Reduced Rank Regression: With Applications to Quantitative Structure-Activity Relationships
Heinz Schmidli · 1995
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, ?, .- 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. (R)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.