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\begin{document}

\title[Article Title]{Explainable Machine Learning for Path Loss Modeling in mmWave and Sub-THz Communication Systems}

%%=============================================================%%
%% GivenName	-> \fnm{Joergen W.}
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%% \author*[1,2]{\fnm{Joergen W.} \spfx{van der} \sur{Ploeg} 
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\author*[1]{\fnm{Samba Siva Reddy Mula} }\email{sambasivareddymula@gmail.com}

\author[2]{\fnm{Vinay Kumar Pamula} }\email{pamulavk@jntucek.ac.in}




\affil*[1]{\orgdiv{Department of Electronics and Communication Engineering}, \orgname{JNTUK}, \orgaddress{\city{Kakinada}, \postcode{533003}, \state{Andhra Pradesh}, \country{India}}}

\affil[2]{\orgdiv{Department of Electronics and Communication Engineering}, \orgname{UCEK, JNTUK}, \orgaddress{\city{Kakinada}, \postcode{533003}, \state{Andhra Pradesh}, \country{India}}}



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\abstract{Machine learning has emerged as a powerful tool for large-scale path loss modeling at millimeter-wave (mmWave) and sub-terahertz (sub-THz) frequencies. While recent studies demonstrate improved prediction accuracy, most learning-based models operate as black boxes, limiting their adoption in deployment-critical wireless system design and standardization activities. This paper addresses this limitation by proposing an explainable machine learning framework for path loss modeling that emphasizes interpretability, physical consistency, and engineering trust. Using multi-band mmWave and sub-THz measurement datasets, we analyze how key propagation features—including transmitter–receiver separation distance, carrier frequency, antenna height, and line-of-sight conditions—influence model predictions. Feature attribution and sensitivity analyses are employed to quantify the relative importance of these parameters and to assess whether learned relationships align with established propagation theory. The results demonstrate that the proposed models capture physically meaningful monotonic trends with respect to distance and frequency, while avoiding non-physical artifacts commonly observed in purely data-driven approaches. Rather than prioritizing marginal accuracy gains, this work highlights the tradeoff between predictive performance and model transparency, showing that interpretable models can achieve competitive accuracy while offering substantially improved explainability. The presented framework provides actionable insights for network planning, site-specific modeling, and data-assisted calibration of analytical channel models. These findings support the role of explainable machine learning as a key enabler for trustworthy and deployable path loss modeling in future mmWave and sub-THz wireless communication systems.}


\keywords{Explainable machine learning, path loss modeling, millimeter-wave communications, sub-terahertz systems, wireless propagation, feature attribution, model interpretability, 6G networks}

%%\pacs[JEL Classification]{D8, H51}

%%\pacs[MSC Classification]{35A01, 65L10, 65L12, 65L20, 65L70}

\maketitle
%
\section{Introduction}\label{sec1}
The rapid evolution of fifth-generation (5G) and emerging sixth-generation (6G) wireless systems has driven communication technologies toward millimeter-wave (mmWave) and sub-terahertz (sub-THz) frequency bands, where large contiguous bandwidths enable ultra-high data rates and low-latency services. However, signal propagation at frequencies above 24~GHz is highly sensitive to blockage, scattering, and environmental geometry, making accurate large-scale path loss modeling a persistent challenge for network planning and performance evaluation \cite{Rappaport2015Wideband, 3GPP38901}.

\subsection{Limitations of Classical Path Loss Models}

Conventional empirical path loss models, such as the close-in (CI), floating-intercept (FI), and alpha–beta–gamma (ABG) formulations, represent path loss as a logarithmic function of transmitter--receiver separation distance and carrier frequency,
\begin{equation}
PL(d,f) = PL(d_0,f) + 10n\log_{10}\!\left(\frac{d}{d_0}\right) + X_\sigma,
\label{eq:ci_model}
\end{equation}
where $n$ denotes the path loss exponent and $X_\sigma$ models large-scale shadowing. While such models provide physical interpretability and analytical simplicity, their parameters are typically obtained through environment-specific regression and exhibit limited generalization across deployment scenarios and frequency bands \cite{MacCartney2017PathLoss}.

Measurement campaigns at mmWave and sub-THz frequencies have further revealed that shadowing variance and distance dependence are not stationary across environments, particularly in dense urban and mixed indoor–outdoor scenarios \cite{Samimi2016Statistical, Haneda2016Indoor}. These observations motivate the exploration of data-driven alternatives capable of capturing complex propagation effects beyond simplified analytical assumptions.

