Change Point Monitoring in Wireless Sensor Networks Under Heavy-Tailed Sequence Environments

Abstract

In the special case of a heavy-tailed sequence environment, change point monitoring in wireless sensor networks faces many serious challenges, such as high communication overhead, particularly sensitivity to sparse changes, and dependence on strict parameter assumptions. In order to solve these limitations, a distributed robust M-estimator-based change point monitoring (DRM-CPM) method is proposed. This method combines ratio statistics with sliding window technology so that in online detection, there is no need to know the distribution before and after changes in advance. A threshold-triggered communication strategy is introduced, where sensors exchange local statistics only when exceeding predefined thresholds, significantly reducing energy consumption. By means of theoretical analysis, the asymptotic characteristics of the statistics are confirmed, and the robustness of the algorithm to heavy-tail noise and unknown parameters is also proved. Simulation results show that the algorithm is better than the existing methods in terms of empirical size control, empirical power, and communication efficiency, particularly in the face of sparse variation or heavy-tailed data. This framework provides a scalable solution for real-time anomaly monitoring with non-Gaussian data characteristics in industrial and environmental applications.

1. Introduction

In an era of ubiquitous data, a particularly critical challenge is identifying the dynamic changes in those large datasets [1]. Change point monitoring (CPM), has become a key technology to solve this problem. Acting as a “data radar”, CPM effectively identifies shifts in the data distribution, which can reveal possible changes in the system state, and provide the necessary support for risk assessment and informed decision-making. The theoretical roots of the change point can be traced back to Page [2]’s pioneering work; the utility of the change point has since extended to a myriad of fields, encompassing environmental monitoring [3,4], industrial process control [5], financial risk mitigation [6], and biomedical analysis [7]. It is of great significance to conduct a comprehensive study on CPM methods. Finally, the proposed monitoring procedures are evaluated through a practical application.

For sensor networks, the CPM algorithm has gradually evolved into a variety of deployment strategies to adapt to the needs of different applications. In the early days, related research mainly focused on the centralized sensor network architecture because it has the characteristics of direct global information aggregation, which has attracted relatively large attention. In such a configuration, each sensor node will collect data at each time instance, and then transmit these observed data to the fusion center for analysis so that global change point detection and decision-making can be achieved; as shown in Figure 1a, the fusion center will receive and dispose data streams from all sensors. The point of change refers to the point at which the data distribution of the affected subset of sensors deviates from the original statistical pattern. The fundamentals of centralized CPM algorithm design include constructing valid statistics that aggregate information from multiple sensors and set appropriate thresholds for anomaly identification. Although the centralized method is relatively mature in terms of theoretical analysis and algorithm development, its inherent centralized data transmission mode will bring a lot of communication bottlenecks and energy consumption problems. This problem is very serious in large-scale, high-density sensor networks.

To mitigate the limitations of centralized architectures, researchers have shifted their research focus to decentralized distributed sensor network architectures as shown in Figure 1b, while also studying the corresponding distributed change point detection algorithms (distributed CPM). Distributed CPM eliminates the need for a fusion center by equipping each sensor node with the ability to make autonomous decisions. In a distributed network, individual sensor nodes are tasked with identifying change points using only their own collected data. Simultaneously, they participate in a restricted exchange and collaboration of information with nearby nodes.

1.1. Related Works

Over the past few decades, research on centralized CPM algorithms has built a theoretical foundation that emphasizes the importance of optimizing communication and computation at the same time. Traditional methods typically rely on quantitative techniques [8,9], involving local decision transmission strategies [10,11] to reduce communication requirements. Quantization compresses data by regulating precision in transmitted values, while the local decision method reduces the communication rate through limited information exchange. Kurt [12] introduced the threshold triggering mechanism, changing the communication mode from continuous communication to event-triggered transmission, and the asymptotic optimality can be maintained. This idea was further explored in the work of Yang [13], which presented a detection framework combining moving average and moving sum (MOSUM) methods, achieving performance close to an ideal centralized system without needing prior knowledge of the pre- and post-change mean. In addition to communication efficiency, energy considerations have also begun to receive attention. An energy-harvesting wireless sensor network has been studied [14], which presents new challenges to data collection and processing and affects the performance of CPM. Sun and Zou [15] developed lightweight algorithms for cases of unknown posterior distributions and anonymous networks, using parameter estimation to reduce computational complexity. However, these algorithms are primarily designed for centralized architectures, and when used in large, growing wireless sensor networks, they fall short in terms of robustness, scalability, and energy efficiency.

Early investigations on distributed change point detection generally focus on finding a balance between effective test statistics and ensuring effective communication. Braca and Paolo [16] proposed two distributed CPM frameworks: the Fused Consensus CUSUM (FC-CUSUM) algorithm, which approximates cumulative sum (CUSUM) statistics via a distributed consensus mechanism, and a non-cooperative algorithm that relies on independent local CUSUM decisions to trigger global alerts. From theoretical analysis, it can be seen that when changes have an impact on multiple sensors, the effect of the FC-CUSUM algorithm is relatively good, but when the changes are sparse, its effectiveness will be reduced, and the performance of non-cooperative algorithms is contrary to that of the FC-CUSUM algorithm. In order to solve this problem, Liu et al. [17] introduced the C-CUSUM algorithm, which uses dynamic average local CUSUM statistics to achieve reliable detection; while this approach provides some stability in terms of the number of sensors affected by the change, it requires constant communication at a higher frequency for the consensus process to work.

The increasing prevalence of Internet of Things (IoT) applications has placed a significant emphasis on the need to lower energy consumption associated with communication. This urgency stems from the fundamental constraints of wireless sensor networks, where frequent communication, as highlighted by [18], leads to substantial energy consumption. This ultimately shortens deployment lifespans due to the inherent difficulties in battery replacement, as noted by [19]. To address this problem, Hsu et al. [20] proposed a quantified transmission scheme that reduces the communication load by reducing the accuracy of the data, but this method requires an understanding of the entire network structure, which makes it unsuitable for networks that change over time. Gu et al. [21] integrated a threshold-trigger mechanism to develop the ECO-CUSUM algorithm (a request-response and censoring scheme). While this approach significantly improves energy efficiency when changes affect a large number of sensors, it remains sensitive to sparse variations. Fellouris and Sokolov [22] extended the research scope to include any unknown sensor subset, and Jiao et al. [23] combined the C-CUSUM algorithm to propose the TR-CUSUM algorithm, achieving strong robustness with respect to the number of affected nodes.

