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Keywords = Woodbury matrix identity

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12 pages, 302 KiB  
Article
Kernel Geometric Mean Metric Learning
by Zixin Feng, Teligeng Yun, Yu Zhou, Ruirui Zheng and Jianjun He
Appl. Sci. 2023, 13(21), 12047; https://doi.org/10.3390/app132112047 - 6 Nov 2023
Viewed by 1249
Abstract
Geometric mean metric learning (GMML) algorithm is a novel metric learning approach proposed recently. It has many advantages such as unconstrained convex objective function, closed form solution, faster computational speed, and interpretability over other existing metric learning technologies. However, addressing the nonlinear problem [...] Read more.
Geometric mean metric learning (GMML) algorithm is a novel metric learning approach proposed recently. It has many advantages such as unconstrained convex objective function, closed form solution, faster computational speed, and interpretability over other existing metric learning technologies. However, addressing the nonlinear problem is not effective enough. The kernel method is an effective method to solve nonlinear problems. Therefore, a kernel geometric mean metric learning (KGMML) algorithm is proposed. The basic idea is to transform the input space into a high-dimensional feature space through nonlinear transformation, and use the integral representation of the weighted geometric mean and the Woodbury matrix identity in new feature space to generalize the analytical solution obtained in the GMML algorithm as a form represented by a kernel matrix, and then the KGMML algorithm is obtained through operations. Experimental results on 15 datasets show that the proposed algorithm can effectively improve the accuracy of the GMML algorithm and other metric algorithms. Full article
(This article belongs to the Special Issue Machine/Deep Learning: Applications, Technologies and Algorithms)
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<p>Varying <span class="html-italic">t</span> (<span class="html-italic">p</span> is fixed).</p> Full article ">Figure 2
<p>Varying <span class="html-italic">t</span> (<span class="html-italic">p</span> is fixed).</p> Full article ">Figure 3
<p>Varying <span class="html-italic">p</span> (<span class="html-italic">t</span> is fixed).</p> Full article ">
13 pages, 1282 KiB  
Article
A Fast and Powerful Empirical Bayes Method for Genome-Wide Association Studies
by Tianpeng Chang, Julong Wei, Mang Liang, Bingxing An, Xiaoqiao Wang, Bo Zhu, Lingyang Xu, Lupei Zhang, Xue Gao, Yan Chen, Junya Li and Huijiang Gao
Animals 2019, 9(6), 305; https://doi.org/10.3390/ani9060305 - 31 May 2019
Cited by 2 | Viewed by 3738
Abstract
Linear mixed model (LMM) is an efficient method for GWAS. There are numerous forms of LMM-based GWAS methods. However, improving statistical power and computing efficiency have always been the research hotspots of the LMM-based GWAS methods. Here, we proposed a fast empirical Bayes [...] Read more.
Linear mixed model (LMM) is an efficient method for GWAS. There are numerous forms of LMM-based GWAS methods. However, improving statistical power and computing efficiency have always been the research hotspots of the LMM-based GWAS methods. Here, we proposed a fast empirical Bayes method, which is based on linear mixed models. We call it Fast-EB-LMM in short. The novelty of this method is that it uses a modified kinship matrix accounting for individual relatedness to avoid competition between the locus of interest and its counterpart in the polygene. This property has increased statistical power. We adopted two special algorithms to ease the computational burden: Eigenvalue decomposition and Woodbury matrix identity. Simulation studies showed that Fast-EB-LMM has significantly increased statistical power of marker detection and improved computational efficiency compared with two widely used GWAS methods, EMMA and EB. Real data analyses for two carcass traits in a Chinese Simmental beef cattle population showed that the significant single-nucleotide polymorphisms (SNPs) and candidate genes identified by Fast-EB-LMM are highly consistent with results of previous studies. We therefore believe that the Fast-EB-LMM method is a reliable and efficient method for GWAS. Full article
(This article belongs to the Collection Applications of Quantitative Genetics in Livestock Production)