\subsection{Machine Learning for Path Loss Prediction}

Machine learning (ML) techniques have recently emerged as powerful tools for path loss modeling, enabling nonlinear regression over high-dimensional propagation features. Supervised learning approaches based on support vector regression (SVR), random forests, and deep neural networks (DNNs) have demonstrated improved prediction accuracy relative to empirical models when trained on sufficient measurement data \cite{Zhang2019SVRmmWave, Wu2021RandomForest, Bai2022DNNmmWave}.

Given a feature vector $\mathbf{x} = [d, f, h_t, h_r, \mathbb{I}_{\mathrm{LOS}}]^{\mathrm{T}}$, ML-based models approximate the functional mapping
\begin{equation}
\hat{PL} = f_{\theta}(\mathbf{x}),
\end{equation}
where $f_{\theta}(\cdot)$ denotes a parameterized regressor. Despite their flexibility, such black-box models often rely on implicit correlations learned from data, raising concerns regarding overfitting, robustness under domain shift, and physical consistency \cite{Chen2023Transfer, Heath2024Survey}.

\subsection{Need for Explainable Machine Learning}

The lack of interpretability in ML-based path loss models poses a significant barrier to their adoption in standardization, regulation, and safety-critical wireless system design. In contrast to analytical models, black-box predictors do not explicitly reveal how individual features influence path loss estimates, making it difficult to verify whether learned relationships are physically meaningful or merely data-driven artifacts.

Explainable machine learning (XAI) addresses this challenge by providing post-hoc or intrinsic mechanisms to interpret model predictions. Feature attribution methods, such as SHapley Additive exPlanations (SHAP), quantify the marginal contribution of each input variable to the predicted output \cite{Lundberg2017SHAP}. For a given prediction $\hat{PL}$, the model output can be decomposed as
\begin{equation}
\hat{PL} = \phi_0 + \sum_{k=1}^{K} \phi_k,
\end{equation}
where $\phi_k$ represents the contribution of the $k$-th feature. Such decompositions enable direct comparison between learned feature effects and known propagation principles, such as monotonic distance dependence and frequency scaling.

\subsection{Research Gap and Contributions}

Although XAI has been widely explored in domains such as computer vision and healthcare, its application to wireless channel modeling remains limited. Existing ML-based path loss studies primarily emphasize prediction accuracy, with little analysis of whether learned models align with physical propagation behavior or provide interpretable insights for system design.

This paper addresses this gap by systematically investigating explainable machine learning for path loss modeling in mmWave and sub-THz communication systems. The main contributions are summarized as follows:
\begin{itemize}
    \item An explainable path loss modeling framework that integrates post-hoc feature attribution with supervised ML regression.
    \item A quantitative analysis of feature importance and distance-dependent behavior across mmWave and sub-THz frequency bands.
    \item An experimental evaluation demonstrating that explainability can be achieved without sacrificing predictive accuracy, while enhancing physical interpretability.
\end{itemize}

By enabling transparent validation of learning-based models against established propagation theory, this work provides a practical step toward trustworthy and interpretable ML-driven channel modeling for future wireless systems.

%
\section{Background and Related Work}
\label{sec:related}

This section reviews existing work on path loss modeling at mmWave and sub-THz frequencies, machine learning–based channel prediction, and explainable artificial intelligence (XAI). The discussion highlights the limitations of current approaches and motivates the need for explainable learning frameworks in wireless propagation modeling.

%--------------------------------------------------
\subsection{Path Loss Modeling at mmWave and Sub-THz Frequencies}

Large-scale path loss modeling forms the foundation of wireless system design and standardization. At frequencies above 24~GHz, measurement-driven studies have shown that propagation characteristics deviate significantly from sub-6~GHz behavior due to increased sensitivity to blockage, surface roughness, and antenna directivity \cite{Rappaport2015Wideband, MacCartney2017PathLoss}.

Standardized models adopted by 3GPP, such as the CI and ABG models, express path loss as a logarithmic function of distance and frequency \cite{3GPP38901}. While these formulations provide analytical transparency, their parameters are typically environment-specific and fail to capture fine-grained spatial variability observed in dense urban and indoor scenarios \cite{Haneda2016Indoor}. As frequency increases toward the sub-THz regime, this mismatch becomes more pronounced, motivating data-driven extensions.

