Characterization of High-Speed Steels—Experimental Data and Their Evaluation Supported by Machine Learning Algorithms

Abstract

X-ray diffractograms of high-speed steels are analyzed using machine learning algorithms to accurately classify various heat treatments. These differently heat-treated steel samples are also investigated by dilatometric analysis and by metallographic analysis in order to label the samples accordingly. Both agglomerative hierarchical clustering and t-distributed stochastic neighbor embedding are employed to automatically classify preprocessed X-ray datasets. The clusters obtained by this procedure agree well with the labeled data. By supervised learning via a support vector machine, hyperplanes are constructed that allow separating the clusters from each other based on the X-ray measurements. The exactness of these hyperplanes is analyzed by cross-validation. The machine learning algorithms used in this work are valuable tools to separate different microstructures based on their diffractograms. It is demonstrated that the separation of martensitic, bainitic, and pearlitic microstructures is possible based on the diffractograms only by means of machine learning algorithms, while the same problem is error-prone when looking at the diffractograms only.

1. Introduction

Heat treatment plays a crucial role in determining the microstructure and thermomechanical properties of materials. This is particularly important for high-speed steels (HSS), as they require precise heat treatment to achieve desirable characteristics such as high red hardness, toughness, and exceptional cutting performance. High-speed steels are widely used in various industries, including the production of high-performance saws, drilling tools, and machining tools [1]. Microstructural changes as they occur during austenitization and subsequent cooling of high-speed steels are investigated by dilatometry, metallography, and X-ray diffraction (XRD). Dilatometry is a powerful experimental technique for characterizing solid–solid phase transformations in steels, as it enables the in situ observation of phase transformations by measuring changes in length during cooling.

XRD is generally used to investigate the crystal structure and phase composition. It is worth remarking that XRD is a complementary technique compared with optical microscopy and dilatometry. For instance, determining the accurate amount of austenite by metallographic investigations frequently comes with unacceptably large errors. However, the amount of austenite is deducible from XRD measurements. It is difficult to distinguish between the amounts of martensite, bainite, and ferrite from XRD measurements in a quantitative manner when evaluated by the Rietveld method [2].

However, distinguishing between these body-centered phases is possible by metallographic investigations and dilatometry [2], ultimately leading to clear experimental findings. However, in this study, the dilatometer is used solely for heat treatment, and phase labeling is based exclusively on metallographic results. A comprehensive examination of the final microstructures is thereby provided.

It is concluded that by combining X-ray diffraction (XRD), metallography, and possibly dilatometry, a robust approach for analyzing microstructural changes can be applied. This enables the identification and characterization of different phases and microstructural properties in high-speed steels. Further details can be found in [2].

Machine learning is increasingly being recognized as a powerful tool in materials research, enabling the extraction of quantitative relationships between atomic-scale structures, which are influenced by heat treatment and material properties; see e.g., [3]. We investigate in this work whether a separation of martensitic, bainitic, and pearlitic microstructures based on diffractograms is possible using machine learning algorithms. To this end, we explore agglomerative hierarchical clustering and t-SNE as representatives of unsupervised learning algorithms, along with supervised learning through a Support Vector Machine. Through the application of an unsupervised hierarchical clustering algorithm, it becomes possible to compare diffractograms resulting from different heat treatments by a certain metric. Another unsupervised learning approach is t-distributed stochastic neighbor embedding (t-SNE) [4,5]. Its strength is to reduce a high-dimensional dataset to, e.g., two dimensions only. In this two-dimensional space, similar diffractograms are located in the neighborhood, whereas diffractograms from strongly different microstructures are far apart. It is presented in this paper how similarities and differences in the microstructure influence the diffractograms by means of these unsupervised machine learning methods.

Additionally, employing support vector machines enables the classification of diffractograms based on their microstructures and phase quantities. Principally, all these methods allow the comparison of unknown diffractograms with those from known heat treatments, thereby enabling the accurate characterization of the material. By analyzing diffractograms with these machine learning tools, different microstructures such as martensite, bainite, and pearlite can be separated based on the diffractograms alone, whereas an accurate assignment of these phases by only analyzing the diffractograms with the Rietveld method is difficult and prone to errors. We will demonstrate that even hierarchical clustering, as an unsupervised learning algorithm, can effectively cluster crystallographic microstructures from the diffractograms. In addition, separating these clusters is possible with supervised learning, in particular with the method of support vector machines.

