Determination of the Condition of Railway Rolling Stock Using Automatic Classifiers

Abstract

Efficient maintenance is paramount for rail transport systems to avoid catastrophic accidents. Therefore, a method that enables the early detection of defects in critical components is crucial for increasing the availability of rolling stock and reducing maintenance costs. This work’s main contribution is the proposal of a methodology for analyzing vibration signals. The vibration signals, obtained from a bogie axle on a test bench, are decomposed into intrinsic functions, to which classical signal processing techniques are then applied. Finally, decision trees are employed to characterize the axle’s state, yielding excellent results.

1. Introduction

It is an acknowledged fact within the field of rail operation that “the economic efficiency and competitiveness of railway transport depends on the safety, availability and maintenance” [1], so researchers, manufacturers, and operators keep in mind that a failure of one the structural elements that suffers high stress such as bogies or wheelsets may result in catastrophic effects both for rolling stock and people. The Eschede rail accident [2], a serious accident in Germany in 1998 that went to court for negligent homicide, was a case in point. The accident was caused by a broken wheel, and both the wheel manufacturer and the railway company were charged. Considering the high standards of maintenance, the authors aim to integrate new procedures and equipment guided by Industry 4.0 [3], in order to increase both security and maintenance efficiency.

Advancements in computing technology have enabled the emulation of physical phenomena, as initially proposed by Feynman in 1982 [4], as well as the reproduction of elements and entities in a completely digital way. Examples are the digital twins used to replicate and test blocks and mechanical assemblies such as bearings [5] or to monitor fruit quality evolution [6].

Advances in computer hardware and software have made it possible to process and store large data more efficiently and quickly. This has also made it easier to handle large datasets in a shorter period of time [7,8], allowing complex calculations to be executed with decreasing time consumption. This makes Artificial Intelligence (AI) viable. That is the name given to methods created to copy how the human brain functions [9], describing the creation of artificial models and computational algorithms that resemble human learning and reproduce human skills.

1.1. Machine Learning

In 1955, Artificial Intelligence (AI) was first introduced as a concept by McCarthy et al. [10], who stated that “If a machine can do a job, then an automatic calculator can be programmed to simulate the machine”, so the basic concepts of AI, such as the use of language and neural networks, were introduced for the first time for computer science.

Once the concept of AI has been defined, its functioning is described [11], and both the objectives and the methodologies used to achieve them are identified, from problem solving to communication, so that the concept of learning is introduced as part of any developed and recognized intelligence, which then describes the concept of machine learning.

Having described machine learning as a concept and as a method, the concept of learning is considered to take into account the fact that machine learning often requires large amounts of data and computational power to achieve impressive results in areas such as autonomous cars and image recognition [12]. However, this approach has its limitations, as it lacks the ability to reason logically and identify causal relationships. The development of machine learning systems and methodologies usually starts with data collection and labeling. The next steps are analysis and algorithm selection until a trained model can be developed and implemented [13].

It is necessary to address the three main points to be taken into account, machine learning, machine teaching, and how the human being is involved in the whole process; therefore, whether the human being is involved in the machine learning process [14] is considered to define different approaches [12], such as active learning [15], in which the system has full control over teaching and learning processes, interactive machine learning [16], in which humans interactively deliver information in a more focused, frequent and incremental way compared to traditional machine learning, and a combination of machine learning and machine teaching [13].

1.2. Vibration Analysis: Condition Monitoring Tool

Vibration signal analysis is one of the most widely used techniques for inspecting mechanical components under operational conditions [17], from the first models [18], as it allows testing in a wide range of elements and situations [19], such as railway infrastructure, general-purpose equipment, or rolling stock.

Many authors studying vibration signals evolution for condition monitoring apply their techniques to rolling elements to analyze bearing faults [20]. Some of them have dealt with vibration analysis performed on railway systems [21]; others focus on ground or track disturbance induced by rolling stock traffic [22].

