A Hybrid Regression and Machine Learning-Based Multi-Output Predictive Modeling of Cutting Forces and Surface Roughness in Rotational Turning of C45 Steel

Abstract

Rotational turning is a hybrid machining process that combines features of milling and conventional turning, resulting in altered chip formation and force generation mechanisms. Despite its technological relevance, the predictive modeling of cutting forces and surface roughness in rotational turning has received little attention. This study applies and evaluates a hybrid regression and machine learning modeling for the multi-output prediction of three cutting force components and two surface roughness parameters during rotational turning of normalized C45 steel. The input variables are tool inclination angle, depth of cut, feed, and cutting speed. Three modeling approaches are compared: stepwise polynomial regression, Gaussian Process Regression, and Random Forest regression, using repeated five-fold cross-validation with ten repetitions. The results show that Gaussian Process Regression provides the highest predictive accuracy for most outputs, particularly for axial and radial forces and roughness parameters, while stepwise regression achieves comparable performance for tangential force with greater interpretability. Random Forest regression exhibits lower accuracy under the structured experimental design. The study demonstrates that combining interpretable regression with probabilistic machine learning enables the accurate prediction of process responses in rotational turning. The proposed methodology represents a novel, statistically validated approach for multi-output modeling of this machining process and supports future applications in process optimization and adaptive manufacturing systems.

1. Introduction

The transformation of manufacturing toward intelligent, data-driven production systems has significantly altered the methodological approach of machining research [1,2]. Traditional deterministic modeling approaches are increasingly complemented by data-centric techniques capable of capturing nonlinear interactions and high-dimensional parameter functions [3]. Within this context, the integration of artificial intelligence methods into machining process analysis has become a central research direction, particularly for the prediction of cutting forces, surface integrity, and energy consumption [4,5,6].

Rotational turning is a machining process in which the cutting tool rotates while engaging a rotating workpiece [7,8]. Unlike conventional longitudinal turning, where the tool makes no rotational movement and advances axially, rotational turning modifies the effective cutting kinematics and chip formation geometry. The inclination angle of the tool introduces a tangential component of motion relative to the cutting edge, influencing chip thickness evolution, contact length, and thermal distribution in the cutting zone. These altered mechanics can potentially reduce tool wear concentration, distribute cutting load more uniformly, and improve process stability [9,10,11]. From an industrial perspective, rotational turning offers potential advantages over conventional longitudinal turning in applications involving difficult-to-machine materials or high productivity requirements. The rotating tool distributes wear along the cutting edge, which can extend tool life and reduce localized thermal damage. In addition, the modified kinematics can lower peak cutting forces and improve process stability, enabling higher material removal rates without exceeding machine tool limits. These characteristics make rotational turning attractive for automated production environments and for components where dimensional accuracy and surface integrity are critical. However, the need for specialized tooling, a more complex setup, and limited availability of predictive models has restricted widespread industrial adoption compared to conventional turning. Consequently, improving the predictive capability for process responses is essential for facilitating practical implementation.

C45 steel (AISI 1045) is a widely used medium-carbon structural steel, valued for its balanced strength and toughness after normalization [12,13]. However, its machinability presents several challenges that complicate process optimization. The relatively high carbon content promotes the formation of built-up edge and adhesive wear [14], particularly at moderate cutting speeds, leading to unstable cutting conditions and fluctuating force signals. Additionally, the combination of strength and ductility can generate high cutting temperatures and significant tool–chip contact stresses, accelerating flank wear and altering effective tool geometry over time [13]. These effects may lower surface integrity through plastic deformation, micro-tearing, or smearing, especially under high feed conditions [15]. In rotational turning, where the tool inclination alters the effective rake and stress distribution, these material-related effects can become more pronounced, causing nonlinear variations in force components and surface quality. The combination of strain-hardening tendencies, temperature-dependent flow stress, and heterogeneous chip morphology makes C45 steel particularly highlight the need for accurate regression and machine-learning-based approaches [16,17,18].

Accurate prediction of cutting forces in rotational turning is essential. Cutting forces determine spindle load, structural deformation of the machine tool, and potential chatter amplitude [19,20]. These directly influence tool life, dimensional accuracy, and energy efficiency. Surface roughness, in turn, affects functional performance of machined components, including fatigue resistance, tribological behavior, and assembly compatibility. The capability to predict both forces and roughness simultaneously enables multi-objective optimization and supports digital twin development [21,22,23]. The relationship between process parameters and machining responses is fundamentally nonlinear [24,25,26]. Depth of cut and feed rate directly determine uncut chip cross-sectional area, while tool inclination modifies effective rake and clearance conditions [27,28]. Cutting speed influences thermal softening and frictional behavior [29,30]. These variables interact in complex ways that may not be fully represented by simple linear or quadratic models.

Polynomial regression models remain widely used in machining research due to their interpretability and explicit analytical structure [31,32,33]. Stepwise regression (SW) enhances classical polynomial modeling by automatically selecting statistically significant terms based on information criteria [34,35]. However, polynomial expansions are limited in their ability to represent highly nonlinear behavior or localized interactions.

Machine learning (ML) methods offer alternative modeling concepts [36,37,38,39]. Gaussian Process Regression provides a non-parametric Bayesian framework capable of modeling smooth nonlinear functions with quantified uncertainty. By employing automatic relevance determination kernels, the relative importance of input variables can be inferred directly from the model. Ensemble-based approaches such as Random Forest regression construct multiple decision trees and aggregate their predictions, thereby reducing variance and improving generalization performance. Despite their growing adoption, several methodological issues persist in machining-related machine learning studies. First, performance comparison between models is often conducted using single train-test splits, which can produce misleading conclusions due to sampling variability [40]. Second, data leakage can occur if preprocessing steps are applied to the entire dataset prior to cross-validation [41]. Third, interpretability is frequently neglected, limiting practical applicability in industrial environments where engineers require physical insight [42].

The integration of machine learning into machining research has intensified over the past decade, driven by the broader transition toward data-centric manufacturing and Industry 4.0 [43,44]. In turn, machine learning has primarily been applied to predict cutting forces, surface roughness, tool wear progression, energy consumption, and process stability. While early studies relied predominantly on Artificial Neural Networks and Support Vector Regression, recent research increasingly incorporates ensemble methods, Gaussian processes, and hybrid physics-informed approaches.

Initial applications of machine learning in turning focused on Artificial Neural Networks (ANN) trained to predict surface roughness from cutting parameters [45,46,47,48]. These models demonstrated improved predictive capability over classical empirical regression, particularly when nonlinear effects were present. However, many early works relied on single train-test splits, small datasets, and limited statistical validation, often reporting only deterministic values without uncertainty quantification. While such models captured nonlinearities effectively, interpretability was limited. Moreover, overfitting was frequently observed when the dataset size was small relative to model complexity. Support Vector Regression (SVR) emerged as a popular alternative due to its ability to manage nonlinear relationships with controlled complexity through kernel functions [49]. Studies comparing SVR to ANN frequently reported similar accuracy but improved generalization stability with SVR [35,50]. Despite these advances, many early studies lacked systematic cross-validation strategies and rarely performed statistical comparisons between models. Consequently, conclusions regarding model superiority were often dataset-dependent and not generalizable.

With increasing computational power, ensemble methods such as Random Forest (RF) and Gradient Boosting became popular in machining applications [51,52]. These models are particularly effective for heterogeneous datasets and complex feature interactions. In operations involving multiple sensors, Random Forest has been applied successfully to tool wear monitoring and chatter detection [53,54]. The noise tolerance and minimal parameter tuning requirements make it attractive for industrial implementation. However, in structured design-of-experiments datasets, ensemble tree methods sometimes demonstrate inferior performance compared to smooth regression models [55,56]. This occurs because tree-based partitioning approximates smooth relationships with piecewise constant segments, which may not efficiently represent polynomial-like response surfaces. Comparative studies in turning have shown that while Random Forest performs well in high-noise or sensor-rich environments, its advantage diminishes in controlled factorial experiments with low measurement variability.

Gaussian Process Regression (GPR) has gained attention in machining research due to its probabilistic framework and ability to provide predictive uncertainty [57,58]. GPR has been used for force prediction under varying tool engagement conditions, demonstrating high accuracy with relatively small datasets. Several recent studies highlight the suitability of GPR for modeling cutting forces and surface roughness in small-to-medium-sized experimental datasets [59,60]. Unlike neural networks, GPR does not require extensive hyperparameter tuning, and unlike ensemble trees, it preserves the smoothness of response surfaces. However, the literature reveals that statistical comparison between GPR and polynomial regression remains limited [36,61]. Many publications compare GPR primarily against ANN or SVR rather than against carefully constructed quadratic regression models.

Response Surface Methodology remains widely used in machining research due to its transparency and physical interpretability [62,63,64]. Quadratic regression models are particularly common in studies. While often categorized as traditional statistical modeling rather than machine learning, polynomial regression provides a strong baseline for structured Design of Experiments (DoE) datasets. In fact, in many controlled machining experiments, quadratic models achieve high predictive accuracy. However, the literature rarely positions polynomial regression within a formal machine learning benchmarking framework. Statistical hypothesis testing between polynomial regression and modern ML models is uncommon. As a result, the true performance gap between classical and contemporary approaches is not always clearly quantified.

In the most recent period, machining research has increasingly explored hybrid modeling strategies that combine data-driven learning with physical constraints [65,66]. Physics-informed machine learning approaches incorporate prior knowledge of process mechanics—such as force models [67], thermal relationships [68], or chip formation theory [69]—to improve generalization and reduce the dependence on large training datasets. Studies in milling and turning have demonstrated that hybrid models can achieve higher accuracy under extrapolation conditions compared to purely data-driven methods [66,70], particularly when process variables interact nonlinearly or when measurement noise is present. Recent works also emphasize uncertainty-aware prediction, digital twins, and real-time monitoring frameworks [71,72], where probabilistic models such as Gaussian processes play a central role. Despite these advances, applications specifically addressing rotational turning remain uncommon, and systematic comparisons between interpretable regression, probabilistic models, and ensemble learning under controlled experimental conditions are still limited. This gap highlights the need for carefully validated benchmarks for complex turning processes.

