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Related Concept Videos

State Space Representation01:27

State Space Representation

735
The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
735
State Space to Transfer Function01:21

State Space to Transfer Function

684
The conversion of state-space representation to a transfer function is a fundamental process in system analysis. It provides a method for transitioning from a time-domain description to a frequency-domain representation, which is crucial for simplifying the analysis and design of control systems.
The transformation process begins with the state-space representation, characterized by the state equation and the output equation. These equations are typically represented as:
684
Transfer Function to State Space01:23

Transfer Function to State Space

966
State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...
966
Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

424
Nonlinear systems often require sophisticated approaches for accurate modeling and analysis, with state-space representation being particularly effective. This method is especially useful for systems where variables and parameters vary with time or operating conditions, such as in a simple pendulum or a translational mechanical system with nonlinear springs.
For a simple pendulum with a mass evenly distributed along its length and the center of mass located at half the pendulum's length,...
424
Linear time-invariant Systems01:23

Linear time-invariant Systems

1.1K
A system is linear if it displays the characteristics of homogeneity and additivity, together termed the superposition property. This principle is fundamental in all linear systems. Linear time-invariant (LTI) systems include systems with linear elements and constant parameters.
The input-output behavior of an LTI system can be fully defined by its response to an impulsive excitation at its input. Once this impulse response is known, the system's reaction to any other input can be...
1.1K
SFG Algebra01:16

SFG Algebra

416
In Signal Flow Graph (SFG) algebra, the value a node represents is determined by the sum of all signals entering that node. This summed value is then transmitted through every branch leaving the node, making the SFG a powerful tool for visualizing and analyzing control systems.
Each node in an SFG corresponds to a variable, and the interactions between nodes are represented by branches with associated gains. When multiple branches lead into a node, the value at that node is the sum of the...
416

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Updated: Apr 12, 2026

Application of Granger Causality Analysis of the Directed Functional Connection in Alzheimer's Disease and Mild Cognitive Impairment
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Granger causality for state-space models.

Lionel Barnett1, Anil K Seth1

  • 1Sackler Centre for Consciousness Science, School of Engineering and Informatics, University of Sussex, Brighton BN1 9QJ, United Kingdom.

Physical Review. E, Statistical, Nonlinear, and Soft Matter Physics
|May 15, 2015
PubMed
Summary
This summary is machine-generated.

State-space models offer a robust solution for Granger causality analysis, overcoming limitations posed by moving average components common in complex systems. This method enhances causal inference accuracy across diverse scientific fields.

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Area of Science:

  • Complex Systems Analysis
  • Time Series Analysis
  • Statistical Inference

Background:

  • Granger causality is crucial for inferring causal interactions in stochastic systems.
  • Autoregressive (AR) modeling, a common method for Granger causality, is confounded by moving average (MA) components in data.
  • MA components arise from common data processing steps like filtering, downsampling, and noise observation.

Purpose of the Study:

  • To develop a robust method for Granger causality analysis unaffected by MA components.
  • To demonstrate the efficacy of state-space (SS) models for calculating Granger causality.
  • To extend Granger causality analysis to non-linear, non-stationary, and non-homoscedastic data.

Main Methods:

  • Utilized state-space (SS) models, equivalent to autoregressive moving average (ARMA) models.
  • Calculated Granger causality directly from SS model parameters via a discrete algebraic Riccati equation.
  • Investigated both conditional and unconditional Granger causality in time and frequency domains.

Main Results:

  • Granger causality estimation using SS models is not degraded by MA components.
  • SS-derived Granger causality estimators exhibit greater statistical power and reduced bias compared to AR estimators.
  • The SS approach provides a framework for relaxing linearity, stationarity, and homoscedasticity assumptions.

Conclusions:

  • State-space models provide a superior and more versatile approach to Granger causality analysis.
  • This method significantly improves causal inference in systems with MA components, common in various scientific disciplines.
  • The SS framework opens new research avenues for Granger causality in complex, real-world data.