Vibration signal analysis is a common testing ground for a wide range of mechanical analyses used to determine bearing failures [23], and other authors perform analysis for axle components [24,25].

1.3. Empirical Mode Decomposition (EMD)

In light of the necessity to develop effective diagnostic techniques, considerable reliance has been placed on signal modeling [25]; thus, signal analytical models are considered the optimal choice compared to full dynamic simulations [26]. It is therefore imperative to develop models that have been empirically tested with real signals. There is limited knowledge about the quantitative effect of their parameters.

The use of envelope analysis for diagnostic purposes has been a customary practice for a long time, owing to the fact that the spectrum of the raw signal frequently exhibits an insufficiency of diagnostic information concerning faults [27]; consequently, novel techniques for enhancing the efficiency of analysis have been developed.

Empirical Mode Decomposition (EMD) was first proposed by Huang et al. in 1998 [28]. The methodology employed is to use envelopes delimited by local maxima and minima, and these envelopes are connected by a cubic spline line. The local maxima are designated as the upper envelope, while the local minima are assigned to the lower envelope. This ensures that all data is enclosed between these envelopes. Subsequent to the definition of the envelopes, the next step is to determine the conditions to develop the so-called intrinsic mode function:

“An intrinsic mode function (IMF) is a function that satisfies two conditions: (1) in the whole data set, the number of extrema and the number of zero crossings must either equal or differ at most by one; and (2) at any point, the mean value of the envelope defined by the local maxima and the envelope defined by the local minima is zero.”

The aim of the EMD technique is to decompose the vibration signals into subsignals (IMFs). Theoretically, each IMF is directly related to the sources that generated it [28]. The purpose of the application of the EMD technique in this study is to ease the identification of the condition of rolling stock, the failure of which can lead to catastrophic accidents with a high probability of human casualties. This is done by decomposing the vibration signals into several IMFs, which can then be measured to obtain a reliable estimate. A method called “bivariate modal decomposition” has been developed [29]. The use of this instrument in this project enables the decomposition of complex signals, thus providing a sophisticated analytical framework.

Since its introduction in 1998, the EMD technique has undergone significant development in a number of disciplines, with widespread use in a variety of applications, especially in the field of prediction [30], although its utilization is extensive in the identification of faults [31], satellite navigation [32], river stages [33], biomedicine [34], or dam deformation [35].

As mentioned above, the EMD technique is based on signal decomposition [36], which allows features to be extracted for analysis [22]. Once a guide to bearing diagnostics has been established [27], the EMD technique has been widely used, for example, in the analysis of wind turbine generator bearings [37].

The complete EMD process is depicted in Figure 1. The upper figure shows the original signal, the middle figure represents the location of the maxima of both the upper (blue) and lower (red) envelopes, and the lower figure represents the first IMF.

1.4. Decision Trees

Machine learning models can be classified into four categories: supervised, semi-supervised, unsupervised, and reinforcement learning, each of which includes several subcategories [38], and among the supervised models are those used for classification and regression [39].

Decision trees have been widely used for classification and regression analysis. While these models are effective, they often suffer from over-complexity and over-fitting and therefore need some degree of simplification [40].

As examples, among the uses of decision trees is fault diagnosis using discrete wavelets [41], and the use of decision trees for fault diagnosis using discrete wavelets and classification combined [42].

The extent to which decision trees are still suitable for classification purposes has been explored, and the feasibility of their use for railway rolling stock characterization has been tested in recent years; the present work is an adaptation of those results combined with the EMD methodology [43].

1.5. Approaches and Challenges

Regarding the other approaches that have been taken to achieve the goal of optimizing the analysis of incoming data, Gafni et al. [44] identify federated learning as a tool for the future, for, among other purposes, using information received from different sources, which, in many cases, may not be fully reliable and/or significant, to a certain extent; Hu et al. [45] address complexity not only with different data sources, but also with different research objectives and data improvement. The amount of data gives rise to a number of challenges, but it also creates an area of interest in the form of decentralized data, as has been pointed out by Yang et al. [46], but the quality of the information, and thus the effectiveness of the analysis, is influenced by the use of data from different sources and the methods of data collection.

