Application of Online Automated Segmentation and Evaluation Method in Anomaly Detection at Rail Profile Based on Pattern Matching and Complex Networks

Abstract

In steel rail production, complex deformations can induce non-uniform changes in cross-sectional profiles along the rail’s length, resulting in unevenness and safety implications. It is essential to perform dimensional testing to ascertain compliance with standard requirements. Currently, profile inspection results are manually evaluated, posing efficiency challenges and a lack of standardized criteria. To address this challenge, this paper proposes an online automatic steel rail segmentation and evaluation method (online-ASE) based on pattern matching and complex networks to enable automatic rail profile assessment. This method initially utilizes offline high-dimensional time series data for conducting Toeplitz Inverse Covariance-based Clustering (TICC) training and constructs a standard quality characterization pattern library through distinct inverse covariance structures between abnormal and normal high-dimensional quality characterization indicators of steel rails. When applied online, the Viterbi shortest path dynamic programming algorithm is utilized to match steel rail data with the pattern library, swiftly identifying anomalous rail segments. Additionally, the algorithm computes the contribution of steel rail quality parameters to the segmentation results using complex network betweenness centrality, thereby explaining the reasons for segment formation. These explanations provide a reference basis for subsequent steel rail repairs. Finally, the effectiveness of the proposed method is validated using real-world steel rail data from a specific steel factory in China.

1. Introduction

As a crucial structural element in rail transportation, the quality of rails significantly impacts train safety and rail longevity. Non-uniform variations in rail cross-sectional dimensions result in rail irregularities, causing damage to both the track and trains. Moreover, this condition generates excessive noise, affecting train speed, operational stability, and passenger comfort. With the advancement of high-speed rail technology, the smoothness of the tracks has become increasingly crucial for the trains. This has raised higher standards for track production, including improved flatness, dimensional accuracy, and precise hundred-meter length requirements. However, despite the prevalent use of online detection equipment in most rail production processes, the determination of cross-sectional dimensions continues to rely on manual assessment. This approach is associated with challenges such as reduced efficiency and accuracy, as well as higher rates of misjudgments and errors due to unclear causes of anomalies. Therefore, there is a need for an online automatic evaluation method that fully utilizes current detection data to determine the abnormal profile of steel rails in real-time, segment normal and abnormal sections, and evaluate the reasons for the occurrence of abnormal profile. Such an approach can not only enhance the efficiency and accuracy of manual measurements but also guide subsequent production by analyzing the root causes, ultimately improving the yield rate of rails.

In the context of data-driven and intelligent manufacturing, data mining methods have witnessed significant growth, revealing patterns that align with practical experience through pure data analysis. Numerous researchers have been dedicated to the study of time series data segmentation, aiming to uncover distinct patterns within entire time series and have successfully applied these methods across multiple domains including industrial,^1,2) agricultural,³⁾ financial,⁴⁾ and social sciences.⁵⁾ However, rail cross-sectional data comprise multiple interconnected parameters, dynamically evolving over time, often exhibiting frequent anomalies at their ends. Utilizing mean values like K-means or breakpoint detection methods such as Binary Segmentation (Binseg),⁶⁾ Dynamic Programming (Dynp)⁷⁾ alone is not sufficiently accurate for this practical problem. This paper abstracts it as a time-series segmentation problem, simultaneously considering parameter interdependencies and temporal dynamics, proposing an Online-ASE approach for detecting anomalies at rail ends. Firstly, we use a substantial amount of training data to establish a standard segmentation pattern library offline. Then, we match the real time data with this pattern library to construct an online segmentation model. Finally, we apply complex network methods to analyze the coupling relationships between various rail parameters and interpret the segmentation results, achieving online automatic segmentation and evaluation of steel rail anomalies at both ends.

The remaining sections of the paper are structured as follows: Section 2 introduces the issue of anomalies at rail ends and discusses relevant works on high-dimensional time-series segmentation and result interpretation. Section 3 provides a detailed exposition of the proposed method. Section 4 evaluates the performance of the proposed method using a real dataset from a steel factory. Finally, conclusions are drawn in Section 5.

2. Related Works

2.1. Rail Production and Abnormal Ends Issues

Rail production is a complex process that requires multi-stage processing of steel billets, such as heating, water descaling, rough rolling, finishing rolling, rail profile measurement, straightening, verification, etc. The production process is shown in Fig. 1. The steel rail is rolled from specialized rail shaped holes, as shown in Fig. 2(a). To ensure strong bending resistance, the cross-sectional structure adopts an “I” shaped, consisting of three parts: rail head, rail waist, and rail base, as demonstrated in Fig. 2(b). Partial dimensions are shown in Fig. 2(c).

Fig. 1. Rail production process. (Online version in color.)

Fig. 2. Rail shaped holes and rail cross-section. (a) The red line area is prone to deformation during rolling of rail shaped holes. (b) Rail cross-section. (c) Partial relevant profile parameters of the cross-section. (Online version in color.)

