Spatial modeling and monitoring considering long-range dependence

We define multi-self-similar random fields, that is, random fields that are self-similar component-wise. We characterize them, relate them to stationary random fields using a Lamperti-type transformation and study these stationary fields. We also extend the notions of local stationarity and local stationarity reducibility to random fields. Our work is motivated by applications arising from climatological and environmental sciences. We illustrate these new concepts with the fractional Brownian sheet and the Lévy fractional Brownian random field.

Integration of Information and Communication Technology (ICT) in modern smart grids (SGs) offers many advantages including the use of renewables and an effective way to protect, control and monitor the energy transmission and distribution. To reach an optimal operation of future energy systems, availability, integrity and confidentiality of data should be guaranteed. Research on the cyber-physical security of electrical substations based on IEC 61850 is still at an early stage. In the present work, we first model the network traffic data in electrical substations, then, we present a statistical Anomaly Detection (AD) method to detect Denial of Service (DoS) attacks against the Generic Object Oriented Substation Event (GOOSE) network communication. According to interpretations on the self-similarity and the Long-Range Dependency (LRD) of the data, an Auto-Regressive Fractionally Integrated Moving Average (ARFIMA) model was shown to describe well the GOOSE communication in the substation process network. Based on this ARFIMA-model and in view of cyber-physical security, an effective model-based AD method is developed and analyzed. Two variants of the statistical AD considering statistical hypothesis testing based on the Generalized Likelihood Ratio Test (GLRT) and the cumulative sum (CUSUM) are presented to detect flooding attacks that might affect the availability of the data. Our work presents a novel AD method, with two different variants, tailored to the specific features of the GOOSE traffic in IEC 61850 substations. The statistical AD is capable of detecting anomalies at unknown change times under the realistic assumption of unknown model parameters. The performance of both variants of the AD method is validated and assessed using data collected from a simulation case study. We perform several Monte-Carlo simulations under different noise variances. The detection delay is provided for each detector and it represents the number of discrete time samples after which an anomaly is detected. In fact, our statistical AD method with both variants (CUSUM and GLRT) has around half the false positive rate and a smaller detection delay when compared with two of the closest works found in the literature. Our AD approach based on the GLRT detector has the smallest false positive rate among all considered approaches. Whereas, our AD approach based on the CUSUM test has the lowest false negative rate thus the best detection rate. Depending on the requirements as well as the costs of false alarms or missed anomalies, both variants of our statistical detection method can be used and are further analyzed using composite detection metrics.

Chapter 16 - Long range dependence in heavy tailed stochastic processes

In recent years, empirical studies on network traffic both in various network configurations including local area networks (LAN) and wide area networks (WAN) convincingly show that the actual traffic exhibit self-similarity and long range dependence (LRD), which are very different from that predicted by traditional telecommunication traffic models, such as Poisson process. Self-similarity in nature brings about the long range dependent burstiness of network traffic. The experimental evidences reveal that the heavy tailness is the key cause for the self-similarity of the network traffic. We present two distinctive predictors based on α-stable innovation for the LRD traffic. The two predictors can minimize the dispersion according to the minimum dispersion criteria with infinite variance. The final predicted values are obtained by combining the previous two individual predicted values. The predicted results for the actual traces show that the two individual predictors are precise and effective, the last compound predictors can enhance the final predicted accuracy.

Wireless traffic and packet-based traffic in general possess heavy-tail marginal distributions and long-range dependence (LRD). The tail parameter /spl alpha/ of a heavy-tail distribution controls the variability of its realizations. Several traffic models, statistical test and resource management algorithms rely on the accurate estimation of the tail parameter. Conventional estimators for the tail parameter only work well when the data is short range dependent. In this paper we propose a new method to estimate the tail parameter from LRD data. This is achieved by utilizing the wavelet transform and extreme value theory. The algorithm is then applied to two stochastic processes that possess both heavy-tail marginal distributions and LRD. Results from our simulation show that the proposed method gives good estimates of /spl alpha/. We then estimate the tail parameter from a recently collected IEEE 802.11b traffic trace.