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<p>Statistical powers for two different simulation experiments using the three different methods (Fast-EB-LMM, EMMA and EB). The heritabilities were set as 0.10, 0.05, 0.05, 0.15, 0.05 and 0.05 for QTN 1–6 in the first simulation experiment and 0.35, 0.35and 0.50 for Trait 1–3 in the second simulation experiment. (<b>A</b>) No polygenic background was simulated in the first simulation experiment. (<b>B</b>) A polygenic background was simulated in the first simulation experiment. (<b>C</b>) An epistatic background was simulated in the first simulation experiment. (<b>D</b>) Statistical power comparison in the second simulation experiment.</p> Full article ">Figure 2
<p>Statistical powers of six simulated QTNs from the first simulation experiment plotted against Type 1 error (in a log10 scale) for the three GWAS methods (Fast-EB-LMM, EMMA and EB). (<b>A</b>) No polygenic background. (<b>B</b>) With polygenic background. (<b>C</b>) With epistatic background.</p> Full article ">Figure 3
<p>Manhattan and quantile-quantile (QQ) plots of genome-wide association studies for CW and BW in the Chinese Simmental beef cattle population. (<b>A</b>,<b>B</b>) are Manhattan and QQ plots for CW; (<b>C</b>,<b>D</b>) are Manhattan and QQ plots for BW.</p> Full article ">
6416 KiB  
Article
Progressive Line Processing of Kernel RX Anomaly Detection Algorithm for Hyperspectral Imagery
by Chunhui Zhao, Weiwei Deng, Yiming Yan and Xifeng Yao
Sensors 2017, 17(8), 1815; https://doi.org/10.3390/s17081815 - 7 Aug 2017
Cited by 17 | Viewed by 4114
Abstract
The Kernel-RX detector (KRXD) has attracted widespread interest in hyperspectral image processing with the utilization of nonlinear information. However, the kernelization of hyperspectral data leads to poor execution efficiency in KRXD. This paper presents an approach to the progressive line processing of KRXD [...] Read more.
The Kernel-RX detector (KRXD) has attracted widespread interest in hyperspectral image processing with the utilization of nonlinear information. However, the kernelization of hyperspectral data leads to poor execution efficiency in KRXD. This paper presents an approach to the progressive line processing of KRXD (PLP-KRXD) that can perform KRXD line by line (the main data acquisition pattern). Parallel causal sliding windows are defined to ensure the causality of PLP-KRXD. Then, with the employment of the Woodbury matrix identity and the matrix inversion lemma, PLP-KRXD has the capacity to recursively update the kernel matrices, thereby avoiding a great many repetitive calculations of complex matrices, and greatly reducing the algorithm’s complexity. To substantiate the usefulness and effectiveness of PLP-KRXD, three groups of hyperspectral datasets are used to conduct experiments. Full article
(This article belongs to the Special Issue Analysis of Multispectral and Hyperspectral Data)
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Figure 1
<p>The imaging principle of push broom scanner.</p> Full article ">Figure 2
<p>Schematic representation of data acquisition and processing: (<b>a</b>) The process of the parallel causal sliding windows; (<b>b</b>) The causal sliding rectangle window at <math display="inline"> <semantics> <mrow> <msubsup> <mstyle mathvariant="bold-italic" mathsize="normal"> <mi>L</mi> </mstyle> <mi>n</mi> <mi>m</mi> </msubsup> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msubsup> <mstyle mathvariant="bold-italic" mathsize="normal"> <mi>L</mi> </mstyle> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> <mi>m</mi> </msubsup> </mrow> </semantics> </math>.</p> Full article ">Figure 3
<p>Recursive update process of the kernel Gram matrix inversion.</p> Full article ">Figure 4
<p>(<b>a</b>) Cuprite Airborne Visible/Infrared Imaging Spectrometer sensor (AVIRIS) image; (<b>b</b>) Spatial positions of A, B, C, K, and M.</p> Full article ">Figure 5
<p>Synthetic dataset. (<b>a</b>) Synthetic image; (<b>b</b>) Ground truth map.</p> Full article ">Figure 6
<p>Pavia center hyperspectral image scene.</p> Full article ">Figure 7
<p>(<b>a</b>) Pavia Center dataset; (<b>b</b>) Ground truth map.</p> Full article ">Figure 8
<p>AVIRIS Hyperspectral Image.</p> Full article ">Figure 9
<p>(<b>a</b>) San Diego airport dataset; (<b>b</b>) Ground truth map.</p> Full article ">Figure 10
<p>Area under the curve (AUC) of the progressive line processing Kernel-RX detector (PLP-KRXD) with a different polynomial kernel parameter <math display="inline"> <semantics> <mi>d</mi> </semantics> </math> on three datasets: Synthetic dataset, Pavia Center dataset, and San Diego Airport dataset.</p> Full article ">Figure 11
<p>Receiver operation characteristic (ROC) curves obtained by the Kernel-RX detector (KRXD) and the PLP-KRXD on three HSI datasets. (<b>a</b>) Synthetic dataset; (<b>b</b>) Pavia Center dataset; (<b>c</b>) San Diego Airport dataset.</p> Full article ">Figure 11 Cont.
