Applied Multivariate Statistics in Geohydrology and Related Sciences

ISBN-13: 9783642803307 / Angielski / Miękka / 2011 / 248 str.

Charles E. Brown

Opis

It has been evident from many years of research work in the geohydrologic sciences that a summary of relevant past work, present work, and needed future work in multivariate statistics with geohydrologic applications is not only desirable, but is necessary. This book is intended to serve a broad scientific audience, but more specifi cally is geared toward scientists doing studies in geohydrology and related geo sciences.lts objective is to address both introductory and advanced concepts and applications of the multivariate procedures in use today. Some of the procedures are classical in scope but others are on the forefront of statistical science and have received limited use in geohydrology or related sciences. The past three decades have seen a significant jump in the application of new research methodologies that focus on analyzing large databases. With more general applications being developed by statisticians in various disciplines, multivariate quantitative procedures are evolving for better scientific applica tion at a rapid rate and now provide for quick and informative analyses of large datasets. The procedures include a family of statistical research methods that are alternatively called "multivariate analysis" or "multivariate statistical methods."

Spis treści

I Introduction to General Statistical and Multivariate Concepts.- 1 General Concepts.- 1.1 Concepts.- 1.2 Definitions.- 1.3 Introduction to Statistical Concepts.- 1.3.1 Meaning of Statistics.- 1.3.2 Probability and Randomness.- 1.4 Meaning of Multivariate Analysis.- 1.5 Univariate to Multivariate Mathematical Generalizations.- 1.6 Nature of References.- 1.7 Related Texts and Publications.- 1.8 Processing Data.- 1.9 Summary.- 1.10 Supplemental Reading.- 2 Introduction to Multivariate Statistical Procedures.- 2.1 Concept.- 2.2 Definitions.- 2.3 Purpose and Rationale.- 2.4 Development of Multivariate Quantitative Procedures.- 2.5 Selection of Methods.- 2.6 Scales of Measurement.- 2.7 Testing for Normality.- 2.7.1 Coefficient of Skewness.- 2.7.2 Coefficient of Kurtosis.- 2.7.3 Significance Tests — Normality.- 2.8 Normal Distribution.- 2.9 Central Limit Theorem.- 2.10 Significance Tests.- 2.10.1 Chi-Square Goodness of Fit and Probability Plats.- 2.10.2 Significance Tests-Homogeneity of Variances.- 2.11 Division of Procedures: Variable-Directed or Individual Directed Tests.- 2.12 Sampling Design in Multivariate Procedures.- 2.13 Sample Survey Design.- 2.14 Standardization.- 2.15 The Statistical Model.- 2.16 Random Variables.- 2.17 Description of Multivariate Quantitative Procedures.- 2.18 General Significance Testing for Multivariate Data.- 2.19 Summary.- 2.20 Supplemental Reading.- II Variable-Directed Procedures Based on Normal Distribution Assumptions.- 3 Correlation.- 3.1 Concept.- 3.2 Definitions.- 3.3 Objective of Procedures.- 3.4 Problems.- 3.5 Significance Tests for Correlations.- 3.6 Interpretation.- 3.7 Numerical Example on Cave Lithology.- 3.8 Numerical Example on Carbonate-Rock Geohydrology.- 3.9 Numerical Example on Fluvial Sediment Geochemistry.- 3.10 Numerical Example — Discussion.- 3.11 Summary.- 3.12 Supplemental Reading.- 4 Factor Analysis.- 4.1 Concept.- 4.2 Matrices.- 4.3 Definitions.- 4.4 Contrasts Between Factor and Principal Components Analyses.- 4.5 Significance Tests — Factor Analysis Decision Tests.- 4.5.1 Significance Tests — Barlett’s Test of Sphericity.- 4.5.2 Generalizations About Correlations and Factor Analysis.- 4.6 Determining Significance of Output.- 4.6.1 Significance Tests — Factor Loadings.- 4.6.2 Significance Tests — Amount of Explained Variation.- 4.7 Extraction Techniques for Factors.- 4.8 Other Significance Tests — Comparisons of Solutions and Datasets.- 4.9 Factor Analysis Procedure.- 4.10 Numerical Example on Water Chemistry.- 4.11 Numerical Example on Sediment Geochemistry.- 4.12 Numerical Example on Drainage Basin Properties.- 4.13 Numerical Example on Detrital Sediment Characteristics.- 4.14 Summary.- 4.15 Supplemental Reading.- 5 Canonical Correlation.- 5.1 Concept.- 5.2 Definitions.- 5.3 Overview of Method.