Despite the rapid growth of machine learning applications in machining research, several methodological gaps remain. First, statistical comparisons between classical polynomial regression and modern machine learning models in turning are limited. Second, many studies emphasize predictive accuracy without systematically linking model behavior to mechanistic machining principles. Third, comprehensive evaluations that simultaneously address multiple force components and surface roughness within a unified and statistically validated framework are rare. Furthermore, such a structured and statistically rigorous multi-model comparison has not yet been reported for rotational turning.

The present study addresses these limitations through a structured regression and machine learning framework supported by repeated cross-validation and formal statistical testing. Three different modeling methods are compared under identical validation conditions: interpretable stepwise quadratic polynomial regression, probabilistic Gaussian Process Regression with automatic relevance determination, and ensemble-based Random Forest regression. Mechanical outputs—tangential, axial, and radial cutting forces—and surface quality indicators—the arithmetic mean roughness and the average peak-to-valley roughness—are modeled within the same experimental domain to ensure consistency and comparability in rotational turning. Beyond predictive benchmarking, the study integrates permutation-based feature importance analysis with mechanistic interpretation, enabling a direct connection between statistical relevance and established machining theory. Performance differences are evaluated using repeated five-fold cross-validation, paired statistical hypothesis testing, effect size quantification, and overfitting index analysis to ensure unbiased comparison and generalization assessment.

By systematically combining interpretability, probabilistic modeling, ensemble learning, and statistical validation, this work contributes a methodologically rigorous perspective on AI integration in turning processes. The results clarify when advanced machine learning methods provide tangible advantages over structured polynomial models and establish a reproducible framework for future data-driven machining studies.

The primary objective of this study is to establish a validated predictive framework for multi-output modeling of cutting forces and surface roughness in rotational turning, and to determine the relative effectiveness of interpretable regression and advanced machine learning techniques for this purpose. Specifically, the study seeks to answer the following research questions: (i) which modeling approach provides the highest predictive accuracy for individual mechanical and surface quality outputs under identical experimental conditions; (ii) whether probabilistic and ensemble-based methods offer measurable advantages over structured polynomial regression for a controlled machining dataset; and (iii) how the statistical relevance of input parameters relates to established physical mechanisms of chip formation and force generation. It is hypothesized that models capable of capturing nonlinear interactions and uncertainty will outperform purely deterministic polynomial formulations for most outputs, while interpretable regression may remain competitive for responses dominated by primary chip load effects.

2. Materials and Methods

2.1. Process Kinematics of Rotational Turning

Rotational turning is a turning process where both the workpiece and the tool rotate; the tool rotation generates a circular feed (circular feed denotes the relative movement of the cutting edge along the workpiece surface produced by rotational motion rather than by linear translation). This procedure differs primarily from conventional longitudinal turning in its kinematic structure and chip formation mechanism. In longitudinal turning, the workpiece rotates with a rotational speed (n_w), while the cutting tool performs a linear feed motion parallel to the axis of rotation. Rotational speed represents the number of revolutions per unit time, typically expressed in revolutions per minute, while linear feed motion is the translational displacement of the tool relative to the workpiece during machining. The cutting edge does not rotate about its own axis and maintains a fixed geometric orientation relative to the workpiece. Material removal is therefore affected by a steady-state engagement condition defined by feed per revolution and depth of cut. In contrast, rotational turning introduces an additional rotational degree of freedom of the cutting tool. As shown in Figure 1, which illustrates the chip removal process during rotational turning, the workpiece performs the primary cutting motion through rotation (n_w), while the cutting tool simultaneously rotates around its own axis with rotational speed (n_t).

The axis of tool rotation is arranged parallel to the symmetry axis of the workpiece. The cutting edge is helical (it follows a screw-like, spiraling geometry along the longitudinal axis of the tool rather than being straight), and its rotation generates a circular feed motion instead of a completely linear one. Consequently, the active cutting zone continuously changes its spatial position relative to the tool surface.

Figure 2 presents the kinematic and geometric configuration of the process. The combination of workpiece rotation (n_w) and tool rotation (n_t) produces a complex relative velocity field at the tool-workpiece interface. The main cutting speed (v_c) component is determined by the circumferential speed of the workpiece, while the rotary motion of the tool generates the effective feed movement. Unlike longitudinal turning, where feed (f) is imposed as a translational displacement, in rotational turning, the circular feed results from the rotation of the helical cutting edge itself.

During the initial stage of engagement (the period where the cutting edge is in contact with and actively removing material from the workpiece), the cutting edge progressively penetrates the workpiece material as it rotates. The chip cross-sectional area increases until the cutting edge reaches the geometric plane defined by the helix axis and the symmetry axis of the workpiece. At this position, the chip cross-section becomes constant, and continuous chip formation begins. This steady-state material removal condition is characteristic of turning operations, although its kinematic origin differs from that of conventional longitudinal turning. The geometric relationship between workpiece length (L_w) and the projection length of the cutting edge onto the base plane (Lt) determines whether additional axial feed is required. The axial projection of the helical cutting edge onto the base plane represents the maximum cylindrical length that can be machined using only the circular feed generated by the rotation of the tool. If L_w ≤ Lt, the circular feed generated by tool rotation is sufficient to machine the entire cylindrical surface. However, when L_w > Lt, the circular motion alone cannot cover the complete surface. In this case, an additional axial feed velocity (v_t,a) must be applied. This axial motion must be precisely synchronized with the rotational feed to ensure uniform surface generation and prevent overlapping or uncut regions. The synchronization condition affects the effective feed per revolution and directly influences the resulting surface topography. In the present study, the additional axial feed velocity was set to zero; therefore, the effective feed motion was generated completely by the rotational movement of the helical cutting tool.

Depth of cut (a_p)—defined as the radial difference between the machined (r_w) and unmachined surfaces (R_w)—is controlled by adjusting the axis distance (a_w) between the rotational axis of the tool and the workpiece centerline, where a_w equals the sum of the helix radius of the tool (r_t) and the radius of the machined surface (r_w). Depth of cut represents the thickness of the material layer removed in a single pass, measured perpendicular to the machined surface. This radial offset defines the engagement width of the cutting edge and therefore determines the uncut chip width. The feed per revolution (f), which is the distance that the tool advances relative to the workpiece during one complete revolution of the workpiece, influences the instantaneous chip thickness and theoretical surface profile height. The tool inclination angle (λ_s), which is the angle between the cutting edge and a reference plane perpendicular to the cutting velocity vector, influencing the chip flow direction and load distribution along the edge, modifies the effective rake and clearance angles along the active cutting edge and alters the direction of resultant cutting force components. Rake angle describes the orientation of the tool face that guides chip flow, while clearance angle prevents contact between the tool flank and the machined surface. In the present study, λ_s is treated as a categorical geometric parameter representing discrete tool configurations rather than a continuously adjustable variable. The combined effects of a_p, f, λ_s, and v_c produce nonlinear interactions in both force generation and surface formation. Because the chip thickness distribution varies along the helical cutting edge and because the engagement condition evolves continuously during rotation, the process cannot be described by simple linear superposition. Quadratic interaction terms are therefore physically justified when constructing regression-based surrogate models.

Compared to longitudinal turning, rotational turning offers potential advantages in distributed tool wear, modified chip flow direction, and altered heat partitioning due to the moving contact zone. However, the additional rotational degree of freedom introduces geometric coupling effects that must be captured in the modeling framework. The explicit consideration of kinematic synchronization, geometric projection length, and axis offset ensures that the present study provides a reproducible and mechanically consistent description of the investigated process.

2.2. Experimental Design, Measurement System and Data Acquisition

The detailed description of the experimental setup, measurement configuration, and boundary conditions has been designed for rotational turning and reported in a previous study [75], and only the parameters directly relevant to the present modeling analysis are summarized here for completeness. Figure 3 shows the experimental setup: the machining center with the data acquisition and surface-measurement equipment (a), and a detailed view of the workpiece and the tool-holding system (b), highlighting their relative positions during cutting.

The experiments were carried out on a Perfect-Jet MCV-M8 machining center (manufacturer: Ping Jeng Machinery Industry, Taichung City, Taiwan), which provided the required rotational synchronization between the workpiece and the cutting tool. All experiments were performed under wet machining conditions using a flooded cooling system. A 5 percent emulsion of Rhenus TS 25 semi-synthetic coolant–lubricant (manufacturer: Rhenus Lub GmbH & Co KG, Mönchengladbach, Germany) was applied at a volumetric flow rate of Q = 2 m³/h. The selected fluid is designed for high-performance machining applications, providing effective heat dissipation and boundary lubrication. The applied flow rate ensured continuous coverage of the tool–workpiece interface and reliable chip evacuation, thereby reproducing typical industrial flooded lubrication conditions and minimizing thermal and frictional fluctuations during cutting.

The experiments were conducted on normalized C45 steel with a hardness of 220 HV. The normalized condition ensures a homogeneous ferrite–pearlite microstructure and stable mechanical properties, providing reproducible cutting behavior and minimizing variability due to material inhomogeneity. C45 steel grade is widely used in mechanical engineering applications, making it representative for industrial turning operations. Its moderate carbon content enables stable chip formation without excessive brittleness or severe built-up edge formation within the selected cutting parameter range. No additional heat treatment was applied prior to cutting to avoid microstructural gradients that could influence force generation and surface formation. The cylindrical workpieces with a to be machined length of 12 mm, and a diameter of 40 mm were clamped directly into the main spindle of the machining center to provide the primary rotational motion.