Previous studies on Sound Event Localization and Detection (SELD) have primarily focused on CNN- and Transformer-based designs. While CNNs possess local receptive fields, making it difficult to capture global dependencies over long sequences, Transformers excel at modeling long-range dependencies but have limited sensitivity to local time–frequency features. Recently, the VMamba architecture, built upon the Visual State Space (VSS) model, has shown great promise in handling long sequences, yet it remains limited in modeling local spatial details. To address this issue, we propose a novel state space model with an attention-enhanced feature fusion mechanism, termed FFMamba, which balances both local spatial modeling and long-range dependency capture. At a fine-grained level, we design two key modules: the Multi-Scale Fusion Visual State Space (MSFVSS) module and the Wavelet Transform-Enhanced Downsampling (WTED) module. Specifically, the MSFVSS module integrates a Multi-Scale Fusion (MSF) component into the VSS framework, enhancing its ability to capture both long-range temporal dependencies and detailed local spatial information. Meanwhile, the WTED module employs a dual-branch design to fuse spatial and frequency domain features, improving the richness of feature representations. Comparative experiments were conducted on the DCASE2021 Task 3 and DCASE2022 Task 3 datasets. The results demonstrate that the proposed FFMamba model outperforms recent approaches in capturing long-range temporal dependencies and effectively integrating multi-scale audio features. In addition, ablation studies confirmed the effectiveness of the MSFVSS and WTED modules.

The most important desidered characteristic for today's networks is quality of service (QoS). Currently DiffServ-based MPLS networks are able to the development of efficient methods for QoS provisioning. This article analysis, through computer simulation, the performance of WFQ schedulers in a DiffServ node loaded by long range dependence (LRD) traffic. The adopted process for the traffic generation consists in the aggregation of the traffic produced by several TCP connections transmitting files with a size that is modeled by heavy tail distribution. This distribution was obtained, from the Internet, by one-week long measurements. The objective is to observe the bandwidth distribution obtained with the WFQ scheduler, as well as the latency, i.e., the file transfer average delay, in an environment with different priorities. Simulations were carried out with the network simulator NS-2, and several simulation results are presented and analyzed.

Co-citation (number of nodes linking to both of a given pair of nodes) is often used heuristically to judge similarity between nodes in a complex network. We investigate the relation between node similarity and co-citation in the context of the Spatial Preferred Attachment (SPA) model. The SPA model is a spatial model, where nodes live in a metric space, and nodes that are close together in space are considered similar, and are more likely to link to one another.Theoretical analysis of the SPA model leads to a measure to estimate spatial distance from the link information, based on co-citation as well as the degrees of both nodes. Simulation results show this measure to be highly accurate in predicting the actual spatial distance.

Video deblurring is still an unsolved problem due to the challenging spatio-temporal modeling process. While existing convolutional neural network (CNN)-based methods show a limited capacity of effective spatial and temporal modeling for video deblurring. This paper presents VDTR, an effective Transformer-based model that makes the first attempt to adapt pure Transformer for video deblurring. VDTR exploits the superior long-range and relation modeling capabilities of Transformer for both spatial and temporal modeling. However, it is challenging to design an appropriate Transformer-based model for video deblurring due to the complicated non-uniform blurs, misalignment across multiple frames and the high computational costs for high-resolution spatial modeling. To address these problems, VDTR advocates performing attention within non-overlapping windows and exploiting the hierarchical structure for long-range dependencies modeling. For frame-level spatial modeling, we propose an encoder-decoder Transformer that utilizes multi-scale features for deblurring. For multi-frame temporal modeling, we adapt Transformer to fuse multiple spatial features efficiently. Compared with CNN-based methods, the proposed method achieves highly competitive results on both synthetic and real-world video deblurring benchmarks, including DVD, GOPRO, REDS and BSD. We hope such a Transformer-based architecture can serve as a powerful alternative baseline for video deblurring and other video restoration tasks. The source code will be available at <uri xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">https://github.com/ljzycmd/VDTR</uri> .

