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Spatial autocorrelation, of which Geary’s c has traditionally been a popular measure, is fundamental to spatial science. This paper provides a new perspective on Geary’s c. We discuss this using concepts from spectral graph theory/linear algebraic graph theory. More precisely, we provide three types of representations for it: (a) graph Laplacian representation, (b) graph Fourier transform representation, and (c) Pearson’s correlation coefficient representation. Subsequently, we illustrate that the spatial autocorrelation measured by Geary’s c is positive (resp. negative) if spatially smoother (resp. less smooth) graph Laplacian eigenvectors are dominant. Finally, based on our analysis, we provide a recommendation for applied studies.

To analyze and synthesize signals on networks or graphs, Fourier theory has been extended to irregular domains, leading to a so-called graph Fourier transform. Unfortunately, different from the traditional Fourier transform, each graph exhibits a different graph Fourier transform. Therefore to analyze the graph-frequency domain properties of a graph signal, the graph Fourier modes and graph frequencies must be computed for the graph under study. Although to find these graph frequencies and modes, a computationally expensive, or even prohibitive, eigendecomposition of the graph is required, there exist families of graphs that have properties that could be exploited for an approximate fast graph spectrum computation. In this work, we aim to identify these families and to provide a divide-and-conquer approach for computing an approximate spectral decomposition of the graph. Using the same decomposition, results on reducing the complexity of graph filtering are derived. These results provide an attempt to leverage the underlying topological properties of graphs in order to devise general computational models for graph signal processing.

Reed–Xiaoli detector (RXD) is recognized as the benchmark algorithm for image anomaly detection; however, it presents known limitations, namely the dependence over the image following a multivariate Gaussian model, the estimation and inversion of a high-dimensional covariance matrix, and the inability to effectively include spatial awareness in its evaluation. In this work, a novel graph-based solution to the image anomaly detection problem is proposed; leveraging the graph Fourier transform, we are able to overcome some of RXD’s limitations while reducing computational cost at the same time. Tests over both hyperspectral and medical images, using both synthetic and real anomalies, prove the proposed technique is able to obtain significant gains over performance by other algorithms in the state of the art.

The rapid advancements in spatially resolved transcriptomics (SRT) enable the characterization of gene expressions while preserving spatial information. However, high dropout rates and noise hinder accurate spatial domain identification for understanding tissue architecture. We present DeepGFT, a method that simultaneously models spot-wise and gene-wise relationships by integrating deep learning with graph Fourier transform for spatial domain identification. Benchmarking results demonstrate the superiority of DeepGFT over existing methods. DeepGFT detects tumor substructures with immune-related differences in human breast cancer, identifies the complex germinal centers accurately in human lymph node, and accurately reveals the developmental changes in 3D Drosophila data.

Gearboxes operate in challenging environments, which leads to a heightened incidence of failures, and ambient noise further compromises the accuracy of fault diagnosis. To address this issue, we introduce a fault diagnosis method that employs singular value decomposition (SVD) and graph Fourier transform (GFT). Singular values, commonly employed in feature extraction and fault diagnosis, effectively encapsulate various fault states of mechanical equipment. However, prior methods neglect the inter-relationships among singular values, resulting in the loss of subtle fault information concealed within. To precisely and effectively extract subtle fault information from gear vibration signals, this study incorporates graph signal processing (GSP) technology. Following SVD of the original vibration signal, the method constructs a graph signal using singular values as inputs, enabling the capture of topological relationships among these values and the extraction of concealed fault information. Subsequently, the graph signal undergoes a transformation via GFT, facilitating the extraction of fault features from the graph spectral domain. Ultimately, by assessing the Mahalanobis distance between training and testing samples, distinct defect states are discerned and diagnosed. Experimental results on bearing and gear faults demonstrate that the proposed method exhibits enhanced robustness to noise, enabling accurate and effective diagnosis of gearbox faults in environments with substantial noise.

