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Network Topology Identification using PCA and its Graph Theoretic Interpretations
TL;DR: It is shown that identification is equivalent to learning a model which captures the approximate linear relationships between the different variables comprising $\mathbf{X}$ such that A_n is full rank (highest possible) and consistent with a network node-edge incidence structure.
Abstract: We solve the problem of identifying (reconstructing) network topology from steady state network measurements. Concretely, given only a data matrix $\mathbf{X}$ where the $X_{ij}$ entry corresponds to flow in edge $i$ in configuration (steady-state) $j$, we wish to find a network structure for which flow conservation is obeyed at all the nodes. This models many network problems involving conserved quantities like water, power, and metabolic networks. We show that identification is equivalent to learning a model $\mathbf{A_n}$ which captures the approximate linear relationships between the different variables comprising $\mathbf{X}$ (i.e. of the form $\mathbf{A_n X \approx 0}$) such that $\mathbf{A_n}$ is full rank (highest possible) and consistent with a network node-edge incidence structure. The problem is solved through a sequence of steps like estimating approximate linear relationships using Principal Component Analysis, obtaining f-cut-sets from these approximate relationships, and graph realization from f-cut-sets (or equivalently f-circuits). Each step and the overall process is polynomial time. The method is illustrated by identifying topology of a water distribution network. We also study the extent of identifiability from steady-state data.
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TL;DR: The proposed method involves the application of principal component analysis and its graph-theoretic interpretation to infer the steady state network topology from smart meter energy measurements and is demonstrated through simulation on randomly generated networks and on IEEE recognized Roy Billinton distribution test system.
Abstract: In a power distribution network, the network topology information is essential for an efficient operation. This network connectivity information is often not available at the low voltage (LV) level due to uninformed changes that happen from time to time. In this paper, we propose a novel data-driven approach to identify the underlying network topology for LV distribution networks including the load phase connectivity from time series of energy measurements. The proposed method involves the application of principal component analysis and its graph-theoretic interpretation to infer the steady state network topology from smart meter energy measurements. The method is demonstrated through simulation on randomly generated networks and also on IEEE recognized Roy Billinton distribution test system.
169 citations
Cites methods from "Network Topology Identification usi..."
...Our approach integrates PCA and its graph-theoretic interpretation (as shown in [22]) to identify the network topology from time series of energy measurements, based on the principle of energy conservation....
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TL;DR: In this article, a pinch retrofit method was proposed to identify network structural changes sequentially to meet the retrofit target, and a case study highlighted the benefits of the new approach.
33 citations
Cites background from "Network Topology Identification usi..."
...204 describes how the edges are incident on nodes [27]....
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06 Jul 2016
TL;DR: In this paper, the authors proposed a new data driven approach to the problem based on Principal Component Analysis (PCA) and its Graph Theoretic interpretations, using energy measurements in equally timed short intervals, generated from smart meters.
Abstract: Consumers with low demand, like households, are generally supplied single-phase power by connecting their service mains to one of the phases of a distribution transformer. The distribution companies face the problem of keeping a record of consumer connectivity to a phase due to uninformed changes that happen. The exact phase connectivity information is important for the efficient operation and control of distribution system. We propose a new data driven approach to the problem based on Principal Component Analysis (PCA) and its Graph Theoretic interpretations, using energy measurements in equally timed short intervals, generated from smart meters. We propose an algorithm for inferring phase connectivity from noisy measurements. The algorithm is demonstrated using simulated data for phase connectivities in distribution networks.
15 citations
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TL;DR: The proposed method involves the application of Principal Component Analysis (PCA) and its graph-theoretic interpretation to infer the topology from smart meter energy measurements to identify the underlying network topology including the load phase connectivity from time series of energy measurements.
