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Related Concept Videos

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...
Regression Toward the Mean01:52

Regression Toward the Mean

Regression toward the mean (“RTM”) is a phenomenon in which extremely high or low values—for example, and individual’s blood pressure at a particular moment—appear closer to a group’s average upon remeasuring. Although this statistical peculiarity is the result of random error and chance, it has been problematic across various medical, scientific, financial and psychological applications. In particular, RTM, if not taken into account, can interfere when researchers try to extrapolate results...
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...
Prediction Intervals01:03

Prediction Intervals

The interval estimate of any variable is known as the prediction interval. It helps decide if a point estimate is dependable.
However, the point estimate is most likely not the exact value of the population parameter, but close to it. After calculating point estimates, we construct interval estimates, called confidence intervals or prediction intervals. This prediction interval comprises a range of values unlike the point estimate and is a better predictor of the observed sample value, y. 
The...
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:
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.

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Related Experiment Videos

A fast method to approximately train hard support vector regression.

Yongping Zhao1, Jianguo Sun

  • 1ZNDY of Ministerial Key Laboratory, Nanjing University of Science & Technology, Nanjing, China.

Neural Networks : the Official Journal of the International Neural Network Society
|September 7, 2010
PubMed
Summary
This summary is machine-generated.

This study introduces a novel greedy algorithm to train Hard Support Vector Regression (HSVR) models, mitigating overfitting caused by noisy data. The new method improves training efficiency and reduces support vectors compared to existing software.

Related Experiment Videos

Area of Science:

  • Machine Learning
  • Computational Statistics

Background:

  • Hard Support Vector Regression (HSVR) is prone to overfitting due to noise, as it lacks bounds on Lagrange multipliers.
  • This limitation can lead to infinitely magnified multipliers, compromising model stability and performance.

Purpose of the Study:

  • To propose a greedy stagewise algorithm for approximately training HSVR.
  • To address the overfitting issue in HSVR by introducing an implicit regularization technique.

Main Methods:

  • A greedy stagewise algorithm is developed, selecting samples with maximal predicted discrepancy at each iteration.
  • Sample weights are updated once per iteration to prevent excessive magnification, acting as an early stopping and regularization mechanism.

Main Results:

  • The proposed algorithm demonstrates improved training time and a reduced number of support vectors compared to LIBSVM2.82.
  • Experimental results on synthetic and real-world datasets validate the algorithm's effectiveness in handling noisy data.

Conclusions:

  • The novel greedy algorithm effectively trains HSVR, offering a robust solution against overfitting.
  • This approach provides a regularization effect, enhancing model performance and efficiency in regression tasks.