2. Experimental

2.1. Materials

The experiments are conducted using high-speed steel samples (HS1-5-1-8, Böhler Edelstahl Grade S504SF, voestalpine Böhler Edelstahl, Kapfenberg, Austria) produced by means of an electric arc furnace (EAF). The investigated samples are cylinders with a diameter of $5\mathrm{mm}$ and a length of $10\mathrm{mm}$.

The measured chemical composition of the samples is provided in

Table 1. Chemical composition for a conventional melted high-speed steel HS1-5-1-8.

Component	C	Cr	Mo	W	V	Co	Si	Mn	Ni	Fe
Mass fraction in %	$0.8$	$4.0$	$4.9$	$1.0$	$1.0$	$7.9$	$0.3$	$0.2$	$0.2$	bal.

.

The samples undergo a soft annealing process at a temperature of 700 °C for several hours, preparing them for subsequent dilatometer experiments.

2.2. Dilatometry and Heat Treatment

Soft-annealed high-speed steel samples (Böhler grade S504 SF) are heat treated using a quenching dilatometer through induction heating. The samples are heated with a constant heating rate of $9.6\mathrm{K}{\mathrm{s}}^{-1}$ starting from room temperature and reaching a maximum temperature of ${\vartheta }_{\max }$ = 1170 °C. Once the maximum temperature ${\vartheta }_{\max }$ is reached, it is held at this maximum temperature for a duration of 300 s. Subsequently, the samples undergo an exponential cooling process. The cooling is characterized by the $t_{8/5}$ time, i.e., the time interval where the sample temperature is between 800 °C and 500 °C. This parameter $t_{8/5}$ is varied across a wide range, from $40\mathrm{s}$ to $40\mathrm{ks}$ (approximately $11.1\mathrm{h}$).

The microstructure consists of pure martensite or martensite–bainite or bainite or bainite–pearlite or almost pure pearlite depending on the applied cooling rates.

The samples obtained after the dilatometer experiments, during which the specified heat treatment is applied, are further examined using X-ray diffraction (XRD) and metallographic analysis. Microstructure analysis is conducted using light microscopy (LIMI). Light microscopic images of the dilatometer samples are then taken using an Olympus BX3M (Olympus Corporation, Tokyo, Japan) to determine the phase composition. Representative microstructures from LIMI are presented in Figure 1. Prior austenite grain size is about 55 µm. The width of the martensitic laths is in the order of 0.5 µm, and the lateral length is in the order of a few µm.

Both characterization techniques (XRD and LIMI) enable the observation of various properties, including the crystal structure, crystal defects, phase composition, and microstructural features.

2.3. X-Ray Diffraction Measurements

The X-ray diffraction (XRD) experiments are conducted using a parallel beam geometry. The required mirror for parallel beam geometry suppresses K$\beta$. As a result, the emission profile contains only K${\alpha }_1$ and K${\alpha }_2$. A “D8 Advance diffractometer” (Bruker axs, Karlsruhe, Germany) equipped with a position-sensitive detector (PSD) is employed to realize a counting time for one diffractogram within 30 min.

The diffraction patterns are acquired with Cr-K$\alpha$ radiation and not with Cu-K$\alpha$ radiation. This choice is made to avoid strong fluorescence of ferrous samples, which occurs when exposed to Cu-K$\alpha$ radiation. The high attenuation length of the Cr-K$\alpha$ radiation compared with Cu-K$\alpha$ radiation is another advantage of the Cr-K$\alpha$ source.

The diffractograms are recorded over a $2\theta$ range from $2{\theta }_{\mathrm{start}}=49.{0000}^{\circ }$ (corresponding to a $d_{\max }$ of 2.761 Å) to $2{\theta }_{\mathrm{end}}=145.{3942}^{\circ }$ (corresponding to a $d_{\min }$ of 1.199 Å) with a step size of $\mathsf{\Delta }2\theta =0.{0354}^{\circ }$. Further details can be found in [2,6].