Floodplains are receptors of sediments from upland municipal areas, waste dumpsites, and adjacent rivers polluting floodplain farmlands used for both rainfed and irrigation continuous cropping. This study was therefore undertaken to study the distribution of heavy metals (HMs) and their potential pollution prediction using spatial modelling in the floodplain farmland around municipal waste dumpsite in Yola Adamawa, State Nigeria. Soil samples were collected systematically at 6 points (0, 20, 40, 60, 80 and 100 m) along three traverses from the dumpsite at interval of 20 m. Mean values of iron (Fe) and chromium (Cr) across the study area were significantly higher than at the dumpsite, while Cu and Zn were significantly higher at dumpsite and decreased as spatial distance increased away from dumpsite. Clay significantly influenced spatial distribution of HMs’ Fe and Cr, while sand and soil pH had negatively effects on HMs. Increase in total organic carbon, available phosphorus (P), exchangeable calcium (Ca) and magnesium (Mg) increased concentration of manganese (Mn), copper (Cu), lead (Pb), chromium (Cr) and zinc (Zn). Among the HMs, copper had the prediction model with highest r - value and was obtained using the square root of spatial distance {Cu = 225.397 - 1.5328(SD) ½} with the highest R-Squared value of 40.98%, though less than 50 %. The overall means of contamination factor (CF) indicated a decreasing order of Pb &gt; Zn &gt; Cd &gt; Cu &gt; Mn &gt; Cr &gt; Fe. The enrichment factor (EF) index had moderate risk for all the metals with only Cr having low risk to the floodplain. Assessment of HMs using CF, EF and potential ecological risks (PER) models had similar trend of correlations with spatial distances from dumpsite, hence were considered to be more appropriate models to use for predicting HMs pollution compared to index of geo-accumulation (Igeo) model in the study area.

Due to the tractability in analytical methods exponential burst distributions are more common in service modeling literature. However, previous traffic measurements findings argue heavy tailed distributions such as Pareto represents burst sizes in WWW browsing and Cauchy distributed burst sizes in e-mail sessions. The effect of long range dependence (LRD) introduced by these heavy tailed distributions results in self similarity in the aggregate traffic. The impact of self similarity is significant in the area of queue delay performance characteristics. This work addresses the issues of service modeling in medium access control (MAC) performance compared to traditional exponential session distributions against the above mentioned distributions in WWW browsing and e-mail sessions applied to an example GPRS MAC system. The results conclude high delay in Cauchy e-mail connection sizes and Pareto WWW burst sessions compared to exponential connections/bursts. Also, once looking in to the matter it can be seen that this delay is mainly due to the congestion caused by different types of bursts/connection distributions. Therefore the bursty traffic such as WWW browsing which demonstrates self similarity at the aggregate level influence the MAC protocol performance greater than in traditional traffic models.

The analysis and modeling of computer network traffic is a daunting task considering the amount of available data. This is quite obvious when considering the spatial dimension of the problem, since the number of interacting computers, gateways and switches can easily reach several thousands, even in a local area network (LAN) setting. This is also true for the time dimension: Willinger and Paxson (see Ann. Statist., vol.25, no.5, p.1856-66, 1997) cite the figures of 439 million packets and 89 gigabytes of data for a single week record of the activity of a university gateway in 1995. The complexity of the problem further increases when considering wide area network (WAN) data. In light of the above, it is clear that a notion of importance for modern network engineering is that of invariants, i.e., characteristics that are observed with some reproducibility and independently of the precise settings of the network under consideration. In this tutorial article, we focus on two such invariants related to the time dimension of the problem, namely, long-range dependence, or self-similarity, and heavy-tail marginal distributions.

A packet switched network such as the Internet provides only the best-effort service for users. Recent studies of the characteristics of packet arrival on a packet switched network have shown a self-similar property with long-range dependence, and also show a heavy-tailed distribution. This paper investigates end-to-end performance of the Internet empirically. It focuses on the inter-arrival process of packets and the property of packet loss between end-to-end hosts. We show that both the inter-arrival process and loss events are self-similar with a long-range dependence property. The inter-arrival process of packets has a heavy-tailed distribution which sums up the difference of all queues in intermediate routers.

We now consider a stochastic approximation scheme in the framework of Chap. 2, but with noise models that go beyond simple martingale difference noise, viz. noise with long-range dependence and heavy tails. This scenario has become important because of the many situations where such models are more natural, e.g., Internet traffic and finance.

A class of continuous-time models is developed for modelling data with heavy tails and long-range dependence. These models are based on the Green function solutions of fractional differential equations driven by Lévy noise. Some exact results on the second- and higher-order characteristics of the equations are obtained. Applications to stochastic volatility of asset prices and macroeconomics are provided.