<p>Receiver operation characteristic (ROC) curves obtained by the Kernel-RX detector (KRXD) and the PLP-KRXD on three HSI datasets. (<b>a</b>) Synthetic dataset; (<b>b</b>) Pavia Center dataset; (<b>c</b>) San Diego Airport dataset.</p> Full article ">Figure 12
<p>The detection maps of the KRXD and the PLP-KRXD on the three HSI datasets. (<b>a</b>) Synthetic dataset; (<b>b</b>) Pavia Center dataset; (<b>c</b>) San Diego Airport dataset.</p> Full article ">Figure 13
<p>The three-dimensional (3D) plots of the detection results obtained by the KRXD and the PLP-KRXD on three HSI datasets. (<b>a</b>) Synthetic dataset; (<b>b</b>) Pavia Center dataset; (<b>c</b>) San Diego Airport dataset.</p> Full article ">Figure 14
<p>Real-time progressive procedures of PLPKRXD on the three HSI datasets. (<b>a</b>–<b>f</b>) Synthetic dataset; (<b>g</b>–<b>l</b>) Pavia Center dataset; (<b>m</b>–<b>r</b>) San Diego Airport dataset.</p> Full article ">
9748 KiB  
Article
Real-Time Anomaly Detection Based on a Fast Recursive Kernel RX Algorithm
by Chunhui Zhao, Xifeng Yao and Bormin Huang
Remote Sens. 2016, 8(12), 1011; https://doi.org/10.3390/rs8121011 - 11 Dec 2016
Cited by 10 | Viewed by 4809
Abstract
Abstract: Real-time anomaly detection has received wide attention in remote sensing image processing because many moving targets must be detected on a timely basis. A widely-used anomaly detection algorithm is the Reed-Xiaoli (RX) algorithm that was proposed by Reed and Yu. The [...] Read more.
Abstract: Real-time anomaly detection has received wide attention in remote sensing image processing because many moving targets must be detected on a timely basis. A widely-used anomaly detection algorithm is the Reed-Xiaoli (RX) algorithm that was proposed by Reed and Yu. The kernel RX algorithm proposed by Kwon and Nasrabadi is a nonlinear version of the RX algorithm and outperforms the RX algorithm in terms of detection accuracy. However, the kernel RX algorithm is computationally more expensive. This paper presents a novel real-time anomaly detection framework based on the kernel RX algorithm. In the kernel RX detector, the inverse covariance matrix and the estimated mean of the background data in the kernel space are non-causal and computationally inefficient. In this work, a local causal sliding array window is used to ensure the causality of the detection system. Using the matrix inversion lemma and the Woodbury matrix identity, both the inverse covariance matrix and estimated mean can be recursively derived without extensive repetitive calculations, and, therefore, the real-time kernel RX detector can be implemented and processed pixel-by-pixel in real time. To substantiate its effectiveness and utility in real-time anomaly detection, real hyperspectral data sets are utilized for experiments. Full article
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Figure 1

Figure 1
<p>Local causal sliding array windows at <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">r</mi> <mi>n</mi> </msub> </mrow> </semantics> </math> and <math display="inline"> <semantics> <mrow> <msub> <mi mathvariant="bold">r</mi> <mrow> <mi>n</mi> <mo>+</mo> <mn>1</mn> </mrow> </msub> </mrow> </semantics> </math>.</p> Full article ">Figure 2
<p>Pavia University hyperspectral image scene.</p> Full article ">Figure 3
<p>(<b>a</b>) The smaller image scene; (<b>b</b>) ground truth of the smaller image scene.</p> Full article ">Figure 4
<p>Pavia Center hyperspectral image scene.</p> Full article ">Figure 5
<p>(<b>a</b>) The smaller image scene; (<b>b</b>) ground truth of the smaller image scene.</p> Full article ">Figure 6
<p>San Diego Airport hyperspectral image scene.</p> Full article ">Figure 7
<p>(<b>a</b>) The smaller image scene; (<b>b</b>) ground truth of the smaller image scene.</p> Full article ">Figure 8
