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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...

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.
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Residuals and Least-Squares Property

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The vertical distance between the actual value of y and the estimated value of y. In other words, it measures the vertical distance between the actual data point and the predicted point on the line
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Linear Approximation in Frequency Domain

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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.
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Generalization, Discrimination, and Extinction

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Generalization, discrimination, and extinction are key concepts in operant conditioning that influence how behaviors are learned and maintained.
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Lagrange Multipliers: Two Constraints

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Updated: Jul 7, 2026

Cross-Modal Multivariate Pattern Analysis

Cross-Modal Multivariate Pattern Analysis

Published on: November 9, 2011

The multilayer perceptron as an approximation to a Bayes optimal discriminant function.

D W Ruck¹, S K Rogers, M Kabrisky

¹Sch. of Eng., US Air Force Inst. of Technol., Wright-Patterson AFB, OH.

IEEE Transactions on Neural Networks

|January 1, 1990

Summary

Multilayer perceptrons trained with backpropagation approximate Bayes optimal discriminant functions. This demonstrates their ability to estimate a posteriori probability functions for classification tasks.

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Last Updated: Jul 7, 2026

Cross-Modal Multivariate Pattern Analysis

Cross-Modal Multivariate Pattern Analysis

Published on: November 9, 2011

Area of Science:

Machine Learning
Artificial Intelligence
Pattern Recognition

Background:

Multilayer perceptrons (MLPs) are fundamental neural network architectures.
Classification tasks require optimal decision boundaries.
Backpropagation is a standard algorithm for training MLPs.

Purpose of the Study:

To demonstrate that MLPs trained via backpropagation approximate the Bayes optimal discriminant function.
To show that MLP outputs estimate a posteriori probabilities.
To generalize this finding across various MLP configurations.

Main Methods:

Theoretical analysis of multilayer perceptrons.
Application of the backpropagation training algorithm.
Mathematical proof for classification accuracy.

Main Results:

MLPs trained with backpropagation approximate the Bayes optimal discriminant function for both two-class and multi-class problems.
The outputs of the trained MLPs were shown to approximate a posteriori probability functions.
The theoretical results hold for any number of network layers and activation functions.

Conclusions:

Multilayer perceptrons are theoretically capable of achieving Bayes optimal classification.
The backpropagation algorithm enables MLPs to learn and represent class posterior probabilities.
This provides a strong theoretical foundation for using MLPs in classification.