Abstract: In a power distribution network, the network topology information is essential for an efficient operation of the network. This information of network connectivity is not accurately available, at the low voltage level, due to uninformed changes that happen from time to time. In this paper, we propose a novel data--driven approach to identify the underlying network topology including the load phase connectivity from time series of energy measurements. The proposed method involves the application of Principal Component Analysis (PCA) and its graph-theoretic interpretation to infer the topology from smart meter energy measurements. The method is demonstrated through simulation on randomly generated networks and also on IEEE recognized Roy Billinton distribution test system.
12 citations
Cites methods from "Network Topology Identification usi..."
...Our approach integrates PCA and its graphtheoretic interpretation (as shown in [22]) to identify the network topology from time series of energy measurements, based on the principle of energy conservation....
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TL;DR: The proposed EACCC scheme achieves optimal connections among users in a vast and complex network, due to strong mutual communication, and can be implemented easily in a realistic scenario.
Abstract: End users’ quality of experience (QoE) is one of the most crucial requirements to be considered in device-to-device communication. Users’ QoE is affected by the ratio of the amount of received data to the amount of shared data. The ratio should be approximate to 1. To achieve this, we propose an evolutionary algorithm-based cooperative content caching and communication (EACCC) scheme. This scheme works by finding the maximum 1-factor of a directed weighted graph, where each edge’s weight corresponds to the mean opinion score of the connection between two users. Finding the maximum 1-factor of graph-based communication is a problem that is difficult to solve optimally, but an evolutionary algorithm can quickly converge to a high-quality solution. Each user provides an opinion score of incoming and outgoing connection, and the maximum 1-factor corresponds to the scheme that maximizes overall QoE. The proposed EACCC scheme achieves optimal connections among users in a vast and complex network, due to strong mutual communication. The simulation results verify that our proposed scheme outperforms traditional QoE optimization schemes. Moreover, our scheme can be implemented easily in a realistic scenario.
7 citations
Cites background from "Network Topology Identification usi..."
...It is crucial to carefully derive necessary conditions to reconstruct content sharing based on matrix interaction in the network [27], [28]....
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References
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01 Jan 1999
TL;DR: In this article, non-negative matrix factorization is used to learn parts of faces and semantic features of text, which is in contrast to principal components analysis and vector quantization that learn holistic, not parts-based, representations.
Abstract: Is perception of the whole based on perception of its parts? There is psychological and physiological evidence for parts-based representations in the brain, and certain computational theories of object recognition rely on such representations. But little is known about how brains or computers might learn the parts of objects. Here we demonstrate an algorithm for non-negative matrix factorization that is able to learn parts of faces and semantic features of text. This is in contrast to other methods, such as principal components analysis and vector quantization, that learn holistic, not parts-based, representations. Non-negative matrix factorization is distinguished from the other methods by its use of non-negativity constraints. These constraints lead to a parts-based representation because they allow only additive, not subtractive, combinations. When non-negative matrix factorization is implemented as a neural network, parts-based representations emerge by virtue of two properties: the firing rates of neurons are never negative and synaptic strengths do not change sign.
9,604 citations
TL;DR: This work introduces a new method called sparse principal component analysis (SPCA) using the lasso (elastic net) to produce modified principal components with sparse loadings and shows that PCA can be formulated as a regression-type optimization problem.
Abstract: Principal component analysis (PCA) is widely used in data processing and dimensionality reduction. However, PCA suffers from the fact that each principal component is a linear combination of all the original variables, thus it is often difficult to interpret the results. We introduce a new method called sparse principal component analysis (SPCA) using the lasso (elastic net) to produce modified principal components with sparse loadings. We first show that PCA can be formulated as a regression-type optimization problem; sparse loadings are then obtained by imposing the lasso (elastic net) constraint on the regression coefficients. Efficient algorithms are proposed to fit our SPCA models for both regular multivariate data and gene expression arrays. We also give a new formula to compute the total variance of modified principal components. As illustrations, SPCA is applied to real and simulated data with encouraging results.