A key point to consider is that current algorithms generally rely on strict parameter assumptions. The log-likelihood ratio statistic given in Equation (1), which is the basis of these algorithms, requires precise provisions for density parameters before and after changes. Although historical data can be used to estimate before changes, the inherent unobservability of post-change parameters introduces a risk of detection failure. In order to solve this problem, Liu  et al. [24] proposed a recursive method to estimate the average value. However, this method has to pay a price in terms of statistical efficiency. Although Yang et al. [13] relied on the MOSUM method to eliminate the dependence on the changed parameters, it can only be used in centralized systems:

$$\begin{array}{c}\hfill L_{t,i}=log{\displaystyle \frac{f_1\left(y_{t,i}\right)}{f_2\left(y_{t,i}\right)}},\end{array}$$

(1)

In reality, the data in the real world is becoming more and more complex, and time series will show very prominent spikes and heavy tails in many practical application scenarios. Such phenomena are common in many fields such as seismology [25], network traffic monitoring [26], material fault detection [27], and mining operations [28]. In detecting change points in autocorrelation coefficients, the ratio statistic method developed by Zhang et al. [29] is noteworthy, especially for its transformation technique toward the mean, which can be adopted in related contexts. In these fields, data modeling based on heavy-tail distribution will have a better effect. However, most prior studies have focused on Gaussian or sub-Gaussian series; extending the analytical framework of time series to heavy-tailed series holds key theoretical significance and can provide more reliable support for risk management and decision-making in related fields.

In order to solve the above challenges, this paper draws inspiration from the framework proposed in [30,31], and then proposes a method for online detection of change points based on ratio statistics. By integrating the sliding window method, the proposed method can carry out the effective real-time monitoring of data flow. No prior information about the distribution before and after the change is required. To adapt to complex real-world scenarios, we incorporate the M-estimation technique, which extends the algorithm’s applicability beyond standard Gaussian or sub-Gaussian sequences to include heavy-tailed sensor data streams. This improvement makes the algorithm significantly improve its robustness when processing scenarios involving sharp tail and heavy-tail data.

1.2. Contributions

In this paper, we propose a distributed change point detection (CPM) algorithm, termed the distributed robust M-estimator-based change point detection (DRM-CPM). The main features of the proposed method are summarized as follows:

• Synergy of Low Energy and Robustness: In wireless sensor networks, achieving robustness usually comes at the cost of high communication overhead. DRM-CPM breaks this trade-off through a distributed event-triggered architecture. Unlike centralized robust methods that require transmitting raw data, each sensor node i exchanges only local statistical information $V_{t,i}$ with its immediate neighbors j, and strictly only when necessary (when $V_{t,i}>\eta$). This significantly reduces energy consumption by avoiding the quadratic path loss of long-range transmission while maintaining the statistical rigor required for heavy-tailed detection.
• Robustness to Change Sparsity: A key achievement is the algorithm’s consistent performance across various levels of change sparsity. As demonstrated in the subsequent analysis, the detection remains satisfactory regardless of whether the change is local or global, addressing the challenge of sparse change capture.
• Unknown Change Parameters: A DRM-CPM online detection algorithm based on ratio-type statistics is developed. The algorithm does not need to know the pre- and post-change distributions. Addressing the problem of change point detection under unknown probability density functions is one of the advantages of the proposed algorithm.
• Robustness to Heavy-tailed Data: The primary theoretical contribution is breaking the reliance on Gaussian assumptions inherent in traditional distributed detection methods. By generalizing distributed CPM to heavy-tailed sequences via bounded M-estimation, the algorithm resolves the detection failure issue in non-Gaussian environments—a capability that standard energy-efficient protocols typically lack.
• Detection Accuracy and Robustness: The DRM-CPM algorithm obtains high detection accuracy and shows robust, rapid and precise change point monitoring performance for different kinds of data sequences.

1.3. Organization

The structure of the paper is as follows. Section 2 models the distributed CPM problem. Section 3 elaborates on the proposed algorithm. Section 4 establishes the theoretical properties of the utilized statistics. Section 5 presents the simulation studies. A real-world application using wind speed data is detailed in Section 6. Finally, Section 7 concludes the paper and discusses potential future work.

2. Materials and Methods

Consider a network comprising N sensors, where each sensor observes data $y_{t,i}$ at each time instant $t=1,2,\dots ,n$, representing dynamically evolving data with specific temporal characteristics. To establish a rigorous foundation for the proposed test, we explicitly define the following three fundamental assumptions: first, all sensor nodes within the network are assumed to maintain time synchronization; second, the anomaly is modeled as a location shift occurring at an unknown time instant $\left[n\tau \right]$, which affects an unknown and time-varying subset of sensors $S_t\subseteq \{1,\dots ,N\}$ while the noise distribution remains stationary; and third, the innovation process ${\epsilon }_{t,i}$ follows a heavy-tailed mixed sequence characterized by a tail index $\kappa$. Consequently, under these explicit conditions, the data model can be represented as follows:

$$\begin{array}{c}\hfill y_{t,i}={\mu }_i+{\delta }_{t,i}I\{t>\left[n\tau \right]\}+{\epsilon }_{t,i},t=1,2,\dots ,n,1\le i\le N,\end{array}$$

(2)

where ${\mu }_i$ is the unknown pre-change location parameter, ${\delta }_{t,i}$ is the location shift, and $I\{·\}$ is the indicator function. The innovation process ${\epsilon }_{t,i}$ follows a heavy-tailed mixed sequence with tail index $\kappa$. After the time instant $\left[n\tau \right]$, the location parameter of the S sensor’s observations shifts from ${\mu }_S$ to ${\mu }_S+{\delta }_S$. Propose the following hypothesis-testing questions:

$$\begin{array}{cc}& H_0:{\delta }_{t,i}=0,\forall t>0,i\in \left[N\right],\hfill \\ & H_1:\left\{\begin{array}{cc}{\delta }_{t,i}=0,\hfill & t=1,2,\dots ,\left[n\tau \right],\hfill \\ {\delta }_{t,i}\ne 0,\hfill & t=\left[n\tau \right]+1,\dots ,n,\hfill \end{array}\right.i\in S,\hfill \end{array}$$

where the null hypothesis $H_0$ posits the absence of any change point, while the alternative hypothesis $H_1$ posits the existence of a single location change point occurring at time $\left[n\tau \right]$, and the set of affected sensors S is fixed but unknown. When $S=N$, this means all sensors experience a change, referred to as a global change.