2.4. Quantification of Phases

X-ray diffractograms are analyzed by the Rietveld method via a fundamental parameter model using Topas 3.0 of Bruker axs [7]. Measurements are conducted across the complete $2\theta$ range using LaB₆ (NIST 660a) as a reference material to determine and account for the instrument-caused peak broadening. Furthermore, the Double-Voigt model is used to differentiate between the size and strain components of defects found in the high-speed steel samples. A detailed description of this model can be found in [8]. An extensive Rietveld analysis of these samples can be found in [2]. The quantification of retained austenite and carbides (${\mathrm{M}}_2\mathrm{C}$, $\mathrm{MC}$), and possibly trace amounts of ${\mathrm{M}}_6\mathrm{C}$, is also achieved using the Rietveld method. The mass fraction of austenite is presented as a function of the $t_{8/5}$ time in

Table 2. Retained austenite after cooling as a function of $t_{8/5}$—cooling time for the steel HS1-5-1-8.

Number	${\mathit{t}}_{8/5}$ Time [s]	Mass Fraction of Austenite · 100
1	40	$13.1$
2	110	$16.6$
3	310	$24.8$
4	500	$24.4$
5	800	$26.5$
6	2300	$26.3$
7	6500	$29.4$
8	9000	$10.3$
9	18,000	0
10	40,000	0

.

By means of the Rietveld method alone, it is difficult, if not impossible, to distinguish between martensite, bainite, and ferrite in pearlite due to their similar crystallographic structures and only slight variations in carbon content. Complementary methods to distinguish between these phases are the microstructural analysis by LIMI and dilatometric measurements. In the case of the LIMI method, differential etching using picric acid is applied. Different phases are etched to different degrees, and the resulting contrasts become visible during the LIMI investigation. Details about this method can be found in [9]. Dilatometry signals are evaluated on the basis of known thermal expansion coefficients of the face-centered austenite and of all body-centered phases, respectively. After austenitization, the slope of the temperature-dependent dilatometer curve is defined by the thermal expansion coefficient of austenite. Nearly identical coefficients of thermal expansion for the body-centered microstructures martensite, bainite, and ferrite can be assumed. However, martensite, bainite, and ferrite in pearlite are distinguishable due to different temperatures of transformation start and transformation end. This information is provided by the dilatometer experiment, which is evaluated in the form of a Time Temperature Transformation (TTT) diagram to be compared with TTT diagrams from literature. Verification of the presence of a phase is then conducted by LIMI analysis. Thereby, the temperature-dependent phase fractions of retained austenite, martensite, bainite, and ferrite within pearlite are determined. A detailed description and results of the complete phase quantification can be found in [2]. It is shown that the phase fraction determined by both micrographs and dilatometry gives the same results within the error margin. The sum of the volume fractions of martensite, bainite, and pearlite must equal unity. The volume fractions of martensite, bainite, and pearlite, as taken from [2], are presented in

Table 3. Relative volume fractions of martensite, bainite, and pearlite as a function of $t_{8/5}$ cooling time for HS1-5-1-8, including labeling.

Number	${\mathit{t}}_{8/5}$ time [s]	Volume Fraction of Martensite · 100 (Label)	Volume Fraction of Bainite · 100 (Label)	Volume Fraction of Pearlite · 100 (Label)
1 (martensite)	40	$100(1)$	$0(-1)$	$0(-1)$
2 (martensite)	110	$100(1)$	$0(-1)$	$0(-1)$
3 (martensite)	310	$93(1)$	$7(-1)$	$0(-1)$
4 (martensite)	500	$62(1)$	$38(-1)$	$0(-1)$
5 (bainite)	800	$40(-1)$	$60(1)$	$0(-1)$
6 (bainite)	2300	$0(-1)$	$100(1)$	$0(-1)$
7 (bainite)	6500	$0(-1)$	$95(1)$	$5(-1)$
8 (bainite)	9000	$0(-1)$	$70(1)$	$30(-1)$
9 (pearlite)	18,000	$0(-1)$	$29(-1)$	$71(1)$
10 (pearlite)	40,000	$0(-1)$	$6(-1)$	$94(1)$