As a measurement instrument for rail dimensions, the rail profile meter is capable of real-time measurements of 24 quality process parameters, including rail height, rail head width, rail base width and others, along the length direction of the rail. These parameters serve as quality evaluation indicators for assessing the cross-sectional quality of the rail, with some of the parameters listed in Table 1. Due to the complex shape variations, significant cross-sectional changes, and non-uniform deformation of the rails before and after rolling, coupled with the interplay of various process parameters, the rail heads and tails undergo uncontrollable deformations. This results in the cross-sectional dimensions and their related relationships at the rail ends failing to meet the specifications, the irregular parts need to be cut off using a flying shear device. Figure 3 illustrates the fluctuation curve of rail cross-sectional dimensions along the rolling direction. The horizontal axis represents the rail’s length, while the vertical axis represents partial profile characteristics, highlighting the non-uniformity of cross-sectional dimensions along the entire length of the rail.

Table 1. Partial evaluation indicators for rail quality.

Indicator parameters	Target (mm)	Upper/lower limit (mm)
Rail height	152	152.6/151.4
Cross-sectional area	65	66.3/64.6
Upper flange thickness	10.5	11.1/10.1
Lower flange thickness	10.5	11.1/10.1
Upper web	93.5	94.1/93
Lower web	93.5	94.1/93
Base width	132	133/131
…	…	…

Fig. 3. The fluctuation curve of the rail’s cross-sectional dimensions along the rolling direction. (Online version in color.)

The real-time acquisition of data through a rail profile meter facilitates the monitoring of quality metric fluctuations. However, there exists no online assessment of steel rail quality. Owing to the multi-stage, multi-parameter, and continuous production attributes of steel rail manufacturing, parameters exhibit interdependencies. The present reliance solely on singular feature analysis for manual steel rail quality assessment proves to be inaccurate. Therefore, it is necessary to analyze the coupled variations in variable relationships, automatically segment normal and abnormal sections, and provide qualitative and quantitative automatic evaluations of the segmentation results. This is essential for guiding subsequent production and enhancing the compliance rate of the rails.

2.2. High-dimensional Time Series Segmentation Method

In the field of time series analysis, traditional time series segmentation methods typically focus on univariate (one-dimensional) time series. Extensive research has been conducted in academia to advance one-dimensional time series segmentation techniques.⁸⁾ M. Basseville et al.⁹⁾ proposed a method for partitioning time series using significant change points, from which methods based on thresholds,¹⁰⁾ extrema,^11,12) Bayesian statistics,¹³⁾ and other approaches have been developed to infer the positions of change points and segment time series. However, change point segmentation methods disrupt the temporal continuity of data. Therefore, subsequence segmentation methods have gradually gained attention in the academic community. L Martí et al.¹⁴⁾ propose a top-down segmentation algorithm that applies linear regression on the time series and performs a statistical test to evaluate if the linear regression and the time series are statistically equal. J. Duan¹⁵⁾ introduced a method can automatically estimate the subsequence pattern lengths and optimize the subsequence segmentation to produce satisfying subsequence clustering.

While the aforementioned change point detection and subsequence segmentation methods have yielded satisfactory results in handling one-dimensional time series data, many real-world applications involve high-dimensional data. Simple extensions of one-dimensional time series segmentation methods are not always easy to implement in such cases. Currently, research in the field of high-dimensional time segmentation is actively ongoing to address this challenge.^16,17) X. Song et al.¹⁾ introduced a novel time series segmentation method based on the Fuzzy C-Means algorithm (FCM) for analyzing time series data from Tunnel Boring Machines (TBMs). However, this method treated sample point data as the subject of study and did not consider the temporal continuity of the data. Y. Matsubara¹⁸⁾ introduced the AutoPlait method for automatically mining and understanding the evolution of time series states. It employs Hidden Markov Models (HMM) to learn patterns within time series data, analyze the evolution of data states. This method primarily focuses on pattern extraction and evolution but does not account for inter-variable relationships. In contrast to the methods discussed above, TICC simultaneously takes into consideration variations in inter-variable relationships and the temporal continuity of data. This approach employs the inverse covariance matrix to establish the graphical dependency structure for each subsequence. It also introduces Toeplitz matrix constraints to govern the structure within specific time windows, thereby fostering the grouping of adjacent window sequences into same cluster and facilitating the clustering of all window sequences.¹⁹⁾ The overall optimization problem of TICC is:

  

$$argmi{n}_{\mathrm{\Theta }\in T,P}{\sum }_{i=1}^K\left[{\Vert \lambda \circ {\mathrm{\Theta }}_i\Vert }_1+{\sum }_{X_t\in P_i}\left(-\mathrm{𝓁}\mathrm{𝓁}\left(X_t,{\mathrm{\Theta }}_i\right)+\beta 𝕀\left\{X_{t-1}\notin P_i\right\}\right)\right]$$(1)

In this context, Θ denotes the inverse covariance, T represents a collection of Toeplitz symmetric block matrices, P designates the allocation of clusters for the inverse covariance matrix, ||λºΘ_i||₁ serves as a one-norm penalty for the Hadamard product to ensure sparsity in the inverse covariance. −ℓℓ(X_t,Θ_i) refers to the log-likelihood values, signifying the likelihood of X_i being assigned to cluster i. 𝕀{X_t−1∉P_i} is used as an indicator function to check if adjacent points are assigned to the same cluster, and β serves as a smoothing penalty encouraging adjacent subsequences to be assigned to the same cluster, acting as a parameter enforcing temporal consistency.