To ensure the asymptotic validity of the proposed test, we impose the following assumptions on the innovation process ${\epsilon }_{t,i}$, the M-estimation function ${\psi }_M(·)$, and the sensor network topology.

Assumption 1. 

The distribution $F_{\epsilon }$ of $\left\{{\epsilon }_t\right\}$ lies within the stable domain of attraction, with a tail index $\kappa \in (0,2]$. When $\kappa >1$, we have $E\left({\epsilon }_t\right)=0$. When $\kappa \le 1$, $\left\{{\epsilon }_t\right\}$ follows a symmetric distribution.

Assumption 2. 

For each sensor $i\in \{1,\dots ,N\}$, let ${\left\{{\epsilon }_{t,i}\right\}}_{t\ge 1}$ denote the local noise sequence. We assume that $\left\{{\epsilon }_{t,i}\right\}$ is a strictly stationary α-mixing sequence uniformly over i. Specifically, for some finite $\chi >0$ and ${\chi }^{\prime }>0$, there exists a constant $C(\chi ,{\chi }^{\prime })>0$, independent of i, such that

$$\sum _{e=0}^{\infty }{(e+1)}^{1/2}{\alpha }_i{\left(e\right)}^{\chi /(2+\chi +{\chi }^{\prime })}\le C(\chi ,{\chi }^{\prime }),$$

where ${\alpha }_i\left(e\right)$ ($e=1,2,\dots$) are the α-mixing coefficients of the i-th sensor. This condition ensures that the dependence between the sequence elements decays sufficiently fast as the time lag increases, and this decay rate is consistent across the entire network.

Assumption 3. 

The M-estimation function ${\psi }_M(·)$ satisfies the following conditions:

(1) 

$E\left[{\psi }_M\left({\epsilon }_t\right)\right]=0$;

(2) 

$E\left[\right|{\psi }_M\left({\epsilon }_t\right){|}^{2+p}]<\infty$, for some $p>0$;

(3) 

$0<E\left[{\psi }_M^{\prime }\left({\epsilon }_t\right)\right]<\infty$ and for some $\beta >1$, $E\left[\right|{\psi }_M^{\prime }\left({\epsilon }_t\right){|}^{\beta }]<\infty$;

(4) 

$0<{\sigma }^2\left({\psi }_M\right)=E\left[{\psi }_M^2\left({\epsilon }_t\right)\right]+2{\sum }_{j=1}^{\infty }E\left[{\psi }_M\left({\epsilon }_t\right){\psi }_M\left({\epsilon }_{t+j}\right)\right]<\infty$.

Assumption 4. 

The influence function ${\psi }_M(·)$ is a bounded, non-decreasing, odd function with a unique root at zero, and ${\psi }_M^{\prime }(·)$ is Lipschitz continuous.

Remark 1. 

The boundedness of ${\psi }_M$ (Assumption 4) ensures that the moment conditions in Assumption 3 are satisfied for ${\psi }_M\left({\epsilon }_t\right)$, even if the raw innovations have infinite variance.

Assumption 5. 

In order to improve the detection performance, the sensors communicate with their neighbors via a static undirected graph to exchange information, and then integrate the information with weighted consensus matrix $\mathbf{W}$. If there is a communication link between sensor i and sensor j, generally $\mathbf{W}$ satisfies:

(1) 

$w_{ij}>0$ (the $(i,j)$-th entry of $\mathbf{W}$ is positive);

(2) 

$w_{ij}=w_{ji}\ge 0$, indicating that the weight of information flow from sensor j to sensor i is the same as that from sensor i to sensor j;

(3) 

${\sum }_{j=1}^N{w}_{ij}=1$, for all i, representing a row stochastic normalization.

where denoted by ${\mathcal{N}}_i$ is the set of neighbors of sensor i, i.e., ${\mathcal{N}}_i=\{j\in \left[N\right]:w_{ij}>0\}$.

The goal of the paper is to detect the change at time $\left[n\tau \right]$ with high accuracy, low false alarm rate, small detection delay and low communication cost, where $\left[n\widehat{\tau }\right]$ is the time at which the algorithm issues an alarm. According to the tradition in hypothesis testing, we use two performance metrics to evaluate the algorithm: empirical size (Size) and empirical power (Power). Size is the probability of incorrectly rejecting the null hypothesis $H_0$, i.e., the probability of Type I error. Power is the probability of correctly rejecting the null hypothesis $H_0$ under the alternative hypothesis $H_1$ so as to avoid Type II error. The estimation is accurate if the alarm time $\left[n\widehat{\tau }\right]$ is in a neighborhood of true change point $\left[n\tau \right]$.

$$\begin{array}{c}\hfill \mathrm{Size}:=P(\mathrm{Reject}{H}_0\mid H_0),\\ \hfill \mathrm{Power}:=P(\mathrm{Reject}{H}_0\mid H_1).\end{array}$$

To evaluate the detection performance, we use the expected detection delay (EDD), which quantifies the expected time elapsed between the occurrence of the change and the issuance of an alarm:

$$\mathrm{EDD}:=E(\left[n\widehat{\tau }\right]-\left[n\tau \right]),$$

The communication cost is measured by $\zeta$, defined as the long-term average communication rate per sensor:

$$\zeta =\underset{n\to \infty }{lim}{\displaystyle \frac{1}{n}}\sum _{t=1}^T{\displaystyle \frac{M_t}{{\sum }_{i\in \left[N\right]}\#{\mathcal{N}}_i}},n>\tau ,$$

where $M_t$ denotes the number of communications at time t, and $\#{\mathcal{N}}_i$ represents the cardinality of set N.

