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Related Concept Videos

Linear Approximation in Time Domain01:21

Linear Approximation in Time Domain

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, the...
Linear time-invariant Systems01:23

Linear time-invariant Systems

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 calculated...
Linear Approximation in Frequency Domain01:26

Linear Approximation in Frequency Domain

Linear systems are characterized by two main properties: superposition and homogeneity. Superposition allows the response to multiple inputs to be the sum of the responses to each individual input. Homogeneity ensures that scaling an input by a scalar results in the response being scaled by the same scalar.
In contrast, nonlinear systems do not inherently possess these properties. However, for small deviations around an operating point, a nonlinear system can often be approximated as linear.
Linearization and Approximation01:26

Linearization and Approximation

Linearization is a mathematical technique used to approximate complex, nonlinear functions with simpler linear models in the vicinity of a chosen reference point. The method is based on the idea that, although a function may be difficult to evaluate exactly, its behavior near a specific input value can often be closely approximated by the tangent line at that point. This approach is particularly useful when small deviations from a known value are involved.Consider the square root function, for...
Linear Approximations01:23

Linear Approximations

For a differentiable function of two variables, linear approximation estimates values near a known point by replacing the curved surface with its tangent plane. Consider the function\begin{equation*}f(x,y)=x^2+3y^2\end{equation*}near the point (2, 1). The exact value at this point is f(2, 1) = 22 + 3(1)2 = 4 + 3 = 7.The linear approximation of f(x, y)) near (a, b) is\begin{equation*}L(x,y)=f(a,b)+f_x(a,b)(x-a)+f_y(a,b)(y-b)\end{equation*}First, compute the partial derivatives: fx(x, y) = 2x and...
Application of Linearization and Approximation01:29

Application of Linearization and Approximation

A drone flying through complex terrain often relies on more than one sensing method to estimate small changes in altitude. Along with direct measurements, air pressure provides a useful indirect indicator of vertical movement. Atmospheric pressure decreases as altitude increases, and this relationship is commonly described using an exponential model. Although accurate, converting pressure measurements into altitude values requires calculations that are too complex to perform repeatedly during...

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Related Experiment Videos

A hybrid linear-neural model for time series forecasting.

M C Medeiros1, A Veiga

  • 1Department of Electrical Engineering, Catholic University of Rio de Janeiro, Rio de Janeiro, Brazil.

IEEE Transactions on Neural Networks
|February 6, 2008
PubMed
Summary

This study introduces a novel neural coefficient smooth transition autoregressive (NCSTAR) model for analyzing nonlinear time series. The NCSTAR model enhances forecasting by integrating neural networks for time-varying parameters and smooth regime transitions.

Related Experiment Videos

Area of Science:

  • Time Series Analysis
  • Machine Learning
  • Econometrics

Background:

  • Nonlinear time series analysis presents challenges for traditional linear models.
  • Existing models like TAR and STAR have limitations in handling complex transitions.
  • Neural networks offer powerful capabilities for modeling complex data patterns.

Purpose of the Study:

  • To propose a novel Neural Coefficient Smooth Transition Autoregressive (NCSTAR) model.
  • To leverage neural networks for modeling time-varying parameters in autoregressive models.
  • To enhance the analysis and forecasting of nonlinear time series data.

Main Methods:

  • A linear model framework with parameters controlled by a neural network is employed.
  • The neural network output induces smooth and multivariate thresholds for regime partitioning.
  • The proposed NCSTAR model is compared to Threshold Autoregressive (TAR) and Smooth Transition Autoregressive (STAR) models.

Main Results:

  • The NCSTAR model naturally incorporates linear multivariate thresholds and smooth transitions.
  • The formulation offers advantages over traditional TAR and STAR models.
  • Neural network integration facilitates the selection of optimal initial values for training.

Conclusions:

  • The NCSTAR model provides a flexible and powerful approach for nonlinear time series analysis.
  • This method effectively captures complex dynamics and transitions between different regimes.
  • The integration of neural networks enhances the performance and applicability of autoregressive models.