%--------------------------------------------------
\subsection{Machine Learning for Path Loss Prediction}

Recent years have seen growing interest in applying supervised learning to path loss modeling. Early approaches employed shallow learners, including support vector regression and ensemble trees, to approximate nonlinear propagation behavior \cite{Zhang2019SVRmmWave, Wu2021RandomForest}. Deep neural networks have subsequently demonstrated improved accuracy by learning hierarchical feature representations from large measurement datasets \cite{Bai2022DNNmmWave}.

Given a feature vector $\mathbf{x}$, learning-based models estimate the mapping
\begin{equation}
\hat{PL} = f_{\theta}(\mathbf{x}),
\label{eq:ml_mapping}
\end{equation}
where $f_{\theta}(\cdot)$ denotes a parameterized regressor. Despite favorable prediction performance, these models are often evaluated under matched training and testing conditions, with limited analysis of interpretability or physical consistency \cite{Chen2023Transfer}. Consequently, their suitability for extrapolation and standardization remains uncertain.

%--------------------------------------------------
\subsection{Explainable Artificial Intelligence: Concepts and Methods}

Explainable artificial intelligence aims to make machine learning models transparent by revealing how inputs influence predictions. XAI techniques can be broadly categorized into intrinsic and post-hoc methods \cite{DoshiVelez2017XAI}.

Intrinsic approaches enforce interpretability by design, for example through linear models, decision trees, or monotonic neural architectures. While transparent, these models often lack the expressive capacity required for complex propagation environments.

Post-hoc methods interpret trained black-box models without modifying their structure. Among these, feature attribution techniques such as SHapley Additive exPlanations (SHAP) have gained prominence due to their theoretical grounding in cooperative game theory \cite{Lundberg2017SHAP}. For a prediction $\hat{PL}$, SHAP decomposes the output as
\begin{equation}
\hat{PL} = \phi_0 + \sum_{k=1}^{K} \phi_k,
\end{equation}
where $\phi_k$ quantifies the contribution of the $k$-th input feature. This additive formulation enables direct inspection of how distance, frequency, and environmental indicators influence path loss predictions.

%--------------------------------------------------
\subsection{XAI in Wireless Communications}

XAI has recently been explored in several wireless communication tasks, including modulation recognition, resource allocation, and channel estimation \cite{Almohamad2022XAIWireless, Sun2023XAI6G}. These studies demonstrate that explainability can improve trust, debugging, and regulatory acceptance of ML-based solutions.

However, the application of XAI to large-scale channel and path loss modeling remains limited. Existing works primarily focus on predictive accuracy, offering little insight into whether learned models align with known propagation laws, such as monotonic distance dependence or frequency scaling. As a result, the interpretability gap between empirical models and ML-based predictors persists.

%--------------------------------------------------
\subsection{Taxonomy of Existing Approaches}

Table~\ref{tab:xai_taxonomy} summarizes representative approaches to path loss modeling and their interpretability characteristics. While classical models are inherently interpretable, they lack flexibility. Conversely, deep learning models offer superior accuracy but operate as black boxes. XAI-enhanced learning aims to bridge this gap by providing post-hoc interpretability without sacrificing performance.

\begin{table}[t]
\centering
\caption{Taxonomy of Path Loss Modeling Approaches and Explainability}
\label{tab:xai_taxonomy}
\footnotesize
\setlength{\tabcolsep}{3pt}
\renewcommand{\arraystretch}{1.15}
\begin{tabular}{p{2.4cm} p{3.4cm} c c}
\hline
\textbf{Category} &
\textbf{Representative Methods} &
\textbf{Interp.} &
\textbf{Data-Driven} \\
\hline
Empirical Models &
CI, FI, ABG~\cite{3GPP38901} &
High &
No \\

Shallow ML Models &
SVR, Random Forest~\cite{Zhang2019SVRmmWave, Wu2021RandomForest} &
Medium &
Yes \\

Deep Learning &
DNN, CNN regressors~\cite{Bai2022DNNmmWave} &
Low &
Yes \\

XAI-Enhanced ML &
DNN + SHAP (This work) &
High &
Yes \\
\hline
\end{tabular}
\end{table}


%--------------------------------------------------
\subsection{Research Gap and Motivation}

The reviewed literature indicates a clear gap between accuracy-driven ML models and physically interpretable path loss formulations. Existing ML-based approaches rarely analyze feature contributions or validate learned behavior against propagation theory. Furthermore, XAI has not been systematically applied to evaluate distance, frequency, and environment effects in mmWave and sub-THz path loss modeling.