Three tools with different inclination geometries were employed in the experiments, which have helical cutting edges suitable for rotational turning. The cutting tools were mounted onto the dynamometer, which was fixed on the machine table to enable accurate measurement of the cutting force components. Tool 1 was a Fraisa P5300682 solid carbide end mill (manufacturer: FRAISA Hungária Kft., Sárospatak, Hungary) made of HM MG10 grade, with a diameter of 20 mm and an inclination angle of 30°. Tool 2 was a Sandvik 1P341-1600-XB end mill (manufacturer: Sandvik Coromant, Sandviken, Sweden) of grade 1630, with a diameter of 16 mm and an inclination angle of 45°. Tool 3 was a Sandvik R215.38-20050-AC38L end mill (manufacturer: Sandvik Coromant, Sandviken, Sweden) of grade 1620, featuring a diameter of 20 mm and an inclination angle of 50°. The selected tools represent industrially relevant solid carbide end mills with distinct inclination geometries, allowing systematic investigation of the influence of cutting edge orientation on force generation and surface formation. Their tough carbide grades and standard diameters ensure stable performance under flooded machining conditions while maintaining comparability between configurations. To ensure controlled and reproducible engagement conditions, only one cutting edge was active in each configuration. In the case of Tool 2, three out of four cutting edges were ground down, while for Tool 3, five out of six edges were removed to guarantee single-edge contact. This preparation eliminates multi-edge interference, simplifies force component analysis, and ensures that chip formation corresponds to a single, well-defined cutting geometry. Although slight differences in tool diameter exist between configurations, these variations are inherently coupled to the categorical inclination parameter λ_s and therefore do not introduce multicollinearity within the regression framework. The inclination angle λ_s is treated as a discrete geometric variable representing distinct tool configurations rather than a continuously adjustable parameter. During the experiments, the conditions of the different cutting tools were continuously monitored, and tool replacement was performed when the wear reached the predefined wear criterion established in the preliminary tests. All tools were manufactured from fine-grain cemented carbide substrates designed for machining steels, providing high hot hardness, compressive strength, and resistance to plastic deformation at elevated temperatures. The cutting edges featured standard peripheral end-mill geometry with positive rake characteristics suitable for stable chip evacuation in continuous cutting. The tools were clamped in precision collet holders to ensure concentricity and minimal runout, and the holder assembly was rigidly connected to the dynamometer to preserve measurement reliability. The overhang length was kept constant across all configurations to maintain comparable structural stiffness and dynamic behavior.

Depth of cut (a_p) was varied at 0.1 mm, 0.2 mm, and 0.3 mm. Cutting speed (v_c) was investigated at 150, 200, and 250 m/min. Feed (f) values were selected as 0.1, 0.2, 0.4, 0.6, and 0.8 mm/rev. However, the experimental matrix was not constructed as a complete three-factor full factorial design. Instead, the feed levels were systematically evaluated along predefined engagement paths to isolate the influence of chip load and surface speed under controlled boundary conditions. Specifically, all feed values were tested at the combinations a_p,1-v_c,1, a_p,2-v_c,1, a_p,3-v_c,1, a_p,1-v_c,2 and a_p,1-v_c,3 (where 1, 2, and 3 indicate the lower, middle, and higher values, respectively). This structured arrangement allows independent analysis of feed effects at constant cutting speed while also enabling assessment of cutting speed influence at the lowest depth of cut. The rationale behind this selection was to ensure stable machining conditions across the entire feed range while limiting excessive tool load and thermal effects at higher engagement combinations. By avoiding simultaneous high values of a_p, f and v_c the design maintains process stability and prevents early tool degradation, while still capturing the dominant nonlinear interactions required for predictive modeling. Each parameter combination was repeated three times to reduce measurement noise and improve the statistical reliability of the dataset. The selected parameter ranges correspond to finishing to medium machining conditions commonly applied in industrial turning of medium-carbon steels, where dimensional accuracy and surface quality are primary objectives. The upper limits of feed and cutting speed were constrained by the power capacity and rigidity of the machine tool as well as by the thermal and mechanical load tolerance of the rotary cutting tool. Similarly, the chosen depth-of-cut range reflects practical engagement levels that allow stable chip formation without excessive deflection or vibration in rotational turning configurations. Extreme combinations were intentionally excluded because they could induce chatter, rapid tool wear, or process instability, which would compromise both measurement reliability and comparability between trials. Conversely, very low parameter values were avoided because they produce discontinuous chip formation and size-effect-dominated cutting mechanics that are not representative of typical production conditions. Overall, the selected ranges ensure industrial relevance while maintaining safe operation, repeatability, and compatibility with the kinematic constraints of the rotational turning setup used in this study.

Cutting forces were measured using a Kistler 9257A three-component dynamometer (manufacturer: Kistler Instrumente AG, Winterthur, Switzerland). The force signals generated by the piezoelectric dynamometer were conditioned using three Kistler 5011 single-channel charge amplifiers (manufacturer: Kistler Instrumente AG, Winterthur, Switzerland), which converted the charge output into proportional voltage signals. These analog voltage signals were acquired via a NI-9215 analog input module installed in a cDAQ-9171 chassis (manufacturer: National Instruments (NI) Corporation, Austin, TX, USA) and transmitted to the data acquisition computer. The measurement signals were sampled at a frequency of 1000 Hz, recorded, and stored using NI LabVIEW 2018 SP1 software (developer: National Instruments (NI) Corporation, Austin, TX, USA) for subsequent processing and analysis. The dynamometer measures force components F_x, F_y, and F_z in the machine coordinate system. The F_z component is oriented parallel to the symmetry axes of both the workpiece and the tool and is directed from the spindle toward the machine table. The F_x and F_y components act within the plane of the machine table; their orientation is defined such that at the midpoint of the machined cylindrical surface, F_x is parallel to the tangential cutting force, while F_y is parallel to the thrust force. This coordinate alignment is illustrated in Figure 3b. To obtain physically meaningful cutting force components, a coordinate transformation was applied [75]. The tangential cutting force F_c corresponds to the main cutting direction, aligned with workpiece rotation. The feed force F_f acts parallel to the feed direction, while the radial force F_p acts normal to the machined surface in the radial direction. A transformation depending on tool orientation converts the measured machine-coordinate force vectors (F_x, F_y, and F_z) into tool-coordinate force vectors (F_c, F_f, and F_p). The transformation accounts for the angular alignment between machine axes and tool geometry. For each cutting pass, steady-state intervals (the continuous chip removal phase) were identified by excluding entry and exit transients. Within the steady-state region, maximum values of F_c(t), F_f(t), and F_p(t) were extracted for analysis (where t is the time parameter).

The surface topography parameters evaluated were the arithmetic mean roughness (R_a) and the average peak-to-valley roughness (R_z). R_a describes the average absolute deviation of the surface profile from the mean line over the evaluation length, providing a stable and widely accepted indication of the overall texture of a machined surface. Because R_a averages the height deviations across the entire profile, it is relatively insensitive to isolated anomalies and therefore offers a representative measure of the general machining quality. For this reason, R_a is the primary parameter required by international standards and is commonly specified on engineering drawings to define surface finish requirements in manufacturing. R_z, in contrast, characterizes the vertical distance between the highest peaks and lowest valleys within the sampling lengths. This parameter is more sensitive to extreme profile features such as feed marks, tool wear effects, or local defects that may not significantly influence the R_a value. R_z therefore provides information relevant to functional behavior, particularly in applications where surface contact mechanics, friction, or lubrication conditions are influenced by peak-to-valley structure. The combined evaluation of R_a and R_z enables the distinction between surfaces that may have similar average roughness but differ substantially in their peak morphology or defect severity. The present study focuses on these parameters because they remain the most widely specified surface finish indicators in industrial practice and international standards. The objective of this research is predictive benchmarking of modeling approaches rather than detailed topographical morphology characterization. While three-dimensional areal parameters could provide additional insight into surface structure, the selected 2D metrics ensure comparability with conventional turning literature and engineering drawing specifications. Extension to 3D parameters represents a logical direction for future investigation. It should be noted that R_a and R_z characterize surface roughness only and are not intended to provide a comprehensive assessment of surface integrity or damage. Phenomena such as microcracking, plastic deformation, residual stresses, phase transformations, white layers, or subsurface alterations cannot be reliably detected using line roughness parameters alone. Therefore, no direct conclusions regarding these surface damage mechanisms are drawn from these metrics in the present work. The parameters are used exclusively to quantify the geometric quality of the machined surface profile in accordance with common industrial practice. More advanced characterization of surface integrity would require complementary methods, such as areal (3D) topography parameters, microstructural analysis, hardness measurements, or residual stress evaluation, which fall beyond the scope of this study.

Surface roughness was assessed using a Mitutoyo SurfTest SJ-301 portable 2D surface roughness tester (manufacturer: Mitutoyo, Kawasaki, Kanagawa, Japan). Measurements were performed along three generatrices on each machined surface, and the acquired profiles were post-processed using AltiMap Premium 6.2.7487 software (Altimet, Thonon-les-Bains, France). The cut-off length was selected according to the relevant standard [76], applying 0.25 mm for feed of 0.1 mm/rev, 0.8 mm for feeds of 0.2 and 0.4 mm/rev and 2.5 mm for feeds of 0.6 and 0.8 mm/rev. For each machining setup, mean values calculated from three measurements were used to reduce random variability.

