix (R) is obtained by normalizing equation (3) based on its variance:

with is the autocorrelation function at lag k and is the time series variance. The relationship between C and R is summarized as:

Since for all each element is obtained by normalizing with the variance , yielding a correlation value within .To analyze the relationship between singular value sequences and autocorrelation behavior, it is crucial to consider implications from Definition 2.1, point (iii), and Lemma 2.3. Given that H is a normal matrix, the following theorem applies in the SVD process.

Theorem 2.4.

Proof. (→)

Similarly, using the same approach, we obtain .According to Lemma 2.3, and have the same and both are self-adjoint, thus

Considering both can be unitarily diagonalized as follows.

with the same eigenvalues and possibly diferent unitary matrices UandV it is obtained that:

Thus,

with and being unitary matrices, if we move one of these unitary matrices to the right-hand side, it is obtained that:

Let with U being a unitary matrix that diagonalizes . Based on (4), it is obtained that . If is considered and

is chosen, it will be checked whether are orthogonal for . It should be noted that:

Since

Additionally,

based on the unitary property of U. Since forms an orthonormal set, a unitary matrix

can be constructed. Thus, for a matrix H , a diagonal matrix a unitary matrix and exist. □

Referring to Lemma 2.2, Lemma 2.3and2.4, there are general consequences that apply to the H as follows:

Corollary 2.5.The bounds for in are

The commonly used value of is often about half the length of the time series, namely . However, in practice, for very large , it is believed that selecting according to Corollary 2.5 is not efective. Therefore, to explore the possibility of an optimal limit on time series data, especially with the AR(1) and AR(2) model through a sequence of singular values on SVD, the following procedure is developed for further analysis.

The simulation process of AR(1) and AR(2) model is executed through an algorithm designed to generate time series data closely resembling the actual AR process. The generated simulation data undergoes analysis using the SVD technique, which decomposes the data matrix into fundamental components, facilitating insights into the intrinsic structure of the data. The entire simulation process is implemented using the R programming language, structured into three procedural components.

The autocorrelation behavior and from SVD in the simulated data model are analyzed numerically through ACF and PACF. Figure 2 and Figure 3 illustrate the correlogram patterns for and AR(2) models, respectively. The ACF plot exhibits a mix of damped exponential and sinusoidal waves, forming an organized pattern. However, ACF provides limited insight into autocorrelation across lag intervals and is more useful for identifying the MA model order 'q' . In contrast, PACF helps determine the AR model order 'p'.

PACF reveals partial correlations after accounting for shorter time intervals. In an AR model, PACF typically cuts of beyond a certain lag. For simulated AR(1) data, this cutofoccurs after lag 1 unless . Ensuring data accuracy becomes challenging as past values heavily influence predictions. Thus, alternative models or external factors beyond the AR(1) model may need further investigation.

The ACF and PACF in the AR(2) model simulation data is analyzed to observe correlogram patterns with varying values of and , rather than correlating it with parameter magnitudes. When and are both positive (Data A), the ACF pattern is more regular and systematic, starting with positive autocorrelation followed by negative values, repeating in a consistent manner. Conversely, when both parameters are negative (Data B), the ACF pattern alternates between positive and negative values. Further examination shows that when is positive and is negative (Data C), the ACF pattern resembles Data A. On the other hand, when is negative and is positive (Data D), the ACF pattern resembles Data B but is more systematic. The PACF correlogram, however, shows a cut-of pattern after lag 2, corresponding to the signs of and .

Interestingly, the PACF patterns in the simulated AR(1) and AR(2) models show a cut-of pattern similar to the scree plot pattern observed in the SVD shown in Figure 4. In the AR(1) model, the PACF cut-of pattern resembles the ’elbow in the SVD scree plot, especially when , the cut-of appears after the lag/λ^∗. Conversely, when , the variance contribution from the is more evenly distributed, the ’elbow’ becomes less distinct, indicating weaker and more dispersed correlations within the data. Thus, the data structure in the AR(1) model changes with diferent values, highlighting the importance of selecting an appropriate value to accurately capture significant patterns in the AR(1) model.

In the simulated AR(2) model data, and contribute the most significantly to explaining the variance in the data compared to other . This ’elbow’ pattern is similar to the PACF pattern, which shows a cut-of after the lag. The cut-of pattern observed in the PACF, which mirrors the pattern in the SVD, suggests that these methods can complement each other for a comprehensive view of TSA. Meanwhile, convergence is evaluated by determining if the remains consistent or varies with the addition of more L. As L increases, there is a greater absorption of data information. The represents the maximum number of to be considered in the analysis. This helps in understanding the data structure and ensures that the SVD analysis captures enough information from the simulated AR(1) and AR(2) model data. Observations focus on changes in variance proportions explained by , as shown in Figure 5 and Figure 6.

In the AR(1) model, the focus will be on for λ^∗ (Figure 5), while in the model, the focus will be on and denoted by and respectively (Figure 6). The sequence decreases monotonically with as the size of L increases. This allows for the selection of such that for . By choosing determining for the simulated data can be done more eficiently. If is chosen as 5% or 1%, the for each simulated and AR(1) and AR(2) model data can be summarized as shown in Table 3 and Table 4

Referring to Table 3, it can be inferred that to efectively use the SVD scree plot pattern in the matrix H for simulated AR (1) model data with selecting or for and between 10 and 18 for would be appropriate. For example, in the case of the model with , if is chosen, the optimal bound is 7 with . Meanwhile, for , the optimal bound is 18 with . Similarly, for , if is chosen, the size is 6 with , and is chosen, the size is 13 with

In the simulation of the AR(2) data model, the plots of and based on Figure 6 exhibit similar patterns, both showing convergence to 0 as L approaches with the sequence decreasing monotonically. For example, referring to Table 4, in Data is set at 5%, the optimal boundary size is with , and with . In contrast, when is set at 1%, the optimal boundary size is with , and with

Matrix-based analysis becomes particularly relevant when applied to established TSA methods such as the Box-Jenkins approach, particularly within the AR model framework. In AR(1) and AR(2) models, the PACF exhibits a characteristic cut-of after lag 1 for AR(1) and lag 2 for AR(2), indicating that PACF values remain significant up to the corresponding model order before declining sharply toward zero. This pattern closely resembles the scree plot in SVD, where the largest singular values decrease markedly after the dominant components, forming an ”elbow” shape. In SVD, only the first few singular values retain most of the information, analogous to how PACF retains significant values only up to the AR model order.

This similarity suggests that PACF and the SVD scree plot can complement each other in identifying key structures in time-series data. PACF helps determine the order of an AR model by identifying where the partial correlations are cut of, while the scree plot in SVD reveals the number of significant components in the decomposition of the data structure. Furthermore, SVD sensitivity to AR parameters depends on the magnitude of due to stationarity requirements, but it is not afected by the sign of the parameter . This can be explained by considering that stationarity depends on the modulus of the characteristic polynomial roots of the AR model rather than the individual coeficient signs.

Other findings indicate that the lower and upper limits of Corollary 2.5 are not efective in the adhesion process of HM (H), which requires an analytical study of the optimal boundary that improves eficiency by selecting a specific ε. It is hypothesized that an optimal limit of may be suficient for time series data. The numerical results obtained in this study suggest promising directions for future research by revealing potential patterns and relationships in time series data. While these findings provide valuable insights, reinforcing them with analytical proofs would enhance their theoretical foundation. Therefore, future studies could focus on developing formal mathematical validations to complement these results. By integrating the SVD approach with trajectory matrices into existing TSA methods, this framework can evolve to be more adaptive and objective, strengthening its applicability in TSA.