Act as a senior IEEE journal reviewer and rigorous...

Act as a senior IEEE journal reviewer and rigorously revise the Methodology section of my manuscript to ensure it meets the highest academic standards. Perform the following tasks:

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1. Academic Audit Criteria

a. Mathematical Correctness

• Verify all equations for formal rigor (e.g., convergence proofs in Appendix).
• Check consistency between equations and pseudocode.

b. Logical Coherence

• Ensure hierarchical flow:
  • Problem formalization → Bayesian framework → Uncertainty fusion → Threshold design.
• Validate that assumptions are explicitly stated and justified.

c. Novelty & Contribution Clarity

• Highlight technical innovations against prior work.
• Remove redundant derivations of known methods.

d. Reproducibility

• Specify hyperparameters.
• Clarify signal preprocessing steps.

e. Notation Consistency

• Ensure variables (e.g., $\sigma_t^2$, $\Gamma_t$) are defined before use and match Appendix/Experiments.
• Replace ambiguous symbols (e.g., $X$ for signals) with domain-standard notation (e.g., $s(t)$).

f. Citation Support

• Ground methodological choices in literature.

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2. Revision Tasks

• Enhance Derivations: Add missing steps for critical equations.
• Fix Logical Gaps.
• Strengthen Academic Tone: Replace informal analogies with formal definitions (e.g., "dynamic" → "time-varying Lipschitz continuity").
• Annotate Changes: Use LaTeX comments (% Review: [Reason]) to explain key modifications.

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Input Manuscript:

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\begin{document}

\title{Bayesian LSTM-Based Detection of Sudden Anomalies in Urban Wireless Signals}

\author{
Bin~Li\textsuperscript{o},~\textit{Member, IEEE},
Tengyu~Guo\textsuperscript{o},~\textit{Student Member, IEEE}, 
and~Ruonan~Zhang\textsuperscript{o},~\textit{Member, IEEE}
}

\markboth{IEEE Internet of Things Journal, Vol. 01, No. 02, August 2025}%
{Li \MakeLowercase{\textit{et al.}}: Bayesian LSTM-Based Detection of Sudden Anomalies in Urban Wireless Signals}

\maketitle

\begin{abstract}
Short-lived anomalies in urban wireless signals can severely disrupt IoT applications, causing sporadic packet losses, high latency, or even complete outages. Traditional threshold-based methods often fail to adapt to the complex, rapidly fluctuating conditions found in dense city deployments. To address these challenges, we propose a \textbf{Bayesian LSTM-based} approach that detects diverse types of burst anomalies (e.g., impulsive spikes, narrowband jammers, repeated micro-spikes) while explicitly quantifying model uncertainty via \textbf{Monte Carlo (MC) dropout}. By modeling a variational approximation of the weight posterior, our approach yields a predictive variance that enables a dynamic thresholding strategy, which automatically adjusts to channel variability and signal-level features. Experiments on a synthetic yet realistic bursty urban wireless dataset---augmented with a small real-world dataset---demonstrate consistent improvements in F1-score and lower calibration error compared to deterministic LSTM baselines, a CNN-LSTM hybrid, a parameter-matched transformer-based model, and classical anomaly detection methods (e.g., one-class SVM, isolation forest). Additionally, we show that a small number of MC samples (e.g., $N=5$) strikes a desirable balance between robust anomaly detection and real-time feasibility on resource-limited edge devices. Our findings suggest that this Bayesian framework can serve as a practical, uncertainty-aware solution for next-generation smart city networks. We provide pseudo-code for data synthesis in the Appendix and plan to release our source code under an open-source license for full reproducibility.
\end{abstract}

\begin{IEEEkeywords}
Bayesian LSTM, anomaly detection, urban wireless signals, Monte Carlo dropout, uncertainty quantification, edge computing.
\end{IEEEkeywords}

\section{Introduction}
\label{sec:intro}

Urban wireless communication infrastructures are becoming increasingly dense and heterogeneous, driven by the explosive growth of Internet of Things (IoT) devices, 5G small-cell deployments, Wi-Fi hotspots, and interconnected sensor networks. In such crowded environments, \emph{short-lived or burst anomalies} in the received signal—ranging from impulsive interference spikes to narrowband jammers—can trigger abrupt changes in link quality. These transient events often lead to severe performance degradation, including sporadic packet failures, elevated latency, or even localized communication outages~\cite{traditional_RL}. Such disruptions are especially detrimental for mission-critical IoT applications (e.g., public safety, industrial automation, or intelligent transportation systems) that rely on consistent, low-latency data delivery.

Compared to rural or less-congested scenarios, urban wireless environments exhibit higher interference levels and more complex multipath fading. Moreover, as urban networks continue to scale (e.g., city-wide sensor deployments), the \emph{dynamic range} of short-burst interference events expands. Traditional threshold-based anomaly detectors often falter in these conditions because they assume relatively static noise floors or slowly varying channels. In practice, city deployments encounter highly non-stationary interference, with momentary bursts that may originate from a variety of devices such as Bluetooth beacons, microwaves, rogue access points, or malicious jammers.

Recently, many municipal governments and industrial stakeholders have reported that unpredictable spikes in crowded frequency bands, such as the 2.4\,GHz ISM band, can undermine IoT services and cause undesired operational costs. For example, a smart traffic system relying on real-time data may suffer from packet drops of over 10\% at random intervals, thereby destabilizing critical services like traffic light scheduling. These real-world observations highlight the urgency of developing sophisticated, adaptive detection schemes that can recognize brief but impactful anomalies without overwhelming the system with false alerts.

\subsection{Challenges in Urban Wireless Anomaly Detection}
\label{sec:challenges}

Detecting short-burst anomalies in urban wireless signals poses multiple challenges. First, the noise floor itself can fluctuate considerably due to factors such as weather conditions, user mobility, building reflections, and overlapping transmissions from a multitude of devices. Hence, any static threshold set to detect an anomaly in one environment may fail in another setting or at a different time of day. Second, the variety of interference sources is large, ranging from periodic micro-spikes emanating from sensor beacons, to impulsive bursts caused by radar pulses, to continuous narrowband jammers. A one-size-fits-all approach based on simple amplitude checks or spectral scans tends to exhibit poor generalization across these diverse anomaly types.

Third, acquiring extensive labeled data of \emph{all} possible short-burst anomalies is challenging. Such rare events often require significant effort to capture, especially if they stem from sporadic or illegal transmissions. Consequently, supervised models risk overfitting if they lack mechanisms to quantify uncertainty about unseen anomaly patterns. Traditional machine learning pipelines relying on handcrafted features also struggle to adapt to new or evolving interference profiles. Finally, real-time detection on resource-limited edge devices adds another layer of complexity: the algorithm must be both robust and computationally efficient, meeting tight latency constraints (often under a few milliseconds) while consuming minimal power and memory.

\subsection{Proposed Solution: Bayesian LSTM with MC Dropout}
To address these issues, we propose a \textbf{Bayesian LSTM}-based anomaly detector that leverages MC (Monte Carlo) dropout for approximate Bayesian inference~\cite{gal2016dropout}. By treating dropout as a variational approximation of the weight posterior and \emph{keeping dropout active at inference time}, we can sample multiple forward passes, effectively capturing epistemic (model) uncertainty. This approach enables the model to produce not just a classification (normal vs.\ anomaly), but also a predictive \emph{variance}, indicating its confidence. High variance often suggests that the current input lies in a region of the feature space not well represented by the training data or includes novel forms of interference.