2.3. Multi-Output Predictive Modeling Using Regression and Machine Learning

All predictive modeling procedures were implemented in MATLAB R2025b (developer: MathWorks Inc., Natick, MA, USA), with additional functionality provided by the Statistics and Machine Learning Toolbox. All models reported are static, data-driven mappings from process inputs to outputs and do not involve numerical integration of dynamical equations; therefore, no integration time step was required. The implemented approaches operate on independent observations and produce instantaneous predictions without simulating time evolution of the machining process. The dataset was imported from a structured file into a table format. The predictor variables were tool inclination (λ_s), depth of cut (a_p), cutting speed (v_c), and feed per revolution (f). The response variables were tangential force (F_c), axial force (F_f), radial force (F_p), arithmetic surface roughness (R_a), and maximum height roughness (R_z). The inclination parameter (λ_s) was encoded as a categorical variable to correctly represent discrete tool geometries in regression models.

Prior to modeling, the dataset underwent systematic validation to ensure numerical stability and reproducibility. Missing values were checked using the “ismissing” function. No imputation procedures were implemented because the dataset was complete. Basic descriptive statistics, including mean, standard deviation, minimum, and maximum values, were computed for all predictors and outputs to verify physical plausibility. Near-zero variance of predictors was assessed using the coefficient of variation. Multicollinearity among predictors was evaluated using the Variance Inflation Factor. A linear model was fitted against the remaining predictors for each predictor (i), and VIF_i was computed as [77]:

$${VIF}_i={\displaystyle \frac{1}{1-{R_i}^2}},$$

(1)

where R_i² is the coefficient of determination of that auxiliary regression. Outlier detection was performed using standardized Z-scores. Observations with an absolute Z-score greater than 3 were examined. No automatic removal was performed to avoid distortion of physically valid high-engagement conditions. Normality of output variables was assessed using the Jarque–Bera test implemented via “jbtest” function. This step informed the inclusion of non-parametric modeling approaches.

To estimate generalization performance, repeated K-fold cross-validation was applied (where K denotes the number of folds). In this study, the dataset was partitioned using 5-fold cross-validation. In each repetition, the data were randomly divided into five mutually exclusive folds of approximately equal size. For each fold i (let i denote the fold index: i = 1 …5):

• the training set consisted of all observations excluding fold i;
• the test set consisted of fold i.

This process was repeated for all five folds. To reduce variability due to random partitioning, the entire 5-fold procedure was repeated 10 times using independent random splits. A fixed random seed (“rng(1)” function) was used for reproducibility. Within each repetition, identical training and test partitions were applied to all models to enable paired statistical comparison.

Three regression approaches were evaluated independently for each output variable:

• Stepwise Polynomial Regression (SW);
• Gaussian Process Regression (GPR);
• Random Forest Regression (RF).

These were selected to represent fundamentally different regression paradigms with increasing model flexibility and complexity. Stepwise polynomial regression provides an interpretable parametric baseline capable of capturing quadratic effects and pairwise interactions typical in machining processes. Gaussian Process Regression represents a non-parametric probabilistic framework suitable for modeling smooth nonlinear relationships with limited datasets. Random Forest regression introduces an ensemble-based method capable of capturing complex, non-additive interactions without predefined functional structure. The comparison therefore enables systematic evaluation of interpretability, flexibility, and generalization behavior within a unified modeling framework. Additional machine learning methods such as Artificial Neural Networks and Support Vector Machines were not included primarily due to considerations of dataset size, interpretability, and comparability with physics-based understanding. Neural networks typically require substantially larger datasets to avoid overfitting and to achieve stable training, particularly when multiple outputs and nonlinear interactions are present. In small-to-medium experimental datasets common in machining research, neural models may exhibit high variance and sensitivity to hyperparameter selection. Support Vector Machines, while effective for nonlinear regression, introduce kernel selection and parameter tuning complexities that can obscure the physical interpretation of model behavior. The selected models cover a spectrum from transparent parametric modeling (SW), through probabilistic nonlinear regression with uncertainty quantification (GPR), to high-capacity ensemble learning (RF), thereby providing a balanced methodological comparison suitable for the available data volume.

Stepwise regression was implemented using the “stepwiselm” function. The upper model specification was quadratic, including linear terms, squared terms, and pairwise interaction terms. The inclination λ_s was specified as a categorical variable. Model selection was guided by the Akaike Information Criterion [77]:

$$AIC=2k-2\mathrm{l}\mathrm{n}\left(L\right),$$

(2)

where k is the number of parameters and L is the likelihood of the model. Terms were iteratively added or removed to minimize AIC.

Gaussian Process Regression was implemented using the “fitrgp” function. An automatic relevance determination (ARD) squared exponential kernel was applied. Predictors were standardized within the modeling procedure (Standardize = true). Hyperparameters were optimized through maximum likelihood estimation on the training data.

Random Forest regression was implemented using the “fitrensemble” function with bootstrap aggregation, and the following settings were applied:

• bagging method;
• number of learning cycles: 100.

Each tree was trained on a bootstrap sample of the training dataset. Predictions were obtained by averaging individual tree outputs.

For each of the three predictor methods, model performance was evaluated solely on the test sets. The following metrics were computed:

• Coefficient of Determination (R²);
• Root Mean Square Error (RMSE);
• Mean Absolute Error (MAE);
• Normalized RMSE (NRMSE);
• Overfitting index to assess generalization stability, which was defined as:

$$OverfitIndex={R^2}_{train}-{R^2}_{test},$$

(3)

where R²_train denotes the average coefficient of determination on the training set, and R²_test denotes the average coefficient of determination on the validation/test set.

For each output, cross-validated R² (coefficient of determination) values across all repetitions and folds were aggregated into paired samples. Pairwise comparisons between models were conducted using paired t-tests. Effect size was quantified using Cohen’s d, with the mean difference between paired performances in the numerator and the standard deviation of these differences in the denominator. This procedure allows statistically accurate evaluation of whether performance differences are significant and practically meaningful. Conventionally, d ≈ 0.2 is interpreted as small, d ≈ 0.5 as moderate, and d ≥ 0.8 as large.

After cross-validation, each model type was retrained on the full dataset for each output variable. Predicted values were computed for diagnostic visualization. Measured-versus-predicted plots were generated to assess the global match between experimental and modeled values.

Feature importance was evaluated using permutation analysis on the fully trained models. For each predictor variable, its values were randomly permuted while all other variables remained unchanged. The increase in mean squared error relative to the baseline model was computed. Importance values were normalized to sum to unity to allow comparability between predictors.

2.4. Overview of the Modeling Workflow

Figure 4 provides a simplified overview of the methodological structure applied in this study. While the previous subsections explain each analytical step individually, the flowchart highlights how these components are integrated into a coherent predictive-modeling framework. The diagram emphasizes the sequential progression from data collection to model-based interpretation, and illustrates the dependencies among preprocessing, model training, validation, and performance assessment.

The workflow also makes explicit two important conceptual layers of the study. First, model development is coupled with iterative validation and comparative evaluation, ensuring that predictive accuracy is assessed in a statistically accurate manner. Second, the inclusion of a feature importance stage and a physical interpretation component underlines that the purpose of the modeling is not only to obtain accurate predictions but also to relate statistical outcomes to the machining mechanisms. This dual focus—prediction and interpretation—is a central aspect of the contribution of the current study.

In summary, the flowchart serves as a visual synthesis of the methodological logic, helping the reader understand how the individual analytical elements interact to form a unified and transparent modeling strategy.

3. Results

This section presents the experimental and modeling results. First, the measured cutting force components and surface roughness parameters are reported numerically and interpreted with respect to the applied machining conditions. Second, the statistical integrity of the experimental dataset is examined through the methods described earlier. Finally, the predictive performance of the evaluated regression and machine learning methods is analyzed separately for each output variable in relation to machining mechanics.

3.1. Presentation of Measured Cutting Forces and Surface Roughness

The complete measured values for all experimental runs are summarized in Appendix A. In the present subsection, the observable tendencies of the measured force components and surface roughness parameters are described.

Across the entire dataset, the cutting force component (F_c) exhibited a clear and systematic increase with both feed (f) and depth of cut (a_p) for all inclination angles (λ_s). At the lowest engagement level, defined by a_p = 0.1 mm and f = 0.1 mm/rev, F_c remained below approximately 100 N for all tools and cutting speeds. With increasing feed, a definite growth was observed. For example, at λ_s = 30° and a_p = 0.1 mm, F_c increased from roughly 50–90 N at f = 0.1 mm/rev to more than 220–255 N at f = 0.8 mm/rev. When a_p was increased to 0.3 mm at v_c = 200 m/min, F_c reached values above 700 N at f = 0.8 mm/rev for all inclination angles. The highest recorded values exceeded 750 N in several cases, indicating substantial mechanical load under maximum chip cross-sectional area conditions. The passive force component (F_p) followed a similar monotonic trend with increasing feed and depth of cut, although its absolute magnitude remained significantly lower than F_c. At moderate engagement, F_p values typically ranged between 70 N and 150 N. Under high engagement conditions, especially at a_p = 0.3 mm and f = 0.8 mm/rev, F_p exceeded 300 N for λ_s = 45°, depending on repetition. The feed direction force (F_f) also increased markedly with feed. For instance, at λ_s = 50° and a_p = 0.3 mm, F_f rose from approximately 70–75 N at f = 0.1 mm/rev to values above 390–409 N at f = 0.8 mm/rev. In general, the relative contribution of F_f became more pronounced at higher inclination angles and larger chip loads. The effect of cutting speed (v_c) on the force components was less dominant than that of feed or depth of cut within the investigated range. For identical a_p and f combinations at a_p = 0.1 mm, moderate reductions in F_c were often observed when v_c increased from 150 to 200 or 250 m/min. However, this tendency was not uniformly linear across all inclination angles and feeds. The primary force trends remained affected by chip cross-sectional area rather than cutting speed variation. From a mechanistic viewpoint, the observed force increases can be interpreted through two complementary aspects: material removal amount and frictional interaction at the tool-workpiece interface. First, the uncut chip cross-sectional area—approximately proportional to the product of feed and depth of cut—determines the volume of material that must be plastically deformed and sheared per revolution. As this removal amount increases, the shear plane area expands, requiring higher energy input and resulting in larger cutting forces. This effect explains the dominant influence of feed and depth of cut on F_c, F_f, and F_p. Second, increasing chip load also enlarges the contact length between the chip and the rake face as well as between the tool flank and the newly generated surface. The larger contact area increases frictional resistance and normal pressure, contributing additional force components beyond shear deformation. Variations in inclination angle modify the orientation and distribution of these contact zones, which can amplify the feed-direction force and passive force at higher chip loads. The comparatively weaker influence of cutting speed within the investigated range can be attributed to competing thermal and frictional effects: higher speeds tend to reduce material strength due to temperature rise, while simultaneously increasing sliding velocity at the interface. Consequently, the combined force response reflects the balance between material removal requirements and frictional conditions rather than speed alone.

