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Linear Approximations

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...

Linearization and Approximation

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 Domain

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...

Application of Linearization and Approximation

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...

Newton’s Method

Newton’s Method

Newton’s Method is a powerful iterative technique for approximating the roots of real-valued, differentiable functions, particularly when analytical solutions are impractical. This approach is widely used in scientific computing, engineering, and finance, where equations may be too complex for traditional algebraic methods to handle. The method relies on an iterative process that refines an initial estimate using the function’s derivative to approach the true solution progressively.

Linear Approximation in Frequency Domain

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.

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A robust backpropagation learning algorithm for function approximation.

D S Chen¹, R C Jain

¹Artificial Intelligence Lab., Michigan Univ., Ann Arbor, MI.

IEEE Transactions on Neural Networks

|January 1, 1994

Summary

This summary is machine-generated.

A new robust backpropagation (BP) algorithm enhances neural network learning by approximating underlying data patterns instead of interpolating training samples. This method improves convergence and resists errors in training data.

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Area of Science:

Artificial Intelligence
Machine Learning
Neural Networks

Background:

Multilayer feedforward neural networks use backpropagation (BP) for learning.
Nonlinear modeling can lead to interpolation of training data.
Erroneous data can cause oscillations in learned mappings.

Purpose of the Study:

To derive a robust BP learning algorithm resistant to noise and gross errors.
To improve the approximation capabilities of neural networks.

Main Methods:

Developed a robust BP algorithm inspired by robust statistics (Huber and Hampel).
Modified the objective function's shape iteratively.
Utilized a nonlinear cascade of affine transformations for functional approximation.

Main Results:

The robust BP algorithm approximates underlying mappings, avoiding interpolation of training samples.
Demonstrated resistance to gross errors in training data.
Showed improved convergence rates due to suppressed influence of incorrect samples.

Conclusions:

The robust BP algorithm offers a superior alternative to conventional BP for noisy datasets.
This approach enhances the reliability and efficiency of neural network training.
The method effectively handles outliers and improves model generalization.