Our framework leverages this estimated predictive uncertainty to drive a \emph{dynamic thresholding} scheme that adjusts to changing noise conditions and different levels of channel degradation. The threshold becomes higher in regions of elevated model or signal-level uncertainty, mitigating the risk of raising false alarms for borderline anomalies. Conversely, in stable conditions with low uncertainty, the threshold is lowered to increase sensitivity to subtle transient anomalies. We further integrate additional signal-level features (e.g., subband energy) to refine the threshold decision, thereby combining both data-driven and domain knowledge about the wireless environment.

\subsection{Contributions and Paper Organization}
\noindent \textbf{Contributions of this work are fourfold:}
\begin{itemize}
    \item We introduce a \textbf{Bayesian LSTM} approach for short-burst anomaly detection in dense urban wireless signals, employing MC dropout to approximate the network weight posterior.
    \item We propose a \textbf{dynamic thresholding} mechanism guided by both signal-level and model-derived variances, reducing overconfidence in non-stationary interference scenarios.
    \item We present an extensive \textbf{empirical evaluation} on a mixed dataset (synthetic urban channel simulations augmented with real-world measurements), demonstrating superior performance (F1-score, calibration error) compared to deterministic LSTM, CNN-LSTM hybrids, transformer-based methods, and classical anomaly detection approaches.
    \item We discuss \textbf{edge deployment feasibility}, including memory footprint ($\approx 240$\,KB in float32) and sub-2\,ms latency on embedded hardware, making the solution suitable for resource-constrained environments.
\end{itemize}

The rest of this paper is organized as follows. Section~\ref{sec:related} surveys the relevant literature in wireless anomaly detection, Bayesian deep learning, and edge optimizations. Section~\ref{sec:methodology} details our redesigned Bayesian LSTM-based detection architecture, including MC dropout placement, the variational loss formulation, and the unified dynamic threshold. In Section~\ref{sec:exp}, we present experimental results on both synthetic and real-world datasets, comparing performance against multiple baselines and classical anomaly detectors. Section~\ref{sec:discussion} provides insights into threshold tuning, interpretability of uncertainty, and real-world deployment. Finally, Section~\ref{sec:conclusion} concludes the paper and outlines future directions, including extending to overlapping anomalies and augmenting real-world validations.

\section{Related Work}
\label{sec:related}

\subsection{Uncertainty-Aware Methods in Wireless Networks}
The growing complexity of modern wireless networks has spurred interest in methods that explicitly manage uncertainty. Bayesian techniques have long been recognized for incorporating prior knowledge and quantifying confidence in parameter estimates~\cite{bayesian_wsns,bayesian_mimo}. In traditional wireless sensor networks, for example, Bayesian filters refine estimates of sensor measurements or link characteristics in the face of noise and fading. However, many legacy Bayesian methods focus on slowly varying processes and are not optimized for short, transient interference spikes in dense deployments.

More recently, Bayesian deep learning has emerged as a powerful framework for modeling uncertainties in neural network parameters and outputs~\cite{gal2016dropout}. Within the wireless domain, researchers have applied approximate Bayesian inference to tasks such as channel estimation and dynamic spectrum access~\cite{bayesian_mimo}. Nonetheless, much of this work targets slowly varying signals, overlooking the rapid time scales associated with impulsive anomalies. Additionally, real-time constraints on edge devices often limit the complexity of Bayesian methods—full-blown Markov Chain Monte Carlo (MCMC) can be too costly. MC dropout, by contrast, offers a tractable means to approximate the parameter posterior in real time without drastically altering the inference pipeline.

\subsection{Deep Learning for Anomaly Detection}
Deep learning approaches have gained prominence in anomaly detection for time-series data, including network traffic monitoring, industrial IoT fault detection, and sensor data analytics~\cite{cnn_sigdetect,lstm_det_1,attention_time_series}. Convolutional neural networks (CNNs) excel at extracting local patterns (e.g., short spikes in the signal amplitude), while recurrent networks like LSTMs can model longer-term temporal dependencies. Hybrid CNN-LSTM architectures combine these strengths by capturing both local structure and temporal evolution~\cite{cnn_lstm_hybrid}.

Despite their success, these deep models usually produce \emph{deterministic} outputs, which can lead to overconfidence—especially in underrepresented regions of the input space. Such overconfidence can inflate false alarms or suppress valid anomalies when encountering novel interference profiles. Moreover, deterministic models commonly operate with fixed thresholds for anomaly detection, which can fail under abrupt shifts in noise levels or interference patterns.

\subsection{Bayesian Models for IoT Anomaly Detection}
An expanding body of work has explored Bayesian neural networks for uncertainty-aware anomaly detection in IoT systems~\cite{bayesian_iot1,bayesian_health}. For example, Bayesian autoencoders can detect gradual changes in multivariate sensor readings, while Bayesian RNNs have been studied for fault detection in healthcare IoT environments. However, these models typically focus on longer-duration anomalies (e.g., equipment drifts or user behavioral changes), rather than the short, high-intensity bursts typical of dense urban interference scenarios. In addition, full Bayesian methods can be computationally prohibitive on embedded hardware. MC dropout mitigates these limitations by performing multiple stochastic forward passes with modest overhead.

\subsection{Limitations of Existing Threshold-Based Approaches}
Threshold-based schemes remain popular for anomaly detection in wireless networks due to their simplicity. Yet, static or semi-static thresholds are particularly vulnerable to false alarms in urban settings, where signals often exhibit large, rapid fluctuations. Adaptive thresholds that track local statistics can offer some benefits, but they struggle with non-stationary fading and diverse interference sources. Heuristic schemes often require manual tuning to ensure generality across various channels or device densities.

In contrast, our proposed approach combines a data-driven Bayesian representation with a \emph{dynamic threshold} policy that leverages model-derived and signal-level uncertainties. The threshold thus becomes a flexible, context-aware boundary that responds to the model’s confidence, as well as additional signal-level cues like subband energy.

\subsection{Edge Computing and Real-Time Constraints}
Finally, the shift toward edge computing in IoT systems necessitates models that are both accurate and lightweight~\cite{trans-edge}. The emergence of embedded GPUs (e.g., NVIDIA Jetson) and specialized AI accelerators makes it possible to run moderately sized deep networks on the edge, yet memory and energy remain prime concerns for large-scale deployments. In this paper, we use a single-layer LSTM architecture with around 60k parameters, carefully balancing performance and resource consumption. Empirical tests confirm that our Bayesian LSTM can operate within sub-2\,ms latency on modern embedded hardware, validating its real-time applicability in dense IoT networks.

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

In this section, we introduce a novel uncertainty-aware mathematical framework to detect \emph{burst anomalies} in dense urban wireless environments. Unlike conventional techniques with static thresholds, we propose a \textbf{unified dynamic threshold} driven by both \emph{signal-level uncertainty} (to model impulsive interference, narrowband jammers, and short micro-spikes) and \emph{model uncertainty} from a \emph{Bayesian LSTM}. We present all formulations in detail (\S\ref{subsec:signal_uncertainty}--\S\ref{subsec:dyn_threshold}), culminating in our final anomaly decision rule (\S\ref{subsec:decision_rule}). Proofs of the main theoretical results appear in the Appendix (see \S\ref{appendix:proofs}) with cross-references to the equations below.
\begin{figure*}[t]
  \centering
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  % ----- PREPROCESSING NODE -----
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        text width=3cm,
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    \textbf{Preprocessing:}\\
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    subband energy ($\eta_t$),\\
    background deviation\\
    ($|x_t - \widehat{\mu}_{\text{bg}}|^2$)
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  % ----- BAYESIAN LSTM NODE -----
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    \textbf{Bayesian LSTM:}\\
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    $\{\hat{y}_t^{(j)}\}_{j=1}^N$\\[4pt]
    \emph{\scriptsize
    Predictive Mean $\overline{y}_t$,\\
    Predictive Variance $\sigma_t^2$}
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    $\widehat{z}_t = \begin{cases}
      1, & \alpha_t \ge \Gamma_t\\
      0, & \text{otherwise}
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  \end{tikzpicture}
  \caption{Proposed Bayesian LSTM anomaly detection pipeline. After preprocessing (STFT and feature scaling), MC dropout produces multiple stochastic forward passes. The system fuses model variance with signal-level metrics to form a dynamic threshold, enabling final anomaly decisions.}
  \label{fig:bayes_pipeline}
\end{figure*}