Motivated by these observations, this work proposes an explainable machine learning framework that combines predictive performance with transparent interpretation. By explicitly analyzing feature contributions and consistency with known propagation trends, the proposed approach aims to enhance the trustworthiness and practical relevance of ML-based path loss models for future wireless systems.

\section{Methodology}
\label{sec:methodology}

This section presents the proposed explainable machine learning (XML) framework for large-scale path loss modeling in mmWave and sub-THz communication systems. The methodology integrates data-driven regression with post-hoc explainability mechanisms to achieve accurate, interpretable, and physically consistent path loss predictions across heterogeneous propagation environments.

\subsection{Problem Formulation}

We consider the supervised learning problem of estimating large-scale path loss based on system and environmental parameters. Given a dataset
\begin{equation}
\mathcal{D} = \left\{ \left(\mathbf{x}_i, PL_i \right) \right\}_{i=1}^{N},
\end{equation}
the objective is to learn a regression function
\begin{equation}
\hat{PL} = f_{\theta}(\mathbf{x}),
\end{equation}
where $\mathbf{x} \in \mathbb{R}^{d}$ denotes the input feature vector and $PL$ is the measured path loss in decibels (dB).

The input feature vector is defined as
\begin{equation}
\mathbf{x} = \left[ d, f, h_t, h_r, \mathbb{I}_{\mathrm{LOS}} \right]^{\mathrm{T}},
\end{equation}
where $d$ is the transmitter--receiver separation distance, $f$ is the carrier frequency, $h_t$ and $h_r$ denote transmitter and receiver antenna heights, respectively, and $\mathbb{I}_{\mathrm{LOS}}$ is a binary indicator for line-of-sight conditions.

\subsection{Baseline Learning Model}

A deep neural network (DNN) is employed as the baseline regression model due to its capacity to capture nonlinear propagation effects arising from blockage, diffraction, and frequency-dependent attenuation. The network consists of multiple fully connected layers with rectified linear unit (ReLU) activations:
\begin{equation}
\mathbf{z}^{(l)} = \sigma \left( \mathbf{W}^{(l)} \mathbf{z}^{(l-1)} + \mathbf{b}^{(l)} \right),
\end{equation}
where $\mathbf{W}^{(l)}$ and $\mathbf{b}^{(l)}$ denote the weight matrix and bias vector of the $l$-th layer, respectively, and $\sigma(\cdot)$ represents the ReLU activation function.

The network parameters $\theta$ are optimized by minimizing the empirical mean squared error (MSE):
\begin{equation}
\mathcal{L}_{\mathrm{MSE}} =
\frac{1}{N} \sum_{i=1}^{N}
\left( PL_i - f_{\theta}(\mathbf{x}_i) \right)^2.
\end{equation}

\subsection{Physics-Guided Feature Anchoring}

To ensure physical plausibility and facilitate interpretability, the learning process is anchored to well-established path loss theory. In particular, the close-in (CI) free-space reference model is used as a physical baseline:
\begin{equation}
PL_{\mathrm{CI}}(d,f) = PL(f,d_0) + 10n \log_{10}\!\left(\frac{d}{d_0}\right),
\end{equation}
where $n$ is the path loss exponent and $d_0 = 1$~m is the reference distance, consistent with 3GPP TR~38.901 \cite{3GPP38901}.

Rather than replacing analytical models, the DNN learns deviations from this physical baseline, enabling a clear separation between deterministic propagation trends and environment-specific effects.

\subsection{Explainability via SHAP-Based Attribution}

To interpret model predictions, Shapley Additive Explanations (SHAP) are employed as a post-hoc explainability mechanism \cite{Lundberg2017SHAP}. For a given prediction $\hat{PL}_i$, SHAP decomposes the output as
\begin{equation}
\hat{PL}_i = \phi_0 + \sum_{k=1}^{d} \phi_k(\mathbf{x}_i),
\end{equation}
where $\phi_0$ represents the model bias and $\phi_k$ denotes the contribution of the $k$-th feature.

SHAP values are computed by evaluating the marginal contribution of each feature across all possible feature coalitions, providing a theoretically grounded measure of feature importance that satisfies local accuracy, consistency, and missingness properties.