1.6. Aim of the Study

The present work explores the use of decision trees combined with EMD as an alternative method to classify and categorize the condition of railway rolling stock. This is achieved by analyzing the different stages of component deterioration, with the aim of determining the initial stage of mechanical fault and improving overall predictive maintenance, which offers results comparable to those obtained by Ruiz et al. [47], whose work was based on the use of Support Vector Machine, and Bustos et al. [48], who used EMD and K-Nearest Neighbor to perform classification under similar conditions.

The methodology applied in this work is frequently used in bearing condition analysis, as is demonstrated by the extensive literature referred to in this introduction. The literature mainly refers to the methodology used for rolling bearings and has been presented more broadly in terms of both elapsed time and the range of methodologies.

1.7. Structure of the Paper

The structure of the document is as follows: The following section details the methodology and techniques applied throughout the research process. Section 3 details the experimental system, the test bed, and the signal processing. Section 4 shows the results of the model training. Section 5 contains the conclusions of this work, as well as suggestions for future research.

2. Proposed Methodology

This section describes the methodology and techniques applied in the treatment of vibration signals in detail. The analysis process starts from two considerations or starting points: firstly, the definition of the bogie under study and its characteristics as a specific mechanical system to be analyzed, and secondly, the working conditions during the study process and the description of the test bench. The second starting point is the measurement process, which covers the equipment used, the sensors used, their description and positioning, and the speed at which the tests were performed.

The next stage of the process was the extraction of the vibration data and its conversion into a MATLAB^® R2024b format for signal processing. The selection and extraction of the raw data from the database were performed according to the measurement conditions. The vibration data in MATLAB^® format was pre-processed for each accelerometer. Finally, the consolidated data were processed in both the time and frequency domains, and the results obtained were analyzed.

Once the collection of vibration signals was completed and prior to their conversion, the EMD technique was applied to each signal. The main aim of this technique is to identify the intrinsic oscillatory modes of any given dynamic signal by its characteristic time scales. The data was then decomposed according to these modes. This is an iterative method for decomposing a dynamic signal into a set of IMFs based on three assumptions [28]:

• For any analyzed signal, the number of extrema is at least two: one maximum and one minimum.
• The characteristic time scale is defined by the time lapse between the extrema.
• If the given signal does not have extrema but does have inflection points, the data can be differentiated to disclose the extrema.

It is important to note that the execution of the algorithm results in the extraction of two loops. The first loop is known as the sieving process, which is executed iteratively until the extracted signal meets the two IMF conditions. The function of the outer loop is to extract all the IMFs from the input signal. The algorithm works as follows:

• For any given signal x(t), identify all extrema.
• Connect all related extrema with a cubic spline curve, so an upper envelope e_max(t) is obtained by connecting all maxima and a lower envelope e_min(t) by connecting all minima. Computing the mean between envelopes results in m₁(t), as shown in Equation (1):

$$m_1\left(t\right)={\displaystyle \frac{e_{max}\left(t\right)+e_{min}\left(t\right)}{2}}$$

(1)

3.

To obtain the first IMF candidate h₁(t), the mean is subtracted from the input signal, as shown in Equation (2):

$$h_1\left(t\right)=x\left(t\right)-m_1\left(t\right)$$

(2)

4.

When the obtained h₁(t) does not meet IMF conditions, h₁(t) is taken as the input data, and Steps 1–3 are repeated until h_1n(t) meets IMF conditions, as shown in Equation (3):

$$h_{1n}\left(t\right)=h_{1\left(n-1\right)}\left(t\right)-m_1\left(t\right)$$

(3)

5.