\subsection{Overview of Signal Model and Bursty Anomalies}
\label{subsec:signal_uncertainty}

We assume a baseband representation of the received signal in discrete time, denoted by $x[t]$ for $t = 1,2,\dots,T$. Let $\Delta t$ be the sampling interval. For clarity, we drop the bracket notation and write $x_t\equiv x[t]$. Formally, we represent the received signal as
\begin{equation}
  x_{t} \;=\; h_{t}\, s_{t} \;+\; \nu_{t}, 
  \label{eq:3.1}
\end{equation}
where $h_{t}$ is a (potentially complex-valued) fading coefficient, $s_{t}$ is the transmitted symbol or subband signal, and $\nu_{t}$ denotes additive noise. In dense urban settings, $h_t$ may exhibit fast time selectivity due to multipath and mobility, while $\nu_t$ incorporates background thermal noise and unmodeled interference.

\subsubsection{Impulsive Interference Representation}
\label{sec:impulsive_interference}
We define \textbf{impulsive bursts} as abrupt amplitude excursions. Let $\mathcal{I}_t$ be an impulsive disturbance:
\begin{equation}
  \mathcal{I}_t \;=\; A_t \,\delta_t,
  \label{eq:3.2}
\end{equation}
where $A_t > 0$ is the random amplitude, and $\delta_t\in\{0,1\}$ indicates whether the impulse is active at time $t$. We further assume
\begin{equation}
  A_t \;\sim\; \mathrm{HeavyTail}(\alpha),
  \quad
  \Pr(\delta_t = 1)\;=\;\rho.
  \label{eq:3.3}
\end{equation}
When $\delta_t=1$, a high-amplitude spike emerges. The overall instantaneous received signal becomes
\begin{equation}
  x_{t} \;=\; h_{t}\, s_{t} \;+\; \nu_{t} \;+\; \mathcal{I}_t.
  \label{eq:3.4}
\end{equation}

\subsubsection{Narrowband Jammers}
\label{sec:narrow_jammers}
We next consider \textbf{narrowband jammers} occupying a limited subband (indexed by $f\in[f_{\mathrm{lo}}, f_{\mathrm{hi}}]$). Denote the subband energy of $x_t$ by
\begin{equation}
  \eta_t \;=\;\int_{f_{\mathrm{lo}}}^{f_{\mathrm{hi}}} \bigl|\mathcal{F}\{ x_\tau \} (f)\bigr|^2 \, df,
  \label{eq:3.5}
\end{equation}
where $\mathcal{F}\{\cdot\}$ denotes the discrete-time Fourier transform over a local window near $t$. A narrowband jamming process $\mathcal{J}_t$ is then modeled as:
\begin{equation}
  \mathcal{J}_t \;=\; g(t)\,\mathbf{1}\{\eta_t \ge \zeta\},
  \label{eq:3.6}
\end{equation}
where $g(t)$ is a (possibly time-varying) amplitude function and $\zeta>0$ is a threshold indicating significant subband power. When $\mathcal{J}_t$ is active, the received signal’s subband region is dominated by the jammer energy.

\subsubsection{Micro-Spike Trains}
\label{sec:microspikes}
\textbf{Micro-spikes} are a distinct type of short interference bursts emitted periodically by certain IoT devices (e.g., sensor beacons). Let $\mathcal{M}_t$ represent a micro-spike train:
\begin{equation}
  \mathcal{M}_t \;=\;\sum_{k=-\infty}^{\infty} \phi \,\mathbf{1}\{t = k\,T_0\},
  \label{eq:3.7}
\end{equation}
where $T_0$ is the micro-spike spacing, and $\phi$ is the typical amplitude. Because these bursts repeat periodically, they are distinct from the random impulses $\mathcal{I}_t$. In practice, we may observe a superposition of all three anomalies:
\begin{equation}
  x_t \;=\; h_{t}\,s_{t}\;+\;\nu_{t}\;+\;\mathcal{I}_t\;+\;\mathcal{J}_t\;+\;\mathcal{M}_t.
  \label{eq:3.8}
\end{equation}

\subsection{Comparative Timeline of Anomaly Types}
Figure~\ref{fig:three_anomalies} illustrates typical time profiles for impulsive spikes, narrowband jammers, and micro-spikes, clarifying their durations and periodicities.

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    \end{groupplot}
  \end{tikzpicture}
  
  \caption{Illustration of three distinct anomaly types in the time domain. 
  \textbf{Left:} A single impulsive spike with steep rise/fall. 
  \textbf{Center:} A narrowband jammer with sustained elevated amplitude in a subband. 
  \textbf{Right:} A micro-spike train occurring at regular intervals.}
  \label{fig:three_anomalies}
\end{figure*}

\subsection{Uncertainty Measures}
\label{subsec:uncert_measures}

We now define two major sources of uncertainty for anomaly detection:

\subsubsection{Signal-Level Uncertainty}
Denote by $\mathcal{U}_s(t)$ the instantaneous \emph{signal-level uncertainty} that captures multi-anomaly contributions in \eqref{eq:3.8}. One possible definition is:
\begin{equation}
  \mathcal{U}_s(t) \;=\;
  \underbrace{\bigl|x_t - \widehat{\mu}_{\text{bg}}\bigr|^2}_{\text{Deviation from background mean}}
  \;+\;
  \underbrace{\kappa \,\max_{f\in\mathcal{B}} \bigl|\mathcal{F}\{ x_\tau \}(f)\bigr|^2}_{\text{Max subband magnitude squared}},
  \label{eq:3.9}
\end{equation}
where $\widehat{\mu}_{\text{bg}}$ is an empirical background mean (over a sliding window with no anomalies), $\kappa>0$ is a scaling factor, and $\mathcal{B}\subset\mathbb{R}$ a relevant band. The first term detects amplitude spikes, and the second identifies strong spectral peaks (narrowband jammers or micro-spikes).

\subsubsection{Model (Bayesian) Uncertainty}
\label{subsec:bayesian_lstm}
We also incorporate \emph{model uncertainty} via a Bayesian LSTM. Let $\mathbf{W}$ be the trainable weight parameters, with prior $p(\mathbf{W})$. During inference, we sample from a variational posterior $q(\mathbf{W}\mid \theta)\approx p(\mathbf{W}\mid x_{1{:}T})$. For each time $t$, the LSTM outputs $\hat{y}_t\in[0,1]$ (an anomaly score), but we keep the dropout active to approximate sampling from $q(\mathbf{W}\mid \theta)$. Formally:
\begin{equation}
  \hat{y}_t^{(j)} \;=\; \mathcal{F}\bigl(x_{1{:}t}; \mathbf{W}^{(j)}\bigr), 
  \quad j=1,\dots,N.
  \label{eq:3.10}
\end{equation}
\textit{Predictive mean} and \textit{variance} can be computed as
\begin{equation}
  \overline{y}_t \;=\; \frac{1}{N}\sum_{j=1}^{N} \hat{y}_t^{(j)},
  \label{eq:3.11}
\end{equation}
\begin{equation}
  \sigma_t^2 \;=\; \frac{1}{N}\sum_{j=1}^{N} \Bigl(\hat{y}_t^{(j)} - \overline{y}_t\Bigr)^2.
  \label{eq:3.12}
\end{equation}
Here, $\overline{y}_t$ serves as an ensemble anomaly score, whereas $\sigma_t^2$ quantifies epistemic (model) uncertainty.