3. Distributed Change Point Detection Method

To address the aforementioned challenges and reduce the communication overhead, this study proposes a M-estimator-based ratio-type statistic, building upon the statistic introduced in [30,32] and incorporating a sliding window technique; this approach is then extended for the online monitoring of sensor data. This score-based formulation ensures a Brownian motion limit rather than a Stable one, effectively circumventing the bottleneck of unknown tail indices. Following the construction above, the local statistic for each sensor is computed as specified in Equation (3):

$$\begin{array}{cc}\hfill V_{t,i}\left(s\right)& ={\displaystyle \frac{{\displaystyle \underset{\left[ns\right]-\left[nh\right]+1\le \left[ns\right]\le \left[ns\right]+\left[nh\right]}{max}\left|\sum _{t=\left[ns\right]-\left[nh\right]+1}^{\left[ns\right]}{\psi }_M\left(y_{t,i}-{\widehat{\mu }}_i\left({\psi }_M\right)\right)\right|}}{\underset{\left[ns\right]-\left[nh\right]+1\le \left[nv\right]\le \left[ns\right]}{max}\left|{\displaystyle \sum _{t=\left[ns\right]-\left[nh\right]+1}^{\left[nv\right]}}{\psi }_M\left(y_{t,i}-{\widehat{\mu }}_{1,i}\left({\psi }_M\right)\right)\right|+\underset{\left[ns\right]+1\le \left[nv\right]\le \left[ns\right]+\left[nh\right]}{max}\left|{\displaystyle \sum _{t=\left[nv\right]+1}^{\left[ns\right]+\left[nh\right]}}{\psi }_M\left(y_{t,i}-{\widehat{\mu }}_{2,i}\left({\psi }_M\right)\right)\right|}}.\hfill \end{array}$$

(3)

where $\left[nh\right]=m/2$ represents half of the sliding window size m, and $\left[nv\right]$ is a dummy index representing the candidate split points within the current sliding window. The maximization over $\left[nv\right]$ is performed to capture the local fluctuation of the score function. As new streaming data arrives, the window slides forward by one unit to achieve online monitoring, where ${\widehat{\mu }}_i\left({\psi }_M\right)$ is the M-estimator of $\mu$, obtained from the observations $y_{\left[ns\right]-\left[nh\right]+1,i},\dots ,y_{\left[ns\right]+\left[nh\right],i}$:

$${\widehat{\mu }}_i=\underset{\beta \in \mathbb{R}}{argmin}\sum _{t=\left[ns\right]-\left[nh\right]+1}^{\left[ns\right]+\left[nh\right]}\rho (y_{t,i}-\beta ),$$

where $\rho$ is a loss function such that ${\rho }^{\prime }={\psi }_M$. Equivalently, the M-estimator in the above equation can also be defined as the solution to the following equation:

$$\sum _{t=\left[ns\right]-\left[nh\right]+1}^{\left[ns\right]+\left[nh\right]}{\psi }_M(y_{t,i}-\beta )=0.$$

Similarly, ${\widehat{\mu }}_{1,i}\left({\psi }_M\right)$ and ${\widehat{\mu }}_{2,i}\left({\psi }_M\right)$ are the M-estimators of the pre-change and post-change location parameter, respectively, computed from the observations $y_{\left[ns\right]-\left[nh\right]+1,i},\dots ,y_{\left[ns\right],i}$ and $y_{\left[ns\right]+1,i},\dots ,y_{\left[ns\right]+\left[nh\right],i}$, respectively, obtained as the solutions to

$$\sum _{t=\left[ns\right]-\left[nh\right]+1}^{\left[ns\right]}{\psi }_M(y_{t,i}-\beta )=0,\sum _{t=\left[ns\right]+1}^{\left[ns\right]+\left[nh\right]}{\psi }_M(y_{t,i}-\beta )=0.$$

The function ${\psi }_M$ can take various forms. When ${\psi }_M\left(x\right)=x,x\in \mathbb{R}$, it reduces to the classical least squares (LS) procedure. If ${\psi }_M\left(x\right)=xI\left\{\right|x|\le K\}+Ksgn\left(x\right)I\left\{\right|x|>K\}$, it is equivalent to the standard Huber loss function. And ${\psi }_M\left(x\right)=xI\left\{\right|x|\le K\}+0I\{\left|x\right|>K\}$ is a more aggressive truncation loss function, where the parameter K is a fixed tuning parameter that determines the robustness and efficiency of the M-estimator.

The preceding content focuses on change point detection implemented at a single sensor. The following section will discuss a collaborative algorithm for distributed wireless sensor networks to achieve energy-efficient and robust online change point detection. To illustrate the communication architecture, a schematic diagram is provided in Figure 2.

Each sensor locally performs a ratio-type test and computes its local ratio statistic $V_{t,i}$. This paper employs a threshold-based communication strategy to regulate the communication frequency. Specifically, an elevated statistic (i.e., $V_{t,i}>\eta$) signals a potential anomaly, identifying the sensor as a candidate node; collectively, these nodes form the set $S_t$, which estimates the subset of affected nodes at time t:

$$\begin{array}{c}\hfill S_t=\{i\in \left[N\right]:V_{t,i}>\eta \}.\end{array}$$

Sensors within $S_t$ proactively initiate data exchange requests with all their neighbors to fuse the neighbors’ statistical information. Upon receiving the data, neighboring nodes fuse their own statistical information with the received information to update their monitoring data. Conversely, sensors with local statistic values below the threshold assist their neighbors in completing data exchange tasks. After exchanging data, each sensor updates its decision statistic ${\tilde{V}}_{t,i}$ based on the following consensus rule:

$$\begin{array}{c}\hfill {\tilde{V}}_{t,i}=\left\{\begin{array}{cc}\sum _{j\in {\mathcal{N}}_i}{w}_{ij}{V}_{t,j}+w_{ii}{V}_{t,i},\hfill & i\in S_t\hfill \\ \sum _{j\in {\mathcal{N}}_i\cap S_t}{w}_{ij}{V}_{t,j}+\sum _{j\in {\mathcal{N}}_i\cap S_{t,c}}{w}_{ij}{V}_{t,i},\hfill & i\in S_{t,c}\hfill \end{array}\right.\end{array}$$

(4)

where $S_{t,c}$ is the complement of $S_t$, and ${\mathcal{N}}_i$ represents the neighborhood of sensor i. The algorithm raises an alarm for sensor i when ${\tilde{V}}_{t,i}>b$. According to Olfati-Saber et al. [33], a distributed consensus is achieved asymptotically on a connected graph, with the convergence rate determined by the algebraic connectivity. Therefore, under Assumptions 3 and 5, the proposed distributed monitoring algorithm is theoretically guaranteed to converge.