. For improved clarity, the results were obtained solely from etched micrographs. It is worth noting that the output of the micrographs corresponds to areas that align with the volume fractions (see Table 3). The dilatometry results, which present changes in length, are proportional to one-third of the volume changes. For fast cooling (black curve $t_{8/5}=110\mathrm{s}$), martensitic transformation occurs. Already for the second curve ($t_{8/5}=300\mathrm{s}$), we see that there is a bainite transformation before the onset of the martensitic transformation. For $t_{8/5}=800\mathrm{s}$, more bainite occurs before martensite. Austenite transforms to upper and lower bainite for $t_{8/5}=2300\mathrm{s}$ (blue curve). An increasing amount of pearlite is detected for $t_{8/5}=6500\mathrm{s}$ and $t_{8/5}=$ 18,000$\mathrm{s}$ (orange and orange-dotted curve).

The samples (1–10) listed in Table 3 have to be assigned to their classes. The samples with numbers (1–4) belong to class “martensite”, numbers (5–8) belong to class “bainite”, and samples (9–10) belong to “pearlite”. All samples are labeled by the following scheme called the one against all strategy [10]: The material is allocated to its class by the highest volume fraction of martensite or bainite or pearlite. The material gets “1” for this class and “−1” for the other two classes. For example, in sample 5, the volume fraction of bainite, which belongs to the bainite class due to the highest volume fraction of bainite, gets “−1” in the martensite column, “1” in the bainite column, and “−1” in the pearlite column. Each sample can only belong to one class.

The phase fractions determined by dilatometry, see Figure 2, are in good agreement with those obtained from the micrographs. The phase fractions in Table 3 are derived from the different colors of the etched phases. For labeling the data, the phase fractions obtained from LIMI are used.

The differentiation of the individual microstructures, martensite, bainite, and pearlite can be done by applying machine learning algorithms on the diffractograms of the differently heat-treated samples, as will be shown in this work. For the sake of simplicity, the focus will be solely on the phase fractions as a distinctive feature.

3. Machine Learning Algorithms

Three main types of machine learning algorithms are commonly used in practical applications: unsupervised learning, supervised learning, and reinforcement learning; details can be found, e.g., in the following textbooks [11,12,13]. In the following, the principles of unsupervised and supervised learning are briefly discussed as these techniques are used in this work:

• Unsupervised learning algorithms do not require labeled data for training. Instead, their objective is to discover patterns and structures within the data by clustering similar data points and identifying relationships between them; details can e.g., be found in [13]. The t-SNE algorithm used is described in Appendix A. Details of the agglomerative hierarchical clustering algorithm are presented in Appendix B.
• Supervised learning algorithms utilize labeled datasets, where each data point is assigned a label standing for a category. These labeled examples consist of input data along with their corresponding desired outputs. Through regression techniques, supervised learning algorithms analyze the labeled data to find the best-fitting model. Once the model is trained, it can be used to classify and assign new data to their categories [13]. Detailed information about the support vector machine can be found in Appendix C. The robustness of the support vector machine results is analyzed through cross-validation. The method is presented in Appendix D.

However, before the machine learning algorithms come into play, the experimental data have to be prepared and preprocessed as described in the next section.

3.1. Data Preparation and Preprocessing

Data preparation plays a crucial role in the analysis of diffractograms by ensuring the suitability of the data for clustering. To this aim, interfering properties such as variations in irradiated specimen surfaces, measurement time per step, and total X-ray tube intensity have to be taken into account.

Various methods are proposed in the literature for data preparation [14,15] with two main steps: normalization and mean subtraction. Some commonly used normalization methods are as follows:

• Min-max scaling [16]: This method rescales the data to a specific range, typically between 0 and 1, by subtracting the minimum value and dividing by the range.
• Z-score standardization [17]: This method transforms the data to have a mean of 0 and a standard deviation of 1. This is achieved by subtracting the mean from the original data. Division of the standard deviation of the subtracted data results in Z-score standardization.
• Power transformation [18]: This method adjusts the data distribution to conform more closely to a desired distribution or meet certain statistical assumptions.
• An alternative method for normalizing diffractograms involves dividing each diffractogram by its average value. In the case of a Principal Component Analysis (PCA) analysis, the mean value is subtracted from the normalized data. This newly introduced area-normalized method is related to the Rietveld method, where the quantitative results are normalized by the overall peak areas [19]. This area-normalized method is particularly recommended when the diffractograms undergo PCA for dimensionality reduction, as will be explained in the next subchapter. This reduces the impact of the mentioned problems, such as different irradiated sample areas, variations in the tube intensity, and different measuring times per step for different diffractograms.