<p>The area under the cure (AUC) of the local real-time causal kernel RX detector (LRTC-KRXD) with the changing kernel parameter <math display="inline"> <semantics> <mi>c</mi> </semantics> </math> on the Pavia University (PaviaU) dataset, Pavia Center dataset, and San Diego Airport dataset.</p> Full article ">Figure 9
<p>The receiver operating characteristics (ROC) curves of the LRTC-KRXD with changing local causal sliding array window width on the (<b>a</b>) PaviaU dataset; (<b>b</b>) Pavia Center dataset; (<b>c</b>) San Diego Airport dataset.</p> Full article ">Figure 10
<p>The grayscale results of the LRTC-KRXD with the changing local causal sliding array window width on the PaviaU, Pavia Center, and San Diego Airport datasets.</p> Full article ">Figure 11
<p>The ROC curves of four anomaly detectors on (<b>a</b>) PaviaU dataset; (<b>b</b>) Pavia Center dataset; (<b>c</b>) San Diego Airport dataset.</p> Full article ">Figure 12
<p>(<b>a</b>–<b>l</b>) The grayscale results of the global real-time causal RX detector (GRTC-RXD), local real-time causal RX detector (LRTC-RXD), local kernel RX detector (LKRXD), and LRTC-KRXD on the PaviaU, Pavia Center, and San Diego Airport datasets.</p> Full article ">Figure 13
<p>(<b>a</b>–<b>l</b>) The 3D plots of the GRTC-RXD, LRTC-RXD, LKRXD, and LRTC-KRXD on the PaviaU, Pavia Center, and San Diego Airport datasets.</p> Full article ">Figure 14
<p>Progressive detection procedures of the LRTC-KRXD on three datasets. (<b>a</b>,<b>f</b>,<b>k</b>) 1/5 detection result; (<b>b</b>,<b>g</b>,<b>l</b>) 2/5 detection result; (<b>c</b>,<b>h</b>,<b>m</b>) 3/5 detection result; (<b>d</b>,<b>i</b>,<b>n</b>) 4/5 detection result; (<b>e</b>,<b>j</b>,<b>o</b>) full detection result.</p> Full article ">
1192 KiB  
Article
Global and Local Real-Time Anomaly Detectors for Hyperspectral Remote Sensing Imagery
by Chunhui Zhao, Yulei Wang, Bin Qi and Jia Wang
Remote Sens. 2015, 7(4), 3966-3985; https://doi.org/10.3390/rs70403966 - 1 Apr 2015
Cited by 54 | Viewed by 5904
Abstract
Anomaly detection has received considerable interest for hyperspectral data exploitation due to its high spectral resolution. A well-known algorithm for hyperspectral anomaly detection is the RX detector. A number of variations have been studied since then, including global and local versions for different [...] Read more.
Anomaly detection has received considerable interest for hyperspectral data exploitation due to its high spectral resolution. A well-known algorithm for hyperspectral anomaly detection is the RX detector. A number of variations have been studied since then, including global and local versions for different type of anomalies. Aiming at a real-time requirement for practical applications, this paper extends the concept of global and local anomaly detectors to be real-time detectors. The algorithms exploit the fact that a true real-time detector must produce its output in a causal manner and at the same time as an input comes in. Taking advantage of the Woodbury matrix identity, the global and local real-time detectors can be implemented and processed pixel-by-pixel in real time. Both synthetic and real hyperspectral imagery are conducted to demonstrate their performance. Full article
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Graphical abstract

Graphical abstract
Full article ">Figure 1
<p>The sliding dual window for LRXD is the most commonly used for hyperspectral anomaly detection.</p> Full article ">Figure 2
<p>(<b>a</b>) Causal matrix local window derived from the traditional local window; (<b>b</b>) Causal matrix local windows at r<sub>n</sub> and r<sub>n+1</sub>.</p> Full article ">Figure 3
<p>(<b>a</b>) Three different local windows; (<b>b</b>) causal array window of width w; (<b>c</b>) causal array local windows at r<sub>n</sub> and r<sub>n+1</sub>.</p> Full article ">Figure 3 Cont.