In essence, the DRM-CPM algorithm adopts a “local-to-global” detection strategy. First, each sensor performs preliminary local screening to flag potential anomalies. Candidate sensors then share diagnostic information with neighbors to collaboratively validate changes, suppressing random noise through consensus-based fusion. This approach ensures alarms are triggered only upon network-wide agreement, improving both sensitivity and robustness. See Algorithm 1 for details.

Algorithm 1 Robust ratio-type distributed change point detection (DRM-CPM).

Input:  Sensor data streams $\left\{y_{t,i}\right\}$, consensus matrix $\mathbf{W}$, window size m, threshold $\eta$, and critical value b. Output:  Alarm time $\left[n\widehat{\tau }\right]$ and location of change.   1:  Online detection:   2:  for  $t=1,2,\dots ,n$  do   3:        // Stage 1: Local screening via sliding window   4:        for $i\in \left[N\right]$ do   5:              Compute local ratio statistic $V_{t,i}$ based on m and Equation (3).   6:        end for   7:        Identify candidate set $S_t=\{i\in \left[N\right]:V_{t,i}>\eta \}$.   8:        // Stage 2: Distributed information exchange   9:        for each sensor $i\in S_t$ do 10:               Send $V_{t,i}$ to all neighboring nodes $j\in {\mathcal{N}}_i$. 11:         end for 12:         for $i\in \left[N\right]$ do 13:               if node i receives $V_{t,j}$ from neighboring nodes then 14:                     Retrieve and cache its own local statistic $V_{t,i}$. 15:               end if 16:         end for 17:         // Stage 3: Global fusion and consensus decision 18:         for $i\in \left[N\right]$ do 19:               Update decision statistic ${\tilde{V}}_{t,i}$ based on $\mathbf{W}$ and Equation (4). 20:               if ${\tilde{V}}_{t,i}>b$ then 21:                     Issue an alarm; 22:                     return Alarm time $\left[n\widehat{\tau }\right]=t$. 23:               end if 24:         end for 25:   end for

Complexity Analysis

The computational efficiency of the DRM-CPM algorithm is evaluated per time step t. Specifically, the first stage involves calculating the local ratio statistics $\left\{V_{t,i}\right\}$ for all N sensors based on Equation (3). Since statistics requires performing M-estimation within a sliding window of size m, this stage incurs a time complexity of $O(N·m)$. The subsequent stages, including candidate identification, neighborhood information exchange, and distributed decision updating, have a complexity of $O\left(\right|S_t|·d_{max}+N·d_{max})$, where $d_{max}$ denotes the maximum degree of the network graph. Consequently, the total time complexity per time step is expressed as $O\left(N(m+d_{max})\right)$. Regarding the memory requirements, each sensor only needs to store its local sliding window data and the statistics received from its neighbors, resulting in a per-node storage complexity of $O(m+d_{max})$. This analysis demonstrates that the algorithm is computationally efficient and spatially scalable for large-scale sensor networks.

4. Theoretical Properties

Lemma 1. 

If Assumptions 1–4 hold, and under the null hypothesis $H_0$, then

$$\begin{array}{cc}\hfill {\left[nh\right]}^{1/2}\left(\widehat{\mu }\right({\psi }_M)-\mu )& \stackrel{d}{\to }{\displaystyle \frac{\sigma \left({\psi }_M\right)·[B(s+h)-B(s-h)]}{2E\left({\psi }_M^{\prime }\left({\epsilon }_t\right)\right)}},\hfill \\ \hfill {\left[nh\right]}^{1/2}\left({\widehat{\mu }}_1\right({\psi }_M)-\mu )& \stackrel{d}{\to }{\displaystyle \frac{\sigma \left({\psi }_M\right)[B\left(s\right)-B(s-h)]}{hE\left({\psi }_M^{\prime }\left({\epsilon }_t\right)\right)}},\hfill \\ \hfill {\left[nh\right]}^{1/2}\left({\widehat{\mu }}_2\right({\psi }_M)-\mu )& \stackrel{d}{\to }{\displaystyle \frac{\sigma \left({\psi }_M\right)[B(s+h)-B\left(s\right)]}{hE\left({\psi }_M^{\prime }\left({\epsilon }_t\right)\right)}}.\hfill \end{array}$$

where $B(·)$ denotes a standard Brownian motion.

Theorem 1. 

Under the null hypothesis, assuming $y_{t,i}$ are derived from the model (2) and Assumptions 1–4 hold, as $2\left[nh\right]\to \infty$, we have

$$\begin{array}{cc}\hfill & V_n\stackrel{d}{\to }\underset{s-h\le s\le s+h}{sup}{\displaystyle \frac{\mid B\left(s\right)-B(s-h)-\frac{B(s+h)-B(s-h)}{2}\mid }{C+D}},\hfill \\ \hfill C=& \underset{s-h+1<v\le s}{sup}|B\left(v\right)-B(s-h)-{\displaystyle \frac{v-s+h}{h}}(B\left(s\right)-B(s-h))|,\hfill \\ \hfill D=& \underset{s<\nu \le s+h}{sup}|B(s+h)-B\left(\nu \right)-{\displaystyle \frac{s+h-\nu }{h}}(B(s+h)-B\left(s\right))|.\hfill \end{array}$$

Remark 2. 

Theorem 1 establishes that under the null hypothesis, the limiting distribution of $V_n$ involves the supremum of Brownian motion functionals. Crucially, due to the use of bounded M-estimation, this limiting distribution is Gaussian based, distinct from the Lévy stable nature of the raw heavy-tailed noise. This implies that the asymptotic behavior is invariant to the specific tail index of the noise. As a consequence, at any fixed significance level, there exists a unique critical value, making the test robust to nuisance parameters and heavy-tailed outliers. This is an appealing feature of the ratio-type test for real data analysis where the tail index of the noise is typically unknown.

The detailed proofs of Lemma 1 and Theorem 1 are provided in Appendix A.