However, the TICC method can only perform offline modeling and lacks the ability to retrospectively evaluate the results. Therefore, this paper introduces an online segmentation and evaluation algorithm. It leverages TICC to construct a standard pattern library for normal and abnormal steel rail data. Real-time data is then matched online with the pattern library using the Viterbi algorithm, and the evaluation of results is achieved through betweenness centrality, enabling the online segmentation and evaluation of steel rail conditions from start to finish.

2.3. Method for Interpreting Model Results

With the exponential growth of artificial intelligence and industrial data, intelligent methods such as machine learning and deep learning have found wide applications in industrial anomaly diagnosis²⁰⁾ and quality prediction.^21,22) However, most models exhibit “black-box” characteristics, offering little to no explanation for their results. This issue has significant implications for certain practical applications, particularly in fields where safety requirements are stringent, such as medicine, intelligent transportation, and industrial anomaly diagnosis. The end anomalies prevalent in rail production necessitate the application of intelligent methods for automated segmentation and handling. However, once segmentation results are obtained, there arises a necessity to explain the reasons behind these results and validate whether they can effectively guide subsequent industrial operations.

Currently, common methods for interpreting model results can be broadly categorized into intrinsic interpretability and post-hoc interpretability. Intrinsic interpretability methods refer to the design of models that inherently possess good interpretability, including Linear Regression (LR), Naive Bayes, and Decision Trees (DT), among others. On the other hand, post-hoc interpretability methods involve the use of interpretable techniques to explain pre-designed models and provide decision rationales. This category includes Shapley Additive Explanation (SHAP),^23,24) Local Interpretable Model-Agnostic Explanation (LIME),^25,26) and complex networks,^27,28) among others. The online-ASE model developed in this paper lacks the ability to explain its results by itself, necessitating the use of post-hoc interpretability methods to analyze the model’s outcomes. It’s important to emphasize that both SHAP and LIME constitute direct mappings from the original data to the final results, measuring the degree to which the original data variables affect the results. However, the online-ASE model in this paper is primarily utilized for the automated segmentation of data, necessitating an explanation for why the model generates specific segmentation results and whether differences are observed between different segments, rather than analyzing how the original data influences the segmentation results. In the realm of complex networks, nodes serve as representations of variables, while edges depict the relationships between these variables. Through methods rooted in graph theory and matrix theory, it becomes feasible to elucidate the interconnections among variables. Centrality analysis stands as a valuable tool in the realm of complex network research, with researchers introducing a plethora of centrality metrics designed to gauge the significance of nodes within networks.^29,30,31) These metrics are instrumental in aiding researchers in the examination and explication of the functional roles of nodes within networks. Among these metrics, betweenness centrality emerges as an method that employs the calculation of shortest paths to measure node centrality within a graph of relationships. This methodology facilitates the quantification of the relative importance of diverse nodes across the entire network.

Based on the synthesis of prior research, this paper intends to employ complex network construction to represent the topological structure of relationships between variables. This will be used to conduct a qualitative analysis of model segmentation results, distinguishing between normal and anomalous segments. Simultaneously, by calculating the betweenness centrality values for all variables within the network, the influence of variables in different network contexts will be analyzed, facilitating a quantitative interpretation of the segmentation results. By combining this interpretation and evaluation with actual industrial process data, we aim not only to validate the effectiveness of the online-ASE model but also to offer valuable guidance for adjusting subsequent industrial operations.

3. Online Automatic Segmentation and Evaluation (Online-ASE) of Steel Rails Based on Pattern Matching and Complex Networks

In rail production, the shape changes before and after rolling affect the deformation at the rail’s ends, necessitating their removal. With the evolution of data-driven techniques, we can establish an automated segmentation model using extensive data to achieve online segmentation of anomalous rail segments. Simultaneously, for the purpose of assessing the accuracy of segmentation model results and providing more effective guidance for subsequent operations, there is a need to interpret these outcomes. Considering these aspects, this paper introduces an Online-ASE method. Firstly, this approach creates standard pattern libraries for each category offline using the TICC algorithm. Subsequently, upon the arrival of real-time data, it employs the Viterbi algorithm to match this data with the pattern libraries, thus achieving online segmentation. Finally, the method assesses the segmentation results using complex networks and betweenness centrality metrics while incorporating actual process parameters, thereby offering both qualitative and quantitative interpretations of the segmentation results. The principles of this method are depicted in Fig. 4, and the pseudo-code for the algorithm is presented in Table 2.