The prepared data can then be used for dimensionality reduction, e.g., by PCA.

3.2. Dimensionality Reduction by PCA

Principal Component Analysis (PCA) is a widely used technique for dimensionality reduction in a dataset X.

In many cases, datasets, e.g., diffractograms, contain highly correlated features (e.g., two very similar austenite peaks). To address this issue and improve computational efficiency, dimensionality reduction methods are applied. Reducing the dimensionality not only enhances computational efficiency but also improves the predictive power of models by eliminating irrelevant features such as noise or background.

Several algorithms are available for computing principal components; frequently, Singular Value Decomposition (SVD) is used to execute PCA [20].

The output of PCA consists of two transformation matrices, namely the coefficient matrix W and the scores matrix U (details can be found in [21]):

• The orthogonal coefficient matrix W, also known as the loading matrix, represents the weights or coefficients that define the principal components in terms of the original variables.
• The orthogonal scores matrix U represents the original data expressed in the new principal component space (the rotated coordinate system).

The diffractogram is transformed by linearly mapping the data vector X via W into the transformation vector T containing the uncorrelated variables.

$$\mathit{T}=\mathit{X}\mathit{W}$$

(1)

The transformed data T exhibit the following favorable characteristics:

• The variables become uncorrelated.
• The first few principal components, consisting of data in the rotated coordinate system (scores) and their weights (coefficients or loadings), capture the most crucial information, while the remaining components represent less significant variations or noise. The importance of individual components can be determined by the absolute values of their corresponding eigenvalues. Higher-order principal components can often be disregarded, leading to reduced dimensionality. The lost information corresponds to the sum of the ignored absolute eigenvalues in relation to the sum of all absolute eigenvalues.
• The original dataset can be reconstructed by multiplying T with the inverted matrix ${\mathit{W}}^{-1}$.

Only the first L entries of the principal components in the vector T are significant, resulting in the approximation:

$${\mathit{T}}_L=\mathit{X}{\mathit{W}}_L$$

(2)

where only the first L columns contain non-zero entries.

4. Results and Discussion

4.1. Preprocessing of Diffractogram Data

The following figures depict typical diffractograms of the three different microstructures, martensite, bainite, and pearlite. The diffractogram of a martensitic microstructure is presented in Figure 3, the diffractogram of a bainitic microstructure in Figure 4, and the diffractogram of a pearlitic microstructure in Figure 5. The different $t_{8/5}$ time characterizes the cooling rate, which is responsible for the evolution of the different microstructures.

The widths of the martensite peaks are determined by the cooling rate-dependent amount of carbon dissolved in tetrahedral sites. In addition, the cooling rate-dependent dislocation density also influences the peak widths. The austenite peaks depend on the interstitially dissolved amount of carbon in austenite. Details can be found in [2,22,23].

Diffractogram data are normalized by its average intensity value (area-weighted normalization), and the mean value is subtracted. Subsequently, PCA is performed on the normalized data. By means of principal component analysis, a system of transformed diffractograms (coefficients) is generated, from which the existing diffractograms can be reconstructed. The color-coded points indicate how much of each principal component (PCA1 or PCA2 or PCA3) is needed to represent the given diffractogram. The PCA scores (PCA1, PCA2, and PCA3) are prefactors that are multiplied with the coefficients. Figure 6 shows a section of this system. Principal component coefficients, or loadings, are the “weights” of the diffractograms. The first three coefficients (loadings) of PCA and the last coefficient (tenth) of PCA are presented in Figure 6. Whereas the first coefficient plotted versus $2\theta$ contains a high amount of information, and the second and third coefficients are also required, the tenth coefficient contains only random noise.