<p>(<b>a</b>) Three different local windows; (<b>b</b>) causal array window of width w; (<b>c</b>) causal array local windows at r<sub>n</sub> and r<sub>n+1</sub>.</p> Full article ">Figure 4
<p>(<b>a</b>) Cuprite AVIRIS image scene; (<b>b</b>) spatial positions of A, B, C, K, and M; (<b>c</b>) spectra of the five signatures extracted from the scene in (b).</p> Full article ">Figure 5
<p>(<b>a</b>) A set of 25 simulated panels; (<b>b</b>) 100th band of synthetic hyperspectral image.</p> Full article ">Figure 6
<p>AVIRIS LCVF subscene.</p> Full article ">Figure 7
<p>Detection results of the same set of target panels with different image sizes: (<b>a</b>) image size of 50 × 50; (<b>b</b>) image size of 100 × 100; (<b>c</b>) image size of 200 × 200.</p> Full article ">Figure 8
<p>Detection results using global RX detector and its real-time version: (<b>a</b>) 100th band of original data; (<b>b</b>) grayscale result of K-RXD; (<b>c</b>) grayscale result of R-RXD; (<b>d</b>) grayscale result of GRTCRXD; (<b>e</b>) db scale result of GRTCRXD.</p> Full article ">Figure 9
<p>(<b>a</b>) 3D plot of global K-RXD; (<b>b</b>) 3D plot of global R-RXD; (<b>c</b>) 3D plot of original GRTCRXD; (<b>d</b>) 3D plot of GRTCRXD in db scale.</p> Full article ">Figure 10
<p>Grayscale detection results of GRTCRXD for TI: (<b>a</b>) no panels detected; (<b>b</b>) row 1 panels detected; (<b>c</b>) row 2 panels detected; (<b>d</b>) row 3 panels detected; (<b>e</b>) row 4 panels detected; (<b>f</b>) row 5 panels detected; (<b>g</b>) final detection result.</p> Full article ">Figure 11
<p>Progressive procedures of GRTCRXD in either grayscale or db scale. Background suppression changes with the processing.</p> Full article ">Figure 12
<p>Detection results of real hyperspectral LCVF image using global RX detector and its real-time version: (<b>a</b>) 100th band of LCVF image; (<b>b</b>) grayscale result of K-RXD; (<b>c</b>) grayscale result of R-RXD; (<b>d</b>) grayscale result of GRTCRXD; (<b>e</b>) db scale result of GRTCRXD.</p> Full article ">Figure 13
<p>GRTCRXD detection results for LCVF: (<b>a</b>) vegetation appears; (<b>b</b>) vegetation and cinder detected; (<b>c</b>) anomaly just appears; (<b>d</b>) anomaly detected; (<b>e</b>) final detection result.</p> Full article ">Figure 14
<p>LRXD results using different local window size with image size 200 × 200:(<b>a</b>) window size 1/15 (inner window size 1 and outer window size 15); (<b>b</b>) window size 3/15; (<b>c</b>) window size 5/15; (<b>d</b>) window size 5/17; (<b>e</b>) window size 7/17.</p> Full article ">Figure 15
<p>LRXD results using different local window size with image size 100 × 100:(<b>a</b>) window size 1/15 (inner/outer window); (<b>b</b>) size 3/15; (<b>c</b>) size 5/15; (<b>d</b>) size 5/17;(<b>e</b>) size 7/17.</p> Full article ">Figure 16
<p>Local real-time anomaly detection experiments: (<b>a</b>) synthetic data scene; (<b>b</b>) global real-time causal RXD anomaly detection results; (<b>c</b>) local real-time causal array RXD anomaly detection results with array window width 15 × 15, (d) local real-time causal array RXD anomaly detection results with array window width 21 × 21.</p> Full article ">Figure 17
<p>Gray scale detection results of local real-time causal array RXD: (<b>a</b>) no panels detected; (<b>b</b>) row 1 panels detected; (<b>c</b>) row 2 panels detected; (<b>d</b>) row 3 panels detected; (<b>e</b>) row 4 panels detected; (<b>f</b>) row 5 panels detected; (<b>g</b>) final detection result.</p> Full article ">Figure 18
<p>CPT (Computing time) for global detectors. (<b>a</b>) TI image; (<b>b</b>) TE image; (<b>c</b>) LCVF image.</p> Full article ">Figure 19
<p>CPT (Computing time) for local detectors. (<b>a</b>) TI image; (<b>b</b>) TE image; (<b>c</b>) LCVF image.</p> Full article ">
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