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.

Partial directed coherence: a new concept in neural structure determination

Parameter estimation when the measurement information may be incomplete is discussed. An ARX model is used as a basic system representation. The presentation covers both missing output and missing input. First reconstruction of the missing values is discussed. The reconstruction is based on a state-space formulation of the system, and is performed using Kalman filtering or fixed-interval smoothing formulas. Several approaches to the identification problem are presented, including a new method based on the EM (expectation maximization) algorithm. The different approaches are tested and compared using Monte Carlo simulations. The choice of method is always a tradeoff between estimation accuracy and computational complexity. According to the simulations the gain in accuracy using the EM method can be considerable if many data are missing. >

A Recursive EM Algorithm for Identification of ARX-Models Subject to Missing Data

A classification approach is proposed for finding ranges of process inputs that result in corresponding ranges of a process profit function using deep learning. Two deep learning tools are used to ...

Deep Learning for Classification of Profit-based Operating Regions in Industrial Processes

Multivariable dynamical processes are characterized by complex cause and effect relationships among variables. Reconstruction of these causal connections from data, especially based on the concept of Granger causality (GC), has attracted significant attention in process engineering with applications to interaction assessment, topology reconstruction, and fault detection. The standard practice for reconstruction of GC networks has been along the parametric route that deploys vector autoregressive (VAR) models but without giving due importance to the structural characteristics of data generating process (DGP). In this work, we first demonstrate that the presence of a mismatch between model structure and DGP leads to biased and/or inefficient estimates of causality measures and, hence, the strengths (weights) of causal connections. This issue is further aggravated for small sample sizes wherein additionally spurious causal relationships are detected. In this respect, we present, second, a systematic methodol...

Efficient Reconstruction of Granger-Causal Networks in Linear Multivariable Dynamical Processes

Reconstruction of process topology from cause-effect analysis of measurements finds applications in root-cause analysis, identification of disturbance propagation pathways, estimation of fault propagation times, and interaction assessment. A widely used technique based on the notion of Granger causality (GC), but is well-suited only for stationary stochastic processes. The GC-based measures and methods, while being useful in certain cases, can be highly restrictive and produce misleading results in engineering applications since changes in process variables are frequently deterministic. The lack of sufficient excitation and presence of measurement errors further restricts their applicability. In this respect, we present (i) a new definition of direct causality for deterministic linear time-invariant (LTI) dynamical systems, and (ii) a novel causality detection method based on delay estimation from noisy multivariable measurements. Efficient estimates of time delays are obtained from recently developed non-parametric frequency domain method based on partial coherence and Hilbert transform relations. In addition to the ability of handling deterministic variations, the proposed causality detection method is well-suited to handle low excitation signals and does not require the specification of any model structure. Case studies involving data from synthetic and benchmark processes are presented to illustrate the utility and efficacy of the proposed method.

A Novel Causality Method for Reconstruction of Process Topology in Multivariable LTI Dynamical Systems

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.

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

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.

Piyush Agarwal

Papers

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

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