Then h_1n(t) is renamed c₁, becoming the first IMF component.

6.

The residue is calculated, as shown in Equation (4):

$$r_1\left(t\right)=x\left(t\right)-c_1\left(t\right)$$

(4)

7.

The residue is used as the new input signal, and steps 1–6 are repeated until no more IMFs can be extracted, following the procedure specified by Equations (5) and (6):

$$c_k\left(t\right)=r_{k-1}\left(t\right)-m_k\left(t\right)$$

(5)

$$r_k\left(t\right)=r_{k-1}\left(t\right)-c_k\left(t\right)\left(r_0\left(t\right)=x\left(t\right)\right)$$

(6)

Finally, the input signal given x(t) is decomposed into a set of N_E components c_k(t) that are the IMFs and residue as shown in Equation (7):

$$x\left(t\right)= \sum _{k=1}^{N_E}{c}_k\left(t\right)+r_N\left(t\right)$$

(7)

As said in the Introduction, in this research, the bivariate EMD algorithm developed by Rilling et al. [29] is used to obtain IMFs.

Once IMFs are obtained from all input signals, a set of EMD-based parameters is computed, so for any extracted IMF, the average PSD and spectral power of each accelerometer are calculated, using Equation (8):

$$S^{C_k}\left(f\right)={\displaystyle \frac{∆t}{N}}{\left|\left.X^{C_k}\left(f\right)\right|\right.}^2; {PSD}^{C_k}=S^{C_k}(f)$$

(8)

where ∆t is the sample time, N is the number of data points in the signal, $X^{C_k}$(f) is the Fourier transform of the IMF $c_k\left(t\right)$, and ${PSD}^{C_k} \mathrm{or} S^{C_k}$ is the power spectral density of IMF $c_k\left(t\right)$.

Equation (9) is used to compute the average PSD of every set of IMFs:

$${\overline{PSD}}^{C_k}={\displaystyle \frac{\sum _{j=1}^{n_s}{PSD}_j^{C_k}}{n_s}}$$

(9)

where ${\overline{PSD}}^{C_k}$ is the average PSD of the processed IMF, ${PSD}^{C_k}$ is the power spectral density of the IMF c_k(t) extracted from the jth signal x(t), and n_s is the number of recorded signals.

Then, the spectral or signal power is computed. According to [17], the signal power P equals the integral (or sum, for the discrete case) over the frequency range of the general distribution function S or PSD, as shown in Equations (10) and (11):

$$P=\sum _{f=0}^{N-1}S\left(f\right)∆f=\sum _{f=0}^{N/2}{S}_{one}\left(f\right)∆f$$

(10)

$$S_{one}\left(f\right)=\left\{\begin{array}{c}2S\left(f\right)\to f=1\dots {\displaystyle \frac{N}{2}}-1\\ S\left(f\right)\to f=0, f={\displaystyle \frac{N}{2}}\end{array}\right.$$

(11)

where ∆f is the frequency interval, N is number of data points in the signal, and S(f) is the signal PSD.

3. Experimental System

To perform the project, a series of vibration signals were obtained from a railway axle, with the following conditions: (i) no faults and (ii) three levels of faults. Subsequent to the testing of the system, the vibration signals were obtained and stored for further processing using two common analysis techniques: signal envelope and power spectral density (PSD). Afterwards, the signal characteristics in the time and frequency domains were extracted using MATLAB^® R2024b software.

This project research is based on Empirical Mode Decomposition (EMD), so each vibration signal is decomposed into its significant components, the intrinsic mode functions (IMFs), using MATLAB^®, and for each IMF, for both the time and frequency domains, its characteristics were extracted.

Finally, decision tree algorithms were tested, with the aforementioned features being utilized as predictors. The subsequent details provide a comprehensive overview of this process.

3.1. Obtaining Vibration Signals: Equipment

A test bench for bogies was utilized in order to acquire vibration signals (see Figure 2). In this instance, a type Y-21 bogie was subjected to testing, with the rear axle disregarded in the context of travel direction.