Fig. 4. Schematic diagram of the online-ASE method. (Online version in color.)

Table 2. Pseudocode of online-ASE algorithm.

Algorithm 2: online-ASE.
Input: K, ω, λ and β as defined in (1).
Output: segmentation result P and Betweenness centrality CB.
1 Estimating the class assignment matrix P and the inverse covariance Θ from Eq. (1).
2 Compute the original mean EO and the stacked mean Es, as defined in Eqs. (3) and (4).
3 Establishing pattern library {ω,K,EO,Es,Θ,P}.
4 Utilizing the Viterbi algorithm to match the parameters {EO–new,Es–new,Θnew}of new data with the pattern library {EO,Es,Θ,P}, resulting in the category result Pnew.
5 Calculate the Betweenness centrality, as defined in (2), to represent variable importance and assess the classification results.

In this section, we will provides an in-depth exploration of the proposed method. Specifically, Section 3.1 elaborates on the offline construction of standard pattern libraries for various quality representations through the application of the TICC method. Section 3.2 explicates the process of matching with the pattern libraries using the Viterbi algorithm to accomplish online segmentation. In Section 3.3, we elucidate the evaluation and interpretation of segmentation results utilizing network structures.

3.1. Building the Standard Pattern Library Offline

Taking into account that when the system operates under similar or identical operational modes, there is a high degree of consistency in the coupling relationships among variables. Segmentation of every dataset offline would result in computational and time inefficiencies. To tackle this challenge, the TICC method can be utilized to calculate distinct coupling relationships between high-dimensional quality indicators of normal and abnormal rail, thus establishing a standard pattern library.

Given the observed data X={x₁,x₂,...,x_i...x_n}, where x_i∈ℝ^m. Since adjacent sample points exhibit temporal consistency, we initially select a window length ω. This transforms the raw data X into stacked data $X^{\prime }=\{X_1^{\prime },X_2^{\prime }\mathrm{...},X_i^{\prime }\mathrm{...},X_N^{\prime }\}$, where N=n−ω+1, each $X_i^{\prime }=\{x_{i-\omega +1},\mathrm{...},x_i\}$ is a ω×m matrix, with ω≪n. Using the input observed data X, we construct the model offline using the TICC algorithm, resulting in the standard pattern library. The specific steps are outlined as follows:

Step 1: Initializing model parameters. This step involves the initialization of model parameters, which includes the number of clusters K, window length ω, regularization parameter λ, and smoothing parameter β. The number of clusters K can be determined based on expert knowledge or Bayesian Information Criterion (BIC). The choice of window length ω is made considering the actual data length. The original data X is transformed into stacked data X′ according to the window length ω.

Step 2: Computation of mean values for original and stacked Data. In this step, we calculate the mean values for both the original data E_O and the stacked data E_S. E_O is derived from averaging the observations in columns (ω−1)×n to ω×n, whereas E_S is the result of averaging the observations in columns 1 to ω×n. Here, ω represents the window size, n signifies the dimensionality of the observed variables, and a indicates the number of rows in the matrix. E_O and E_S serve as indicators of the fluctuation characteristics of the original and stacked data, providing auxiliary information for category features. The calculation formulas are presented below:

  

$$E_O=\left[\frac{1}{a}{\sum }_{i=1}^a{x}_{i,\left(w-1\right)n},\frac{1}{a}{\sum }_{i=1}^a{x}_{i,\left(w-1\right)\left(n+1\right)},\mathrm{...},\frac{1}{a}{\sum }_{i=1}^a{x}_{i,wn}\right]$$(2)

  

$$E_S=\left[\frac{1}{a}{\sum }_{i=1}^a{x}_{i,1},\frac{1}{a}{\sum }_{i=1}^a{x}_{i,2},\mathrm{...},\frac{1}{a}{\sum }_{i=1}^a{x}_{i,wn}\right]$$(3)

Step 3: Computation of initial clustering results. We perform clustering on the stacked data using a Gaussian Mixture Model to obtain the initial clustering results. This initial clustering facilitates subsequent computations of the topological Toeplitz inverse covariance matrix and optimization iterations.

Step 4: Computation by plugging the parameter outcomes {K,λ,β,X′,E_O,E_S,P_original} from the previous three steps into Eq. (1). Through this iterative process, we calculate the topological Toeplitz inverse covariance matrices, denoted as Θ_i, for each category. Simultaneously, we obtain the clustering results represented as P. It is essential to highlight that there exists a mutual correspondence between the category structures in Θ and the clustering results.