\subsection{Bayesian LSTM Recurrence}
\label{subsec:lstm_recur}
For completeness, let us specify the LSTM equations with dropout-based sampling. At each time step $t$,
\begin{figure}
  \centering
  \includegraphics[
    width=0.5\textwidth,
  ]{LSTM cell.pdf}
  \caption{Units in long short-term memory networks.} 
  \label{fig1}
  \end{figure}
\begin{equation}
  \mathbf{i}_t = \sigma\bigl(\mathbf{M}_x\mathbf{W}_{xi} x_t + \mathbf{M}_h\mathbf{W}_{hi} \mathbf{h}_{t-1} + \mathbf{b}_i\bigr),
  \label{eq:3.13}
\end{equation}
\begin{equation}
  \mathbf{f}_t = \sigma\bigl(\mathbf{M}_x\mathbf{W}_{xf} x_t + \mathbf{M}_h\mathbf{W}_{hf} \mathbf{h}_{t-1} + \mathbf{b}_f\bigr),
  \label{eq:3.14}
\end{equation}
\begin{equation}
  \mathbf{o}_t = \sigma\bigl(\mathbf{M}_x\mathbf{W}_{xo} x_t + \mathbf{M}_h\mathbf{W}_{ho} \mathbf{h}_{t-1} + \mathbf{b}_o\bigr),
  \label{eq:3.15}
\end{equation}
\begin{equation}
  \tilde{\mathbf{c}}_t = \tanh\bigl(\mathbf{M}_x\mathbf{W}_{xg} x_t + \mathbf{M}_h\mathbf{W}_{hg} \mathbf{h}_{t-1} + \mathbf{b}_g\bigr),
  \label{eq:3.16}
\end{equation}
\begin{equation}
  \mathbf{c}_t = \mathbf{f}_t \odot \mathbf{c}_{t-1} \;+\; \mathbf{i}_t \odot \tilde{\mathbf{c}}_t,
  \label{eq:3.17}
\end{equation}
\begin{equation}
  \mathbf{h}_t = \mathbf{o}_t \odot \tanh(\mathbf{c}_t),
  \label{eq:3.18}
\end{equation}
where $\mathbf{i}_t,\mathbf{f}_t,\mathbf{o}_t$ are input, forget, and output gates, $\tilde{\mathbf{c}}_t$ is the candidate cell state, and $\mathbf{M}_x,\mathbf{M}_h$ are random dropout masks (Bernoulli), drawn anew at each forward pass. The final output layer for anomaly scoring is typically
\begin{equation}
  \hat{y}_t^{(j)} \;=\;\sigma\!\bigl(\mathbf{w}_{\mathrm{out}}^T \mathbf{h}_t + b_{\mathrm{out}}\bigr).
  \label{eq:3.19}
\end{equation}

\subsection{Unified Dynamic Threshold}
\label{subsec:dyn_threshold}

Given the signal-level uncertainty $\mathcal{U}_s(t)$ in \eqref{eq:3.9} and model uncertainty $\sigma_t^2$ in \eqref{eq:3.12}, we propose a unified threshold function:
\begin{equation}
  \Gamma_t \;=\; \mathcal{G}\Bigl(\mathcal{U}_s(t),\,\sigma_t^2;\,\boldsymbol{\beta}\Bigr),
  \label{eq:3.20}
\end{equation}
where $\boldsymbol{\beta} = (\beta_0,\beta_1,\dots,\beta_5)$ is a parameter vector. One specific instantiation is:
\begin{IEEEeqnarray}{rCl}
  \Gamma_t &=&
  \beta_0
  \;+\;\beta_1 \,\mathcal{U}_s(t)
  \;+\;\beta_2\, \sigma_t^2 
  \nonumber\\
  &&+\;\beta_3 \,\exp\bigl(-\beta_4 \,\|\mathbf{h}_t - \mathbf{h}_{t-1}\|^2\bigr)
  \;+\;\beta_5 \,\frac{1}{1+\sigma_t^2}.
  \label{eq:3.21}
\end{IEEEeqnarray}
High $\mathcal{U}_s(t)$ or $\sigma_t^2$ raises the threshold, preventing spurious alarms under large or uncertain conditions. Meanwhile, rapid changes in the hidden state $\mathbf{h}_t$ (captured by $\|\mathbf{h}_t - \mathbf{h}_{t-1}\|^2$) may transiently adjust the threshold depending on $\beta_3$ and $\beta_4$.

\subsubsection{Combined Variance Perspective}
For computational simplicity, one may define a \textit{combined variance} $\varsigma_t^2$:
\begin{equation}
  \varsigma_t^2 \;=\; \sigma_t^2 \;+\; \lambda_0\, \widetilde{\eta}_t,
  \label{eq:3.22}
\end{equation}
where $\widetilde{\eta}_t$ is a normalized subband energy (similar to \eqref{eq:3.5}), and $\lambda_0\ge 0$. Substituting $\varsigma_t^2$ into \eqref{eq:3.21} yields an alternative threshold function:
\begin{equation}
  \Gamma_t^\prime \;=\; \beta_0 + \beta_1 \,\mathcal{U}_s(t) + \beta_2\, \varsigma_t^2 \;+\;\dots
  \label{eq:3.23}
\end{equation}
\begin{algorithm}[!t]
  \caption{Dynamic Threshold Fusion}
  \label{alg:threshold_fusion}
  \begin{algorithmic}[1]
  \REQUIRE MC samples $N$, LSTM output $\hat{y}_t^{(j)}$, signal measure $\mathcal{U}_s(t)$, hyperparameters $\{\beta_i\}$, $\upsilon_0$
  \STATE \textbf{Compute predictive mean and variance:}
  \begin{align*}
  &\overline{y}_t \leftarrow \frac{1}{N}\sum_{j=1}^N \hat{y}_t^{(j)}, \quad
  \sigma_t^2 \leftarrow \frac{1}{N}\sum_{j=1}^N (\hat{y}_t^{(j)} - \overline{y}_t)^2
  \end{align*}
  \STATE \textbf{Compute raw anomaly score:} 
  $\alpha_t \leftarrow \overline{y}_t + \upsilon_0\,\mathcal{U}_s(t)$
  \STATE \textbf{Compute dynamic threshold:}
  \begin{align*}
  \Gamma_t \leftarrow \beta_0 + \beta_1\,\mathcal{U}_s(t)
  \;+\;\beta_2\,\sigma_t^2
  \end{align*}
  \STATE \textbf{Anomaly decision:}
  \begin{align*}
  \widehat{z}_t \leftarrow \begin{cases}
      1, & \text{if } \alpha_t \ge \Gamma_t, \\
      0, & \text{otherwise.}
  \end{cases}
  \end{align*}
  \RETURN $\widehat{z}_t$
  \end{algorithmic}
  \end{algorithm}

\subsection{Overall Anomaly Score and Decision}
\label{subsec:decision_rule}
We define a \textit{raw anomaly score} $\alpha_t$:
\begin{equation}
  \alpha_t \;=\; \overline{y}_t \;+\; \upsilon_0 \,\mathcal{U}_s(t),
  \label{eq:3.25}
\end{equation}
where $\upsilon_0\in [0,1]$ merges the Bayesian LSTM prediction $\overline{y}_t$ with the signal-level measure $\mathcal{U}_s(t)$. Then the final decision is:
\begin{equation}
  \widehat{z}_t \;=\;\begin{cases}
    1, & \text{if }\alpha_t \;\ge\; \Gamma_t, \\
    0, & \text{otherwise}.
  \end{cases}
  \label{eq:3.26}
\end{equation}
Equations \eqref{eq:3.20}--\eqref{eq:3.26} define a flexible, context-aware detection boundary that dynamically adapts to changes in model confidence and observed signal fluctuations.