5. Simulations

To evaluate the performance of the proposed algorithm, we conducted experiments on synthetic datasets and provided a comparative analysis against benchmark approaches, specifically MOSUM and TR-CUSUM, in centralized and distributed frameworks, respectively. Additionally, by varying the parameters, including the magnitude of change $\delta$, tail index $\kappa$, and window size m, the influence of different parameter settings on the detection performance is investigated. All experimental results, namely, the empirical size (Size), empirical power (Power), expected detection delay (EDD), etc., are based on 2000 Monte Carlo repetitions.

In this experiment, a randomly generated geometric graph is used to generate the distributed sensor network topology. That is, 50 sensor nodes are randomly scattered in a $1\times 1$ unit square region, and the communication radius is set to be 0.3 to construct the neighborhood. Following the consensus weight design in [34], the largest connected component is adopted, with all edge weights being uniformly distributed. The affected sensor is randomly selected as the center, and all sensors located within a circular region of radius $r=0.3$ centered at this sensor are considered affected. The number of sensors within this affected region ranges approximately from 14 to 17. The following data generation model is used to simulate the data sequences for each sensor node:

$$y_{t_i}=\left\{\begin{array}{cc}{\epsilon }_{t,i},\hfill & 1\le t\le \left[n\tau \right],\hfill \\ {\delta }_i+{\epsilon }_{t,i},\hfill & \left[n\tau \right]+1\le t\le n,\hfill \end{array}\right.$$

where ${\epsilon }_{t,i}$ are heavy-tailed random variables. Unless otherwise specified, the parameters are set as follows: the threshold parameter $K=1.345$ for the Huber function was determined through numerical simulations based on previous studies [35,36] and corresponds to 95% asymptotic efficiency under the standard normal distribution; sample size $n=1000$; tail index $\kappa =0.6,0.8,1.2,1.6,2$; magnitude of change $\delta =0.5,1,1.5$; window size $m=2\left[nh\right]=30,40,50,60,70,80,100$; and affected region radius $r=0.1,0.2,0.3,0.5,0.7$.

5.1. Asymptotic Critical Values

Experiment I determines the threshold and critical value through Monte Carlo simulations. In the TR-CUSUM algorithm, the critical value must be estimated by fitting no-change sample data during the monitoring process, which significantly increases the algorithm’s computational complexity; moreover, the TR-CUSUM algorithm does not provide explicit guidance for selecting the threshold $\eta$. In view of this, based on the theoretical proof that the test statistic converges to a functional of Brownian motion under the null hypothesis (Theorem 1), the threshold $\eta$ and critical value b are set according to the significance level $\alpha =0.1$ and $\alpha =0.05$, the corresponding quantile of the limiting distribution, respectively. These significance levels are chosen to rigorously control the false alarm rate while ensuring the detection results maintain standard statistical confidence. In practice, the algorithm is robust to the choice of $\eta$ within a reasonable range of significance levels, as the global aggregation mechanism effectively compensates for the information filtered by local thresholds. This ensures a stable trade-off between communication efficiency and detection sensitivity without requiring sensitive manual tuning. This approach is consistent with the critical value selection method used in the MOSUM algorithm; however, the threshold $\eta$ setting in MOSUM remains somewhat subjective. Additionally, since the limiting distribution under the null hypothesis is a function of window size, we investigate the thresholds and critical values under $m=40,50,60,70$. A larger window size m enhances estimation accuracy and reduces false alarms by increasing local samples, while a smaller m minimizes detection delay; detailed experimental analysis of this trade-off is provided in later sections. The specific results are summarized in

Table 1. Thresholds and critical values.

$\mathit{\kappa }$	$\mathit{m}=40$		$\mathit{m}=50$		$\mathit{m}=60$		$\mathit{m}=70$
	$\mathit{\eta }$	$\mathit{b}$	$\mathit{\eta }$	$\mathit{b}$	$\mathit{\eta }$	$\mathit{b}$	$\mathit{\eta }$	$\mathit{b}$
2.0	1.9135	2.0624	1.8336	1.9597	1.7764	1.8917	1.7553	1.8837
1.6	1.9010	2.0685	1.8177	1.9411	1.7654	1.8801	1.7587	1.8776
1.2	1.9081	2.0438	1.8292	1.9440	1.7728	1.8962	1.7496	1.8794
0.8	1.8849	2.0068	1.8047	1.9493	1.7624	1.8730	1.7532	1.8719
0.4	1.9209	2.0822	1.7937	1.9464	1.7484	1.8798	1.7431	1.8848

.

According to the experimental results presented in Table 1, it can be observed that the proposed change point detection algorithm maintains relatively stable threshold and critical values across different tail index distributions. This observation is highly consistent with the recent findings [37,38], further demonstrating the robustness of the proposed method in handling complex and diverse data environments.

5.2. Performance Evaluation

In this section, the empirical size and power of the proposed distributed robust M-estimator-based change point detection (DRM-CPM) method are compared with those of the benchmark algorithms, MOSUM and TR-CUSUM, to demonstrate the effectiveness of the proposed approach.

Experiment II: Under the null hypothesis $H_0$, the empirical size is evaluated by setting the sample size $n=1000$ and the window size $m=2\left[nh\right]=40,50,60,70$, and then analyzing the empirical rejection rates of different change point detection methods under heavy-tailed noise. According to the tail index $\kappa$, the sequences exhibit different distributional characteristics: when $\kappa =2$, the sequence exhibits the characteristics of a Gaussian distribution; when $\kappa <2$, the tail of the sequence becomes heavier as $\kappa$ decreases, leading to more frequent extreme events. The specific results are presented in

Table 2. Empirical sizes under $H_0$.