- 5.4 Significance Tests-Importance of Wilks’ Lambda (?).- 5.5 Significance Tests — Bartlett’s Test Using Chi-Square Variable.- 5.6 Canonical Correlation Procedure.- 5.7 Numerical Example on Sediment and Soil Pollution.- 5.8 Summary.- 5.9 Supplemental Reading.- 6 Multiple Regression.- 6.1 Concept.- 6.2 Definitions.- 6.3 Overview of Methods.- 6.4 Multiple Regression Procedure.- 6.5 Stepwise Multiple Regression.- 6.6 Significance Tests — Addressing Multicollinearity.- 6.7 Multiple Regression Applications.- 6.8 Numerical Example on Water Yield.- 6.9 Numerical Example on Streamflows.- 6.10 Numerical Example on Geomorphic Variables.- 6.11 Numerical Example on Sampling Sites.- 6.12 Multivariate Multiple Regression.- 6.13 Nonlinear Regression.- 6.14 Multivariate Models with Qualitative (Dummy) Variables.- 6.14.1 Multivariate Models with One Qualitative Variable.- 6.14.2 Multivariate Models with Multiple Qualitative Variables.- 6.15 Summary.- 6.16 Supplemental Reading.- 7 Multivariate Analysis of Variance.- 7.1 Concept.- 7.2 Definitions.- 7.3 Overview of Analysis of Variance.- 7.3.1 Procedure.- 7.3.2 Repeated Measures Design in Analysis of Variance.- 7.3.3 Two-Way Multivariate Analysis of Variance Procedure.- 7.3.4 Multiple Factor Analysis of Variance (Factorial Designs).- 7.4 Overview of Manova.- 7.4.1 Canonical Variates Analysis.- 7.5 Multivariate Analysis of Variance (MANOVA) Procedure.- 7.6 Numerical Example on Water Geochemistry.- 7.7 Numerical Example on Mine Tailings Hydrology.- 7.8 Profile Analysis and Significance Tests.- 7.9 Profile Analysis — Discussion.- 7.10 Comparison of Means and Simultaneous Tests on Several Linear Combinations.- 7.11 Calculating the Confidence Interval for the Difference of Two Means.- 7.11.1 Simultaneous Confidence Intervals.- 7.11.2 The Bonferroni Method of Multiple Comparisons.- 7.12 Summary.- 7.13 Supplemental Reading.- 8 Multivariate Analysis of Covariance.- 8.1 Concept.- 8.2 Definitions.- 8.3 Overview of Analysis of Covariance.- 8.3.1 Procedure.- 8.3.2 One-Way Model with Multiple Covariates.- 8.3.3 M-Way Model with Covariates.- 8.4 Special Tests on Covariance Matrices.- 8.5 Multivariate Analysis of Covariance Procedure.- 8.6 Numerical Example — Tests on Covariance Matrices.- 8.7 Summary.- 8.8 Supplemental Reading.- III Variable-Directed Techniques not Based on Normal Distribution Assumptions.- 9 Principal Components.- 9.1 Concept.- 9.2 Definitions.- 9.3 Overview and Objectives.- 9.4 Principal Components Analysis Procedure.- 9.5 Defining Sedimentary Processes.- 9.6 Numerical Example on Ground Water Geochemistry.- 9.7 Numerical Example on Properties of Carbonate Rock Aquifers.- 9.8 Uses of PCA.- 9.8.1 Generalizations.- 9.8.2 Other Uses of PCA.- 9.8.2.1 Deriving Discriminant Functions.- 9.8.2.2 For Multiple Regression.- 9.8.2.3 For Analysis of Variance.- 9.9 Summary.- 9.10 Supplemental Reading.- IV Individual-Directed Techniques Based on Normal Distribution Assumptions.- 10 Multiple Discriminant Analysis.- 10.1 Concept.- 10.2 Definitions.- 10.3 Overview.- 10.4 Parameters for Classification.- 10.5 Jackknifing Classification.- 10.6 Similarities with Other Methods.- 10.7 Discriminant Function Coefficients.- 10.8 Discriminant Scores and Probabilities.- 10.9 Discriminant Analysis Procedure.- 10.10 Numerical Example on Ground-Water Sources.- 10.11 Numerical Example on Hydrogeochemistry of Carbonate Terrains.- 10.12 Numerical Example on Stream Water Chemistry.- 10.13 Numerical Example on Oil Field Chemistry.- 10.14 Summary.- 10.15 Supplemental Reading.- V Individual-Directed Techniques not Based on Normal Distribution Assumptions.- 11 Cluster Analysis.- 11.1 Concept.- 11.2 Definitions.- 11.3 Overview.- 11.4 Cluster Analysis Procedure.- 11.5 Numerical Example on Properties of Mineralized Waters.- 11.6 Numerical Example on Stream Water Quality.- 11.7 Variable Clustering.- 11.8 Integrated Cluster Analysis.- 11.9 Numerical Example on Carbonate Rock Data.- 11.10 Summary.- 11.11 Supplemental Reading.- 12 Multiple Logistic Regression.- 12.1 Concept.- 12.2 Overview.- 12.3 Logistic Regression.- 12.4 Log-Linear Models.- 12.5 Probit Models.- 12.6 Log-Linear Models for Contingency Tables.- 12.7 Numerical Example on Geochemical Data.- 12.8 Relation to Discriminant Analysis.- 12.9 Multinomial Response Models.