The following points constitute the primary aspects of the test bench:

The loading system is operated by hydraulic cylinders. These are installed on the bogie over the front axle, which is the moving axle. The system is used to assess the movement of the axle using two rollers positioned on each of the axle wheels.

The arrangement of three uniaxial accelerometers was implemented to facilitate the measurement of acceleration in the three spatial directions: longitudinal, axial, and vertical. These components were installed on the axle box cover in order to acquire the vibration signals from the bearing and the wheelset.

The scheme shown in Figure 3 represents the directions of space in which the sensors are arranged and the axle defect determined by the depth e in mm.

3.2. Obtaining Vibration Signals: Experiment Procedures

The sensors used in this work were CMSS-RAIL-9100 piezoelectric accelerometers manufactured by SKF (AB SKF, Göteborg, Sweden) for industrial use, which are insensitive to electromagnetic fields and radiation. The measurements could be enabled under harsh conditions, and accelerometer details are listed below (see

Table 1. Accelerometer technical details.

Parameter	Value
Sensitivity (±20%)	10.2 mV (m/s2)
Acceleration range	±490 m/s2
Frequency range (±3 dB)	0.52 Hz to 8 kHz
Temperature range	−54 °C to 121 °C
Resonance frequency	25 kHz
Amplitude linearity	±1%
Transverse sensitivity	≤7%

).

Four type of test conditions, which were characterized by the parameter e, failure depth in mm, were arranged (see Figure 3, right scheme): undamaged axle (D0, 0.0 mm), damaged Axle Defect 1 (D1, 5.7 mm), Defect 2 (D2, 10.9 mm) and Defect 3 (D3, 15.0 mm).

Tests were performed at 50 km/h and 10 Tm (50% of maximum axle load), the sampling frequency was set at 12,800 Hz, and the sampling time was set at 1.28 s, with 16,384 data points per measure.

3.3. Testing Development

The database located in the MAQLAB Laboratory was managed by condition monitoring software that stored the acceleration and speed data received from the measurement system, as well as the date, the accelerometer identifier, and other useful information, in the database.

Once vibration signals were obtained, they were processed using the EMD technique so that each signal was decomposed into its IMFs using MATLAB^®, and then the following parameters were extracted for each IMF:

Time domain:

• RMS (RMS)
• Peak value (PK)
• Kurtosis (KUR)
• Crest factor (FCR)
• Standard deviation (DST)
• Skewness (SKW)
• Shape factor (SFR)

Frequency domain and PSD:

• RMS (RMS-f)
• Peak value (PF-f)
• Peak value frequency (FR)
• Spectral power (P)

One of the signals used in this work is represented in the time domain in Figure 4.

3.4. Signal Processing

As outlined in Section 3.3, the first step was to obtain the primary hyperparameters for each IMF. These parameters were subsequently used as predictors to train models using decision tree algorithms.

As described in Section 2, the processing of the vibration signals used the EMD technique, which ensures the decomposition of each signal into its constituent IMFs employing MATLAB^® software.

This process was completed by obtaining up to nine IMFs for each signal. It should be noted that some signals can be further decomposed into ten IMFs; however, for the purposes of this study, nine IMFs were used to preserve a homogeneous analysis.

The process of signal analysis is described below:

• For any given signal x(t), all IMFs are obtained.
• Up to 9 IMFs are obtained for every signal.
• The following groups are considered according to the IMF amount used in each case: 3 IMF, 4 IMF, 5 IMF, 6 IMF, 7 IMF, 8 IMF, and 9 IMF.
• For each IMF, the hyperparameters referred to were obtained.
• For each group, the model was trained.

During the process of EMD decomposing the signals into their IMF components, using MATLAB^® software, it was found that the maximum number of components common to all signals was nine, although some signals reached up to ten components. In this study, the first nine IMFs of all signals were considered to homogenize the study.