\subsection{Bayesian Objective and Training}
\label{subsec:bayesian_training}
Given time-series $\mathbf{x}$ and corresponding anomaly labels $\mathbf{z}\in\{0,1\}$, our training objective is a negative \emph{variational lower bound} plus a calibration penalty. Specifically, define the approximate posterior $q(\mathbf{W}\mid \theta)$ and prior $p(\mathbf{W})$. The negative Evidence Lower Bound (ELBO) is:
\begin{equation}
  \mathcal{L}_{\mathrm{ELBO}} \;=\; -\sum_{t=1}^T \log p(z_t \mid x_{1{:}t}, \mathbf{W})
  \;+\;
  \mathrm{KL}\bigl(q(\mathbf{W}\mid \theta)\,\|\,p(\mathbf{W})\bigr).
  \label{eq:3.27}
\end{equation}
We add a regularization term that penalizes unbounded predictive variance:
\begin{equation}
  \mathcal{R}_{\mathrm{var}} \;=\; \lambda_1\sum_{t=1}^T \sigma_t^2,
  \label{eq:3.28}
\end{equation}
where $\sigma_t^2$ is as in \eqref{eq:3.12}, and $\lambda_1>0$ is a small weighting constant. The final objective becomes
\begin{equation}
  \mathcal{J} \;=\; \mathcal{L}_{\mathrm{ELBO}} + \mathcal{R}_{\mathrm{var}},
  \label{eq:3.29}
\end{equation}
minimized with respect to variational parameters $\theta$ and any threshold parameters $\boldsymbol{\beta}$.

\subsubsection{Threshold Parameter Optimization}
\label{sec:threshold_opt}
We may optimize the vector $\boldsymbol{\beta}$ in \eqref{eq:3.20} or \eqref{eq:3.21} either by grid search or by including it in the gradient-based training:
\begin{equation}
  \boldsymbol{\beta}^\ast \;=\;\arg\min_{\boldsymbol{\beta}} \Bigl[\mathcal{J}(\theta,\boldsymbol{\beta}) + \lambda_2 \|\boldsymbol{\beta}\|^2\Bigr],
  \label{eq:3.30}
\end{equation}
where $\lambda_2>0$ is a small regularization constant to avoid overfitting the threshold parameters.

\subsection{Summary}
We have introduced a comprehensive \emph{uncertainty-aware} framework for burst anomaly detection. The synergy between $\mathcal{U}_s(t)$ in \eqref{eq:3.9} and the Bayesian variance $\sigma_t^2$ in \eqref{eq:3.12} underpins a robust, dynamic threshold $\Gamma_t$ in \eqref{eq:3.20}, culminating in the final decision rule \eqref{eq:3.26}. In the next section, we experimentally demonstrate its advantages over purely deterministic or static-threshold approaches. Proofs of stability, existence of solutions, and additional derivations for the above equations are deferred to Appendix~\ref{appendix:proofs}.

\section{Experiments and Results}
\label{sec:exp}

In this section, we present an extensive evaluation of our proposed Bayesian LSTM-based anomaly detection framework. We compare against both \emph{classical thresholding methods} and state-of-the-art (SOTA) deep learning baselines, provide \emph{ablation studies} on our key components (dynamic thresholding, MC Dropout), and detail additional \emph{robustness tests} (varying SNR, adversarial jamming). Moreover, we report comprehensive metrics (accuracy, F1, \textbf{Brier Score}, \textbf{ECE}), show \emph{statistical significance} via confidence intervals and $p$-values, analyze \emph{failure cases}, and discuss how our simulations comply with \textbf{3GPP TR 38.901} channel models. We also label all simulated results and describe our publicly available code and data repository to ensure reproducibility.

\subsection{Datasets and Experimental Setup}
\label{subsec:datasets}

\subsubsection{Synthetic Urban Wireless Data}
\label{subsec:syntheticdata}
We generate a \textbf{simulated} urban wireless dataset following \emph{3GPP TR 38.901} channel models for dense urban micro-cellular scenarios. Specifically, we incorporate Rayleigh and Rician fading profiles under varying Doppler spreads (up to $120$\,km/h). We superimpose short-burst anomalies of three types (impulsive spikes, narrowband jammers, and micro-spike trains) via our custom anomaly-injection scripts. The default \emph{SNR range} is \{5, 10, 15, 20, 25, 30\}\,dB, reflecting moderate to high interference conditions. 
We randomly inject anomalies at different rates (1--10\% of total samples) to mimic sporadic interference events. Each sample in the time series is labeled ``normal'' or ``anomalous.'' 
All system parameters (path loss exponents, fading speeds, noise floors) follow or extend guidelines from 3GPP TR 38.901, ensuring realistic wide-area conditions. We split this synthetic dataset into training, validation, and testing sets in a 70/15/15 ratio.

\subsubsection{Small Real-World 2.4\,GHz Dataset}
We include a small real-world capture (5\% of total) from a city testbed that monitors the 2.4\,GHz ISM band; this testbed is deployed in compliance with local regulations. The environment exhibits sporadic Wi-Fi interference and occasional Bluetooth collisions. Although relatively small, these measurements help cross-validate our model on practical waveforms. We merge these real traces into the final training/testing pools for integrated evaluation.

\subsubsection{Adversarial Dataset for Robustness Tests}
To test adversarial resilience, we generate an additional scenario where a malicious narrowband jammer adaptively shifts its center frequency to overlap with peak subband energies. We also introduce random phase collisions to confuse the model. A portion of our test data (10\% of samples) includes these adversarial segments; the labels remain binary (anomaly vs.\ normal).

\subsubsection{Implementation Details and Hyperparameters}
We implement all models in PyTorch and train on an NVIDIA RTX 3090. For reproducibility, we provide a Dockerfile with versions of key libraries (PyTorch 1.13, CUDA 11.7). Table~\ref{tab:hyperparams} summarizes key hyperparameters. Our \textbf{source code and synthetic data scripts} (including anomaly injection) will be released under an open-source license in a public repository. 

\begin{table}[ht]
\centering
\caption{Key Hyperparameters and Training Setup}
\label{tab:hyperparams}
\begin{tabular}{@{}ll@{}}
\toprule
\textbf{Parameter} & \textbf{Value/Description} \\
\midrule
LSTM hidden size        & 64 \\
Dropout rate ($p$)      & 0.3 (input + recurrent) \\
MC samples ($N$)        & 5 (default) \\
Optimizer              & Adam ($\eta_0=10^{-3}$) \\
Batch size             & 32 \\
Training epochs        & Up to 15 + early stopping \\
Fading profiles        & 3GPP TR 38.901 (Rayleigh/Rician) \\
SNR levels             & \{5, 10, 15, 20, 25, 30\}\,dB \\
Anomaly injection rate & 1\%--10\% \\
\bottomrule
\end{tabular}
\end{table}

\paragraph*{Bayesian Regularization and KL Divergence}
During training, we monitor the KL divergence term (between the approximate posterior $q(\mathbf{W}\mid\theta)$ and the assumed Gaussian prior) as part of the ELBO objective. Observed KL values remain stable (typically $<0.15$ per mini-batch) in line with Gaussian prior assumptions, confirming that the variational dropout approach is not diverging significantly from the chosen prior.

\subsection{Baselines and Comparative Approaches}
\label{subsec:baselines}

We compare our \textbf{Bayesian LSTM} against the following methods:

\paragraph*{(1) Classical Thresholding}
A simple fixed-threshold rule that flags an anomaly if the signal amplitude (or short-term energy) exceeds a preset level derived from training data statistics. Additionally, we test an \emph{adaptive threshold} variant that dynamically tracks the local mean and flags large deviations.

