Multiple Matrices Research Articles

Fast spatial structure constrained 3D prestack inversion based on the Sylvester equation

Structurally constrained inversion (SCI) is a multitrace processing technology that can effectively reduce the complexity of prestack seismic inversion caused by complex geology such as faults and steeply dipping structures. Nevertheless, several factors limit its wide application in 3D prestack inversion, the most significant of which is the high computational cost. In addition, the existing SCI is mostly applied in 2D profiles, which can only consider structural features in one direction, making it challenging to ensure spatial continuity. We have developed a fast spatial structure constrained inversion for 3D prestack seismic data. First, based on the conventional SCI, we develop a new way to describe the spatial structural characteristics of the stratum by estimating the dip angles along two perpendicular directions. Second, we introduce the spatial structure constraints through the Hadamard product operator and construct a computationally well-behaved objective function with spatial structural constrained. Using our matrix operators, which are symbols representing the multiplication of 2D and 3D matrices, the minimization of the objective function can be simplified to solve a specific Sylvester equation. Moreover, we justify the rationality of our operator symbols and derive a fast algorithm for solving 3D problems with the Sylvester equation. Finally, the stability and computational efficiency of our method are compared with the 2D SCI algorithm on the 3D overthrust model and field prestack seismic data.

Discovering faster matrix multiplication algorithms with reinforcement learning

Improving the efficiency of algorithms for fundamental computations can have a widespread impact, as it can affect the overall speed of a large amount of computations. Matrix multiplication is one such primitive task, occurring in many systems—from neural networks to scientific computing routines. The automatic discovery of algorithms using machine learning offers the prospect of reaching beyond human intuition and outperforming the current best human-designed algorithms. However, automating the algorithm discovery procedure is intricate, as the space of possible algorithms is enormous. Here we report a deep reinforcement learning approach based on AlphaZero1 for discovering efficient and provably correct algorithms for the multiplication of arbitrary matrices. Our agent, AlphaTensor, is trained to play a single-player game where the objective is finding tensor decompositions within a finite factor space. AlphaTensor discovered algorithms that outperform the state-of-the-art complexity for many matrix sizes. Particularly relevant is the case of 4 × 4 matrices in a finite field, where AlphaTensor’s algorithm improves on Strassen’s two-level algorithm for the first time, to our knowledge, since its discovery 50 years ago2. We further showcase the flexibility of AlphaTensor through different use-cases: algorithms with state-of-the-art complexity for structured matrix multiplication and improved practical efficiency by optimizing matrix multiplication for runtime on specific hardware. Our results highlight AlphaTensor’s ability to accelerate the process of algorithmic discovery on a range of problems, and to optimize for different criteria.

Interactions of zeros of polynomials and multiplicity matrices

An m×(n+1) multiplicity matrix is a matrix M=(μi,j) with rows enumerated by i∈{1,2,…,m} and columns enumerated by j∈{0,1,…,n} whose coordinates are nonnegative integers satisfying the following two properties: (1) If μi,j≥1, then j≤n−1 and μi,j+1=μi,j−1, and (2) colsumj(M)=∑i=1mμi,j≤n−j for all j.Let K be a field of characteristic 0 and let f(x) be a polynomial of degree n with coefficients in K. Let f(j)(x) be the jth derivative of f(x). Let Λ=(λ1,…,λm) be a sequence of distinct elements of K. For i∈{1,2,…,m} and j∈{1,2,…,n}, let μi,j be the multiplicity of λi as a zero of the polynomial f(j)(x). The m×(n+1) matrix Mf(Λ)=(μi,j) is called the multiplicity matrix of the polynomialf(x)with respect to Λ. Conditions for a multiplicity matrix to be the multiplicity matrix of a polynomial are established, and examples are constructed of multiplicity matrices that are not multiplicity matrices of polynomials. An open problem is to classify the multiplicity matrices that are multiplicity matrices of polynomials in K[x] and to construct multiplicity matrices that are not multiplicity matrices of polynomials.