The scores of the principal component coefficients are listed in

Table 4. First three of the ten scores of the principal component coefficients (PCA1, PCA2, and PCA3) for the 10 diffractograms with increasing $t_{8/5}$ time.

Number	${\mathit{t}}_{8/5}/\mathbf{s}$	PCA1	PCA2	PCA3
1 (martensite)	40	0.1505	0.4366	−0.3775
2 (martensite)	110	0.1740	0.4370	−0.2785
3 (martensite)	310	0.1487	0.3880	−0.1940
4 (martensite)	500	0.2022	0.3698	0.0199
5 (bainite)	800	0.1785	0.1939	0.2282
6 (bainite)	$2.3·{10}^3$	0.2029	0.1201	0.4385
7 (bainite)	$6.5·{10}^3$	0.2820	0.1762	0.4772
8 (bainite)	$9.0·{10}^3$	0.2785	0.0470	0.3152
9 (pearlite)	$18·{10}^3$	0.5609	−0.2833	0.0319
10 (pearlite)	$40·{10}^3$	0.5825	−0.4070	−0.4118

.

The principal component scores from Table 4 are illustrated in Figure 7. The color-coded points in Figure 7 are scaled by $t_{8/5}$ with the color bar shown on the right side.

From the eigenvalues follows that the first component accounts for 80.0% of the variance, the second component explains 16.8%, and the third component captures 2.3%.

In other words, the 10-dimensional problem (10 diffractograms are used) can be reduced to 2 or 3. Principal components higher than 3 do not contain useful information. In the next section, the preprocessed data are analyzed by unsupervised (e.g., hierarchical clustering and t-SNE) and supervised machine (e.g., support vector machine) learning algorithms.

4.2. Machine Learning Algorithm Applied to XRD Data

4.2.1. Agglomerative Hierarchical Clustering of XRD Data

The analysis of the area-normalized diffractograms by hierarchical clustering results in a dendrogram presented in Figure 8. The distance measure is based on the Euclidean norm. Thereby, the dissimilarity of the diffractograms is computed. As a linkage method, the “ward” method [24] is used for choosing the pair of clusters to merge at each step. The linkage is calculated by the recursive Lance–Williams algorithms [25].

According to the dendrogram in Figure 8, the two samples (samples 1 and 2) with fully martensitic composition exhibit the highest similarity. Samples 3 and 4, which are also predominantly martensitic, cluster closely together. The predominantly bainitic samples (samples 5, 6, 7, and 8) are clustered together. The predominantly pearlitic samples (9 and 10) form a distinct class. In summary, the results obtained from the area-normalized diffractograms show a pattern that clustered together similar microstructures.

Other normalization methods do not yield satisfactory results. An example using Z-score normalization is shown in Figure 9. It is, e.g., evident that samples 2 and 3 have a more different microstructure than samples 2 and 1, but 2 and 3 are clustered together. Further, samples 5 and 7 as well as 6 and 8 are clustered, where a clustering of 5 with 6 and 7 with 8 would better represent reality.

Thus, area normalization is superior to other normalization methods (e.g., Z-score normalization as shown here). The reasons are as follows: Area normalization works since the relative amounts of the phase fractions are considered by this normalization technique. In other words, the areas of the peaks—which are proportional to the phase fractions—are characteristic features of the diffractograms and are not corrupted by this normalization technique. Differences in the diffractograms, including peak widths and peak heights, are analyzed by the Euclidean norm.

In comparing the results of hierarchical clustering using area-normalized data and PCA-transformed data, they are presented in Figure 10. The sorting appears completely random when the complete transformed data are used to construct the dendrogram. In this figure, the algorithm’s challenges with noise become apparent.

An illustrative example involving three factor loadings is presented in Figure 11. As mentioned earlier, the third PCA component contains minimal additional information, representing only 2.3% of the variance. It’s worth noting that this component carries the same weight as the first component, which accounts for 80.0% of the variance. However, the third component also exhibits a significant level of noise, as evident in Figure 6c.

However, when hierarchical clustering is performed using only the first two factor loadings, the resulting dendrogram (Figure 12) is similar to the dendrogram obtained from the normalized XRD data in Figure 8.

