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Related Concept Videos

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...
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 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...
Chebyshev's Theorem to Interpret Standard Deviation01:15

Chebyshev's Theorem to Interpret Standard Deviation

Chebyshev’s theorem, also known as Chebyshev’s Inequality, states that the proportion of values of a dataset for K standard deviation is calculated using the equation:

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Related Experiment Videos

The Chebyshev-polynomials-based unified model neural networks for function approximation.

T T Lee1, J T Jeng

  • 1Dept. of Electr. Eng., Nat. Taiwan Inst. of Technol., Taipei.

IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society
|February 8, 2008
PubMed
Summary
This summary is machine-generated.

We introduce an approximate transformable technique to create Chebyshev-Polynomials-Based (CPB) unified model neural networks. These networks offer faster learning and universal approximation capabilities for feedforward and recurrent neural networks.

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Area of Science:

  • Artificial Intelligence
  • Machine Learning
  • Neural Networks

Background:

  • Conventional feedforward and recurrent neural networks face limitations in learning speed and approximation capabilities.
  • Unified model neural networks offer a potential solution for enhanced performance.
  • Chebyshev polynomials provide a powerful mathematical framework for approximation.

Purpose of the Study:

  • To propose an approximate transformable technique for developing Chebyshev-Polynomials-Based (CPB) unified model neural networks.
  • To establish the relationship between single-layer and multilayer perceptron neural networks using this technique.
  • To demonstrate the universal approximator capability and improved learning speed of CPB unified models.

Main Methods:

  • Developed an approximate transformable technique, encompassing direct and indirect transformations.
  • Utilized Chebyshev polynomial approximation to construct CPB unified model neural networks.
  • Applied the recursive least square method with a forgetting factor as the learning algorithm.

Main Results:

  • Established the equivalence between CPB unified models and functional link networks based on Chebyshev polynomials.
  • Demonstrated that CPB unified models possess universal approximator capabilities.
  • Showcased significantly faster learning speeds compared to conventional feedforward/recurrent neural networks.
  • Derived conditions for optimal least square approximation in single-variable scenarios.

Conclusions:

  • The proposed approximate transformable technique effectively yields CPB unified model neural networks.
  • CPB unified models offer enhanced learning efficiency and universal approximation for neural networks.
  • The method provides a robust approach for functional approximation with reduced computational time.