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Related Concept Videos

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...
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...
Approximate Integration01:24

Approximate Integration

In many practical and theoretical contexts, the exact value of a definite integral may be inaccessible. This limitation typically arises when the antiderivative of a function is either unknown or cannot be expressed in a closed mathematical form. Alternatively, it can occur when a function is defined not by a formula but by a finite set of empirical data points, such as those collected during experiments. In these cases, approximate integration techniques provide a valuable solution.One of the...
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 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.
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...

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Related Experiment Videos

Function approximation using fuzzy neural networks with robust learning algorithm.

W Y Wang1, T T Lee, C L Liu

  • 1Dept. of Electron. Eng., St. John's & St. Mary's Inst. of Technol., Taipei.

IEEE Transactions on Systems, Man, and Cybernetics. Part B, Cybernetics : a Publication of the IEEE Systems, Man, and Cybernetics Society
|January 1, 1997
PubMed
Summary
This summary is machine-generated.

This study introduces a robust fuzzy neural network using B-spline membership functions for accurate function approximation, even with data outliers. The novel algorithm effectively handles erroneous data, reducing learning iterations and improving convergence speed.

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Area of Science:

  • Artificial Intelligence
  • Machine Learning
  • Computational Intelligence

Background:

  • Function approximation is crucial in many scientific and engineering fields.
  • Training data often contains outliers, which can significantly degrade model performance.
  • Existing methods may struggle with noisy datasets, leading to inaccurate approximations.

Purpose of the Study:

  • To develop a robust function approximation method using fuzzy neural networks.
  • To address the challenge of outliers in training data.
  • To improve the convergence speed and accuracy of function approximation models.

Main Methods:

  • Utilized B-spline membership functions (BMF's) within a fuzzy neural network architecture.
  • Developed novel learning rules for weighting values and BMF's based on a robust objective function.
  • Employed a gradient descent method for optimizing the network parameters.
  • Implemented a robust learning algorithm designed to mitigate the impact of erroneous data points.

Main Results:

  • The proposed robust learning algorithm effectively minimizes the influence of outliers during training.
  • Achieved rapid convergence of the approximated function to the desired error tolerance.
  • Demonstrated significant reduction in the number of learning iterations required.
  • Successfully applied the method to both one-dimensional (curves) and two-dimensional (surfaces) function approximation tasks.

Conclusions:

  • The novel fuzzy neural network with BMF's offers an efficient and feasible approach for robust function approximation.
  • The developed robust learning algorithm enhances model performance in the presence of noisy data.
  • The method shows promise for applications requiring accurate function approximation from imperfect datasets.