Figure 5 below illustrates a single signal, obtained from the time domain and decomposed into its nine IMFs, as previously described.

Decision trees were used in the present work using MATLAB^®, which provides three types, Fine Tree, Medium Tree, and Coarse Tree. Fine Tree was selected with characteristics described below:

• Gini index, to ensure that only one class exists on any node on every split.
• Up to 100 divisions.
• Cross validation: all gathered data were distributed over five packs, one used for training, one for analysis, and one for validation.

The training process was as follows:

• The model was trained for every hyperparameter. The obtained results were registered.
• The hyperparameters were combined in groups of two, three, and so on, until all hyperparameters were combined for each class, time and frequency, at each end of the axle.
• Different combinations of hyperparameters belonging to the two classes, time and frequency, were achieved by varying the composition of the constituent groups of hyperparameters undergoing training. This process was continued until all the hyperparameters of the two classes were used for each end of the axle.
• Steps 1 to 3 were performed by taking all the hyperparameters for the axis as a whole without distinguishing the ends.

After the generation of each training-related confusion matrix and ROC curve, subsequent analytical procedures could be initiated.

In this paper, we have considered the combinations of hyperparameters shown, selected because they have given significant results. These combinations are presented for comparison purposes, with the aim of assessing the effectiveness of the method as a whole.

4. Results

As outlined in the preceding sections, the algorithms have been subjected to rigorous testing. The effectiveness of each test has been documented, providing a comprehensive evaluation of the performance of the algorithms, where for each hyperparameter used as predictor, the model is tested for both axle ends, and each of the aforementioned hyperparameters is utilized as a predictor, thereby enabling the model to undergo testing for both the left and right axle ends, in accordance with the direction of travel. This comprehensive evaluation covers a total number of IMFs ranging from three to nine, ensuring a thorough assessment of the model’s capabilities. Any result is an effectiveness percentage.

The first tests have been performed with a single hyperparameter as a predictor. So far, no substantial results have been obtained, as most of the values do not reach 80% efficiency, both in time and frequency.

The initial approximation was conducted by integrating the hyperparameters that functioned as predictors and attained the best efficiency in each instance.

The first combination was performed by taking the predictors with the least effective results in the time domain, which are peak value (PK), skewness (SKW), and shape factor (SFR).

The findings from these experiments have not resulted in a substantial improvement in the efficiency of the model. Notably, the attained efficiency has remained below 70%, indicating a clear deficiency in the process.

The subsequent stage of the experiment was the combination of hyperparameters. The use of these metrics as predictors is indicated in the following efficiency results. These results were obtained by taking the RMS and kurtosis of those combinations that have been referred to. It should be noted that this was done individually for each combination.

As shown in

Table 2. Ratio of percentages obtained for the different IMFs, for combinations (a) and (b).

IMF	3		4		5		6		7		8		9
Side	L	R	L	R	L	R	L	R	L	R	L	R	L	R
c. (a)	69.4	68.6	74.0	78.3	94.1	76.3	89.9	74.2	93.3	75.1	95.5	83.5	95.5	86.9
c. (b)	69.7	70.7	69.4	75.1	73.9	71.0	69.9	72.4	78.6	71.2	86.6	74.4	87.4	76.0

, a range of percentages has been obtained for the various numbers of IMFs (three to nine) on either side of the bogie (left or right), utilizing two distinct combinations of hyperparameters. Each combination corresponds to the following hyperparameters:

(a)

RMS (RMS) and kurtosis (KUR)

(b)

RMS (RMS), kurtosis (KUR), crest factor (FCR), and standard deviation (DST).

• For RMS and kurtosis combined, the results are higher for higher numbers of IMFs used, mostly over 80% overall efficiency, although still inconclusive.
• For RMS, kurtosis, crest factor, and standard deviation, the results are in the 70% range mostly, although some are just below 70%.