Surface roughness parameters (R_a and R_z) demonstrated strong sensitivity to feed. At low feed values of 0.1 mm/rev, R_a typically remained below 0.7 µm for all inclination angles, with minimum measured averages around 0.32–0.34 µm at λ_s = 50° and v_c = 250 m/min. Corresponding R_z values were generally between 2.0 µm and 4.5 µm. As feed increased to 0.6 and 0.8 mm/rev, a substantial decline in surface finish was observed. For λ_s = 45° at a_p = 0.1 mm and f = 0.8 mm/rev, R_a exceeded 4.0 µm and R_z approached or exceeded 17 µm across repetitions. A comparable increase was measured for λ_s = 50°, where R_a values reached approximately 4.0–4.2 µm and R_z exceeded 18 µm in certain high-feed conditions. The influence of depth of cut on surface roughness was less consistent than that of feed. At low feeds, increasing a_p from 0.1 mm to 0.3 mm did not systematically increase R_a; in several cases, comparable or slightly reduced values were observed at intermediate depths. However, under high-feed conditions, the combination of large ap and high f generally produced the highest roughness values. Clear differences were also visible between inclination angles. At low feeds, the 50° tool frequently produced the lowest R_a and R_z values, particularly at higher cutting speeds. In contrast, the 45° configuration exhibited the highest roughness at elevated feeds, with R_a exceeding 4 µm and R_z approaching 18 µm under extreme conditions. The 30° tool showed moderate roughness levels across most parameter combinations and showed relatively stable behavior over repeated runs.

The repetition consistency was generally good for both force and roughness measurements. The three individual R_a and R_z measurements per run showed limited dispersion, and averaged values closely represented the individual observations. Although force magnitudes varied between repetitions, the relative ordering of parameter effects remained consistent. Overall, the measured dataset reveals coherent process behavior: increasing feed and depth of cut led to substantial growth in cutting forces, while surface roughness was primarily affected by feed and tool inclination angle. These experimentally observed tendencies form the basis for the subsequent statistical characterization and predictive modeling analysis.

3.2. Statistical Validation of the Experimental Dataset

Before evaluating the predictive performance of the applied regression and machine learning models, a comprehensive statistical validation of the complete experimental dataset was carried out. The objective of this procedure was to verify data integrity, confirm statistical adequacy of the experimental design, and assess distributional properties that may influence model behavior. First, a missing value analysis was performed for all predictors λ_s, a_p, f, and v_c, as well as for the response variables F_c, F_p, F_f, R_a, and R_z. No missing entries were identified, confirming full dataset completeness and eliminating the need for imputation or case exclusion. Descriptive statistical analysis was then conducted to examine central tendency, dispersion, and value ranges. All predictor variables covered the predefined experimental intervals, confirming correct implementation of the designed parameter matrix. The standard deviations and coefficients of variation indicated sufficient spread in each predictor, excluding near-zero variance conditions that could bias regression coefficients or reduce model sensitivity. Multicollinearity was evaluated using Variance Inflation Factor analysis. All predictors yielded VIF values equal to 1.000, indicating complete statistical independence among λ_s, a_p, f, and v_c. This result confirms the orthogonality of the experimental design and ensures stable coefficient estimation in parametric regression models. Outlier detection was performed using standardized Z-scores with a threshold of |Z| > 3, which result is shown in

Table 1. Number of found outliers in the measured values across all 225 experimental observations.

	Fc	Fp	Ff	Ra	Rz
Number of outliers	4	6	7	0	0

. No outliers were observed among the predictor variables or roughness parameters. A limited number of extreme values were identified in the force components F_c, F_p, and F_f. These values correspond to high-engagement cutting conditions and were therefore retained, as they represent physically meaningful process states rather than measurement errors.

Normality of the response variables was assessed using the Jarque–Bera test. The results indicated statistically significant deviations from normal distribution for all output variables. This non-normal behavior, particularly pronounced for the cutting force components, supports the methodological decision to include non-parametric and ensemble-based machine learning approaches alongside polynomial regression.

Finally, the correlation matrix of the predictor variables confirmed negligible pairwise correlations, further validating the structural independence of the experimental factors. Overall, the validation procedure demonstrates that the dataset is complete, statistically strong, free from multicollinearity, and suitable for both parametric and non-parametric predictive modeling.

3.3. Predictive Modeling and Comparison

To provide a transparent representation of the predictive relationships identified in this study, the final regression formulations obtained for each target variable are summarized in Equations (4)–(8). These expressions correspond to the optimized quadratic Stepwise models for the cutting force components (F_c, F_f, F_p) and the surface roughness parameters (R_a, R_z), expressed as functions of the process variables (I₄₅ and I₅₀ are indicator variables: I₄₅ = 1 if λ_c = 45, else 0; I₅₀ = 1 if λ_c = 50, else 0):

F_c (λ_s, a_p, v_c, f) = 81.33 + 27.06 ∙ I₄₅ − 9.92 ∙ I₅₀ − (305.42 − 47.47 ∙ I₄₅ − 96.22 ∙ I₅₀) ∙ a_p + (31.14 − 28.30 ∙ I₄₅ + 41.64 ∙ I₅₀) ∙ f − 0.1718 ∙ v_c + 2686.41 ∙ a_p ∙ f + 1312.75 ∙ a_p² − 63.92 ∙ f²,

(4)

F_f (λ_s, a_p, v_c, f) = 49.96 − 30.23 ∙ I₄₅ − 51.77 ∙ I₅₀ − (449.97 − 278.96 ∙ I₄₅ − 368.13 ∙ I₅₀) ∙ a_p − (47.11 − 48.42 ∙ I₄₅ − 99.97 ∙ I₅₀) ∙ f + 1316.78 ∙ a_p ∙ f + 854.66 ∙ a_p² − 34.91 ∙ f²,

(5)

F_p (λ_s, a_p, v_c, f) = 4.72 + 94.67 ∙ I₄₅ + 47.78 ∙ I₅₀ − (27.28 + 78.20 ∙ I₄₅ + 325.81 ∙ I₅₀) ∙ a_p + (145.868 − 63.85 ∙ I₄₅ − 82.99 ∙ I₅₀) ∙ f + (0.00981 − 0.157 ∙ I₄₅ − 0.00287 ∙ I₅₀) ∙ v_c + 845.46 ∙ a_p ∙ f − 0.213 ∙ v_c ∙ f + 690.72 ∙ a_p² − 59.32 ∙ f²,

(6)

R_a (λ_s, a_p, v_c, f) = 1.327 − 0.409 ∙ I₄₅ − 0.705 ∙ I₅₀ − 0.00118 ∙ v_c − (4.571 − 3.131 ∙ I₄₅ − 3.674 ∙ I₅₀) ∙ f + 6.475 ∙ f²,

(7)

R_z (λ_s, a_p, v_c, f) = 6.733 − 1.279 ∙ I₄₅ − 2.974 ∙ I₅₀ − 0.00550 ∙ v_c − (16.291 − 11.532 ∙ I₄₅ − 14.116 ∙ I₅₀) ∙ f + 24.934 ∙ f².

(8)

The coefficients were estimated using the full dataset after model structure selection and statistical validation. Presenting these equations enables direct analytical evaluation of the predicted responses and assists potential implementation in process planning or control environments where interpretable parametric models are advantageous. For the Gaussian Process Regression and Random Forest approaches, the predictive models are defined by nonparametric structures that depend on the entire training dataset rather than a compact set of coefficients. In the case of GPR, predictions are obtained through kernel-based covariance functions and training data weights, while Random Forest models consist of large ensembles of decision trees. As a result, these models do not admit a concise closed-form expression comparable to polynomial regression. Due to this inherent structural complexity and the very large number of parameters involved, explicit model representations are not presented in this study.

The predictive performance of the three modeling approaches for each investigated output variable is presented in the rest of this section. The models compared are adaptive quadratic Stepwise regression, Gaussian Process Regression with ARD squared exponential kernel, and Random Forest regression using bagging. All results are based on repeated 5-fold cross-validation with ten repetitions, ensuring statistically strong evaluation. For each output, mean cross-validated R², standard deviation of R², RMSE, MAE, normalized RMSE, mean training R², and overfitting index were computed. The calculated values used in the evaluation are shown in

Table 2. Performance metrics of the regression models for each output variable.