$\mathit{\kappa }$	$\mathit{m}=40$			$\mathit{m}=50$
	DRM-CPM	TR-CUSUM	MOSUM	DRM-textbf	TR-CUSUM	MOSUM
2.0	0.0818	0.0434	0.0218	0.0507	0.0434	0.0365
1.6	0.0683	–	0.0947	0.0661	–	0.2471
1.2	0.0538	–	0.1076	0.0369	–	0.1360
0.8	0.0625	–	0.3864	0.0207	–	0.4625
0.6	0.0982	–	0.3163	0.0254	–	0.2125
$\mathit{\kappa }$	$\mathit{m}=\mathbf{60}$			$\mathit{m}=\mathbf{70}$
	DRM-CPM	TR-CUSUM	MOSUM	DRM-CPM	TR-CUSUM	MOSUM
2.0	0.0456	0.0434	0.0131	0.0205	0.0434	0.0157
1.6	0.0318	–	0.1087	0.0317	–	0.1952
1.2	0.0241	–	0.2392	0.0193	–	0.3164
0.8	0.0337	–	0.1097	0.0146	–	0.3631
0.6	0.0184	–	0.2738	0.0093	–	0.2065

.

The results of the empirical size comparison show that when $\kappa =2$, TR-CUSUM can maintain a relatively stable empirical size close to the significance level, but when $\kappa <2$, the empirical size of this method is obviously different from the significance level, which indicates that the method is completely ineffective in controlling the false alarm rate under the heavy-tail distribution. This is mainly because TR-CUSUM statistics rely on the log-likelihood ratio, which requires accurate knowledge of the probability density function and its parameters before and after the change. Although it may be possible to accurately estimate these parameters for normal distribution (e.g., $\kappa =2$), for a heavy-tail distribution with infinite variance, accurately estimating these parameters becomes challenging, severely limiting the TR-CUSUM method’s usefulness in heavy-tail scenarios.

When $\kappa =2$, the MOSUM can still keep a reasonable error control level as shown in the significance level because of the heavy fluctuation of heavy-tail process. While, when $\kappa <2$, its empirical size is much larger than the significance level, and it is very sensitive to tail index. Since the heavy-tailed data will lead to large fluctuations, it will thus easily cause a false alarm to appear in methods sensitive to extreme values.

In contrast, as shown in Table 2, the empirical size of our proposed DRM-CPM method remains consistently stable around the significance level of 0.05, regardless of the tail index $\kappa$. This demonstrates the excellent robustness and error control capability of our method under heavy-tailed noise conditions. Both the experimental results and the theoretical analysis (Theorem 1) support the effectiveness and practical applicability of the DRM-CPM method for online change point detection in heavy-tailed data.

Experiment III is conducted under the alternative hypothesis $H_1$, with a sample size of $n=1000$. The primary goal is to investigate and discuss the Power performance of different change point methods for heavy-tailed distributions with varying parameters. As shown in Figure 3a–i, the empirical performance of DRM-CPM, D-LAD, TR-CUSUM and MOSUM under different tail indices $\kappa$ when $\delta =1,1.5,2$.

From the numerical results, we can see that TR-CUSUM and MOSUM suffer significant failure when $\kappa <2$, with their empirical power approaching 0. The underlying reasons for this phenomenon have been previously discussed. To further evaluate the performance against robust alternatives, we introduced the Distributed Least Absolute Deviations (D-LAD) algorithm (which minimizes the $L_1$ norm) as a benchmark. The results indicate that while D-LAD achieves comparable power to DRM-CPM at larger window sizes (e.g., $m=60$), its performance degrades notably as the window size decreases (e.g., $m=40$). This decline is attributed to the lower statistical efficiency of the median-based approach in small samples, whereas DRM-CPM maintains higher power by effectively utilizing bounded influence functions. Consequently, DRM-CPM proves to be more suitable for heavy-tailed distributions.

However, when the $\delta$ is small, (e.g., as shown in Figure 3a), the detection power of DRM-CPM becomes bad. It may be caused by the fact that its signal strength when small is relatively weak when heavy tailed, which makes it more difficult to be detected. When increased, (Figure 3b,c), the empirical power of DRM-CPM greatly improves. It shows that DRM-CPM has a strong detection ability in structural changes with a larger magnitude of change.

Based on the results of Experiment III, in Experiment IV, we would like to further explore how different methods will affect the EDD and average communication cost.

Table 3. EDD and $\zeta$ under $H_1$ ($m=50$).

$\mathit{\delta }$	k	EDD			$\mathit{\zeta }$
		DRM-CPM	R-CUSUM	MOSUM	DRM-CPM	TR-CUSUM	MOSUM
	2.0	24.06	10.67	23.86	0.2738	0.2251	0.0758
	1.6	23.81	–	36.72	0.2809	–	0.1010
1	1.2	24.22	–	19.09	0.2902	–	0.1792
	0.8	24.51	–	20.64	0.3021	–	0.2117
	0.6	25.31	–	17.11	0.3277	–	0.2153
	2.0	19.96	5.21	22.57	0.2533	0.2281	0.0531
	1.6	19.88	–	14.95	0.2577	–	0.0932
1.5	1.2	20.19	–	25.27	0.2588	–	0.1076
	0.8	21.07	–	16.59	0.2634	–	0.1645
	0.6	21.09	–	7.21	0.2864	–	0.1032
	2.0	18.97	3.41	25.97	0.2548	0.2289	0.0493
	1.6	18.66	–	37.82	0.2501	–	0.1082
2	1.2	18.59	–	35.08	0.2511	–	0.0625
	0.8	18.64	–	20.63	0.2523	–	0.0994
	0.6	18.83	–	9.58	0.2553	–	0.1133

shows the EDD and $\zeta$ values of different methods under different $\delta$ and $\kappa$.

From the EDD values in Table 3, when $\kappa =2$, the EDD values of MOSUM and DRM-CPM are relatively close. However, EDD of TR-CUSUM is much lower than the other two. This is mainly because, compared with MOSUM and DRM-CPM, TR-CUSUM is more sensitive to the structural change of sequence, which leads to the change point appearing faster when the change happens with known pre- and post-change distributions. Regarding the average communication cost $\zeta$, the values for DRM-CPM and TR-CUSUM are essentially consistent, while MOSUM exhibits a notably lower $\zeta$. However, when $\kappa <2$, the TR-CUSUM method completely fails, resulting in the absence of EDD and $\zeta$ values and rendering them without practical significance. Similarly, the MOSUM method nearly fails within this range, and its EDD and $\zeta$ values lack practical meaning.

DRM-CPM demonstrates relatively stable EDD and $\zeta$ across different magnitudes of change. Nevertheless, the EDD value increases with decreasing $\delta$, indicating that smaller $\delta$ increases the difficulty of change point identification, consequently leading to a longer EDD.