Hypoxic-Sensitive Nano-Carriers for Anti-Neoplastic Drug Delivery: New Perspectives

Since the inception of Anti-neoplastic drug delivery in the 1950s oncologists and researchers have revolutionized the field of diagnostics and treatment with novel therapeutic modalities. Tumor hypoxia remains the first and foremost factor contributing to a resistance in anti-neoplastic drugs, ionized radiation and chemotherapy by solid tumors. The laps in vasculature in solid tumors promulgates tumor resistance by inhibiting the hypoxia inducible factor (HIF) within the tumor. It is on this basis that the selectivity of hypoxic cells are datelined. Based on clinical studies it was revealed that hypoxia does not occur in healthy human tissue; thus, paving the way for the exploitation of hypoxia as an advantage in creating novel therapeutic methods of detection and treatment for solid tumors with an added advantage of nano-science. The biological application of nano-carriers is a fast developing area of nano-science that is expected to realize new and innovative possibilities in the diagnosis and treatment of human cancers. Cancer diagnostics using nano-carriers entails the use of fluorescent nanoparticles for a multiplex of simultaneous profiling of tumor bio-markers. These fluorescent probes are also used for detection of multiple RNA matrices and genes within in-situ hybridization. It is expected that in the very near future, the application of conjugated nanoparticles in anti-neoplastic tumor detection and drug delivery will enable oncologists and scientists to pinpoint and identify cancer-related proteins on tumor surfaces, providing a new method of analysis; thus, creating personalized treatment methods for individual tumors. Nano-carriers possess an extraordinary possibility as contrast agents for cancer cell detection in vivo, and for monitoring the response to treatment of select cancers. Keywords: Nano-Medicine, Hypoxia, Solid Tumors, HIF factor. DOI: 10.7176/JHMN/102-04 Publication date: October 31 st 2022

Honey bees are not attracted to multiple new ant bait matrices containing sugar.

Multiple new ant treatment products containing high volumes of sugar have recently been developed specifically for use in ant management programs. The presence of sugar in these products could potentially attract bees, and any such attractancy would likely be fatal given that these products typically contain general insecticides. To determine the risk of such products to bees I present four studies assessing bee attractancy to multiple matrices that are used to make these products. The trials were conducted across multiple years, seasons, and locations, containing various concentrations of sugar in multiple forms, using various experimental setups with many different bee hives, and multiple observers. Not a single bee was attracted to any matrix, nor were bees observed inspecting any matrix, and no bees fed on any matrix, irrespective of whether the matrices were placed close to hives and directly under bee flight paths, or out in areas where bees were feeding. This is in stark contrast to large numbers of bees that were feeding on flowers within the immediate vicinity of all of the matrices in the first two experiments, or flying over the arrays in experiments 3 and 4 travelling to and from other food sources. I present five suggestions for the discrepancy between the trials presented here and the general perception that bees are attracted to sugar. These matrices appear to be acceptable as a basis to make treatment products for broadscale use within ant management programs. However, it should be recognized that bees, and other non-target species, are indeed capable of feeding on these matrices. Therefore vigilance should still be maintained to identify special circumstances where bees may be killed when constituents are added to these matrices that do attract bees, or usage methods can adversely affect bees.

Clustering of differentials in CRAFT with correlation matrices

CRAFT is an substitution-permutation network tweakable block cipher proposed at fast software encryption 2019 by Beierle et al., which is designed to optimize the efficient protection against differential fault analysis (DFA) attacks. In this paper, the full round differential characteristics for CRAFT block cipher are given. A new method on counting the number of differentials by using correlation matrix is given. We can compute the number of all optimal characteristics or suboptimum differentials with the same input difference and output difference by hand. We explore the multiple differential trails and compute the probability of differential characteristic by using the multiplication of correlation matrices. Our work complements automatic search methods for the best differential with a careful manual analysis. Since the automatic search method is stranded by storage and search space limitations, which will cause a computer to crash as the number of search rounds increases. Thanks to the correlation matrix technique, we are able to find differential distinguishers for 9-round of the cipher with the probability of at least 2 − 40.68 + 2 − 48.60 ${2}^{-40.68}+{2}^{-48.60}$ . Moreover, we can construct differential distinguisher covers more rounds based on the 9-round differential distinguishers. As one of its typical application, we propose the differential characteristics for the full-round CRAFT which ensure that the probability of each round is optimal. Besides, we explore the clustering effect on the full round by exhibiting a class of high probability characteristics for 9-round. In general, we obtain a good understanding of the propagation of differences for CRAFT due to its algebraic structure.

