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Related Concept Videos

Equation of State01:07

Equation of State

The equation of state is an equation that relates physical quantities, such as pressure, volume, temperature, and the number of moles, of a thermodynamics system with each other. The equation relating physical quantities with each other can be a simple mathematical expression or too complicated to express in mathematical form. In either case, a relationship between physical quantities exists. If the equation of state cannot be expressed in a mathematical form, then experimental data and...
Variables and Equations of State01:27

Variables and Equations of State

The physical state of a pure substance can be defined by certain state variables such as volume (V), pressure (p), temperature (T), and amount of substance (n). When two gases are separated by a movable wall, the gas with the higher pressure naturally compresses the gas with the lower pressure. This causes the high-pressure gas to expand and the low-pressure gas to compress until both gases achieve mechanical equilibrium. At this point, their pressures equalize, and the movement of the wall...
Regression Analysis01:11

Regression Analysis

Regression analysis is a statistical tool that describes a mathematical relationship between a dependent variable and one or more independent variables.
In regression analysis, a regression equation is determined based on the line of best fit– a line that best fits the data points plotted in a graph. This line is also called the regression line. The algebraic equation for the regression line is called the regression equation. It is represented as:
State Space Representation01:27

State Space Representation

The frequency-domain technique, commonly used in analyzing and designing feedback control systems, is effective for linear, time-invariant systems. However, it falls short when dealing with nonlinear, time-varying, and multiple-input multiple-output systems. The time-domain or state-space approach addresses these limitations by utilizing state variables to construct simultaneous, first-order differential equations, known as state equations, for an nth-order system.
Consider an RLC circuit, a...
Transfer Function to State Space01:23

Transfer Function to State Space

State-space representation is a powerful tool for simulating physical systems on digital computers, necessitating the conversion of the transfer function into state-space form. Consider an nth-order linear differential equation with constant coefficients, like those encountered in an RLC circuit. The state variables are selected as the output and its n−1 derivatives. Differentiating these variables and substituting them back into the original equation produces the state equations.
In an RLC...
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...

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O-cresol Concentration Online Measurement Based On Near Infrared Spectroscopy Via Partial Least Square Regression
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Published on: November 8, 2019

Equations of states in singular statistical estimation.

Sumio Watanabe1

  • 1Precision and Intelligence Laboratory, Tokyo Institute of Technology, 4259 Nagatsuda Midori-ku, 226-8503 Yokohama Japan. swatanab@pi.titech.ac.jp

Neural Networks : the Official Journal of the International Neural Network Society
|August 25, 2009
PubMed
Summary

This study reveals universal mathematical relations among four errors in singular statistical models, enabling generalization performance prediction. New information criteria can estimate these errors for both regular and singular models.

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Published on: September 5, 2019

Area of Science:

  • Machine Learning
  • Statistical Modeling
  • Information Theory

Background:

  • Singular statistical models, common in machine learning with hierarchical structures or hidden variables, are nonidentifiable with singular Fisher information matrices.
  • These properties prevent standard convergence of posterior distributions and asymptotic normality of estimators, hindering generalization performance prediction.
  • Existing methods struggle to accurately assess generalization performance in these complex models.

Purpose of the Study:

  • To investigate and establish universal mathematical relationships among four key error types in singular statistical models: Bayes generalization error, Bayes training error, Gibbs generalization error, and Gibbs training error.
  • To demonstrate that generalization errors (Bayes and Gibbs) can be reliably estimated using their corresponding training errors.
  • To develop broadly applicable information criteria for model selection and performance evaluation in both regular and singular statistical models.

Main Methods:

  • Analysis of four error metrics: Bayes generalization error, Bayes training error, Gibbs generalization error, and Gibbs training error.
  • Derivation of universal mathematical equations connecting these error terms, applicable across various distributions and models.
  • Development of estimation techniques for generalization errors based on training errors.

Main Results:

  • Proved universal mathematical relations exist among the four studied error types, establishing them as "equations of state" in statistical estimation.
  • Demonstrated that Bayes and Gibbs generalization errors can be effectively estimated using their respective training errors.
  • Proposed novel information criteria applicable to both regular and singular statistical models for performance assessment.

Conclusions:

  • The established universal relations provide a theoretical foundation for understanding and predicting generalization performance in singular statistical models.
  • The proposed estimation methods and information criteria offer practical tools for model selection and evaluation, overcoming limitations of traditional approaches.
  • This work advances the field of statistical learning by providing unified criteria for analyzing complex models.