Step 5: In this phase, we compile a standard pattern repository {K,λ,β,X′,E_O,E_S,Θ,P} based on the results achieved in the four preceding steps.

It is worth noting that Θ_i and P from Step 4 cannot be computed in a single step. Equation (1) poses a complex mixed combinatorial optimization problem where the class allocation matrix P and the inverse covariance matrix Θ are coupled together, resulting in a highly non-convex problem. Obtaining a globally optimal solution is challenging. To address this, we employ the Expectation-Maximization (EM) approach, which decomposes the solution of Eq. (1) into two distinct steps:

E-step: With the clustering structural parameters Θ known, our objective is to compute the precise classification results denoted as P. The objective function is formulated as follows:

  

$$minimaze{\sum }_{i=1}^K{\sum }_{X_t\in P_i}\left(-\mathrm{𝓁}\mathrm{𝓁}\left(X_i^{\prime },{\mathrm{\Theta }}_i\right)+\beta 𝕀\left\{X_{i-1}^{\prime }\notin P_i\right\}\right)$$(4)

M-step: When we have the specific classification results P, we proceed to update the clustering structural parameters Θ:

  

$$\begin{array}{l}minimize-logdet{\mathrm{\Theta }}_i+tr\left(S_i{\mathrm{\Theta }}_i\right)+\frac{1}{\left|P_i\right|}{\Vert \lambda \circ {\mathrm{\Theta }}_i\Vert }_1\\ \mathrm{s}\mathrm{u}\mathrm{b}\mathrm{j}\mathrm{e}\mathrm{c}\mathrm{t}\mathrm{\hspace{0.17em} } \mathrm{t}\mathrm{o}\mathrm{\hspace{0.17em} } {\mathrm{\Theta }}_i\in T\end{array}$$(5)

The E and M steps are iteratively optimized using a combination of Dynamic Programming (DP) and the Alternating Direction Method of Multipliers (ADMM). This approach allows for the concurrent assignment of each time point to a class while updating the clustering structure.

3.2. Online Segmentation Algorithm Based on Pattern Matching

Upon the completion of constructing the standard pattern library, real-time data is subjected to online clustering and segmentation. The question arises of how to correlate the relevant results from real-time data with the standard pattern library established in the preceding section to effectively determine the data’s classification. The Viterbi algorithm, a dynamic programming method, finds widespread utility in solving various problems, such as predicting hidden Markov models, conditional random fields, and probability calculations in Seq2Seq models. It employs dynamic programming concepts to tackle the shortest path problem within a given graph. At each stage of path selection, it stores the minimum total cost accrued from all previous selections up to the current point, as well as the chosen forward path based on the current cost conditions. This algorithmic approach is particularly suitable for the template matching problem while maintaining temporal consistency, as discussed in this section. The algorithm’s principles are illustrated in Fig. 5. The value $\mathrm{𝓁}{\mathrm{𝓁}}_{(X_i^{\prime },{\mathrm{\Theta }}_i)}$ for each node in Fig. 5 represents the likelihood of assigning the i-th windowed data $X_{i-new}^{\prime }$ to the category structure Θ_i. The formula is as follows:

  

$$\begin{array}{l}\mathrm{𝓁}{\mathrm{𝓁}}_{\left(X_i^{\prime },{\mathrm{\Theta }}_i\right)}=-1/2{\left(X_i^{\prime }-{\mu }_i\right)}^T{\mathrm{\Theta }}_i\left(X_i^{\prime }-{\mu }_i\right)\\ \hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}+1/2lgdet{\mathrm{\Theta }}_i-n/2lg\left(2\pi \right)\end{array}$$(6)

Fig. 5. Principle diagram of the viterbi algorithm. (Online version in color.)

Where $\mathrm{𝓁}{\mathrm{𝓁}}_{(X_i^{\prime },{\mathrm{\Theta }}_i)}$ denotes the logarithmic likelihood value of data $X_{i-new}^{\prime }$ being associated with category structure Θ_i, representing the probability of assigning $X_{i-new}^{\prime }$ to Θ_i. Given that each Θ_i corresponds to a specific category K_i, $\mathrm{𝓁}{\mathrm{𝓁}}_{(X^{\prime },{\mathrm{\Theta }}_i)}$ further signifies the likelihood of assigning each $X_{i-new}^{\prime }$ to category K_i.

In the online phase of the online-ASE approach, the Viterbi algorithm is applied to match the relevant results {E_O–new,E_S–new,Θ_new} of real-time data X_new with the pattern library {E_O,E_S,Θ,P}. This procedure calculates the maximum probability assignment path for all instances of $X_{i-new}^{\prime }$ within the time window sequence $X_{new}^{\prime }$. This process yields the most suitable category outcome P_new, thereby enabling real-time clustering segmentation. The specific steps of the online segmentation algorithm based on pattern matching are as follows:

Step 1: Begin by preprocessing real-time data X_new, transforming it into stacked window data $X_{new}^{\prime }$ using the optimal parameters ω computed from the offline model.