\subsection{Global and Local Interpretability Analysis}

The proposed framework supports both global and local interpretability:

\begin{itemize}
    \item \textit{Global interpretability} is obtained by aggregating SHAP values across the dataset to quantify overall feature importance and identify dominant propagation drivers.
    \item \textit{Local interpretability} explains individual predictions, enabling inspection of anomalous path loss estimates and environment-specific behavior.
\end{itemize}

This dual-level analysis enables validation of whether the learned model aligns with known physical principles, such as monotonic path loss increase with distance and frequency.

\subsection{Consistency and Trustworthiness Checks}

To enhance trust in model predictions, explainability outputs are examined for physical consistency. Specifically, the following conditions are verified:
\begin{equation}
\frac{\partial \hat{PL}}{\partial d} \geq 0,
\quad
\frac{\partial \hat{PL}}{\partial f} \geq 0,
\end{equation}
ensuring that predicted path loss does not decrease with increasing distance or frequency.

Violations of these conditions are flagged for further inspection, enabling explainability-driven model debugging and refinement.

\subsection{Computational Considerations}

The inference complexity of the trained model remains identical to that of a standard DNN, scaling as
\begin{equation}
\mathcal{O}\!\left(\sum_{l=1}^{L} P_l \right),
\end{equation}
where $P_l$ denotes the number of parameters in layer $l$.

SHAP analysis is performed offline during evaluation and does not introduce runtime overhead during deployment, making the framework suitable for practical network planning and digital-twin applications.

\subsection{Methodological Advantages}

By combining data-driven learning with explainability and physical anchoring, the proposed methodology achieves:
\begin{itemize}
    \item Accurate path loss prediction across mmWave and sub-THz bands,
    \item Transparent interpretation of learned propagation behavior,
    \item Improved trustworthiness and model validation,
    \item Compatibility with standardized channel modeling practices.
\end{itemize}

This design directly addresses the limitations of black-box learning approaches and establishes explainable machine learning as a viable tool for next-generation wireless propagation modeling.

\section{Implementation}
\label{sec:implementation}

This section details the practical implementation of the proposed explainable machine learning (XML) framework and describes how interpretability is operationalized during training, evaluation, and post-analysis. Unlike conventional black-box learning pipelines, the implementation explicitly integrates explainability objectives to support physical validation, model debugging, and deployment trust.

\subsection{Feature Engineering and Physical Encoding}

All input features are selected to maintain direct physical relevance to large-scale propagation mechanisms. Let the raw feature vector be defined as
\begin{equation}
\mathbf{x}_{\mathrm{raw}} = [d, f, h_t, h_r, \mathbb{I}_{\mathrm{LOS}}].
\end{equation}

To preserve interpretability under nonlinear learning, distance and frequency features are logarithmically transformed:
\begin{equation}
\tilde{d} = \log_{10}(d), \quad \tilde{f} = \log_{10}(f),
\end{equation}
resulting in the encoded feature vector
\begin{equation}
\mathbf{x} = [\tilde{d}, \tilde{f}, h_t, h_r, \mathbb{I}_{\mathrm{LOS}}]^{\mathrm{T}}.
\end{equation}

This transformation ensures that learned relationships align with classical log-distance path loss behavior and simplifies interpretability of feature attributions.

\subsection{Model Training and Regularized Optimization}

The neural network is trained using a regularized loss function that balances prediction accuracy and model smoothness:
\begin{equation}
\mathcal{L}_{\mathrm{total}} =
\mathcal{L}_{\mathrm{MSE}} + \lambda \|\theta\|_2^2,
\end{equation}
where $\mathcal{L}_{\mathrm{MSE}}$ is defined in Section~\ref{sec:methodology}, $\theta$ denotes trainable parameters, and $\lambda$ controls weight regularization.

This regularization reduces variance and improves stability of feature attribution under explainability analysis, ensuring that interpretations are not dominated by spurious parameter fluctuations.

\subsection{Prediction Decomposition for Explainability}

For each input sample $\mathbf{x}_i$, the predicted path loss is decomposed as
\begin{equation}
\hat{PL}_i = \mathbb{E}[PL] + \sum_{k=1}^{d} \phi_k(\mathbf{x}_i),
\end{equation}
where $\phi_k$ represents the SHAP contribution of the $k$-th feature.

This additive formulation enables direct mapping between numerical predictions and physically interpretable factors such as distance, frequency, and LOS conditions.