Increasing the dimensionality in this manner negatively impacts the performance of the hierarchical clustering algorithm. Therefore, it is advisable to perform hierarchical clustering on the normalized diffractograms directly.

In summary, hierarchical clustering yields good results for area-weighted diffractograms, but analyzing PCA data depends significantly on the number of factor loadings considered. As it is difficult to predict how many factor loadings need to be taken into account for a good result, this method is not recommended.

4.2.2. t-SNE Analysis of XRD Data

The area-normalized diffractograms are used to visualize similarity data using the t-SNE method. This approach retains the local structure of the data while also revealing significant global structures such as clusters [5].

Any new dataset should be analyzed with corresponding labeled measurements using t-SNE. Depending on where the new dataset is located afterwards, the existing microstructure can be determined from the diffractogram alone. As demonstrated in Figure 13, martensite, bainite, and pearlite are reasonably separated into different clusters. A summary of the labels can also be found in Table 3.

A certain drawback of this method is that with every analysis, the results are slightly different (e.g., the mapping is rotated). However, this will not compromise the principal outcome.

4.2.3. Support Vector Machines to Separate Phases by Hyperplanes

In this subchapter, the application of support vector machines (SVM) to the task of separating phases through hyperplane classification is explored. The hinge loss function serves as the chosen objective function in this maximum-margin SVM. Raw data from the measurements have to be preprocessed, including a PCA transformation as a prerequisite for the SVM model training. The one-against-all strategy enables the classification of samples into their respective classes based on the decision boundaries determined by the SVM model. Here three distinct microstructural conditions (“martensite”, ”bainite”, and “pearlite”) are separated by means of support vector machines (SVM). The classifications (labels) are taken from Table 4.

Figure 14, Figure 15 and Figure 16 are presented to visualize the hyperplanes in relation to the three classes, demonstrating the selectivity of the SVM model.

It has been established that separation is achievable without any difficulties.

The robustness of the SVM algorithm is determined by the Leave-One-Out Cross-Validation (LOOCV). First, the distances to the hyperplane are computed for all ten datasets. Then, the hyperplane is calculated with one dataset left out. The distance of this altered hyperplane to the left-out dataset is determined. Thereby, a measure for the uncertainty of the hyperplane is computed. The individual altered hyperplanes are presented in Figure 17, Figure 18 and Figure 19.

When the distance of the dataset to the hyperplane is given with a positive sign, then the dataset belongs to the class that is separated from the other classes by the hyperplane. On the contrary, a negative value indicates it belongs to the rest. The distances to the altered hyperplanes with martensite separated from the rest are presented in

Table 5. Calculation distance to hyperplane of martensite to rest by Leave-One-Out Cross-Validation (LOOCV) compared with all data points as a function of $t_{8/5}$ cooling time for HS1-5-1-8.

Number	${\mathit{t}}_{8/5}$ Time [s]	Distance from Hyperplane Calculated from All Points	Distance from Hyperplane Leave-One-Out Cross-Validation
1 (martensite)	40	0.4644	0.4634
2 (martensite)	110	0.3929	0.3930
3 (martensite)	310	0.3018	0.3012
4 (martensite)	500	0.1337	0.0384
5 (bainite)	800	−0.1340	−0.0321
6 (bainite)	2300	−0.3359	−0.3362
7 (bainite)	6500	−0.3314	−0.3314
8 (bainite)	9000	−0.3090	−0.3094
9 (pearlite)	18000	−0.3731	−0.3730
10 (pearlite)	40000	−0.1521	−0.0195

. As evident from Table 5 and Figure 17, the distance of the omitted data point from the hyperplane barely changes, indicating strong robustness.

In the following, the hyperplane that separates bainite from the martensite and pearlite is investigated by the LOOCV technique. The altered hyperplanes are presented in Figure 18. Not all datasets are correctly classified after using the LOOCV test. Datasets 4 and 9 are misclassified, as can be seen in

Table 6. Calculation distance to hyperplane of bainite to rest by Leave-One-Out Cross-Validation (LOOCV) compared with all data points as a function of $t_{8/5}$ cooling time for HS1-5-1-8.