The subsequent execution of the testing procedure involves the integration of the time and frequency hyperparameters as predictors, considering these combinations at both axle endings.

Following the structure set out in Table 2,

Table 3. Ratio of percentages obtained for the different IMFs, for combinations (c) and (d).

IMF	3		4		5		6		7		8		9
Side	L	R	L	R	L	R	L	R	L	R	L	R	L	R
c. (c)	84.2	85.8	81.8	87.7	80.1	76.4	79.6	79.2	80.6	96.7	84.6	78.4	84.5	79.1
c. (d)	83.0	83.9	82.8	88.4	91.7	78.6	86.2	78.5	91.1	80.8	95.1	84.5	95.5	87.3

shows the results from combinations of parameters from both the time and frequency domains:

(c)

RMS, kurtosis, standard deviation, RMS-f, peak value frequency (FR), and spectral Power (P).

(d)

RMS, kurtosis, crest factor, standard deviation, and all considered PSD parameters.

The results obtained demonstrate an overall effectiveness of more than 80%, with only a few exceptions where the value is slightly below this threshold; however, taken as a whole, they do not constitute significantly high values that can be considered acceptable for practical use.

The ensuing phases of the experiment were performed with all the hyperparameters hitherto considered in this work serving as predictors, initially for each end of the axle, both in the time and the frequency domains; the obtained results are shown in

Table 4. Model results. All hyperparameters combined.

IMF	3		4		5		6		7		8		9
Side	L	R	L	R	L	R	L	R	L	R	L	R	L	R
(t)	74.8	74.8	80.2	80.3	94.0	84.2	92.6	82.2	95.9	86.5	95.4	90.3	95.8	92.5
(fr)	81.2	84.3	76.4	85.4	67.7	71.2	69.9	75.1	64.3	68.3	59.6	63.1	57.4	61.4
(t + fr)	82.3	81.3	81.8	84.3	96.2	85.4	90.7	82.9	96.0	87.7	99.6	90.2	99.4	92.6

, where (t) represents hyperparameters in time domain, (fr) in the frequency domain, and (t + fr) all combined.

• The overall efficiency using all hyperparameters considered in this work ranges from 81.3% to as high as over 96%, so the results obtained in these trials are significantly superior; however, it should be noted that these results do not constitute conclusive evidence.

It is important to note at this point that the initial observation is considered in conjunction with the results obtained for the case of five IMFs. This is carried out in order to observe the confusion matrices that have been obtained at both ends: the model accuracy at the left axle end is 96.2% and at the right axle end is 85.4%; both confusion matrices are shown below (see Figure 6). As it was previously mentioned in Section 3.2, four conditions are defined: undamaged axle (D0, 0.0 mm), damaged Axle Defect 1 (D1, 5.7 mm), damaged Axle Defect 2 (D2, 10.9 mm), and damaged Axle Defect 3 (D3, 15.0 mm).

Subsequently, a series of tests were conducted, incorporating a range of combinations of hyperparameters as predictors. However, it is important to note that the distinction between the axle ends is not recognized during this process. Therefore, the combination of predictors was derived by merging the values from both ends of the axle (see

Table 5. Ratio of percentages obtained for the different IMFs, for combinations (I) to (III).

IMF	3	4	5	6	7	8	9
(I)	87.3	87.7	98.5	91.2	96.7	99.0	99.1
(II)	89.4	88.9	95.9	91.7	93.6	99.0	99.0
(III)	88.9	90.5	98.1	95.8	96.6	99.3	99.4

):

(I)

RMS (RMS), kurtosis (KUR), crest factor (FCR), and standard deviation (DST);

(II)

RMS (RMS), kurtosis (KUR), standard deviation (DST), RMS (PSD), peak value frequency (FR), and power (P);

(III)

RMS (RMS), kurtosis (KUR), crest factor (FCR), and standard deviation (DST) plus all PSD parameters.