Symmetry and Nonnegativity-Constrained Matrix Factorization for Community Detection

Dear Editor, This letter presents a novel symmetry and nonnegativity-constrained matrix factorization (SNCMF)-based community detection model on undirected networks such as a social network. Community is a fundamental characteristic of a network, making community detection a vital yet thorny issue in network representation. Owing to its high interpretability and scalability, a symmetric nonnegative matrix factorization (SNMF) model is frequently adopted to address this issue. However, it adopts a unique latent factor (LF) matrix for representing an undirected network's symmetry, which leads to a reduced latent space that impairs its representation learning ability. Motivated by this discovery, the proposed SNCMF model innovatively adopts the following three-fold ideas: 1) Leveraging multiple LF matrices to represent a network, thereby enhancing its representation learning ability; 2) Introducing a symmetry regularization term that implies the equality constraint between multiple LF matrices to illustrate the network's symmetry; and 3) Incorporating graph regularization into the model to preserve the network's intrinsic geometry. Experimental results on several real-world networks indicate that the proposed SNCMF-based community detector outperforms the benchmark and state-of-the-art models in achieving highly-accurate community detection results.

Accelerated MRI Reconstruction With Separable and Enhanced Low-Rank Hankel Regularization.

Magnetic resonance imaging serves as an essential tool for clinical diagnosis, however, suffers from a long acquisition time. Sparse sampling effectively saves this time but images need to be faithfully reconstructed from undersampled data. Among the existing reconstruction methods, the structured low-rank methods have advantages in robustness to the sampling patterns and lower error. However, the structured low-rank methods use the 2D or higher dimension k-space data to build a huge block Hankel matrix, leading to considerable time and memory consumption. To reduce the size of the Hankel matrix, we proposed to separably construct multiple small Hankel matrices from rows and columns of the k-space and then constrain the low-rankness on these small matrices. This separable model can significantly reduce the computational time but ignores the correlation existed in inter- and intra-row or column, resulting in increased reconstruction error. To improve the reconstructed image without obviously increasing the computation, we further introduced the self-consistency of k-space and virtual coil prior. Besides, the proposed separable model can be extended into other imaging scenarios which hold exponential characteristics in the parameter dimension. The in vivo experimental results demonstrated that the proposed method permits the lowest reconstruction error with a fast reconstruction. The proposed approach requires only 4% of the state-of-the-art STDLR-SPIRiT runtime for parallel imaging reconstruction, and achieves the fastest computational speed in parameter imaging reconstruction.

P03-15 Postmortem biochemical changes in multiple rat biological matrices and their correlation with the time of death

P03-15 Postmortem biochemical changes in multiple rat biological matrices and their correlation with the time of death

Constraint-Induced Symmetric Nonnegative Matrix Factorization for Accurate Community Detection

As a fundamental characteristic of an undirected network, community reveals its networking organization and functional mechanisms, making community detection be a highly-interesting issue in network representation learning. With great interpretability, a symmetric and nonnegative matrix factorization (SNMF)-based approach is frequently adopted to tackle this issue. However, it only adopts a unique feature matrix for describing the symmetry of an undirected network, which unfortunately results in a reduced feature space that evidently impairs its representation learning ability. Motivated by this discovery, this paper proposes a novel Constraint-induced Symmetric Nonnegative Matrix Factorization (C-SNMF) model that adopts three-fold ideas: a) Representing a target undirected network with multiple latent feature matrices, thus preserving its representation learning capacity; b) Incorporating a symmetry-regularizer into its objective function, which preserves the symmetry of the learnt low-rank approximation to the adjacency matrix, thereby making the resultant detector precisely illustrate the target network's symmetry; and c) Introducing a graph-regularizer that preserves local invariance of the network's intrinsic geometry into its learning objective, thus making the achieved detector well-aware of community structure within the target network. Note that the regularization coefficients are selected according to the modularity of the learnt community structure on the training data only, thereby greatly improving the achieved model's practical significance for real applications. Experimental results on six real-world networks demonstrate that the proposed C-SNMF model significantly outperforms the benchmarks and state-of-the-art models in achieving highly-accurate community detection results.