Output Variable	Model	R2 (Mean)	Std. Dev. R2	RMSE (Mean)	MAE (Mean)	NRMSE	Train R2 (Mean)	Overfit Index
Fc	SW	0.991	0.0023	14.674	10.58	0.021	0.993	0.0019
	GPR	0.991	0.0022	14.761	10.586	0.021	0.995	0.0036
	RF	0.943	0.0187	38.265	25.084	0.054	0.960	0.0165
Ff	SW	0.986	0.0053	8.932	6.5556	0.023	0.990	0.0034
	GPR	0.994	0.0016	5.871	4.1049	0.015	0.997	0.0025
	RF	0.914	0.0250	22.953	14.406	0.058	0.940	0.0263
Fp	SW	0.919	0.0328	16.931	11.791	0.050	0.944	0.0250
	GPR	0.984	0.0103	7.174	5.1589	0.021	0.996	0.0113
	RF	0.883	0.0385	20.919	14.038	0.061	0.922	0.0395
Ra	SW	0.966	0.0102	0.200	0.1651	0.049	0.971	0.0057
	GPR	0.991	0.0094	0.097	0.064043	0.024	0.997	0.0061
	RF	0.962	0.0120	0.215	0.15037	0.053	0.973	0.0117
Rz	SW	0.949	0.0123	1.004	0.86172	0.060	0.957	0.0078
	GPR	0.986	0.0052	0.518	0.36632	0.031	0.996	0.0093
	RF	0.959	0.0106	0.919	0.68179	0.055	0.972	0.0132

.

To quantify the differences in predictive performance across models, paired t-tests were applied to the fold-level cross-validated R² values. The comparisons in

Table 3. Statistical comparison of regression models based on cross-validated R². ΔR² is the difference between the mean R² of the compared models; CI denotes the 95% confidence interval; t and p correspond to paired t-test results; Cohen’s d indicates effect size.

Comparison Basis	Output Variable	SW vs. GPR	SW vs. RF	GPR vs. RF
ΔR2	Fc	0.0001	0.0483	0.0482
	Ff	−0.008	0.0724	0.0805
	Fp	−0.0658	0.0359	0.1017
	Ra	−0.0251	0.0042	0.0293
	Rz	−0.037	−0.0093	0.0277
CI	Fc	[−0.0003, 0.0005]	[0.0429, 0.0537]	[0.0428, 0.0536]
	Ff	[−0.0093, −0.0067]	[0.0652, 0.0796]	[0.0734, 0.0876]
	Fp	[−0.0739, −0.0578]	[0.0261, 0.0457]	[0.0910, 0.1125]
	Ra	[−0.0284, −0.0219]	[0.0004, 0.0079]	[0.0253, 0.0333]
	Rz	[−0.0405, −0.0336]	[−0.0142, −0.0044]	[0.0243, 0.0312]
paired t-test results	Fc	t(49) = 0.474, p = 0.6373	t(49) = 18.013, p < 0.001	t(49) = 17.951, p < 0.001
	Ff	t(49) = −12.402, p < 0.001	t(49) = 20.229, p < 0.001	t(49) = 22.824, p < 0.001
	Fp	t(49) = −16.522, p < 0.001	t(49) = 7.365, p < 0.001	t(49) = 18.993, p < 0.001
	Ra	t(49) = −15.524, p < 0.001	t(49) = 2.251, p = 0.0289	t(49) = 14.629, p < 0.001
	Rz	t(49) = −21.631, p < 0.001	t(49) = −3.827, p = 0.0004	t(49) = 16.315, p < 0.001
Cohen’s d	Fc	0.067	2.547	2.539
	Ff	−1.754	2.861	3.228
	Fp	−2.337	1.042	2.686
	Ra	−2.195	0.318	2.069
	Rz	−3.059	−0.541	2.307
Result	Fc	GPR ≈ SW	SW > RF	GPR > RF
	Ff	GPR > SW	SW > RF	GPR > RF
	Fp	GPR > SW	SW > RF	GPR > RF
	Ra	GPR > SW	SW > RF	GPR > RF
	Rz	GPR > SW	RF > SW	GPR > RF

report: (1) the difference in mean R² between the two models (ΔR²), which reflects the average performance gap across the cross-validation folds; (2) the 95% confidence interval (CI) for this difference, characterizing its uncertainty; (3) the t-statistic and associated p-value from the paired t-test used to assess whether the difference departs from zero; and (4) Cohen’s d, which expresses the effect size in standardized units.

Permutation-based feature importance analysis was used to quantify the contribution of each input parameter to the prediction of each output variable. This method evaluates the decrease in model performance when a given input is randomly permuted, thereby providing a model-uncertain measure of how strongly each parameter influences the predicted outcome.

Table 4. Permutation-based feature importance values for each output variable across the three regression models (Stepwise, GPR, and Random Forest). The table reports the normalized importance of each input parameter (λ_s, a_p, v_c, f).

Input Variable	Output Variable	SW	GPR	RF
λs	Fc	0.0058	0.0068	0.0040
	Ff	0.1089	0.1040	0.0611
	Fp	0.2162	0.2662	0.1784
	Ra	0.2484	0.2272	0.2000
	Rz	0.2158	0.2364	0.1919
ap	Fc	0.5762	0.4939	0.4894
	Ff	0.4941	0.4785	0.5225
	Fp	0.3914	0.3944	0.378
	Ra	0.0000	0.0008	0.0036
	Rz	0.0000	0.0075	0.0037
vc	Fc	0.0008	0.0008	0.0124
	Ff	0.0000	0.0003	0.0058
	Fp	0.0051	0.0098	0.0174
	Ra	0.0009	0.0082	0.0017
	Rz	0.0012	0.0084	0.0052
f	Fc	0.4172	0.4986	0.4942
	Ff	0.3969	0.4171	0.4106
	Fp	0.3874	0.3296	0.4262
	Ra	0.7506	0.7637	0.7946
	Rz	0.7830	0.7477	0.7992

summarizes the resulting importance values for all output variables and models, reported separately for the Stepwise, GPR, and Random Forest models. The values are presented in a line-by-line format within each cell to assist direct comparison across models and output variables. These values represent averages across all cross-validation repetitions. While minor variability was observed between folds, the ranking of dominant predictors remained consistent.

3.3.1. Tangential Cutting Force

The tangential cutting force (F_c) represents the primary mechanical load component during material removal. Accurate modeling of F_c is essential for power estimation, spindle load prediction, and structural deformation analysis. Both Stepwise regression and Gaussian Process Regression demonstrated high predictive performance for F_c. Mean cross-validated R² values exceeded 0.991 for both models. The difference between them was minimal and statistically insignificant, with p = 0.6373. The corresponding Cohen’s d was small, indicating a negligible practical difference in predictive capability.

The mean RMSE for these two models was approximately 14.7 N. When normalized relative to the observed range of F_c, the NRMSE was approximately 0.021, corresponding to roughly 2.1 percent of the total force variation range. This level of accuracy indicates that the quadratic polynomial structure is sufficient to capture the dominant physical dependencies affecting tangential force. Random Forest regression achieved lower predictive performance, with a mean R² around 0.943. Although this value still indicates a strong correlation, the decrease relative to the other two models was statistically significant, with p < 0.0001 and a large effect size (Cohen’s d > 2.5). This suggests that ensemble tree-based partitioning was less effective in representing the smooth parametric relationships present in the structured experimental dataset. The standard deviation of R² across folds and repetitions remained below 0.005 for Stepwise and GPR, demonstrating high stability.

The overfitting index for both models was below 0.01, indicating minimal difference between training and testing performance. Permutation feature importance revealed that depth of cut and feed collectively accounted for more than 90 percent of predictive relevance. Cutting speed exhibited near-zero importance across all models, while inclination angle contributed modestly. This aligns with classical force theory, where tangential force is approximately proportional to uncut chip cross-sectional area, determined primarily by a_p and f.

The predictive performance of the three machine learning models is illustrated in Figure 5. Each subplot illustrates the model-specific prediction accuracy by plotting individual data points against the ideal 1:1 reference line (red). The closeness of the scatter points to this line indicates the degree of agreement between measured and predicted forces, highlighting the relative performance of the three approaches. Both the SW and GPR models show excellent agreement with the measured cutting forces, with most predictions deviating less than approximately 5–8% across the full range. Their data points align closely with the 1:1 reference line, indicating minimal systematic bias. In contrast, the RF model shows increasing underestimation above roughly 450–500 N, with predictions tending to saturate around 600–650 N. This behavior suggests reduced model sensitivity in the upper force domain, likely caused by limited high-force training samples or insufficient model depth. Overall, SW and GPR demonstrate greater generalization capability compared to RF.

3.3.2. Axial Cutting Force

The axial force component acts parallel to the feed direction and contributes to tool deflection and dimensional deviation. Accurate modeling of F_f is important for predicting axial load and potential vibration excitation. Gaussian Process Regression achieved the highest predictive accuracy for F_f, with a mean cross-validated R² = 0.9942. Stepwise regression achieved R² = 0.9862, while Random Forest achieved R² = 0.9137. All pairwise differences were statistically significant with p < 0.0001. The effect size between GPR and Stepwise was moderate, while the effect size between GPR and Random Forest was large. The mean RMSE for GPR was 5.87 N, significantly lower than that of the other models. The NRMSE was approximately 0.022, indicating low relative prediction error.

The overfitting index for GPR remained below 0.005, demonstrating excellent generalization. Stepwise regression exhibited slightly higher overfitting but remained within acceptable limits. Feature importance analysis indicated that feed was the dominant predictor, followed by depth of cut. Inclination angle contributed more significantly than in F_c modeling, reflecting the sensitivity of axial force to the geometric orientation of the cutting edge. Cutting speed again showed a negligible contribution. The superior performance of GPR suggests that F_f contains nonlinear interactions not fully captured by quadratic polynomial terms. The ARD kernel allows adaptive scaling of input dimensions, effectively modeling subtle curvature in the response surface.

As illustrated in Figure 6 (the plots provide a concise visual measure of prediction accuracy based on the deviation from the red 1:1 reference line), the SW and GPR models provide highly accurate predictions of the feed force, with typical deviations remaining under 5–7% across the studied range. Their predictions align closely with the 1:1 reference line, reflecting minimal error. In contrast, the RF model exhibits systematic underestimation above approximately 250–300 N and saturates near 350–400 N, indicating limited responsiveness to higher force values. Consequently, SW and GPR demonstrate markedly better overall predictive reliability compared to RF.