- 12.10 Summary.- 12.11 Supplemental Reading.- VI Other Approaches to Explore Multivariate Data.- 13 Coefficient of Variation.- 13.1 Concept and Procedure.- 13.2 Numerical Example on Geochemical Data of Karst Aquifer.- 13.3 Summary.- 13.4 Supplemental Reading.- 14 Correspondence Analysis.- 14.1 Procedure.- 14.2 Numerical Example on Ground-Water Mixing Zones.- 14.3 Numerical Example on Water Chemistry.- 14.4 Summary.- 14.5 Supplemental Reading.- 15 Multivariate Probit Analysis.- 15.1 Concept and Procedure.- 15.2 Numerical Example on Biological Data.- 15.3 Numerical Example on Agricultural Data.- 15.4 Summary.- 15.5 Supplemental Reading.- VII Multivariate Measures of Space, Distance, and Time.- 16 Multivariate Time Series Modeling.- 16.1 Concept and Introduction.- 16.2 Definitions.- 16.3 Treatment of Time Series Data.- 16.4 Multivariate Time Series Procedures.- 16.4.1 Types of Multivariate Models.- 16.4.2 Box-Jenkins (ARIMA) Method.- 16.5 Numerical Example on Air Pollution Data.- 16.6 State Space Models.- 16.7 Markov Process Models.- 16.8 Dynamic Regression Models.- 16.9 Multivariate Stochastic Models.- 16.10 Numerical Example on Stochastic Analysis of Hydrologic Data.- 16.11 Multivariate Transfer Functions.- 16.12 Intervention Models.- 16.13 Diagnostics.- 16.14 Numerical Example on Ground-Water Data.- 16.15 Fit of Autoregressive Models.- 16.16 Summary.- 16.17 Supplemental Reading.- 17 Multivariate Spatial Measures.- 17.1 Multidimensional Scaling.- 17.2 Regionalized Variables and Spatial Correlation.- 17.2.1 Introduction.- 17.2.2 Concept of Multivariate Geostatistics.- 17.3 Numerical Example on Soil Pollution Data.- 17.4 Limitations of Linear Model of Co-regionalization.- 17.5 Analysis of Directional Data.- 17.6 Summary.- 17.7 Supplemental Reading.- VIII Multivariate Data Preparation, Plotting, and Conclusions.- 18 Multivariate Data Preparation and Plotting.- 18.1 Introduction.- 18.2 Box Plots — Radon Concentrations.- 18.3 Scatter Plots — Carbonate Rock Data.- 18.4 Blob Plot — Geographic Data.- 18.5 Multidimensional Data Plot — Toxicological Data.- 18.6 Metroglyphs.- 18.7 Chernoff Faces.- 18.8 Andrews Plots.- 18.9 Probability Plots.- 18.10 Histograms — Carbonate Rock Data.- 18.11 Quality Control Graphs-Carbonate Rock Data.- 18.12 Star Diagrams.- 18.13 Simultaneous Confidence Intervals and Bonferroni Intervals.- 18.14 Summary.- 18.15 Supplemental Reading.- 19 Summary and Generalizations of Multivariate Quantitative Procedures.- 19.1 Introduction.- 19.2 Multivariate Methods and Generalizations.- 19.3 Methods Covered.- 19.4 Normality Tests.- 19.5 Multivariate Data Preparation and Plotting.- 19.6 Correlation Analysis.- 19.7 Coefficient of Variation.- 19.8 Factor Analysis.- 19.9 Canonical Correlation.- 19.10 Multiple Linear Regression.- 19.11 Nonlinear Regression.- 19.12 Principal Components Analysis.- 19.13 Correspondence Analysis.- 19.14 Discriminant Analysis.- 19.15 Multivariate Analysis of Variance (MANOVA).- 19.16 Multiple-Factor Analysis of Variance (Factorial Design).- 19.17 Cluster Analysis.- 19.18 Multivariate Analysis of Covariance.- 19.19 Analysis of Covariance.- 19.20 Special Tests on Covariance Matrices.- 19.21 Multivariate Multiple Regression.- 19.22 Logistic Regression.- 19.23 Multivariate Probit Analysis.- 19.24 Multivariate Time Series Analyses.- 19.25 Multidimensional Scaling.- 19.26 Multivariate Spatial Statistics.- 19.27 Nonparametric Multivariate Analysis.- 19.28 Summary.- 19.29 Supplemental Reading.- Appendix Introduction to Numerical Analysis.- A.1 General Concepts.- A.2 Solution of Simultaneous Linear Algebraic Equations.- A.2.1 Sets of Linear Equations.- A.2.2 Calculations in Matrix Algebra.- A.2.3 Definitions and Notation.- A.2.4 Basic Matrix Operations.- A.2.4.1 Comparison of Matrices.- A.2.4.2 Matrix Addition and Subtraction.- A.2.4.3 Matrix Multiplication.- A.2.4.4 Transposition.- A.2.4.5 Partitioning.- A.2.5 Matrix Inversion.- A.2.5.1 Inverse.- A.2.5.2 Computation of Inverse.- A.2.6 Eigenvalues and Eigenvectors of a Matrix.- A.2.7 Matrix Algebra and Solution of Simultaneous Linear Equations.- A.2.8 Direct Methods: Gaussian Elimination.- A.2.9 Indirect Methods: Gauss-Seidel Iteration.- A.3 Multivariate Normal Distribution in Matrix Form.- References.- Author Index.