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Linear least squares

About: Linear least squares is a research topic. Over the lifetime, 2697 publications have been published within this topic receiving 112179 citations.


Papers
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Journal ArticleDOI
TL;DR: A least squares version for support vector machine (SVM) classifiers that follows from solving a set of linear equations, instead of quadratic programming for classical SVM's.
Abstract: In this letter we discuss a least squares version for support vector machine (SVM) classifiers. Due to equality type constraints in the formulation, the solution follows from solving a set of linear equations, instead of quadratic programming for classical SVM‘s. The approach is illustrated on a two-spiral benchmark classification problem.

8,811 citations

Book
01 Apr 1996
TL;DR: Theorems and statistical properties of least squares solutions are explained and basic numerical methods for solving least squares problems are described.
Abstract: Preface 1. Mathematical and statistical properties of least squares solutions 2. Basic numerical methods 3. Modified least squares problems 4. Generalized least squares problems 5. Constrained least squares problems 6. Direct methods for sparse problems 7. Iterative methods for least squares problems 8. Least squares problems with special bases 9. Nonlinear least squares problems Bibliography Index.

3,405 citations

Book
01 Jan 1980
TL;DR: In this paper, the authors present a method to estimate the least squares of a scatterplot matrix using a simple linear regression model, and compare it with the mean function of the scatterplot matrices.
Abstract: Preface.1 Scatterplots and Regression.1.1 Scatterplots.1.2 Mean Functions.1.3 Variance Functions.1.4 Summary Graph.1.5 Tools for Looking at Scatterplots.1.5.1 Size.1.5.2 Transformations.1.5.3 Smoothers for the Mean Function.1.6 Scatterplot Matrices.Problems.2 Simple Linear Regression.2.1 Ordinary Least Squares Estimation.2.2 Least Squares Criterion.2.3 Estimating sigma 2.2.4 Properties of Least Squares Estimates.2.5 Estimated Variances.2.6 Comparing Models: The Analysis of Variance.2.6.1 The F-Test for Regression.2.6.2 Interpreting p-values.2.6.3 Power of Tests.2.7 The Coefficient of Determination, R2.2.8 Confidence Intervals and Tests.2.8.1 The Intercept.2.8.2 Slope.2.8.3 Prediction.2.8.4 Fitted Values.2.9 The Residuals.Problems.3 Multiple Regression.3.1 Adding a Term to a Simple Linear Regression Model.3.1.1 Explaining Variability.3.1.2 Added-Variable Plots.3.2 The Multiple Linear Regression Model.3.3 Terms and Predictors.3.4 Ordinary Least Squares.3.4.1 Data and Matrix Notation.3.4.2 Variance-Covariance Matrix of e.3.4.3 Ordinary Least Squares Estimators.3.4.4 Properties of the Estimates.3.4.5 Simple Regression in Matrix Terms.3.5 The Analysis of Variance.3.5.1 The Coefficient of Determination.3.5.2 Hypotheses Concerning One of the Terms.3.5.3 Relationship to the t -Statistic.3.5.4 t-Tests and Added-Variable Plots.3.5.5 Other Tests of Hypotheses.3.5.6 Sequential Analysis of Variance Tables.3.6 Predictions and Fitted Values.Problems.4 Drawing Conclusions.4.1 Understanding Parameter Estimates.4.1.1 Rate of Change.4.1.2 Signs of Estimates.4.1.3 Interpretation Depends on Other Terms in the Mean Function.4.1.4 Rank Deficient and Over-Parameterized Mean Functions.4.1.5 Tests.4.1.6 Dropping Terms.4.1.7 Logarithms.4.2 Experimentation Versus Observation.4.3 Sampling from a Normal Population.4.4 More on R2.4.4.1 Simple Linear Regression and R2.4.4.2 Multiple Linear Regression.4.4.3 Regression through the Origin.4.5 Missing Data.4.5.1 Missing at Random.4.5.2 Alternatives.4.6 Computationally Intensive Methods.4.6.1 Regression Inference without Normality.4.6.2 Nonlinear Functions of Parameters.4.6.3 Predictors Measured with Error.Problems.5 Weights, Lack of Fit, and More.5.1 Weighted Least Squares.5.1.1 Applications of Weighted Least Squares.5.1.2 Additional Comments.5.2 Testing for Lack of Fit, Variance Known.5.3 Testing for Lack of Fit, Variance Unknown.5.4 General F Testing.5.4.1 Non-null Distributions.5.4.2 Additional Comments.5.5 Joint Confidence Regions.Problems.6 Polynomials and Factors.6.1 Polynomial Regression.6.1.1 Polynomials with Several Predictors.6.1.2 