Step 2: Calculate the mean values E_O–new and E_S–new following Eqs. (2) and (3).

Step 3: Establish the initial clustering results as P_{original–new}.

Step 4: Substituting P_{original–new} into Eq. (5), use the ADMM algorithm to compute the category structure Θ_new for each stacked window data $X_{i-new}^{\prime }$.

Step 5: Compute the likelihood of assigning Θ_new to pattern Θ using Eq. (6), and applying the Viterbi algorithm involves finding the shortest assignment path, which is the optimal selection of allocating all $X_{i-new}^{\prime }$ in $X_{new}^{\prime }$ to their respective categories.

Consequently, the proposed Online-ASE model, following the creation of a standard pattern library in the offline phase, directly addresses the clustering segmentation problem through linear programming by utilizing a Viterbi algorithm-based method to match the standard pattern library. It simplifies the overall optimization objective formula (1) of the offline model TICC, eliminating the necessity for alternating optimizations of the class assignment matrix P and the inverse covariance Θ through DP and ADMM algorithms. The online-ASE model diminishes the number of iterations required during the offline training phase of TICC, resulting in a substantial enhancement of algorithm efficiency.

3.3. Interpretation and Evaluation of Segmentation Results Based on Complex Networks

After offline construction of the standard pattern library and the completion of online segmentation through pattern matching, we obtain the online segmentation results. It is essential to provide a coherent interpretation and evaluation of these online segmentation outcomes. In theory, the segmentation results should demonstrate similarities in intra-segment variable relationships and disparities in inter-segment variable relationships. We employ complex networks to establish network structures for different segments and conduct comparative analyses of changes in variable couplings between segments. These variations in inter-segment variable relationships are used for a qualitative interpretation of the model’s segmentation results.

Moreover, intermediate nodes in the network hold a more substantial influence over nodes at the endpoints of the path, governing and restraining interactions between non-adjacent nodes. Therefore, when a node resides on multiple shortest paths connecting other nodes, it assumes the role of a core node with increased betweenness centrality.³²⁾ Thus, we employ betweenness centrality to quantify the impact of nodes within the network, offering a quantitative explanation of the segmentation results. The definition of betweenness centrality is as follows:

  

$$C_B\left(\nu \right)={\sum }_{s\ne \nu \ne t\in V}\frac{{\sigma }_{st}\left(\nu \right)}{{\sigma }_{st}}$$(7)

Within this context, σ_st(v) stands for the quantity of shortest paths from node s to t that pass through node v, and σ_st represents the total number of shortest paths from s to t.

In Fig. 6, we assume two network structures constructed based on variable correlations within diverse data segments. Nodes in the network correspond to variables, and when represented by the same letter, they signify identical variables. Edges within the network depict relationships between variables. The evaluation of variable influence on distinct networks can be conducted by calculating the betweenness centrality of the same variable in different networks. For instance, utilizing Formula (6), the betweenness centrality of node D in Fig. 6(a) is calculated to yield C_B(D), which equals 9/15. In Fig. 6(b), node D’s betweenness centrality is represented as C_B(D), with a value of 7/19. This observation underscores the variation in the influence of the same variable on other variables within different networks. This quantitative method can effectively expound upon the segmentation results, analyze the influence of all variables across diverse networks, and, subsequently, provide guidance for adjusting different variable parameters in subsequent operations.

Fig. 6. Illustration of a network structure built from diverse variable relationships. (Online version in color.)

Given the variable degrees of coupling in inter-segment variable relationships, it is possible to create a range of distinct network structures. Additionally, the influence of individual nodes on other nodes within the network differs. Therefore, in the online-ASE model we introduce, complex network analysis and betweenness centrality are incorporated to enable the automatic interpretation and evaluation of segmentation results.

4. Results and Discussion

4.1. Data Overview

In this section, we validate the performance of the proposed method using measured data from a rail profile meter. The dataset comprises a substantial amount of cross-sectional profile data for steel rails, with each data point containing 26 features, such as rail height, cross-sectional area, and bottom width, among others (as detailed in Table 1). Any anomaly in any of these 26-dimensional quality indicators would lead to the classification of the steel rail as substandard. Given the presence of missing values in the collected industrial time-series data, data preprocessing is essential. In this study, we eliminate dimensions for which observations were not recorded, resulting in a dataset with 24 dimensions. When observations are zero, they are replaced with the mean of adjacent data points. Z-score standardization is utilized to ensure distinct variations among the parameters.

4.2. Parameter Configuration

Before embarking on model training, it is imperative to configure the class parameters. According to the tenets of steel rail production theory, the number of classes K is established at 2. In situations where determining the class number in advance through expert knowledge is not feasible, alternative methods, such as the Bayesian Information Criterion (BIC) or the elbow method, can be employed. In the dataset employed in this study, the class number determined by both of these methods aligns with empirical knowledge. Furthermore, the sparse regularization parameter λ is set at 0.11 to maintain matrix sparsity. The window length ω is fixed at 5, while the penalty parameter β is set at 5 to enforce temporal consistency and encourage the allocation of adjacent subsequences to the same cluster.