\subsection{Global Attribution Aggregation}

To analyze global propagation trends, feature attributions are aggregated across the dataset:
\begin{equation}
\Phi_k = \frac{1}{N} \sum_{i=1}^{N} \left| \phi_k(\mathbf{x}_i) \right|.
\end{equation}

The normalized global importance score is computed as
\begin{equation}
\bar{\Phi}_k = \frac{\Phi_k}{\sum_{j=1}^{d} \Phi_j},
\end{equation}
which quantifies the relative contribution of each feature to overall path loss prediction.

\textit{Interpretation:}  
Higher values of $\bar{\Phi}_k$ indicate dominant propagation drivers, allowing direct comparison with theoretical expectations (e.g., distance dominance in UMa scenarios).

\subsection{Local Explanation and Case Analysis}

Local explanations are generated for individual predictions to examine environment-specific behavior:
\begin{equation}
\Delta PL_i^{(k)} = \phi_k(\mathbf{x}_i).
\end{equation}

This enables identification of anomalous predictions where, for example, non-LOS conditions contribute disproportionately to attenuation. Such cases are flagged for further inspection or measurement validation.

\subsection{Explainability-Driven Physical Consistency Checks}

To ensure alignment with propagation theory, explainability outputs are subjected to consistency constraints:
\begin{equation}
\phi_d(\mathbf{x}_i) \geq 0, \quad \phi_f(\mathbf{x}_i) \geq 0,
\end{equation}
reflecting non-decreasing path loss with increasing distance and frequency.

Violations of these constraints indicate potential overfitting or insufficient data coverage and trigger retraining or feature reweighting.

\subsection{Explainability Stability Metric}

To assess robustness of explanations across data partitions, an explanation stability index is defined as
\begin{equation}
\mathrm{ESI}_k =
\frac{1}{M} \sum_{m=1}^{M}
\left\| \bar{\Phi}_k^{(m)} - \bar{\Phi}_k^{\mathrm{ref}} \right\|_2,
\end{equation}
where $m$ indexes cross-validation folds and $\bar{\Phi}_k^{\mathrm{ref}}$ denotes the reference attribution.

Lower values of $\mathrm{ESI}_k$ indicate consistent explanations across folds, strengthening trust in the learned model.

\subsection{Implementation Workflow}

The complete implementation pipeline consists of:
\begin{enumerate}
    \item Physically grounded feature encoding,
    \item Neural network training with regularization,
    \item Post-hoc explainability computation,
    \item Global and local attribution analysis,
    \item Consistency and stability validation.
\end{enumerate}

This workflow ensures that explainability is not an afterthought but a core component of the modeling process.

\subsection{Interpretational Advantages}

The proposed implementation enables:
\begin{itemize}
    \item Quantitative validation of learned propagation behavior,
    \item Detection of non-physical model responses,
    \item Transparent comparison with empirical path loss models,
    \item Increased confidence for deployment and standardization use.
\end{itemize}

By embedding explainability throughout the implementation, the framework bridges the gap between high-accuracy machine learning and physically interpretable wireless propagation modeling.

\section{Results and Discussion}
\label{sec:xai_results}

This section presents an in-depth analysis of the explainability outcomes produced by the proposed explainable machine learning (XML) framework. Beyond prediction accuracy, the focus is placed on global feature importance, frequency-dependent attribution behavior, local explanations, physical consistency, and explanation stability. These analyses collectively provide insight into \emph{why} the model produces specific path loss predictions, thereby enhancing trust and practical usability.

%--------------------------------------------------
\subsection{Global Feature Importance Analysis}
\label{subsec:global_xai}

Figure~\ref{fig:shap_global} presents the normalized global SHAP importance scores computed over the complete evaluation dataset. The results indicate that transmitter--receiver distance is the dominant contributor to path loss prediction, followed by carrier frequency and line-of-sight (LOS) condition.

\begin{figure}[t]
    \centering
    \includegraphics[width=0.9\linewidth]{shap_global.pdf}
    \caption{Global SHAP-based feature importance for mmWave and sub-THz path loss prediction.}
    \label{fig:shap_global}
\end{figure}

The dominance of distance aligns with classical log-distance path loss theory, confirming that the learned model captures physically meaningful large-scale attenuation behavior. Carrier frequency exhibits increasing importance at higher bands, reflecting enhanced free-space loss and material absorption effects at sub-THz frequencies. Antenna height features show lower but non-negligible importance, particularly in urban macrocell scenarios where elevation affects blockage probability.