Using the Delta Method to Estimate a Minimum or a Maximum.6.1.3 Fractional Polynomials.6.2 Factors.6.2.1 No Other Predictors.6.2.2 Adding a Predictor: Comparing Regression Lines.6.2.3 Additional Comments.6.3 Many Factors.6.4 Partial One-Dimensional Mean Functions.6.5 Random Coefficient Models.Problems.7 Transformations.7.1 Transformations and Scatterplots.7.1.1 Power Transformations.7.1.2 Transforming Only the Predictor Variable.7.1.3 Transforming the Response Only.7.1.4 The Box and Cox Method.7.2 Transformations and Scatterplot Matrices.7.2.1 The 1D Estimation Result and Linearly Related Predictors.7.2.2 Automatic Choice of Transformation of Predictors.7.3 Transforming the Response.7.4 Transformations of Nonpositive Variables.Problems.8 Regression Diagnostics: Residuals.8.1 The Residuals.8.1.1 Difference Between e and e.8.1.2 The Hat Matrix.8.1.3 Residuals and the Hat Matrix with Weights.8.1.4 The Residuals When the Model Is Correct.8.1.5 The Residuals When the Model Is Not Correct.8.1.6 Fuel Consumption Data.8.2 Testing for Curvature.8.3 Nonconstant Variance.8.3.1 Variance Stabilizing Transformations.8.3.2 A Diagnostic for Nonconstant Variance.8.3.3 Additional Comments.8.4 Graphs for Model Assessment.8.4.1 Checking Mean Functions.8.4.2 Checking Variance Functions.Problems.9 Outliers and Influence.9.1 Outliers.9.1.1 An Outlier Test.9.1.2 Weighted Least Squares.9.1.3 Significance Levels for the Outlier Test.9.1.4 Additional Comments.9.2 Influence of Cases.9.2.1 Cook's Distance.9.2.2 Magnitude of Di .9.2.3 Computing Di .9.2.4 Other Measures of Influence.9.3 Normality Assumption.Problems.10 Variable Selection.10.1 The Active Terms.10.1.1 Collinearity.10.1.2 Collinearity and Variances.10.2 Variable Selection.10.2.1 Information Criteria.10.2.2 Computationally Intensive Criteria.10.2.3 Using Subject-Matter Knowledge.10.3 Computational Methods.10.3.1 Subset Selection Overstates Significance.10.4 Windmills.10.4.1 Six Mean Functions.10.4.2 A Computationally Intensive Approach.Problems.11 Nonlinear Regression.11.1 Estimation for Nonlinear Mean Functions.11.2 Inference Assuming Large Samples.11.3 Bootstrap Inference.11.4 References.Problems.12 Logistic Regression.12.1 Binomial Regression.12.1.1 Mean Functions for Binomial Regression.12.2 Fitting Logistic Regression.12.2.1 One-Predictor Example.12.2.2 Many Terms.12.2.3 Deviance.12.2.4 Goodness-of-Fit Tests.12.3 Binomial Random Variables.12.3.1 Maximum Likelihood Estimation.12.3.2 The Log-Likelihood for Logistic Regression.12.4 Generalized Linear Models.Problems.Appendix.A.1 Web Site.A.2 Means and Variances of Random Variables.A.2.1 E Notation.A.2.2 Var Notation.A.2.3 Cov Notation.A.2.4 Conditional Moments.A.3 Least Squares for Simple Regression.A.4 Means and Variances of Least Squares Estimates.A.5 Estimating E(Y |X) Using a Smoother.A.6 A Brief Introduction to Matrices and Vectors.A.6.1 Addition and Subtraction.A.6.2 Multiplication by a Scalar.A.6.3 Matrix Multiplication.A.6.4 Transpose of a Matrix.A.6.5 Inverse of a Matrix.A.6.6 Orthogonality.A.6.7 Linear Dependence and Rank of a Matrix.A.7 Random Vectors.A.8 Least Squares Using Matrices.A.8.1 Properties of Estimates.A.8.2 The Residual Sum of Squares.A.8.3 Estimate of Variance.A.9 The QR Factorization.A.10 Maximum Likelihood Estimates.A.11 The Box-Cox Method for Transformations.A.11.1 Univariate Case.A.11.2 Multivariate Case.A.12 Case Deletion in Linear Regression.References.Author Index.Subject Index.

3,215 citations

Journal ArticleDOI
TL;DR: In this article, a measure based on confidence ellipsoids is developed for judging the contribution of each data point to the determination of the least squares estimate of the parameter vector in full rank linear regression models.
Abstract: A new measure based on confidence ellipsoids is developed for judging the contribution of each data point to the determination of the least squares estimate of the parameter vector in full rank linear regression models. It is shown that the measure combines information from the studentized residuals and the variances of the residuals and predicted values. Two examples are presented.

2,477 citations


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Performance
Metrics
No. of papers in the topic in previous years
YearPapers
20238
202212
202145
202066
201975
201863