3.3.3. Radial Cutting Force

Radial force is often associated with surface integrity and dimensional accuracy, as it acts perpendicular to the machined surface and influences deflection. Modeling of F_p proved more challenging than F_c and F_f. Gaussian Process Regression achieved a mean R² = 0.9845, while Stepwise regression achieved R² = 0.9186 and Random Forest achieved R² = 0.8828. All pairwise comparisons were statistically significant with p < 0.0001. Effect sizes were large, particularly between GPR and the other two models. The standard deviation of R² for F_p was higher than for F_c and F_f, indicating greater variability across folds. This reflects increased complexity and sensitivity of radial force to parameter interactions.

The overfitting index remained moderate for all models, with GPR maintaining the lowest value. Random Forest showed slightly higher overfitting, suggesting that ensemble partitioning may over-adapt to training subsets under a limited sample size. Feature importance analysis revealed a stronger influence of inclination angle compared to F_c and F_f. In Stepwise and GPR models, λ_s accounted for more than 20 percent of predictive importance. This is physically plausible because radial force is strongly influenced by effective cutting geometry and edge orientation relative to the workpiece. Feed and depth of cut remained significant predictors, while cutting speed again showed minimal importance.

As shown in Figure 7 (representing the accuracy of the model by illustrating the alignment of the data points with the red 1:1 reference line), the GPR model provides the most accurate predictions for the passive force, with deviations predominantly within 5–6% and predictions closely following the 1:1 reference line. The SW model also performs well but exhibits slightly higher scatter, particularly between 150 and 250 N. In contrast, the RF model displays consistent underprediction above approximately 180–200 N, with saturation around 230–260 N, indicating limited responsiveness to higher passive force values. Consequently, GPR demonstrates the strongest overall predictive robustness for F_p.

3.3.4. Arithmetic Mean Roughness

Arithmetic mean roughness reflects average surface height deviation and is widely used as a surface quality indicator. Gaussian Process Regression achieved mean R² = 0.9909 with RMSE = 0.0966 µm. Stepwise regression achieved R² = 0.9798, while Random Forest achieved R² = 0.9767. The difference between Stepwise and Random Forest was statistically insignificant (p = 0.1132). However, GPR significantly outperformed both, with p < 0.0001. The NRMSE for GPR was approximately 0.019, corresponding to less than 2 percent of the observed R_a range.

Overfitting indices were small for all models, indicating stable generalization. Feature importance analysis demonstrated that feed dominated R_a prediction, contributing approximately 75 to 80 percent of total importance. This is consistent with theoretical surface profile formation, where feed determines peak spacing and amplitude. Inclination angle contributed secondary importance, likely due to modification of effective cutting edge trajectory. Depth of cut and cutting speed exhibited negligible influence on R_a within the investigated range.

As shown in Figure 8, where each subplot summarizes the prediction accuracy by comparing measured and estimated values against the red 1:1 reference line, GPR provides the most accurate predictions of the surface roughness R_a, with deviations typically below 5% and point distributions closely following the 1:1 trend. The SW model also performs well but exhibits slightly larger dispersion, especially within the 1–3 µm region. The RF model shows the weakest agreement, characterized by overprediction at low roughness values and underprediction above approximately 3 µm. These trends indicate that GPR provides superior robustness for modeling R_a, while RF requires further optimization to capture roughness variations reliably.

3.3.5. Peak-to-Valley Roughness

Peak-to-valley roughness captures extreme surface deviations and is more sensitive to outliers and local irregularities. Gaussian Process Regression achieved a mean R² = 0.9864 with RMSE = 0.518 µm. Stepwise regression achieved R² = 0.9602, while Random Forest achieved R² = 0.9528. All pairwise comparisons involving GPR were statistically significant. The higher variability in R_z prediction compared to R_a reflects the increased stochasticity of peak-to-valley measures.

Feature importance patterns mirrored those of R_a. Feed remained the dominant predictor, contributing approximately 80 percent importance. Inclination angle contributed moderately, while depth of cut and cutting speed were minor.

As presented in Figure 9, which provides a concise visual measure of prediction accuracy based on the deviation from the red 1:1 reference line, the GPR model provides the most accurate prediction of the peak–valley roughness, with deviations typically below 5% and data points clustering tightly along the 1:1 line. The SW model also captures the trend well, though with moderately increased dispersion in the mid-range. In contrast, the RF model shows systematic deviations, including underprediction above roughly 12 µm and some overestimation at lower values. These results indicate that GPR returns the highest predictive reliability for R_z, while RF requires additional tuning for improved surface roughness modeling.

3.3.6. Stability, Generalization Analysis, Summary of Model Comparison

Across all outputs, Stepwise regression and GPR demonstrated low standard deviation of R² values, indicating consistent performance across repeated folds. Random Forest showed higher variability, suggesting sensitivity to sample partitioning. The overfitting index remained below 0.02 for all models, with GPR typically exhibiting the lowest values. This confirms that the repeated cross-validation framework effectively prevented optimistic bias. The majority of significant differences observed in the present study based on Cohen’s d correspond to large effect sizes, indicating not only statistical significance but also practically meaningful performance gaps.

Overall, Gaussian Process Regression achieved the highest predictive performance for four of five outputs. Stepwise regression matched GPR performance for tangential force and remained competitive for other outputs. Random Forest underperformed relative to the other two models, particularly for force components. Statistical hypothesis testing confirmed that performance differences were not due to random sampling variation. Large effect sizes indicate practically meaningful superiority of GPR in several cases. The structured experimental design likely favored smooth regression-based models, while tree-based partitioning did not exploit the continuous parametric relationships as effectively. Based on the comparison of measured and predicted values shown in the five figures, GPR consistently yields the closest agreement, with deviations typically below 5–6% and minimal scatter. SW also performs reliably but shows moderately larger dispersion, especially in mid-range regions. RF displays the weakest stability, with systematic over- or underprediction and saturation at higher outputs. Overall, GPR demonstrates the strongest generalization, followed by SW, while RF shows clear limitations across all examined responses.

4. Discussion

The results demonstrate that high predictive accuracy can be achieved for both cutting force components and surface roughness parameters using data-driven regression models trained on structured experimental data. However, the statistical advantage of certain models is only part of the interpretation. The deeper question concerns why specific modeling strategies perform better under controlled machining conditions and what this implies for physical understanding, digital process modeling, and industrial implementation.

The tangential force exhibited near-perfect predictability using both Stepwise regression and Gaussian Process Regression. Cross-validated R² values above 0.99 indicate that the dominant force formation mechanisms are highly structured and well captured by smooth functional relationships. From a mechanistic standpoint, F_c is primarily proportional to the uncut chip cross-sectional area, which is defined by feed (f) and depth of cut (a_p). In orthogonal and oblique cutting theory, the tangential force can be approximated as [27,28]:

$$F_c=k_c·a_p·f,$$

(9)

where k_c represents the specific cutting force. Within the investigated parameter range, k_c does not vary strongly with cutting speed, particularly under stable machining conditions and moderate temperature rise. This explains the negligible importance of cutting speed observed in all three modeling approaches. The small contribution of inclination angle λ_s to F_c prediction further confirms that tangential force magnitude is primarily affected by chip load rather than geometric redirection of force components.

The fact that quadratic Stepwise regression achieved performance comparable to GPR suggests that the underlying response surface for F_c is close to quadratic in the investigated domain. This is consistent with mechanistic modeling assumptions and validates the structured design of experiments. In contrast, Random Forest regression underperformed. Tree-based ensemble models partition the feature space into piecewise constant regions. While effective for highly nonlinear or discontinuous relationships, they are less efficient at modeling smooth polynomial-like surfaces with a limited sample size. The dataset represents a structured grid with continuous relationships. In such cases, regression-based models exploit parametric continuity more effectively.

For axial force and radial force, the situation differs slightly. While feed and depth of cut remain dominant, the increased importance of inclination angle in F_p prediction reflects geometric sensitivity of radial load distribution. Radial force is influenced by the orientation of the cutting edge relative to the workpiece surface normal, which affects the projection of resultant cutting force components. The greater performance of GPR for F_f and F_p indicates the presence of nonlinear interactions not fully captured by quadratic polynomial expansion. The ARD squared exponential kernel adapts characteristic length scales for each input dimension, effectively learning the smooth curvature of the response surface without imposing explicit polynomial structure. Importantly, the overfitting indices for GPR remained low, demonstrating that increased flexibility did not lead to poor generalization. This confirms the suitability of repeated cross-validation as a robustness mechanism.

The modeling of R_a and R_z further reinforces the connection between physical surface formation and statistical feature importance. Feed was consistently identified as the dominant predictor for both R_a and R_z. This is expected from theoretical surface generation geometry. In turn, the theoretical arithmetic roughness can be approximated by [27,28]:

$$R_a\approx {\displaystyle \frac{f^2}{32·r_{\epsilon }}},$$

(10)

where r_ε represents the tool nose radius. Although this simplified expression assumes ideal kinematics without material side flow or vibration, it captures the fundamental quadratic dependence of surface amplitude on feed. The small importance of cutting speed suggests that within the selected speed range, no significant transition occurred in built-up edge formation, thermal softening, or dynamic instability. Consequently, surface topography was affected primarily by geometric imprinting rather than thermomechanical effects.