\paragraph*{(2) One-Class SVM and Isolation Forest}
Common shallow outlier detectors. We fit these on normal segments and then classify points exceeding a learned boundary as anomalies.

\paragraph*{(3) Deterministic LSTM}
Same architecture but without MC dropout at inference (dropout is only used during training in feedforward layers). This model provides a direct comparison to highlight the impact of Bayesian inference.

\paragraph*{(4) CNN-LSTM Hybrid}
A reference deep model that first applies temporal CNN filters, then passes features to an LSTM. Commonly reported in top-tier conferences for time-series anomaly detection.

\paragraph*{(5) Transformer (Param-Matched)}
We include a 2-layer transformer encoder (4 heads, dimension 64) with a parameter count of $\approx 60$k, matching our Bayesian LSTM size. Transformers have recently achieved SOTA results in time-series tasks.

\subsection{Evaluation Metrics and Statistical Testing}
\label{subsec:metrics}

We report:
\begin{itemize}
    \item \textbf{Accuracy/F1-Score}: To evaluate classification performance on normal vs.\ anomaly.
    \item \textbf{Expected Calibration Error (ECE)}: Measures how well predicted probabilities align with true frequencies.
    \item \textbf{Brier Score}: A proper scoring rule for probabilistic forecasts, lower is better.
    \item \textbf{Latency and Memory} on an NVIDIA Jetson module to confirm edge feasibility.
\end{itemize}

\noindent\textbf{Statistical Significance and Confidence Intervals:}
Each experiment is repeated $5$ times with different random seeds and data shuffles. We report \emph{mean} plus a $95\%$ confidence interval (CI). Where relevant, we perform a paired two-sided $t$-test comparing Bayesian vs.\ deterministic LSTM. All reported $p$-values are below $0.01$, indicating strong statistical significance for observed improvements.

\subsection{Main Results on Synthetic+Real Dataset}
\label{subsec:mainresults}

Table~\ref{tab:main_results} presents aggregated results on the combined (synthetic + real) test set. All values are averaged over 5 runs, with 95\% CIs in parentheses.

\begin{table*}[ht]
\centering
\caption{Performance Comparison on Mixed Test Set (Synthetic + Real), Showing Mean $\pm$ 95\% CI}
\label{tab:main_results}
\begin{tabular}{lcccccc}
\toprule
\textbf{Model} & \textbf{F1-Score} & \textbf{Accuracy} & \textbf{Brier Score} & \textbf{ECE} & \textbf{Latency (ms)} & \textbf{Params} \\
\midrule
Classical Threshold       & 0.62$\pm$0.02 & 0.78$\pm$0.03 & 0.24$\pm$0.01 & 0.22$\pm$0.01 & --   & -- \\
Adaptive Threshold        & 0.66$\pm$0.03 & 0.80$\pm$0.04 & 0.20$\pm$0.02 & 0.19$\pm$0.01 & --   & -- \\
One-Class SVM             & 0.70$\pm$0.02 & 0.82$\pm$0.02 & 0.18$\pm$0.02 & 0.20$\pm$0.02 & 2.0  & -- \\
Isolation Forest          & 0.71$\pm$0.02 & 0.83$\pm$0.03 & 0.17$\pm$0.02 & 0.18$\pm$0.01 & 2.1  & -- \\
CNN-LSTM Hybrid           & 0.76$\pm$0.02 & 0.85$\pm$0.01 & 0.15$\pm$0.01 & 0.19$\pm$0.02 & 1.3  & 70k \\
Transformer (60k)         & 0.78$\pm$0.03 & 0.86$\pm$0.02 & 0.14$\pm$0.01 & 0.17$\pm$0.01 & 2.3  & 60k \\
Deterministic LSTM        & 0.79$\pm$0.02 & 0.87$\pm$0.02 & 0.12$\pm$0.01 & 0.15$\pm$0.01 & 0.9  & 60k \\
\textbf{Bayesian LSTM (N=5)} & \textbf{0.88$\pm$0.01} & \textbf{0.92$\pm$0.01} & \textbf{0.09$\pm$0.01} & \textbf{0.07$\pm$0.01} & 1.2  & 60k \\
\bottomrule
\end{tabular}
\end{table*}

\noindent\textbf{Key Observations:}
\begin{itemize}
    \item \emph{F1-Score}: Bayesian LSTM surpasses both classical detectors and deep baselines (p-values $<0.01$). 
    \item \emph{Calibration}: It achieves an ECE of $0.07$, outperforming others by a clear margin. The \textbf{Brier Score} is also lowest (0.09).
    \item \emph{Edge Latency}: Multiple MC passes add minor overhead ($1.2$\,ms vs.\ $0.9$\,ms for deterministic). This remains feasible for real-time use.
    \item \emph{Statistical Significance}: The difference between Bayesian LSTM and the next-best method (Det.\ LSTM) is statistically significant ($p<0.01$).
\end{itemize}

\subsection{Ablation Studies}
\label{subsec:ablation}

We conduct targeted ablation experiments to validate the importance of \textbf{(i) MC Dropout} and \textbf{(ii) Dynamic Thresholding}.

\subsubsection{MC Dropout Variations}
Table~\ref{tab:mc_ablation2} compares $N\in \{1,5,10\}$ MC samples. When $N=1$, the inference is effectively deterministic and offers weaker calibration (ECE rises to 0.11). With $N=5$, performance nearly saturates while retaining low latency.

\begin{table}[ht]
\centering
\caption{Effect of MC Sample Count on Bayesian LSTM (Mean $\pm$ 95\% CI)}
\label{tab:mc_ablation2}
\begin{tabular}{lcccc}
\toprule
\textbf{MC Samples} & \textbf{F1-Score} & \textbf{Brier Score} & \textbf{ECE} & \textbf{Latency} \\
\midrule
$N=1$   & 0.81$\pm$0.02 & 0.14$\pm$0.01 & 0.11$\pm$0.01 & 0.9\,ms \\
$N=5$   & 0.88$\pm$0.01 & 0.09$\pm$0.01 & 0.07$\pm$0.01 & 1.2\,ms \\
$N=10$  & 0.89$\pm$0.01 & 0.08$\pm$0.01 & 0.06$\pm$0.01 & 1.6\,ms \\
\bottomrule
\end{tabular}
\end{table}

\subsubsection{Dynamic Threshold vs.\ Static}
We isolate the effect of the proposed dynamic threshold by deactivating it (i.e., using a single global cutoff). In that case, the Bayesian LSTM F1-score drops from $0.88\pm0.01$ to $0.84\pm0.02$ and the Brier score degrades (from $0.09$ to $0.13$). These changes confirm that \emph{dynamic thresholding} helps reduce false alarms in borderline conditions.

\subsection{Robustness Under Varying SNR and Adversarial Attacks}
\label{subsec:robustness}

\subsubsection{Performance Across SNR Levels}
Figure~\ref{fig:snr} plots F1-scores at different SNRs (5--30\,dB). The Bayesian LSTM shows a more graceful degradation under low SNR (5--10\,dB), retaining $\approx 0.80$ F1 vs.\ $\approx 0.70$ for deterministic LSTM or CNN-LSTM. At higher SNR (25--30\,dB), all models converge, but the Bayesian approach still exhibits lower ECE (not shown for brevity).

\subsubsection{Adversarial Jamming Scenario}
When tested on our adversarial dataset (Section~\ref{subsec:datasets}), classical threshold methods exhibit false alarms exceeding $25\%$. In contrast, the Bayesian LSTM’s predictive variance helps identify uncertain patterns around shifting jammers, achieving a stable F1 of $0.83$ vs.\ $0.74$ for deterministic LSTM. The difference is statistically significant ($p<0.01$), confirming better resilience to adversarial interference.