5.3. The Numerical Dependency on Parameters

Experiment V aims to compare the performance of different detection methods under the fixed conditions of $\kappa =2$, $\delta =2$, and $m=50$, while varying the number of affected sensors S. Specifically, the proportion of affected sensors, denoted as r, is set to $0.15,0.24,0.33,0.50,0.70$, corresponding to $3,8,14,36,50$ affected sensors, respectively.

As depicted in Figure 4, the solid lines represent the empirical power (Power) of each method, and the dashed lines represent the EDD. The observation results indicate that when the number of affected sensors is very small, the Power of the DRM-CPM method experiences a certain degree of decrease but remains above $0.9$. Additionally, the TR-CUSUM method also exhibits an increase in EDD when the number of affected sensors is small; however, its performance remains relatively stable under other conditions. Contrarily, the EDD of the MOSUM method rises when the number of affected sensors becomes smaller, and the detection power of this method decreases. Remarkably, the EDD of the DRM-CPM method is remarkably stable against the number of affected sensors. Note that in Figure 4, the Y-axis starts from 0.9 to highlight the differences in high-power scenarios, which visually accentuates minor Monte Carlo fluctuations. This demonstrates that DRM-CPM effectively resists variations in spatial scale, maintaining robust detection timeliness in both local failure and global change scenarios.

Since the sliding window size m is a crucial parameter for the DRM-CPM method, Experiment VI examines its impact on detection performance. In this experiment, with $\delta =2$, we set $m=30,40,50,60,70,80,100$ and examine how the method behaves for different window sizes. The experimental results are shown in Figure 5. The solid lines show the Power for different change point indices $\kappa$, and the dashed lines show the corresponding EDD. The convergence of Power values and the linear predictability of EDD collectively confirm the high operational stability of the proposed method under parameter perturbations.

Observations from the figure reveal that when $m=30$, the Power value significantly decreases. This is mainly because an excessively small m fails to extract sufficient effective change information from the data, thereby increasing the false alarm rate. When $m\ge 50$, the Power tends to stabilize. Furthermore, the experiment demonstrates that the EDD exhibits a near-linear increasing trend with the increase in m. Practically, selecting m requires balancing statistical stability against detection latency. While m must be sufficient (typically $\ge 30$) to satisfy asymptotic normality, it should be kept minimal to reduce the Average Detection Delay (ADD). We recommend performing an offline calibration using stable historical data to identify the minimum m that maintains the target false alarm rate.

6. Empirical Applications

In this section, the effectiveness of the proposed method is evaluated using daily average wind speed data from Atlantic City International Airport in New Jersey, USA, along with five nearby monitoring stations. Covering a full year from 1 November 2022, to 31 October 2023, the data comprises 365 observations in total (source: NCEI NOAA https://www.ncei.noaa.gov/ (accessed on 20 January 2026). Using the R package stable (version 3.4.2) provided by Nolan [39], we estimate the tail index as $\kappa =1.705$, which confirms that the wind speed data exhibits notable heavy-tailed behavior.

Three change point detection methods, DRM-CPM, TR-CUSUM, and MOSUM, were applied to monitor the dataset. The results are shown in Figure 6 below. The dashed line represents the original standardized data, highlighting its heavy-tailed nature. The solid line shows the 7-day moving average, which helps visualize shifts in the location over time. The three methods gave different results: DRM-CPM detected a change point at $k_{\mathrm{DRM}\text{-}\mathrm{CPM}}^{\ast }=230$ (18 June 2023), TR-CUSUM flagged one much earlier at $k_{\mathrm{TR}\text{-}\mathrm{CUSUM}}^{\ast }=108$ (16 February 2023), and MOSUM identified a later change point at $k_{\mathrm{MOSUM}}^{\ast }=263$ (21 July 2023).

To validate the empirical findings, we cross-referenced the detected change points with the National Climate Report from the National Centers for Environmental Information (NCEI/NOAA). The results are highly consistent with recorded extreme weather events in the region. Specifically, NOAA reports confirm that on 8 June, two significant tornadoes (EF-1 and EF-0) struck the southern part of the study area. These were followed on 16 and 17 June by intense straight-line winds with speeds reaching up to 100 mph in neighboring Pennsylvania. Such localized, high-intensity atmospheric disturbances represent abrupt structural shocks to the wind speed distribution. The ability of DRM-CPM to identify these specific time points—amidst the heavy-tailed noise inherent in storm data—demonstrates its superior sensitivity to “regime shifts” compared to TR-CUSUM and MOSUM, which failed to accurately localize these brief but violent transitions.

7. Conclusions

This paper addresses the change point detection problem in distributed sensor networks and proposes a distributed robust M-estimator-based change point detection (DRM-CPM). The core achievement of this study lies in resolving the detection failure issue in heavy-tailed data scenarios by overcoming the traditional dependence on Gaussian assumptions. The algorithm significantly reduces the communication overhead in wireless sensor networks by introducing an event-triggered communication strategy and Ratio-type test statistics, and achieves robust detection of change points with unknown pre- and post-change distribution parameters and heavy-tailed noise. Theoretical analysis and experimental results demonstrate that DRM-CPM maintains a high detection power under different tail indices $\kappa \in (0,2]$ while keeping a low false alarm rate, and its EDD and $\zeta$ are superior to existing methods. Compared with TR-CUSUM and MOSUM algorithms, DRM-CPM fulfills the research objectives set out in the introduction by providing a solution for sparse changes in non-Gaussian environments.

However, the current study is not without its limitations. First, when the magnitude of change $\delta$ is relatively small, the detection delay tends to increase as data fluctuations become more pronounced, which may affect real-time performance in high-noise scenarios. Second, the selection of the sliding window size m currently relies on a manual trade-off between detection sensitivity and timeliness, lacking a fully automated optimization mechanism.

Building upon these findings, future research will focus on several key directions to enrich the proposed framework. We intend to explore adaptive window adjustment strategies that can dynamically respond to varying network conditions. Furthermore, extending the algorithm to multivariate changes and non-homogeneous network environments will be a priority to enhance its applicability and scientific depth in increasingly complex sensing scenarios. The research findings of this paper provide theoretical support and technical references for online anomaly detection in fields such as industrial monitoring and environmental sensing.