3,102 citations
Additional excerpts
...However it is also well known that interpreting these relationships and obtaining physical insights is a hard task [2]....
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...Introductory section has been revised, and a more detailed literature survey has been added....
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TL;DR: This section will review those books whose content and level reflect the general editorial poltcy of Technometrics.
Abstract: This section will review those books whose content and level reflect the general editorial poltcy of Technometrics. Publishers should send books for review to Ejaz Ahmed, Depatment of Mathematics and Statistics, University of Windsor, Windsor, ON N9B 3P4 (techeditor@uwindsoxca). The opinions expressed in this section are those of the reviewers These opinions do not represent positions of the reviewer's organization and may not reflect those of the editors or the sponsoring societies. Listed prices reflect information provided by the publisher and may not be current The book purchase programs of the American Society for Quality can provide some of these books at reduced prices for members. For infbrmation, contact the American Society for Quality at 1-800-248-1946.
2,342 citations
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...Revised version uploaded on Jan 22, 2016....
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01 Oct 1975
TL;DR: Graph Theory and Its Applications to Problems of Society and its Applications to Algorithms and Computer Science.
Abstract: Introductory Graph Theory with ApplicationsGraph Theory with ApplicationsResearch Topics in Graph Theory and Its ApplicationsChemical Graph TheoryMathematical Foundations and Applications of Graph EntropyGraph Theory with Applications to Engineering and Computer ScienceGraphs Theory and ApplicationsQuantitative Graph TheoryApplied Graph TheoryChemical Graph TheoryA First Course in Graph TheoryGraph TheoryGraph Theory with ApplicationsGraph Theory with ApplicationsSpectra of GraphsFuzzy Graph Theory with Applications to Human TraffickingApplications of Graph TheoryChemical Applications of Graph TheoryRecent Advancements in Graph TheoryA Textbook of Graph TheoryGraph Theory and Its Engineering ApplicationsGraph Theory, Combinatorics, and ApplicationsAdvanced Graph Theory and CombinatoricsTopics in Intersection Graph TheoryGraph Theory with Applications to Engineering and Computer ScienceGraph Theory and Its Applications, Second EditionHandbook of Research on Advanced Applications of Graph Theory in Modern SocietyGraph Theory with Applications to Algorithms and Computer ScienceGraph TheoryGraph Theory with Algorithms and its ApplicationsGraph TheoryGraph Theory with ApplicationsGraph Theory ApplicationsHandbook of Graph TheoryGraph Theory and Its Applications to Problems of SocietyBasic Graph Theory with ApplicationsTen
809 citations
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...If the objective is to reduce dimensionality (or obtain a low rank approximation of X), then it suffices to store only the hyperplane on which most of the data resides (U1) and the projection of data onto this hyperplane....
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...Reduced Row Ehelon Form (RREF): A matrix is in RREF iff • It is in row echelon form....
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...Hence, the model obtained is given by: UT2 X ≈ 0 (3)...
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Posted Content•
TL;DR: A modification of the classical variational representation of the largest eigenvalue of a symmetric matrix is used, where cardinality is constrained, and a semidefinite programming-based relaxation is derived for the sparse PCA problem.
Abstract: We examine the problem of approximating, in the Frobenius-norm sense, a positive, semidefinite symmetric matrix by a rank-one matrix, with an upper bound on the cardinality of its eigenvector. The problem arises in the decomposition of a covariance matrix into sparse factors, and has wide applications ranging from biology to finance. We use a modification of the classical variational representation of the largest eigenvalue of a symmetric matrix, where cardinality is constrained, and derive a semidefinite programming based relaxation for our problem. We also discuss Nesterov's smooth minimization technique applied to the SDP arising in the direct sparse PCA method.
572 citations
Additional excerpts
...Some problems in the realm of structured sparsity have also been studied recently [22, 23] where the idea is to incorporate appropriate structural constraints in an optimization framework....
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