\subsection{Failure Case Analysis}
\label{subsec:failures}

We observe that short, \emph{highly overlapping collisions} sometimes mimic impulsive spikes, causing a transient drop in detection accuracy. Specifically, in 2.4\,GHz real captures, certain collisions from multiple Wi-Fi devices can appear pulse-like. While the Bayesian variance does increase in such ambiguous regions, the model may still misclassify some collisions as regular impulsive interference. 
Future solutions might involve \emph{domain-specific features} (e.g., packet header detection) or higher-level knowledge to differentiate collisions from purely impulsive noise more accurately.

\subsection{Link to Theoretical Assumptions}
\label{subsec:theoretical_assump}

Our training objective includes a KL term between the approximate dropout-based posterior and a \emph{Gaussian prior} (\S\ref{sec:methodology}). Empirically, the monitored KL divergence remains within acceptable bounds across epochs (mean $\approx 0.14$ in the final model), indicating that the network’s weight distribution does not stray far from the assumed Gaussian. This supports the viability of the Bayesian treatment for capturing epistemic uncertainty in fast-varying wireless signals.

\subsection{Reproducibility and Open-Source Release}
\label{subsec:reproducibility}
All experiments reported here can be reproduced via:
\begin{itemize}
    \item \textbf{Open-source code} and \textbf{synthetic-data generator}, which includes the 3GPP channel model scripts and anomaly injection references.
    \item A \textbf{Docker container} specifying CUDA/cuDNN versions, Python packages, and hardware settings.
    \item Hyperparameter details in Table~\ref{tab:hyperparams}, plus example configuration files in the repository.
\end{itemize}
We also plan to include the small 2.4\,GHz real dataset (subject to anonymization and municipal privacy regulations) for academic research purposes.

\subsection{Ethics and Compliance}
\label{subsec:ethics}

\paragraph*{Wireless Standards Compliance}
Our \textbf{simulated} data generation and channel models follow 3GPP TR 38.901 guidelines for urban micro-cell conditions. The real-world 2.4\,GHz captures were obtained under standard Wi-Fi testbed conditions in a university laboratory, fully compliant with local spectrum regulations.

\paragraph*{Simulation Labeling and Parameter Justification}
All results except Section~\ref{subsec:datasets}’s real subset are labeled as \emph{simulated}. Parameter choices (e.g., SNR range, anomaly injection rates) reflect realistic but controlled conditions in dense urban scenarios, informed by typical city IoT deployments.

\subsection{Summary of Experimental Findings}
\label{subsec:summary}

In summary, our Bayesian LSTM substantially outperforms deterministic and classical detectors across \textbf{all tested SNR levels}, including adversarial jamming conditions, while offering superior \emph{calibration} (ECE, Brier Score) and robust \emph{statistical significance} ($p<0.01$). Ablation studies highlight the necessity of both \emph{MC Dropout} and \emph{dynamic thresholding}, and failure-case analysis reveals possible confusion between collisions and impulsive noise that can be addressed by future domain-specific features. Finally, our public code/data release ensures reproducibility and transparent validation of the proposed approach.

\section{Discussion}
\label{sec:discussion}

\subsection{Dynamic Threshold Complexity}
Our threshold \eqref{eq:3.21} involves six parameters. While we rely on grid search with moderate ranges (e.g., $\beta_i \in [-1,1]$), deployments with strict constraints may opt for a simpler scheme. A reduced form without exponential and sigmoid-like terms yields slightly lower F1 (around 0.85 vs.\ 0.88) but is easier to tune.

\subsection{Memory Footprint and Edge Viability}
With 64 LSTM hidden units ($\approx$60k parameters), our model occupies $\approx 240$\,KB in float32. This is feasible on devices like an NVIDIA Jetson AGX Xavier or an advanced ARM MCU with hardware floating-point support. Further compression via quantization or pruning is also possible without eliminating MC dropout.

\subsection{Interpretability of Uncertainty}
A significant advantage of our Bayesian approach is the availability of predictive variance estimates. In practice, high variance often flags novel or underrepresented operating conditions. Real-time plots of variance vs.\ time can reveal transitional interference segments. Future work may integrate saliency maps or other explainable AI techniques to pinpoint anomaly causes more precisely.

\subsection{Limitations and Future Directions}
Though we included a small real-world dataset (5\% of total) from a 2.4\,GHz testbed, the limited scale may not capture all urban interference scenarios. Future work will gather larger, more diverse real-world traces (e.g., multi-antenna captures, multiple frequency bands) to further validate cross-environment generalizability. Additionally, \emph{online adaptation} or \emph{continual learning} strategies could be explored: as networks evolve, the Bayesian LSTM could periodically update its parameters or threshold hyperparameters using a small buffer of newly observed signals. Finally, more systematic experiments on multi-class labeling of overlapping anomalies would provide deeper insights into how well the model can disentangle concurrent interference types.

\section{Conclusion}
\label{sec:conclusion}
We presented a \textbf{Bayesian LSTM-based} approach for detecting sudden anomalies in urban wireless signals, accommodating multiple short-burst interference types through MC dropout-based uncertainty quantification. By combining signal-level measures with model-derived variance, we form a dynamic threshold that mitigates both false alarms and missed detections in dense, fast-varying conditions. Experiments on a synthetic but realistic dataset---augmented with preliminary real-world measurements---show that a modest number of stochastic forward passes (e.g., $N=5$) significantly improves detection and calibration relative to deterministic and classical baselines, while preserving real-time feasibility on edge devices.

\paragraph*{Future Extensions}
\begin{itemize}
    \item \textbf{Extensive Real-World Evaluation}: Larger-scale field tests in dense urban environments, possibly leveraging multi-antenna captures in 5G or beyond.
    \item \textbf{Online Adaptation}: Implement continual learning or fine-tuning, updating model parameters and threshold hyperparameters as new data arrive.
    \item \textbf{Alternative Bayesian Approximations}: Explore ensembles or flow-based methods to reduce MC sample counts and enhance uncertainty estimates.
    \item \textbf{Multi-Class Overlapping Anomalies}: Move beyond binary classification to distinguish among multiple interference types concurrently.
\end{itemize}

We believe this Bayesian framework provides a practical, uncertainty-aware solution for next-generation smart city networks, offering both high detection accuracy and low latency in the face of unpredictable interference spikes.

\appendix

\subsection{Synthetic Data Generation Pseudocode}
\label{app:pseudocode}
Below is concise pseudo-code for generating the synthetic bursty urban wireless dataset. Further documentation and code will be released in our public repository upon publication.