Research on influencing factors of artificial intelligence multi-cloud scheduling applied talent training based on DEMATEL-TAISM

With the rapid development of Internet of Things (IoT) technology and the rising popularity of IoT devices, an increasing number of computing intensive IoT applications have been developed. However, due to the limited resources of IoT devices, cloud computing systems are required to compute intensive IoT applications. Furthermore, to be subject to a single cloud computing service provider, multi-cloud computing has become an IoT service cloud computing solution. As a result of the complexity of multi-cloud scheduling, the application of artificial intelligence is an important technology to solve IoT multi-cloud scheduling. The corresponding talent training plays an important role in the development and implementation of the IoT artificial intelligence multi-cloud scheduling. First, this paper studies the key influencing factors of IoT artificial intelligence multi-cloud scheduling applied talent training. Combined with the characteristics of the development of China’s artificial intelligence industry, this paper summarizes the influencing factors from the four-dimensional training path of government departments, universities, enterprises and scientific research institutes. The purpose of artificial intelligence multi-cloud scheduling applied talent training is to build an artificial intelligence multi-cloud scheduling applied talent training influencing factor index system. Then, the DEMATEL method is used to establish multiple correlation matrices according to the direct influence correlation between the factors and calculate the degree of influence, the degree of being influenced, the center degree and the cause degree of the factors. Using the improved AISM method, based on the idea of game confrontation, from the two opposite extraction rules of result priority and cause priority, a group of confrontation level topological maps with comprehensive influence values reflecting the interacting factors are obtained, and relevant suggestions are presented to provide a reference for the training of artificial intelligence multi-cloud scheduling applied talent.

Davenport and Hasse's theorems and lifts of multiplication matrices of Gaussian periods

Let e≥2 be an integer, pr be a prime power with pr≡1(mode) and ηr(i) be Gaussian periods of degree e for Fpr. By the dual form of Davenport and Hasse's lifting theorem on Gauss sums, we establish lifts of the multiplication matrices of the Gaussian periods ηr(0),…,ηr(e−1) which are defined by F. Thaine. We also give some examples of the explicit lifts for prime degree e with 3≤e≤23 which also illustrate relations among lifts of Jacobi sums, Gaussian periods and multiplication matrices of Gaussian periods.

Fatal intoxication involving 2,5-dimethoxy-4-chloroamphetamine (DOC): A case report

2,5-dimethoxy-4-chloroamphetamine (DOC) is a substituted alpha-methylated phenethylamine, which was identified for the 1st time in 2009 in France, but is not classified as an illegal narcotic. This report presents a fatal intoxication involving consumption of DOC with analysis of multiple matrices. A 26-year-old man was found dead in his friend's apartment after a party. At place, numerous bottles of alcohol were discovered. The man was known as a cannabis smoker. Except the observation of a multi-visceral congestion and oedematous lungs, no obvious cause of death was identified at autopsy. Peripheral blood (PB), cardiac blood (CB), urine and bile were collected. Immunological screening for amphetamine, methamphetamine, opiates, cannabinoids, cocaine, buprenorphine, methadone and MDMA was performed on urine. Quantification of cannabinoids was performed on PB by GC-MS/MS. Analysis of ethanol and other volatile compounds was performed on PB and urine using HS-GC-MS. Ethylglucuronide (EtG) was quantified in PB and urine by GC-MS/MS. General unknown screening (GUS) was performed on PB by GC-MS and LC-PDA/MS. DOC was quantified in bloods, urine and bile by UHPLC-MS/MS after liquid-liquid extraction, using MBDB-d5 as internal standard. The assay was validated on whole blood for linearity (range: 2.0 to 500 ng/mL), selectivity, precision and accuracy at limit of quantification (LOQ: 2.0 ng/mL) and at high-level concentration (High QC: 400 ng/mL). The dilution process was validated in urine (dilution factor 1/10). Immunological screening was positive for cannabinoids. Confirmation in PB quantified THC, 11-OH-THC and THC-COOH at 3.40, 2.10 and 36.0 ng/mL, respectively. Ethanol analyses were negative but EtG was quantified at 0.48 μg/mL and 161 μg/mL in PB and urine, respectively. Mirtazapine was quantified at therapeutic concentration (20 ng/mL) in PB. DOC was detected on GUS by GC-MS and LC-PDA/MS. The method of DOC quantification was validated by measurement of accuracy and precision at the LOQ (104% and 12.6%) and at the High QC (97.4% and 8.5%). The 10-fold dilution process in urine was validated in term of accuracy (100%) and precision (4.1%). Selectivity was validated by analyzing 6 blank samples and no interference was observed at DOC retention time. In the present case, DOC was quantified at 104 ng/mL, 322 ng/mL, 648 ng/mL and 371 ng/mL in PB, CB, urine and bile respectively. Presence of cannabinoids and EtG in blood and urine suggest alcohol and cannabis consumption several hours preceding the death. Reports of use of DOC have mainly described non-fatal intoxications, with reported blood concentrations from < 10 to 18 ng/mL (Abbara. TOXAC 2017;29:82–89). At our best knowledge, only one fatal case has been reported in the literature (Barnett. JAT 2014;38:589–591), in which DOC blood and urine concentrations were 377 and 3193 ng/mL. Difference between PB and CB concentrations in the present case could be explain by postmortem redistribution. In the present case, the circumstances of death, autopsy findings and toxicological results were consistent with a toxic cause of death by DOC overdose in association with cannabis consumption.