Number	${\mathit{t}}_{8/5}$ Time [s]	Distance from Hyperplane Calculated from All Points	Distance from Hyperplane Leave-One-Out Cross-Validation
1 (martensite)	40	−0.4829	−0.4831
2 (martensite)	110	−0.3969	−0.3969
3 (martensite)	310	−0.3025	−0.3014
4 (martensite)	500	−0.1147	0.0032 misclassified
5 (bainite)	800	0.1146	0.0740
6 (bainite)	2300	0.3161	0.3167
7 (bainite)	6500	0.3202	0.3172
8 (bainite)	9000	0.1880	0.1875
9 (pearlite)	18000	−0.1153	0.1719 misclassified
10 (pearlite)	40000	−0.5211	−0.5198

. As evident from Table 6 and Figure 18, the distance of the omitted data point from the hyperplane noticeably changes, indicating weak robustness. The hyperplane is nearly parallel to PCA 3. That means that changes in this hyperplane are changes in PCA 3. As mentioned earlier, PCA 3 only accounts for 2.3% of the variance.

The variation of the hyperplane that separates pearlite from the other classes is calculated according to the LOOCV test. The different hyperplanes due to leaving out a distinct dataset are presented in Figure 19.

All datasets are classified correctly for all altered hyperplanes in case pearlite is to be separated from all other classes. The distances of the datasets to the original and to the altered hyperplanes have the same sign for all cases (

Table 7. Calculation distance to hyperplane of pearlite to rest by Leave-One-Out Cross-Validation (LOOCV) compared with all data points as a function of $t_{8/5}$ cooling time for HS1-5-1-8.

Number	${\mathit{t}}_{8/5}$ Time [s]	Distance from Hyperplane Calculated from All Points	Distance from Hyperplane Leave-One-Out Cross-Validation
1 (martensite)	40	−0.2955	−0.2729
2 (martensite)	110	−0.3299	−0.3300
3 (martensite)	310	−0.3431	−0.3424
4 (martensite)	500	−0.4018	−0.4018
5 (bainite)	800	−0.3744	−0.3734
6 (bainite)	2300	−0.4028	−0.4015
7 (bainite)	6500	−0.4262	−0.4256
8 (bainite)	9000	−0.2560	−0.1816
9 (pearlite)	18000	0.2552	0.0868
10 (pearlite)	40000	0.5612	0.5750

). Figure 19 clearly shows that the distance of the omitted dataset from the hyperplane exhibits minimal change, underscoring its strong robustness.

It can be stated that the hyperplanes separating martensite from the rest (bainite and pearlite) and pearlite from the rest (martensite and bainite) have been found to be highly robust during cross-validation. The hyperplane that separates bainite from the rest (martensite and pearlite) appears to be unstable and heavily influenced by individual data points.

5. Conclusions and Outlook

In this study, we conducted a comprehensive analysis of microstructure evolution in high-speed steels by employing materials characterization techniques and machine learning algorithms. Data normalization was crucial to ensure the appropriate application of the machine learning algorithms.

Standard normalization methods such as “Z-Score” or “Min-max” do not provide satisfactory results for the diffraction data analyzed in this work. However, “area normalization”, which is related to data preparation in the Rietveld method, works well.

The main results of this work are summarized as follows:

• After “area-normalization” of the diffraction data, the different material classes “martensite”, “bainite”, and “pearlite” are successfully grouped by agglomerative hierarchical clustering as well as by the t-SNE method.
• PCA is an effective method to eliminate irrelevant information and serves as a prerequisite for the setup of SVMs. Problems occurred using PCA-transformed data in hierarchical clustering because the algorithm is sensitive to noise.
• By means of support vector machines (SVM), different microstructures in steels can be distinguished on the basis of diffractograms only. To maximize SVM effectiveness, the data must be transformed using principal component analysis (PCA).
• LOOCV serves as an appropriate method for assessing the accuracy of the hyperplanes generated by SVM.

The methodology presented in this work appears particularly well suited for in situ observations, where a large number of XRD diffractograms are available. In such cases, the approach can efficiently cluster similar diffractograms.

It is expected that machine learning algorithms investigated in this work on a small scale are applied to monitor the evolution of the microstructure during materials processing in the future.