\begin{algorithm}
  \caption{Synthetic Urban Wireless Data Generation}
  \label{alg:synthetic_data}
  \begin{algorithmic}[1]
  % Time Complexity: O(\texttt{Num\_sequences} \times \texttt{Sequence\_length})
    \REQUIRE{ ~\\
      \vspace{0.1cm}
      \begin{itemize}
        \item \texttt{Num\_sequences}: Number of sequences 
        \item \texttt{Sequence\_length}: Length of each sequence 
        \item \texttt{Fading\_params}:\{Rayleigh/Rician, Doppler speed, ...\}
        \item \texttt{SNR\_levels}: [-5, 0, 5, 10, 15, 20] dB  
        \item \texttt{Anomaly\_types}: \{Impulsive, Narrowband, Micro-spike\}
        \item \texttt{p\_anomaly}: Probability of anomaly
      \end{itemize}
      \vspace{0.1cm}
    }  
  \ENSURE \texttt{dataset\_X}, \texttt{dataset\_Y} (time-series + labels)
  \STATE Initialize empty datasets: \texttt{dataset\_X}, \texttt{dataset\_Y}
  \FOR{$i \gets 1$ \TO \texttt{Num\_sequences}}
      \STATE \texttt{channel = simulate\_fading(Fading\_params, Sequence\_length)} 
      \COMMENT{Simulate $h_t$, see eqs.~(3.1) and (3.4)}
      \STATE \texttt{SNR = random\_choice(SNR\_levels)} 
      \COMMENT{Select an SNR from [-5, 0, 5, 10, 15, 20] dB}
      \STATE \texttt{noise = generate\_awgn(Sequence\_length, SNR)} 
      \COMMENT{Generate $\nu_t$}
      \STATE \texttt{signal = channel + noise}
      \STATE Initialize \texttt{label} array of length \texttt{Sequence\_length} to 0
      \FOR{$t \gets 1$ \TO \texttt{Sequence\_length}}
          \IF{\texttt{random()} < \texttt{p\_anomaly}}
              \STATE \texttt{anomaly\_type = random\_choice(Anomaly\_types)}
              \IF{\texttt{anomaly\_type} = ``Impulsive''}
                  \STATE \texttt{signal[t] += heavy\_tail\_amplitude()} 
                  \COMMENT{Draw $A_t$ from heavy-tailed distribution (eq.~(3.3))}
                  \STATE \texttt{label[t] = 1}
              \ELSIF{\texttt{anomaly\_type} = ``Narrowband''}
                  \STATE \texttt{inject\_jammer(signal, t, ...)}
                  \COMMENT{Implements eq.~(3.6)}
                  \STATE \texttt{label[t : t + duration] = 1}
              \ELSIF{\texttt{anomaly\_type} = ``Micro-spike''}
                  \STATE \texttt{inject\_micro\_spikes(signal, t, intervals=[...])}
                  \COMMENT{Implements eq.~(3.7)}
                  \STATE \texttt{label[t : t + duration] = 1}
              \ENDIF
          \ELSE
              \STATE \texttt{label[t] = 0}
          \ENDIF
      \ENDFOR
      \STATE \texttt{dataset\_X[i] = signal}
      \STATE \texttt{dataset\_Y[i] = label}
  \ENDFOR
  \RETURN \texttt{dataset\_X, dataset\_Y}
  \end{algorithmic}
  \end{algorithm}

\section{Proofs and Derivations}
\label{appendix:proofs}

In this appendix, we provide detailed proofs and derivations for the key equations introduced in Section~\ref{sec:methodology}. All references to equation numbers (e.g., (3.1), (3.10)) refer back to those in the main text.

\subsection{Proof of Signal Representation \eqref{eq:3.4}}
\label{app:proof_eq34}
\noindent \textbf{Lemma A.1.} \emph{(Equivalence of augmented signal model)}  
\newline
\textit{Equation \eqref{eq:3.4} can be seen as a linear superposition of the nominal signal plus impulsive interference under standard baseband assumptions.}
\begin{IEEEproof}
We start with \eqref{eq:3.1}, which states $x_t = h_t\,s_t + \nu_t$. Adding the impulsive term $\mathcal{I}_t$ from \eqref{eq:3.2} yields the augmented form
$$
  x_t \;=\; h_t\,s_t + \nu_t + A_t \,\delta_t.
$$
Because $\delta_t$ is a Bernoulli event with probability $\rho$, we interpret $A_t \delta_t$ as a discrete impulse that overlaps additively with the nominal signal. By reorganizing terms, the expression becomes identical to \eqref{eq:3.4}. The linearity follows from superposition in the baseband representation.
\end{IEEEproof}

\subsection{Stability of Threshold \eqref{eq:3.21}}
\label{app:proof_eq321}
\noindent \textbf{Lemma A.2.} \emph{(Boundedness and continuity)}  
\newline
\textit{The dynamic threshold $\Gamma_t$ in \eqref{eq:3.21} remains bounded and continuous in $t$, assuming $\beta_i\ge 0$ and finite $\|\mathbf{h}_t\|$.}
\begin{IEEEproof}
From \eqref{eq:3.21}, observe that each summand is non-negative if $\beta_i\ge 0$. Specifically,
$$
  \exp\!\bigl(-\beta_4 \|\mathbf{h}_t - \mathbf{h}_{t-1}\|^2\bigr)\;\in\;(0,1],
$$
and $\sigma_t^2\ge 0$ by definition in \eqref{eq:3.12}. Hence $\Gamma_t\ge 0$. Continuity follows because each term (exponential, norm, polynomial) is continuous in $t$, under standard assumptions about the LSTM states $\mathbf{h}_t$.  
\end{IEEEproof}

\subsection{Existence of Optimal $\boldsymbol{\beta}$ in \eqref{eq:3.30}}
\label{app:proof_eq330}
\noindent \textbf{Lemma A.3.} \emph{(Convexity in $\boldsymbol{\beta}$ under mild assumptions)}  
\newline
\textit{If $\mathcal{J}(\theta,\boldsymbol{\beta})$ is convex in $\boldsymbol{\beta}$, and $\|\boldsymbol{\beta}\|^2$ is strictly convex, then there exists a unique global minimizer for \eqref{eq:3.30}.}
\begin{IEEEproof}
By standard arguments in convex optimization, adding a strictly convex $\|\boldsymbol{\beta}\|^2$ penalty ensures strong convexity if the original objective $\mathcal{J}$ is convex in $\boldsymbol{\beta}$. Consequently, a unique global minimum exists in $\boldsymbol{\beta}$-space. Detailed conditions for convexity of $\mathcal{J}$ can be enforced by restricting the threshold function $\Gamma_t$ to linear or monotonic transformations in $\boldsymbol{\beta}$.  
\end{IEEEproof}

\subsection{Regularization of Predictive Variance \eqref{eq:3.28}}
\label{app:proof_eq328}
\noindent \textbf{Lemma A.4.} \emph{(Variance penalty prevents degenerate solutions)}  
\newline
\textit{Including $\sigma_t^2$ in the training cost, as in \eqref{eq:3.28}, avoids the trivial solution $\sigma_t^2 \to\infty$.}
\begin{IEEEproof}
Suppose the network attempts to assign infinite variance for all time points to reduce misclassification risk. Then the penalty term $\sum_t \sigma_t^2$ in \eqref{eq:3.28} imposes unbounded cost, forcing the training to prefer finite $\sigma_t^2$. Hence, no trivial infinite-variance minimizer exists.  
\end{IEEEproof}

\subsection{Derivation of LSTM Recurrence \eqref{eq:3.13}--\eqref{eq:3.19}}
\label{app:proof_lstm}
\noindent \textbf{Lemma A.5.} \emph{(Dropout as variational sampling)}  
\newline
\textit{Applying Bernoulli masks $\mathbf{M}_x,\mathbf{M}_h$ inside the gating equations \eqref{eq:3.13}--\eqref{eq:3.18} corresponds to approximate samples from a known variational distribution $q(\mathbf{W}\mid \theta)$.}
\begin{IEEEproof}
See \cite{gal2016dropout} for the theoretical underpinnings of dropout-based Bayesian inference. In short, each forward pass in the LSTM is equivalent to sampling from a Bernoulli-based variational distribution on the weight matrices. For each gate, we multiply the relevant input or hidden vector by an i.i.d.\ mask with probability $p$. This ensures different (random) subsets of parameters are active in each pass, approximating draws from the posterior.  
\end{IEEEproof}

\vspace{2ex}
These results confirm the rigor and consistency of our uncertainty-aware detection framework. In particular, Lemmas~\ref{app:proof_eq34}--\ref{app:proof_lstm} justify the mathematical foundations for combining (i) a superposition-based signal model, (ii) dynamic threshold design, and (iii) Bayesian LSTM inference for robust burst anomaly detection in urban wireless environments.

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\end{thebibliography}

\end{document}

Output:

• Revised Methodology section only in LaTeX.
• No markdown; use LaTeX comments for annotations.

---