4.3. Online Segmentation Results

The model constructed in this paper yields statistical results for 100 steel rails, as presented in Table 3. These findings align with empirical observations, confirming the accuracy of the methodology. The experimental results indicate an average steel rail length of approximately 105 meters, with an average head anomaly length of 1.8 meters and an average tail anomaly length of 2.3 meters. These results offer valuable insights for the subsequent diagnosis and serve as a foundational reference for anomaly repairs. Moreover, Fig. 7 summarizes the head and tail lengths of 100 rail beams, predominantly concentrated within the 0 to 3 meter range. However, there are notable exceptions (as indicated in the red box in Fig. 7), with some rail beams exhibiting head lengths as long as 6.7 meters and tail lengths of 5.4 meters, which markedly degrades the quality of the rails.

Table 3. Statistics on ends segmentation results of 100 rails.

100 Rails	Mean (m)	Maximum (m)	Minimum (m)
Total length	105.2	109.3	104.1
Head abnormal length	1.8	6.7	0
Middle abnormal length	0.7	1	0
Tail abnormal length	2.3	5.4	0

Fig. 7. Histogram of abnormal ends lengths for 100 rails. (a) Head abnormal length (dm). (b) Tail abnormal length (dm). (Online version in color.)

The model developed in this study can also address the issue of occasional anomalies during the middle stationary phase. For instance, in the experimental dataset, the model identified segmentation points at positions 3, 382, 389, and 1028 for a specific steel rail. As a result, it not only detected anomalies at the beginning and end but also identified an anomaly in the middle section of the same rail, as depicted in Fig. 8(a). Upon examining the original time-series data for the steel rail within the 382–389 interval, which corresponds to the physical space between 38.2 meters and 38.9 meters, an anomalous change was observed in the “base width” parameter, as shown in Fig. 8(b). This alteration led to the presence of an anomaly in the middle section of steel rail. By comparing this with the inverse covariance matrix of the normal data in the adjacent 374–381 interval, changes in the coupling relationships among dimensions within this anomalous interval were identified, as displayed in Figs. 8(c1) and 8(c2). This provides further evidence of the method’s effectiveness.

Fig. 8. Segmentation results of a particular steel rail. (a) Segmentation outcomes; (b) Anomalous original data for base width; (c1) Inverse covariance matrix heatmap for the original data in the 374–381 interval; (c2) Inverse covariance matrix heatmap for the original data in the 381–382 interval. (Online version in color.)

4.4. Explanation of Segmentation Results

Taking a specific rail as a case study, the online-ASE model classifies it into three segments: 1–21, 22–1050, and 1051–1060, as presented in Fig. 9(a). Analysis reveals that the rail’s head anomaly occurs within the first 2.1 meters, while the tail anomaly is situated in the final 0.9 meters, as indicated in Fig. 9(b). Figure 10 visually represents the complex networks and their corresponding betweenness centrality for two distinct data categories: anomalies at both ends and the stable central section. The circles in the figure exhibit different colors, transitioning from deep yellow to light yellow, indicating a change in betweenness centrality values from high to low. The numbers inside the circles are associated with the labels of actual rail profile variables.

Fig. 9. Segmentation results of a specific rail. (a) Segmentation results. (b) Original data from partial dimensions. (Online version in color.)

Fig. 10. Complex network and betweenness centrality distribution for the specific steel rail data. (a) data from the steel rail ends. (b) data from the midsection of the steel rail. (Online version in color.)

Using Fig. 10, we can interpret the segmentation results of the online-ASE model based on the complex network and mediation centrality distribution of steel rail data. It becomes evident that there are substantial differences in the complex network and betweenness centrality distribution between head-and-tail data and midsection data. In particular, Fig. 10(a) highlights the existence of correlations in only a few variables, indicating their higher fluctuations and greater impact on the results. Notably, the 14th attribute (right upper symmetrical) and the 8th attribute (base width) exhibit the highest betweenness centrality, significantly affecting the fluctuations in head-and-tail dimensions. Following closely are head height, rail height, and left upper symmetrical, along with other process parameters. The root cause of this outcome lies in the limited effectiveness of short-stroke control by the vertical rolls during the rolling process, leading to frequent width deviations at the ends sections. In Fig. 10(b), the parameters interact with one another, leading to intermediary centrality values that exhibit relatively stable fluctuations. This coordinated interplay of parameters maintains the stability of coupled relationships, thereby upholding uniform cross-section dimensions along the steel rail length direction. The application of complex network analysis and intermediary centrality allows us to interpret the results of the online automatic segmentation model proposed in this paper. The consistency between the interpretation of the results and basic principles indirectly validates the effectiveness of the proposed method.