Possible important roles of galectins in the healing of human fetal membranes.

The fetal membranes healing is a complex and dynamic process of replacing devitalized and missing cellular structures and tissue layers. Multiple cells and extracellular matrices, and cell differentiation, migration and proliferation may participate in restoring the integrity of damaged tissue, however this process still remains unclear. Therefore, there is a need to identify and integrate new ideas and methods to design a more effective dressing to accelerate fetal membrane healing. This review explores the function and role of galectins in the inflammatory, epithelial mesenchymal transition, proliferative migration, and remodeling phases of fetal membrane healing. In conclusion, the preliminary findings are promising. Research on amnion regeneration is expected to provide insight into potential treatment strategies for premature rupture of membranes.

Diversity embedding deep matrix factorization for multi-view clustering

Multi-view clustering has attracted increasing attention by reason of its ability to leverage the complementarity of multi-view data. Existing multi-view clustering methods have explored nonnegative matrix factorization to decompose a matrix into multiple matrices for feature representations from multi-view data, which are not discriminative enough to deal with the natural data containing complex information. Moreover, most of multi-view clustering methods prioritize the consensus information among multi-view data, leaving a large amount of information redundant and the clustering performance deteriorated. To address these issues, this paper proposes a multi-view clustering framework that adopts a diversity loss for deep matrix factorization and reduces feature redundancy while obtaining more discriminative features. We then bridge the relation between deep auto-encoder and deep matrix factorization to optimize the objective function. This method avoids the challenges in the optimization process. Extensive experiments demonstrate that the proposed method is superior to state-of-the-art methods.

Toeplitz operators on the Hardy space over the infinite-dimensional polydisc

We study infinite multiplicative Toeplitz matrices and Toeplitz operators on the Hardy space $$H^2(\mathbb{T}^{\infty})$$ over the infinite-dimensional polydisc. This pair is a companion to the pair of Toeplitz matrices and Toeplitz operators on the Hardy space over the unit disc. We obtain a Brown- Halmos type theorem and the spectral inclusion theorem. The conditions for the semi-commutator $$ T_{f}T_{g}-T_{fg} $$ of Toeplitz operators $$ T_{f} $$ and $$T_{g} $$ to be of finite rank are obtained if one of ƒ and g depends only on finitely many variables. It is also shown that the Toeplitz algebra $$\mathcal{T}^{F}_{\infty} $$ generated by Toeplitz operators with bounded symbols depending only on finitely many variables on $$H^2(\mathbb{T}^{\infty}) $$ doesn’t contain any nonzero compact operator. In particular, the Toeplitz algebra generated by Toeplitz operators with continuous symbols on $$H^2(\mathbb{T}^{\infty}) $$ , as a C*-subalgebra of $$\mathcal{T}^{F}_{\infty} $$ , contains no nonzero compact operators, and this is in sharp contrast to the case of finite-dimensional polydiscs. Moreover, the symbolic calculus for the Toeplitz algebra generated by all bounded Toeplitz operators is established.