The higher prediction difficulty for R_z compared to R_a reflects the statistical nature of peak-to-valley metrics. R_z is sensitive to isolated extreme events, including micro-chipping, adhesion effects, and stochastic material pull-out. Such events introduce variability, not entirely deterministic with respect to nominal cutting parameters. Therefore, even advanced models show slightly lower R² for R_z than for R_a. The performance advantage of GPR for surface parameters indicates that subtle nonlinear interactions exist, particularly between feed and inclination angle. The kernel-based approach captures these interactions smoothly, while Stepwise regression approximates them only through limited polynomial cross-terms. Interestingly, Random Forest performance for Ra was relatively close to Stepwise regression. This suggests that for monotonic relationships dominated by a single variable, such as feed, even piecewise constant approximations are sufficient.

One of the most notable results is the consistently low feature importance of cutting speed across all outputs. Several physical explanations are plausible within the investigated domain: First, the selected speed range may lie within a stable machining regime where temperature variation does not significantly alter shear flow stress or friction conditions. Second, the material behavior of normalized medium-carbon steel in the investigated range may not exhibit pronounced strain-rate sensitivity, affecting cutting forces at the tested speeds. Third, surface roughness in tangential turning is dominated by kinematic imprinting rather than thermally activated deformation mechanisms. It is important to highlight that these conclusions are valid only within the experimental parameter window. Extrapolation beyond this domain would require additional validation. From a modeling perspective, the low importance of cutting speed reduces effective dimensionality, which partially explains the strong performance of polynomial regression.

A central methodological question concerns the trade-off between interpretability and accuracy. Stepwise regression offers explicit analytical expressions relating outputs to inputs. Coefficients directly quantify effect magnitude and sign. This supports physical interpretability and integration into mechanistic frameworks. Gaussian Process Regression provides superior predictive performance but operates as a non-parametric kernel model. While feature relevance can be inferred from ARD length scales and permutation importance, the model does not yield a compact analytical equation. Random Forest regression provides moderate interpretability through feature importance but lacks a smooth analytical structure. For industrial deployment, interpretability is often critical. In process planning or adaptive control systems, transparent models support operator trust and regulatory compliance. However, in digital twin or AI-assisted monitoring frameworks, predictive accuracy may take precedence over analytical simplicity. The slight but statistically significant improvements of GPR may justify its use when maximal precision is required.

The observed differences—derived from the comparison of measured versus predicted values in all figures—highlight distinct modeling behaviors. GPR outperforms the other methods because its kernel-based structure captures nonlinearities and maintains accuracy even in sparsely sampled regions. SW performs adequately where the underlying relationships are near-linear, but shows larger deviations as nonlinear effects increase. RF exhibits saturation and systematic bias, likely due to limited tree depth, averaging of extremes, and insufficient representation of high-value data. These patterns indicate that both the force- and roughness-related responses contain nonlinear dependencies that benefit from GPR’s flexibility. Consequently, model selection strongly influences predictive reliability, particularly in upper-range mechanical and surface parameters.

The experimental design employed a structure with controlled parameter variation. Such datasets are characterized by low noise, uniform parameter coverage, smooth response surfaces, and limited sample size. Under these conditions, regression-based models tend to outperform high-capacity ensemble methods. In contrast, if the dataset included tool wear progression, material inhomogeneity, coolant variations, or sensor-derived dynamic signals, nonlinear and tree-based models might demonstrate stronger relative performance. Therefore, model selection should consider dataset topology rather than relying on general AI performance rankings.

The OverfitIndex values obtained in the present study range between 0.0019 and 0.0395 across all outputs and models. In practical machine learning applications, differences below 0.02 are generally interpreted as negligible optimism bias, while values below 0.05 are commonly considered indicative of acceptable generalization stability for regression problems of comparable size. Within this framework, the SW and GPR models consistently demonstrate excellent stability, with OverfitIndex values predominantly below 0.01, reflecting nearly identical training and validation performance. The RF model exhibits slightly higher values, particularly for F_p prediction, yet remains below the conservative 0.05 threshold. Importantly, even the maximum observed index of 0.0395 does not suggest severe model variance or memorization effects. These results confirm that the repeated 10-fold cross-validation protocol effectively constrained overfitting, and that reported predictive performances can be regarded as robust and generalizable within the investigated parameter domain.

Repeated 5-fold cross-validation with ten repetitions provided 50 evaluation instances per model and output. This substantially reduces the variance in performance estimates compared to single-split validation. The paired t-test approach enabled direct comparison of models under identical fold partitions, eliminating disruption from sampling differences. Effect size analysis complemented p-values by quantifying practical relevance. Large Cohen’s d values for GPR advantage in F_p modeling confirm not only statistical significance but a substantial magnitude difference. The low standard deviation of R² across repetitions indicates model stability and suggests that the dataset contains a high signal-to-noise ratio.

High-accuracy predictive models of forces and surface roughness enable several industrial applications: real-time force prediction for spindle load management, surface quality estimation without offline measurement, energy consumption estimation through force integration, and adaptive parameter optimization. The demonstrated ability to achieve R² values above 0.9 for major outputs indicates that data-driven surrogate models can reliably replace computationally intensive mechanistic simulations in certain contexts. Moreover, Gaussian Process Regression integrally provides predictive uncertainty estimates through posterior variance. This capability is particularly valuable for digital twin implementations, where confidence intervals support risk-aware decision-making. From a practical standpoint, such models can be integrated into machine tool controllers or supervisory monitoring systems to provide continuous, real-time estimation of process states using readily available input parameters such as feed, depth of cut, cutting speed, and tool configuration. Because the evaluated models operate as static mappings rather than time-dependent simulations, their computational demand during inference is low, enabling deployment on industrial PCs or edge devices. In particular, GPR-based predictions combined with uncertainty bounds can support adaptive control strategies, where machining parameters are adjusted dynamically to avoid excessive forces, chatter onset, or unacceptable surface finish. In production environments, this capability can help predictive maintenance by detecting deviations from expected force signatures that may indicate tool wear or abnormal cutting conditions. Furthermore, integration into process planning software allows the offline optimization of cutting parameters to balance productivity, tool life, energy consumption, and surface quality before machining begins. These applications align with Industry 4.0 concepts of manufacturing systems and digital twins, where data-driven models provide fast, interpretable surrogates of complex physical processes.

Despite strong predictive performance, several limitations must be acknowledged. First, the dataset size remains moderate. Although cross-validation mitigates optimistic bias, larger datasets would strengthen generalization confidence. Second, tool wear progression was not explicitly modeled. All experiments were conducted under controlled conditions with a consistent tool state. Third, extrapolation beyond the investigated parameter domain is not guaranteed. Fourth, only static machining outputs were considered. Dynamic vibration signals and acoustic emissions were not included. Future work should address these aspects and explore hybrid modeling approaches combining mechanistic theory with machine learning.

5. Conclusions

This study presented a structured AI-based framework for predicting cutting force components and surface roughness parameters in rotational turning. Three regression approaches were systematically compared: adaptive quadratic Stepwise regression, Gaussian Process Regression with ARD kernel, and Random Forest ensemble modeling. Model evaluation employed repeated 5-fold cross-validation with statistical hypothesis testing and effect size analysis to ensure strong comparison. Cross-validated R2 values exceeded 0.9 for the principal outputs, while normalized RMSE values remained approximately within 2–3 percent under optimal conditions, demonstrating high predictive capability across both mechanical and surface quality responses.

A central finding is that model performance is strongly influenced by dataset structure. The smooth, well-controlled experimental design favored regression-based approaches over tree-based ensembles. Stepwise regression performed nearly identically to Gaussian Process Regression for tangential force and surface roughness, indicating that quadratic relationships adequately describe these outputs within the investigated parameter domain. In contrast, Gaussian Process Regression provided consistently higher accuracy for feed and passive force components, which exhibited stronger nonlinear behavior. Quantitatively, GPR reduced prediction error relative to Random Forest by approximately 20–35 percent for these components and achieved modest improvements of about 5–10 percent over Stepwise regression in the most nonlinear cases. Feature relevance analysis confirmed that feed and depth of cut dominate force generation, whereas feed is the primary determinant of surface roughness, while cutting speed had limited influence within the examined range. These findings align with established machining theory, supporting the physical plausibility of the models.

From a methodological standpoint, the study demonstrates that reliable model comparison requires not only mean accuracy metrics but also variability assessment, overfitting indicators, and effect size evaluation. The combined statistical validation framework ensures that observed performance differences are meaningful rather than artifacts of data partitioning. From an industrial perspective, the results indicate that accurate surrogate models of cutting forces and surface finish can support real-time monitoring, adaptive parameter selection, and digital manufacturing applications without reliance on computationally expensive simulations.

The principal contributions of this work can be summarized as follows:

• A validated benchmark comparison of three fundamentally different regression paradigms applied to rotational turning data.
• It was shown that quadratic parametric models can achieve accuracy comparable to advanced nonlinear methods for tangential force and surface roughness prediction in structured rotational turning experiments, indicating that complex black-box models are not always necessary when the physical relationships are predictable.
• The work quantified that nonlinear probabilistic modeling provides clear advantages primarily for force components influenced by tool orientation and chip flow redistribution (notably F_f and F_p), where interaction effects cannot be captured adequately by low-order polynomials.
• The results demonstrate that accurate prediction of machining responses is achievable across a wide engagement range, from light finishing conditions to high chip-load regimes, using a single unified modeling approach, supporting its applicability for adaptive process control and digital manufacturing scenarios.

Future research should extend the framework to broader operating conditions, including tool wear progression, different workpiece materials, and more complex tool geometries. Incorporating three-dimensional surface parameters, additional sensor inputs, or hybrid physics-informed machine learning models may further improve predictive robustness. Investigation of online model updating and transferability across machines also represents an important direction toward practical deployment in adaptive manufacturing systems. Overall, the study demonstrates that when experimental design is structured, AI-based regression models can achieve high accuracy while preserving physical interpretability, thereby providing a practical pathway toward data-driven process modeling in advanced machining.