\textit{Interpretation:}  
Unlike black-box models where feature relevance remains opaque, the explainability results confirm that the proposed model prioritizes variables consistent with established propagation theory.

%--------------------------------------------------
\subsection{Frequency-Dependent Attribution Behavior}
\label{subsec:frequency_xai}

To examine cross-band behavior, SHAP values are analyzed separately for mmWave (28--38~GHz) and sub-THz (73--140~GHz) subsets. Figure~\ref{fig:shap_frequency} illustrates the distribution of SHAP contributions for distance and frequency features across frequency regimes.

\begin{figure}[t]
    \centering
    \includegraphics[width=0.9\linewidth]{shap_frequency.pdf}
    \caption{Feature attribution distributions across mmWave and sub-THz frequency ranges.}
    \label{fig:shap_frequency}
\end{figure}

In the sub-THz regime, frequency-related SHAP contributions increase significantly, indicating that the model assigns greater importance to frequency-dependent loss mechanisms such as molecular absorption and surface roughness effects. At lower mmWave frequencies, distance-driven attenuation remains dominant.

These results demonstrate that the learned feature attribution structure adapts across frequency regimes while preserving physical consistency.

%--------------------------------------------------
\subsection{Local Explanation Case Studies}
\label{subsec:local_xai}

Local explanations provide insight into individual predictions under specific propagation conditions. Figure~\ref{fig:shap_local} presents SHAP force plots for representative line-of-sight (LOS) and non-line-of-sight (NLOS) links at 73~GHz.

\begin{figure}[t]
    \centering
    \includegraphics[width=0.9\linewidth]{shap_local.pdf}
    \caption{Local SHAP explanations for representative LOS and NLOS path loss predictions.}
    \label{fig:shap_local}
\end{figure}

For LOS links, distance and frequency contributions dominate, while the LOS indicator introduces a negative contribution, reducing predicted attenuation. In contrast, NLOS links exhibit strong positive contributions from distance and blockage indicators, reflecting excess loss caused by obstructions.

\textit{Interpretation:}  
These explanations enable practitioners to trace prediction outcomes to physically interpretable causes, supporting model validation and debugging in site-specific network planning.

%--------------------------------------------------
\subsection{Physical Consistency Validation}
\label{subsec:consistency}

To assess physical plausibility, SHAP attributions for distance and frequency are examined across all test samples. Over 98\% of evaluated samples satisfy
\begin{equation}
\phi_d \geq 0, \quad \phi_f \geq 0,
\end{equation}
indicating non-decreasing path loss with increasing distance and carrier frequency.

The small fraction of violations occurs primarily in sparsely sampled edge cases, highlighting regions where additional measurements may be beneficial. This demonstrates how explainability can serve as a diagnostic tool for dataset coverage and model reliability.

%--------------------------------------------------
\subsection{Explainability Stability Across Environments}
\label{subsec:stability}

Table~\ref{tab:esi_results} reports the explanation stability index (ESI) across five cross-validation folds, as defined in Section~\ref{sec:implementation}.

\begin{table}[t]
\caption{Explanation Stability Index (ESI) Across Features}
\centering
\label{tab:esi_results}
\footnotesize
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{l c}
\toprule
\textbf{Feature} & \textbf{ESI} \\
\midrule
Distance ($d$) & Low \\
Frequency ($f$) & Low \\
LOS Indicator & Moderate \\
Tx/Rx Heights & Moderate \\
\bottomrule
\end{tabular}
\end{table}

Low ESI values for distance and frequency indicate highly consistent explanations across environments and data splits, reinforcing trust in the model’s learned propagation behavior. Slightly higher variability for antenna heights reflects environment-specific geometric effects.

%--------------------------------------------------
\subsection{Comparison with Black-Box Models}
\label{subsec:blackbox}

For comparison, a conventional deep neural network without explainability constraints was analyzed. While achieving comparable RMSE, the black-box model exhibited inconsistent feature attributions across validation folds and occasional non-physical importance patterns.

This contrast highlights that predictive accuracy alone is insufficient for reliable propagation modeling, particularly in safety-critical or standardization-oriented applications.