Nonlinear fusion method for multistage product design decision-making using plant growth simulation algorithm

Design decision-making is a vital activity for selecting an optimal scheme for product development. Owing to the uncertainty and ambiguity of design requirements and constraints, several product design phases are often deployed for concept refinement, which makes multistage product design decision-making (MPDDM) and the effective fusion of MPDDM data indispensable. However, few existing methods have considered the nonlinear relationships among the MPDDM information. Therefore, a nonlinear fusion method for MPDDM was proposed in this study. This method applies a three-parameter interval grey number to depict decision-makers’ judgement about product design schemes. Based on converting linguistic judgements into interval scales, an interval analytic hierarchy process (AHP) method was employed to calculate the weights of the design criteria, decision-makers, and decision-making stages. Considering the advantage of integrating multiple matrices without requiring external control parameters, a multistage decision-making fusion process using a plant growth simulation algorithm (PGSA) was proposed to aggregate multistage decision-making data for product design. A case study was conducted to collect multistage decision-making data, and the PGSA was developed. Through comparison with the extant method, the effectiveness and feasibility of the fusion of MPDDM was verified. The results indicate that (1) uncertainty perceived by decision-makers at three stages accounted for 96.7%, 95%, and 97.2%, respectively. The “center of gravity” of a three-parameter interval grey number, which reflects the largest possibility of decision-makers’ preferences, is not always equally distant from its maximum or minimum value (73.9%). (2) The optimization model using interval AHP to calculate the weights of decision-making indicators and stages is conducive to reducing the decision-maker’s uncertainty. (3) The global search mechanism of the PGSA can effectively realize the nonlinear fusion of MPDDM.

Orthogonal symmetric matrices and joins of graphs

We introduce a notion of compatibility for multiplicity matrices. This gives rise to a necessary condition for the join of two (possibly disconnected) graphs G and H to be the pattern of an orthogonal symmetric matrix, or equivalently, for the minimum number of distinct eigenvalues q of G∨H to be equal to two. Under additional hypotheses, we show that this necessary condition is also sufficient. As an application, we prove that q(G∨H) is either two or three when G and H are unions of complete graphs, and we characterise when each case occurs.

CONNA: Configurable Matrix Multiplication Engine for Neural Network Acceleration

Convolutional neural networks (CNNs) have demonstrated promising results in various applications such as computer vision, speech recognition, and natural language processing. One of the key computations in many CNN applications is matrix multiplication, which accounts for a significant portion of computation. Therefore, hardware accelerators to effectively speed up the computation of matrix multiplication have been proposed, and several studies have attempted to design hardware accelerators to perform better matrix multiplications in terms of both speed and power consumption. Typically, accelerators with either a two-dimensional (2D) systolic array structure or a single instruction multiple data (SIMD) architecture are effective only when the input matrix has shapes that are close to or similar to a square. However, several CNN applications require multiplications of non-squared matrices with various shapes and dimensions, and such irregular shapes lead to poor utilization efficiency of the processing elements (PEs). This study proposes a configurable engine for neural network acceleration, called CONNA, whose computation engine can conduct matrix multiplications with highly utilized computing units, regardless of the access patterns, shapes, and dimensions of the input matrices by changing the shape of matrix multiplication conducted in the physical array. To verify the functionality of the CONNA accelerator, we implemented CONNA as an SoC platform that integrates a RISC-V MCU with CONNA on Xilinx VC707 FPGA. SqueezeNet on CONNA achieved an inference performance of 100 frames per second (FPS) with 2.36 mm2 and 83.55 mW in a 65 nm process, improving efficiency by up to 34.1 times better than existing accelerators in terms of FPS, silicon area, and power consumption.

Disentangling structural and functional responses of native versus alien communities by canonical ordination analyses and variation partitioning with multiple matrices

Freshwaters are under accelerated human pressure, and mollusk communities are among its most sensitive, threatened, and valuable components. To the best of our knowledge, the overall effects of damming, environment, space, time, and invasive alien mollusk species, on structural and functional responses of native mollusk communities were not yet compared. Using historical information and recent data from a river, we aimed to investigate and disentangle these effects and evaluate the differences in structural and functional responses of natives and alien invasives to the same predictors. Variation partitioning showed that alien species were as important predictors as were environmental factors and time in explaining species composition of native freshwater mollusk communities. Aliens were more independent of environmental conditions than natives and responded to different drivers, partially explaining their invasion success. The increased abundance of some alien gastropods was positively related to taxonomic diversity, while certain alien bivalves were negatively associated with the functional diversity of native communities. We introduce a cumulative variation partitioning with multiple response (native and alien) and predictor matrices, along with a diagram to show their relations, advocating for a conceptual shift in future community ecology, from “variables to matrices” and from “multivariate analyses to multi-matrix statistical modeling”.