• Considering RMS, kurtosis, crest factor, and standard deviation, efficiency improves with an increase in the number of IMFs considered for the study, reaching 99% efficiency.
• The subsequent case examined in this study employs the root mean square (RMS) in both the time and frequency domains, alongside kurtosis, standard deviation, peak value frequency, and power. The outcomes of this analysis are analogous to those previously obtained.
• It is evident from the findings of this study that the results obtained from the hyperparameters in the frequency domain, which include RMS, kurtosis, crest factor, and standard deviation, are marginally higher. The efficiency levels recorded were 88.9% at the lowest for three IMFs, with all others exceeding 90%.

The final stage of the model training process involves the consideration of all the hyperparameters utilized in this study. In this stage, values of both axle ends are taken into account, and results are gathered and displayed in

Table 6. Model results. Global test.

IMF	3	4	5	6	7	8	9
c. global	87.8	90.4	99.3	99.2	99.6	99.7	99.8

below.

Subsequent to the conclusion of the final test, it became apparent that the outcomes demonstrated exceedingly elevated figures. As demonstrated in Figure 1, at the lowest number of IMFs, an efficiency approaching 90% was attained, which signifies a high success rate, as illustrated in Figure 7 below.

For comparison purposes, Figure 8 shows confusion matrices for the cases of five IMFs and nine IMFs, which both hold very similar accuracy.

The confusion matrices shown in Figure 8 above graphically show the approximation of the model’s efficiency for both cases considered (five and nine IMFs). The values of precision (see Figure 9) and accuracy (see Figure 10) for both cases are shown below so that both models can be compared. The ROC curves associated with both cases are also presented.

For the purpose of comparison, the precision and accuracy values for the analyses conducted for five and nine IMFs are presented. On the one hand, it is evident that the precision of the model exceeds 99% in both cases, with a marginal increase observed in the case of nine IMFs.

Finally, the ROC curves associated with each of the two cases compared, for five and nine IMFs, are plotted, showing that, in both cases, the results are close to 100% overall effectiveness (see Figure 11).

5. Conclusions

In an analysis of the results obtained from training decision tree algorithms using vibration signals decomposed by the Empirical Mode Decomposition (EMD) technique into the so-called intrinsic mode function (IMF), it can be concluded that the resulting technique is effective enough to characterize rolling stock equipment, such as axles, which were used for testing in this work.

The first approach was carried out considering the first three IMFs so that a comprehensive analysis could be performed; using only the first two IMFs was discarded as those two subsignals could be considered too close to the original signal, so the first three IMFs were selected as the starting point, taking that consideration into account.

Despite the fact that the 87.8% model accuracy rate does not meet the necessary criteria for practical considerations, this value could potentially be considered for further analysis. The classification of a large number of elements, as shown in the related confusion matrix, is indicative of an accurate and precise analysis (see Figure 7).

When operating a system based on five IMFs, the model accuracy escalates to 99.3% which means that it is a highly reliable and accurate classification methodology; the accuracy ranges from 99.3% for five IMFs to 99.8% for nine IMFs.

Considering how similar the accuracy values of both classes are, we could make some considerations to meet any requirement:

• A classification model based on the analysis of the first five IMFs is sufficiently accurate for operational use and is therefore open to further development. For rapid response, this technique is currently recommended.
• The nine-IMF case offers higher accuracy, so it is recommended for high-precision use. However, the close accuracy of the other classes analyzed in this work could mean that a lower computing effort could be recommended to obtain a high-precision model with less computational expense.

Further consideration of the requirements for the best model should be given, taking into account not only the model results but also the preparation and computation timing costs. However, the high accuracy obtained by the five-IMF class could be considered without any other considerations.

On the basis of this study, further work could be carried out, taking into account other combinations of IMFs, the application of the methodology using different signal types and axle conditions, and tests under operational conditions of rolling stock.