This paper presents the benefits of using the random-walk normalized Laplacian matrix as a graph-shift operator and defines the frequencies of a graph by the eigenvalues of this matrix. A criterion to order these frequencies is proposed based on the Euclidean distance between a graph signal and its shifted version with the transition matrix as shift operator. Further, the frequencies of a periodic graph built through the repeated concatenation of a basic graph are studied. We show that when a graph is replicated, the graph frequency domain is interpolated by an upsampling factor equal to the number of replicas of the basic graph, similarly to the effect of zero-padding in digital signal processing.

Autism spectrum disorder (ASD) is a complex neurodevelopmental condition associated with disrupted brain connectivity. Traditional graph-theoretical approaches have been widely employed to study ASD biomarkers; however, these methods are often limited to static topological measures and lack the capacity to capture spectral characteristics of brain activity, especially in multimodal data settings. This limits their ability to model dynamic neural interactions and reduces their diagnostic precision. To overcome these limitations, we propose a Graph Signal Processing (GSP)-based framework that integrates spectral-domain features with topological descriptors to model brain connectivity more comprehensively. Using publicly available fMRI and EEG datasets, we construct subject-specific connectivity graphs where nodes represent brain regions and edges encode functional interactions. We extract advanced GSP features such as Graph Fourier Transform coefficients, spectral entropy, and clustering coefficients, and combine them using Principal Component Analysis (PCA). These are classified using a Support Vector Machine (SVM) with a radial basis function (RBF) kernel. The proposed model achieves 98.8% classification accuracy, significantly outperforming prior multimodal GSP studies. Feature ablation analysis reveals that spectral entropy contributes most to this improvement, with its removal resulting in a nearly 30% performance drop. Additionally, a 25% sparsity threshold in graph construction was found to maximize both robustness and computational efficiency. These findings demonstrate that incorporating frequency-domain information through GSP enables a more discriminative and biologically meaningful representation of ASD-related neural patterns, offering a promising direction for accurate diagnosis and biomarker discovery.

The electroencephalogram (EEG) source localization has been one of the most critical brain science research issues. Over the past few decades, the most preeminent ways for solving the EEG inverse problem are to employ prior information or regularization, but this approach is intractable to localize the deep active brain source. Here, the graph Fourier transform (GFT) and the bi-directional long-short term memory (BiLSTM) neural network are introduced to solve the EEG inverse problem in a more efficient and robust way. The presented GFT-BiLSTM in this paper not only fully utilizes spatial information of the brain source signal, but also makes full use of the powerful self-learning ability of the BiLSTM. In the GFT-BiLSTM, the GFT is used for the signal decomposition of the source signal to reduce its dimension. The BiLSTM is adopted to learn the mapping relationship between the brain sources and the recorded EEG. The results show that GFT-BiLSTM outperforms other state-of-the-art inverse models in synthetic data. Regardless of whether the activated region is single or multiple, the area under the curve (AUC) corresponding to GFT-BiLSTM can be reliably above 0.96. When the signal-to-noise ratio (SNR) varies, the GFT-BiLSTM exhibits strong robustness with the highest AUC while the lowest localization error (LE). This fully demonstrates the superiority of GFT-BiLSTM when applied to solve the EEG inverse problem.

Geary’s c is a prominent measure of spatial autocorrelation in univariate spatial data. It uses a weighted sum of squared differences. This paper develops Geary’s c for multivariate spatial data. It can describe the similarity/discrepancy between vectors of observations at different vertices/spatial units by a weighted sum of the squared Euclidean norm of the vector differences. It is thus a natural extension of the univariate Geary’s c. This paper also develops a local version of it. We then establish their properties.

Spatial autocorrelation, which describes the similarity between signals on adjacent vertices, is central to spatial science, and Geary’s c is one of the most-prominent numerical measures of it. Using concepts from spectral graph theory, this paper documents new theoretical results on the measure. MATLAB/GNU Octave user-defined functions are also provided.