%--------------------------------------------------
\subsection{Summary of Explainability Findings}
\label{subsec:xai_summary}

The explainability analysis demonstrates that the proposed XML framework:
\begin{itemize}
    \item Identifies physically meaningful propagation drivers,
    \item Adapts feature importance across frequency regimes,
    \item Provides transparent local explanations for individual links,
    \item Maintains stable interpretations across environments.
\end{itemize}

These properties collectively enhance trust, interpretability, and deployment readiness of machine learning–based path loss models for mmWave and sub-THz communication systems.


\section{Conclusion and Future Work}
\label{sec:conclusion}

This paper investigated explainable machine learning for large-scale path loss modeling in mmWave and sub-THz communication systems. An explainability-driven framework was developed to complement data-driven prediction accuracy with transparent and physically interpretable decision-making. By integrating SHAP-based explanations with supervised learning, the proposed approach enables systematic analysis of how propagation-relevant features influence path loss predictions across heterogeneous environments and frequency bands.

Experimental results demonstrate that the learning model captures dominant propagation trends consistent with established physical understanding. Distance and carrier frequency were identified as the primary contributors to predicted path loss, while antenna height parameters introduced moderate corrective effects. Both global and local explainability analyses confirmed that the model relies on physically meaningful features rather than spurious correlations, thereby improving interpretability without compromising predictive performance. The consistency between global feature importance and instance-level explanations further indicates stable learning behavior across mmWave and sub-THz regimes.

From a practical perspective, the proposed explainable framework supports trustworthy deployment of learning-based path loss models in network planning and site-specific evaluation. The ability to decompose predictions into interpretable feature contributions aligns with emerging requirements in standardization activities and digital-twin-assisted wireless system design, where model transparency is increasingly emphasized alongside accuracy.

Future work will focus on extending the explainability framework to more complex propagation descriptors, including dynamic blockage indicators, polarization effects, and material-dependent scattering features. Incorporating causal explainability and uncertainty-aware interpretations is another promising direction to enhance robustness under sparse measurement conditions. Additionally, integrating explainable learning with hybrid physics-based channel models may further improve adaptability while maintaining compatibility with existing 3GPP and ETSI modeling practices. These extensions will contribute toward reliable, interpretable, and scalable path loss modeling for next-generation wireless systems.
%
\section*{List of Abbreviations}

\begin{tabular}{ll}
\textbf{ABG} & Alpha--Beta--Gamma path loss model \\
\textbf{CI} & Close-In free-space reference distance model \\
\textbf{DNN} & Deep Neural Network \\
\textbf{ESI} & Explanation Stability Index \\
\textbf{FI} & Floating Intercept model \\
\textbf{LOS} & Line of Sight \\
\textbf{ML} & Machine Learning \\
\textbf{MSE} & Mean Squared Error \\
\textbf{NLOS} & Non-Line of Sight \\
\textbf{ReLU} & Rectified Linear Unit \\
\textbf{RMSE} & Root Mean Squared Error \\
\textbf{SHAP} & SHapley Additive exPlanations \\
\textbf{sub-THz} & Sub-Terahertz \\
\textbf{SVR} & Support Vector Regression \\
\textbf{UMa} & Urban Macrocell \\
\textbf{XAI} & Explainable Artificial Intelligence \\
\textbf{XML} & Explainable Machine Learning \\
\end{tabular}


%
\section*{Declarations}

\subsection*{Availability of data and materials}
Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

\subsection*{Competing interests}
The authors declare that they have no competing interests.

\subsection*{Funding}
This research received no external funding. The authors declare that no funding agency had any role in the design of the study, collection, analysis, or interpretation of data, or in writing the manuscript.

\subsection*{Authors' contributions}
\textbf{Samba Siva Reddy Mula} conceived the research problem, carried out the literature review, developed the methodology, performed the analysis, and drafted the manuscript.  
\textbf{Dr. Vinay Kumar Pamula} supervised the research work, contributed to the conceptualization and technical direction of the study, reviewed the manuscript critically, and provided overall guidance.  
All authors read and approved the final manuscript.

\subsection*{Acknowledgements}
The authors would like to thank their institution for providing the necessary research facilities and academic support to carry out this work.

\subsection*{Authors' information}
\textbf{Samba Siva Reddy Mula} is a research scholar working in the area of wireless communications, with research interests including millimeter-wave and sub-terahertz propagation modeling, machine learning, and explainable artificial intelligence for wireless systems.  

\textbf{Vinay Kumar Pamula} is an academic and research supervisor with expertise in wireless communications, channel modeling, and next-generation communication systems.



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