Reduced Rank Regression: With Applications to Quantitative Structure-Activity Relationships

ISBN-13: 9783790808711 / Angielski / Miękka / 1995 / 179 str.

H. Schmidli; Heinz Schmidli

Opis

Reduced rank regression is widely used in statistics to model multivariate data. In this monograph, theoretical and data analytical approaches are developed for the application of reduced rank regression in multivariate prediction problems. Both classical and Bayesian inference are discussed.

Spis treści

1. Quantitative Structure Activity Relationships (QSAR).- 1.1. Introduction.- 1.2. Modification of Substances.- 1.3. Physico—Chemical Descriptors.- 1.4. Biological Descriptors.- 1.5. Prediction Model.- 1.6. The Development of an Insecticide: an Example.- 2. Linear Multivariate Prediction.- 2.1. Introduction.- 2.2. Multivariate Prediction.- 2.3. Prediction Criteria.- 2.3.1. Introduction.- 2.3.2. Loss Function.- 2.3.3. Expected Loss.- 2.3.4. Relevant Predictor Space.- 2.3.5. Estimation of the Expected Loss.- 2.3.5.1. Point Predictors.- 2.3.5.2. Density Predictors.- 2.4. Exploratory Graphical Methods.- 2.5. Method and Variable Selection.- 2.5.1. Introduction.- 2.5.2. Method Selection.- 2.5.3. Variable Selection.- 2.6. Assessment of the Goodness of Prediction of the Selected Model.- 3. Heuristic Multivariate Prediction Methods.- 3.1. Introduction.- 3.2. Principal Component Regression.- 3.3. Partial Least Squares.- 3.4. Dimension Selection.- 3.5. Example.- 4. Classical Analysis of Reduced Rank Regression.- 4.1. Introduction.- 4.2. QSAR: Biological Responses.- 4.3. Reduced Rank Regression Models.- 4.3.1. Model.- 4.3.2. Parametrization.- 4.3.3. Reduced Rank Regression or Multivariate Regression ?.- 4.3.4. The Geometry of Reduced Rank Regression Models.- 4.3.5. Likelihood.- 4.3.6. Error Structure.- 4.3.7. Maximum Likelihood Estimation of the Parameters B, ?, µ.- 4.3.8. Maximum Likelihood Estimation of the Parameter A.- 4.3.8.1. Known Error Covariance Matrix.- 4.3.8.2. Error Covariance Matrix Proportional to the Identity Matrix.- 4.3.8.3. Unstructured Error Covariance Matrix.- 4.3.8.4. Diagonal Error Covariance Matrix.- 4.3.9. Asymptotic Distribution of the Predictions.- 4.3.10 Example.- 4.4. Extensions of the Standard Reduced Rank Regression Model.- 4.4.1. Structured Error Covariance Matrix.- 4.4.2. Latent Variable Models.- 4.4.3. Non—normal Errors, Outliers and Robustification.- 4.4.4. Nonlinearities.- 4.4.5. Econometric Models.- 4.5. Prediction Criteria for the Rank Selection of Reduced Rank Regression Models.- 4.5.1. Likelihood.- 4.5.2. Information Criterion for Rank Selection.- 4.5.2.1. Estimation of the Information Criterion.- 4.5.2.2. A Simulation Study.- 4.5.3. Mean Squared Error of Prediction for Rank Selection.- 4.5.3.1. Estimation of the Mean Squared Error of Prediction.- 4.5.3.2. A Simulation Study.- 4.5.4. Example.- 4.6. Variable Selection for Reduced Rank Regression Models.- 4.6.1. Prediction Criteria for Variable Selection.- 4.6.2. A Simulation Study.- 4.6.3. Example.- 5. Bayesian Analysis of Reduced Rank Regression.- 5.1. Introduction.- 5.2. The Reduced Rank Regression Model.- 5.2.1. Likelihood.- 5.2.2. Parametrization.- 5.2.3. Full Conditional Priors.- 5.2.4. Full Conditional Posteriors.- 5.2.5. Structured Error Covariance Matrix.- 5.2.6. Predictive Distribution.- 5.2.7. Rank Determination.- 5.3. Markov Chain Monte Carlo Methods.- 5.3.1. Gibbs Sampling of the Posterior Distribution.- 5.3.2. Gibbs Sampling of the Predictive Distribution.- 5.4. Example.- 6. Case Studies.- 6.1. ®Voltaren: An Anti-Inflammatory Drug.- 6.1.1. Data.- 6.1.2. Analysis.- 6.2. Development of a Herbicide.- 6.2.1. Data.- 6.2.2. Analysis.- 7. Discussion.- A.1 Introduction.- A.2 Multivariate Regression MR.- A.3 Principal Component Analysis PCA.- A.4 Partial Least Squares PLS.- A.5 Canonical Correlation Analysis CCA.- A.6 Reduced Rank Regression with Diagonal Error Covariance Matrix RRR.- A.7 Redundancy Analysis RDA.- A.8 Software.- A.9 Matrix Algebra Definitions.- A.10 Multivariate Distributions.- References.- Main Notations and Abbreviations.