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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...
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...
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...
Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving01:29

Mechanistic Models: Compartment Models in Algorithms for Numerical Problem Solving

Mechanistic models play a crucial role in algorithms for numerical problem-solving, particularly in nonlinear mixed effects modeling (NMEM). These models aim to minimize specific objective functions by evaluating various parameter estimates, leading to the development of systematic algorithms. In some cases, linearization techniques approximate the model using linear equations.
In individual population analyses, different algorithms are employed, such as Cauchy's method, which uses a...

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Related Experiment Videos

Functional approximation by feed-forward networks: a least-squares approach to generalization.

A R Webb1

  • 1Defence Res. Inst., Great Malvern.

IEEE Transactions on Neural Networks
|January 1, 1994
PubMed
Summary

This study introduces a least-squares method for function approximation and generalization, creating radial basis function networks. Adding a regularization term improves generalization for noisy data, enhancing function approximation accuracy.

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Area of Science:

  • Computational mathematics
  • Machine learning
  • Signal processing

Background:

  • Function approximation and generalization are critical in machine learning.
  • Noisy input data presents challenges for accurate function approximation.
  • Radial basis function networks are a common tool for these tasks.

Purpose of the Study:

  • To explore a least-squares approach for function approximation and generalization with noiseless training data.
  • To develop a method that generalizes well to data corrupted by input noise.
  • To investigate the impact of regularization on generalization performance.

Main Methods:

  • Utilized a least-squares approach for function approximation.
  • Developed a generalizer in the form of a radial basis function network for finite training samples.
  • Introduced a regularization term to the error criterion for improved generalization.
  • Applied the method to point-source location using receiver array data.

Main Results:

  • The least-squares approach yields a radial basis function network for finite samples under specific noise conditions.
  • A modified error criterion with a regularization term enhances generalization.
  • The feedforward architecture demonstrated effectiveness in point-source location.

Conclusions:

  • The proposed least-squares method with regularization improves generalization in function approximation tasks.
  • Radial basis function networks are suitable generalizers for this approach.
  • The technique shows promise for applications like signal processing and source localization.