4.5. Analysis of Online Effectiveness

The TICC algorithm, which serves as a method for temporal clustering and segmentation, necessitates multiple iterations for computing variable coupling relationships. This leads to extended computation times, making it unsuitable for real-time online applications. Given the batch-oriented nature of steel rail production, we opted to use the offline TICC method for establishing a standard template library. This approach formed the basis for the creation of the online-ASE algorithm, which guarantees result accuracy and minimizes the requirement for iterative parameter optimization. We conducted the experiments ten times on 100 steel rails and calculated the average results to compare the operational speeds of the TICC and Online-ASE models, as illustrated in Fig. 11. Upon further calculation, it was found that the average segmentation time for the TICC model was 95.98 ± 4.08 seconds per rail, compared to just 24.31 ± 0.83 seconds for the Online-ASE model. Online-ASE significantly outpaced TICC, fulfilling the time-sensitive requirements of industrial steel rail segmentation applications.

Fig. 11. Runtime comparison between TICC and Online-ASE methods. (Online version in color.)

While meeting timeliness requirements, the accuracy of the model is also a consideration. Table 4 compares the effectiveness of the K-means, Binseg, Dynp, TICC, and Online-ASE models. Accuracy, Precision, Recall, and F1_score were chosen as effectiveness indicators. Each method was applied to classify 100 steel rails, and the highest and average classification results for these 100 steel rails were compared for model effectiveness. Bold values indicate the optimal results. The results indicate that TICC performs best in the classification of individual steel rails, while the Online-ASE model demonstrates higher accuracy and stability across 100 steel rail data. The traditional K-means method only considers the distance between sample points and does not account for temporal consistency and the correlation between variables, resulting in poor performance on time series data. The Binseg method focuses more on local features, dividing the data into relatively uniform segments and seeking transitions between segments. However, when steel rail data undergoes transitions, it may still fall within the normal specification range, leading to lower accuracy for this method. The Dynp method attempts to find the globally optimal positions of change points but does not take into account the correlation between variables. Moreover, due to the uneven distribution of categories in the rail dataset, normal class data significantly outweigh abnormal data. Consequently, the aforementioned models are more likely to achieve high accuracy in categories with abundant data, while exhibiting poorer performance in categories with limited data, resulting in an overall lower accuracy and higher precision. TICC involves the recalculation of the covariance structure for all samples and the iterative computation of optimal results for each steel rail, resulting in unstable segmentation effects. As a consequence, it lacks effectiveness advantages in the average experimental results of 100 steel rails. On the contrary, Online-ASE takes into account both the temporal nature of the data and the relationships between variables. It also preserves the variable covariance structure for each category, leading to a model with higher effectiveness and stronger stability.

Table 4. Comparative Effectiveness of Methods.

	K-means	Binseg	Dynp	TICC	Online-ASE
Accuracy (Max)	0.79	0.71	0.71	0.95	0.94
Precision (Max)	0.98	0.97	0.99	0.99	0.96
Recall (Max)	0.78	0.72	0.72	0.98	0.99
F1_score (Max)	0.87	0.82	0.82	0.98	0.97
Accuracy (Ave)	0.43	0.48	0.48	0.47	0.91
Precision (Ave)	0.88	0.93	0.92	0.87	0.93
Recall (Ave)	0.42	0.48	0.48	0.47	0.97
F1_score (Ave)	0.56	0.63	0.62	0.56	0.95

5. Conclusion

This study is driven by the requirement to detect deformations in steel rail production. In cases where the cross-sectional dimensions at the rail ends fail to meet the prescribed standards, they have to be removed. Presently, the determination of dimension compliance is based on manual judgment, characterized by low efficiency, reduced precision, and a lack of standardized evaluation criteria, rendering the interpretation of assessment results difficult. Consequently, this paper presents an online automatic segmentation and result evaluation method based on pattern matching and complex networks.

Our model has established a standard pattern library for both normal and abnormal segments of steel rails. Through matching new steel rail data with this pattern library, we can perform online segmentation of abnormal sections at the rail ends. The application of betweenness centrality from graph theory to interpret the segmentation results of the model unveils variations in the importance of variables across different categories, facilitating the automated evaluation of normal and abnormal rail dimensions along the entire length. The model proposed in this study has been validated using authentic data from a steel manufacturing facility. The findings illustrate that this method can serve as a viable alternative to manual assessment, facilitating automated dimension evaluation and, consequently, enhancing both efficiency and accuracy. This, in turn, leads to reduced material wastage for steel production caused by manual rail cutting based on workers’ experience. Furthermore, the interpretation and analysis of the segmentation results provides valuable insights to guide subsequent production processes, thereby increasing the yield of high-quality steel rails.

Acknowledgments

This work was supported by the Fundamental Research Funds for the Central Universities [Grant numbers: FRF-BD-22-03].

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