Reconstructing Plant Connectivity using Directed Spectral Decomposition

Partial directed coherence: a new concept in neural structure determination

Abstract: Abstract. This paper introduces a new frequency-domain approach to describe the relationships (direction of information flow) between multivariate time series based on the decomposition of multivariate partial coherences computed from multivariate autoregressive models. We discuss its application and compare its performance to other approaches to the problem of determining neural structure relations from the simultaneous measurement of neural electrophysiological signals. The new concept is shown to reflect a frequency-domain representation of the concept of Granger causality.

Reconstruction of missing data in multivariate processes with applications to causality analysis

Abstract: Recovery of missing observations in time-series has been a century-long subject of study, giving rise to two broad classes of methods, namely, one that reconstructs data and the other that directly estimate the statistical properties of the data, largely for univariate processes. In this work, we present a data reconstruction technique for multivariate processes. The proposed method is developed in the framework of sparse optimization while adopting a parametric approach using vector auto-regressive (VAR) models, where both the temporal and spatial correlations can be exploited for efficient data recovery. The primary purpose of recovering the missing data in this work is to develop a directed graphical or a network representation of the multivariate process under study. Existing methods for data-driven network reconstruction are built on the assumption of data being available at regular intervals. In this respect, the proposed method offers an effective methodology for reconstructing weighted causal networks from missing data. The scope of this work is restricted to linear, jointly stationary multivariate processes that can be suitably represented by VAR models of finite order and missing data of the random type. Simulation studies on different data generating processes with varying proportions of missing observations illustrate the efficacy of the proposed method in recovering the multivariate signals and thereby reconstructing weighted causal networks.

Network Topology Identification using PCA and its Graph Theoretic Interpretations

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.

Extended-AUDI method for simultaneous determination of causality and models from process data

Abstract: To the best of our knowledge, there are few methods which can determine both causality and models from process data, although both of them are crucial in practical applications The extended augmented UD identification (EAUDI) is an identification approach which does not need a priori causal relationship between variables in advance In this method, however, the information contained in the augmented information matrix (AIM) is still not fully utilized and yet helpful for causality analysis, namely, whether the values of cross-regressive coefficients are sufficiently weak to be considered as insignificant Based on this, the EAUDI method is further extended to detect causality from process data, and it can also provide models of all connecting paths simultaneously Moreover, hypothesis testing (F-distribution) is proposed to verify the results of this approach (by testing cross-regressive coefficients) The effectiveness of the proposed method is demonstrated by numerical examples

Reconstruction of causal graphs for multivariate processes in the presence of missing data

Abstract: Learning temporal causal relationships between time series is an important tool for the identification of causal network structures in linear dynamic systems from measurements. The main objective in network reconstruction is to identify the causal interactions between the variables and determine the connectivity strengths from time-series data. Among several recently introduced data-driven causality measures, partial directed coherence (PDC), directed partial correlation (DPC) and direct power transfer (DPT) have been shown to be effective in both identifying the causal interactions as well as quantifying the strength of connectivity. However, all the existing approaches assume that the observations are available at all time instants and fail to cater to the case of missing observations. This paper presents a method to reconstruct the causal graph from data with missing observations using sparse optimization (SPOPT) techniques. The method is particularly devised for jointly stationary multivariate processes that have vector autoregressive (VAR) structure representations. Demonstrations on different linear causal dynamic systems illustrate the efficacy of the proposed method with respect to the reconstruction of causal networks.

Investigating Causal Relations by Econometric Models and Cross-Spectral Methods

Abstract: There occurs on some occasions a difficulty in deciding the direction of causality between two related variables and also whether or not feedback is occurring. Testable definitions of causality and feedback are proposed and illustrated by use of simple two-variable models. The important problem of apparent instantaneous causality is discussed and it is suggested that the problem often arises due to slowness in recording information or because a sufficiently wide class of possible causal variables has not been used. It can be shown that the cross spectrum between two variables can be decomposed into two parts, each relating to a single causal arm of a feedback situation. Measures of causal lag and causal strength can then be constructed. A generalisation of this result with the partial cross spectrum is suggested.

Investigating causal relations by econometric models and cross-spectral methods

Abstract: There occurs on some occasions a difficulty in deciding the direction of causality between two related variables and also whether or not feedback is occurring. Testable definitions of causality and feedback are proposed and illustrated by use of simple two-variable models. The important problem of apparent instantaneous causality is discussed and it is suggested that the problem often arises due to slowness in recordhag information or because a sufficiently wide class of possible causal variables has not been used. It can be shown that the cross spectrum between two variables can be decomposed into two parts, each relating to a single causal arm of a feedback situation. Measures of causal lag and causal strength can then be constructed. A generalization of this result with the partial cross spectrum is suggested.The object of this paper is to throw light on the relationships between certain classes of econometric models involving feedback and the functions arising in spectral analysis, particularly the cross spectrum and the partial cross spectrum. Causality and feedback are here defined in an explicit and testable fashion. It is shown that in the two-variable case the feedback mechanism can be broken down into two causal relations and that the cross spectrum can be considered as the sum of two cross spectra, each closely connected with one of the causations. The next three sections of the paper briefly introduce those aspects of spectral methods, model building, and causality which are required later. Section IV presents the results for the two-variable case and Section V generalizes these results for three variables.

Spectral Analysis and Time Series

Abstract: Preface. Preface to Volume 2. Contents of Volume 2. List of Main Notation. Basic Concepts. Elements of Probability Theory. Stationary Random Processes. Spectral Analysis. Estimation in the Time Domain. Estimation in the Frequency Domain. Spectral Analysis in Practice. Analysis of Processes with Mixed Spectra.

New Introduction to Multiple Time Series Analysis

Abstract: This reference work and graduate level textbook considers a wide range of models and methods for analyzing and forecasting multiple time series. The models covered include vector autoregressive, cointegrated, vector autoregressive moving average, multivariate ARCH and periodic processes as well as dynamic simultaneous equations and state space models. Least squares, maximum likelihood, and Bayesian methods are considered for estimating these models. Different procedures for model selection and model specification are treated and a wide range of tests and criteria for model checking are introduced. Causality analysis, impulse response analysis and innovation accounting are presented as tools for structural analysis. The book is accessible to graduate students in business and economics. In addition, multiple time series courses in other fields such as statistics and engineering may be based on it. Applied researchers involved in analyzing multiple time series may benefit from the book as it provides the background and tools for their tasks. It bridges the gap to the difficult technical literature on the topic.

The scientist and engineer's guide to digital signal processing

Abstract: In early 1980s, DSP was taught as a graduate level course in electrical engineering. A decade later, DSP had become a standart part of the ungraduate curriculum.