Statistics: An Introduction Using R
TABLE OF CONTENTS Preface xiii Chapter 1 Introduction ......................................................................................... 1 Scope......................................................................................................2 Mathematical Statistics ........................................................................... 3 Overview ................................................................................................ 3 Data Collection ....................................................................................... 3 Sampling ................................................................................................. 3 Experimental and Observational Studies .................................................. 4 Experiments ............................................................................................ 4 Observational Study ................................................................................ 5 Types of Data ......................................................................................... 5 Terminology and Theory of Inferential Statistics ......................................6 Statistics, Estimators and Pivotal Quantities ............................................. 6 Null Hypothesis and Alternative Hypothesis ............................................ 7 Error .......................................................................................................8 Interval Estimation .................................................................................. 9 Significance............................................................................................. 9 Examples .............................................................................................. 10 Misuse .................................................................................................. 11 Misinterpretation: Correlation ................................................................ 12 History of Statistical Science ................................................................. 12 Etymology ............................................................................................. 14 Origins in Probability Theory ................................................................. 14 Development of Modern Statistics ......................................................... 18 Design of Experiments .......................................................................... 21 vi Bayesian Statistics ................................................................................ 22 Important Contributors to Statistics ....................................................... 24 Applications .......................................................................................... 25 Applied Statistics, Theoretical Statistics and Mathematical Statistics ..... 25 Machine Learning and Data Mining ...................................................... 25 Statistics in Society ............................................................................... 25 Statistical Computing ............................................................................. 25 Statistics Applied to Mathematics or the Arts ........................................ 25 Specialized Disciplines .......................................................................... 26 Chapter 2 Descriptive Statistics ........................................................................ 28 Center ................................................................................................... 28 Mean .................................................................................................... 28 Types of Mean...................................................................................... 29 Distribution of the Sample Mean ........................................................... 34 Arithmetic Mean ................................................................................... 34 Geometric Mean ................................................................................... 37 Harmonic Mean .................................................................................... 45 Median.................................................................................................. 57 Finite Set of Numbers ........................................................................... 57 Mode (Statistics) ................................................................................... 69 Mode of a Sample................................................................................. 69 Comparison of Mean, Median and Mode............................................... 70 Unimodal Distributions .......................................................................... 72 Dispersion ............................................................................................. 72 Measures .............................................................................................. 73 Sources ................................................................................................. 74 A Partial Ordering of Dispersion ........................................................... 75 Variance ............................................................................................... 75 Definition .............................................................................................. 75 Chapter 3 Summarizing Database ...................................................................... 78 Object Databases ................................................................................. 79 Mandatory ............................................................................................ 79 Selecting a Database ............................................................................ 79 File-Based Approaches ......................................................................... 80 Disadvantages of a File-Based Approach .............................................. 80 Database Approaches ........................................................................... 81 Hierarchical Databases ......................................................................... 81 Weaknesses of Hierarchical Databases ................................................ 81 Strengths of RDBMS ............................................................................ 82 Weaknesses .......................................................................................... 82 Network Databases .............................................................................. 82 vii Description of Network Database ......................................................... 82 Strengths of Network Databases........................................................... 83 Weaknesses of Network Databases ..................................................... 83 Relational Databases ............................................................................ 83 Database System vs Files System ......................................................... 83 Approach .............................................................................................. 85 Distributed Database ............................................................................ 87 Architecture .......................................................................................... 87 Database Models .................................................................................. 90 Database Schemes and Database Instances ......................................... 91 Comparison of Spatial Models and Database Models ............................ 92 DBMS-Architecture and Data Independence ....................................... 93 Three-Schemes Architecture ................................................................ 93 Data Independence ............................................................................... 94 Database Languages and Database Interfaces ..................................... 95 Computerized of Databases .................................................................. 95 Types of Databases .............................................................................. 96 Basic Concepts of Transaction Management......................................... 97 Objectives of Distributed Transaction Management ............................... 98 A Model for Transaction Management in a Distributed System ............. 99 Distributed Database System (DDBMSS) .......................................... 100 Background......................................................................................... 101 DBMSS and File Management Systems .............................................. 102 Data Integration in Heterogeneous Database Systems ........................ 103 Measuring Data Quality in Heterogeneous Databases ......................... 105 Collection of Distributed Database ...................................................... 107 Advantages and Disadvantages of Distributed Systems ....................... 107 Management System Distributed Database ......................................... 108 Oracle’s Distributed Database Architecture ........................................ 109 Clients and Servers ............................................................................. 109 Databases and Database Links ........................................................... 111 Distributed Data Processing (DDP) .................................................... 112 Basic Business Data Processing Principles ......................................... 113 Explain the Audit Procedures in an Electronic Data Processing Audit.. 114 Distributed Database Design Concepts ............................................... 115 Objectives of Data Distribution ........................................................... 115 Data Fragmentation ............................................................................ 116 Design Issues for Modelling Behaviour in Distributed Systems ............ 117 Challenges in Going from Analysis to Design....................................... 117 Requirements for Better Conceptual Models ....................................... 119 Data Representation ........................................................................... 120 Data Suit ............................................................................................. 120 Strategies and Testing of Data ............................................................ 120 viii Formal Methods .................................................................................. 121 Description.......................................................................................... 121 Formal Technical Reviews .................................................................. 122 Technical Methods .............................................................................. 123 Reviews .............................................................................................. 126 Technical Views.................................................................................. 128 Process ............................................................................................... 129 Initial Plateau ...................................................................................... 129 Alpha and Beta Representation ........................................................... 129 Customer Involvement ........................................................................ 129 Alpha Product ..................................................................................... 130 Beta Product ....................................................................................... 130 Alpha or Beta Testing Phase ............................................................... 130 Alpha/Beta Usability and Acceptability ............................................... 131 Administration Management in Distributed Database .......................... 131 Site Autonomy .................................................................................... 131 Distributed Database Security ............................................................. 132 Authentication without Passwords ....................................................... 132 Supporting User Accounts and Roles................................................... 132 Centralized User and Privilege Management ....................................... 133 Distributed Transaction Management .................................................. 135 Data Management Subsystem ............................................................. 136 Exception Handling ............................................................................. 137 Components within Transaction Management...................................... 138 Lock Management .............................................................................. 138 Pros and Cons of Concurrency ........................................................... 139 Concurrency Problems ........................................................................ 140 Model of Consistency and Lock Protocols .......................................... 140 Application Development of Distributed Database .............................. 148 Location Transparency........................................................................ 148 SQL and Commit Transparency .......................................................... 149 Replication Transparency .................................................................... 149 Remote Procedure Calls (RPCS) ........................................................ 150 Distributed System .............................................................................. 150 Remote SQL Statements ..................................................................... 150 Distributed SQL Statements ................................................................ 151 Shared SQL for Remote and Distributed Statements ........................... 151 Remote Transactions .......................................................................... 151 Distributed Transactions ...................................................................... 152 Two-Phase Commit Mechanism.......................................................... 152 Database Link Name Resolution ......................................................... 153 Name Resolution when the Global Database Name is Complete ......... 153 Name Resolution when the Global Database Name is Partial .............. 153 ix Name Resolution when no Global Database Name is Specified ........... 154 Terminating the Search for Name Resolution ...................................... 154 Schema Object Name Resolution ........................................................ 154 Example of Global Object Name Resolution: Complete Object Name.. 155 Example of Global Object Name Resolution: Partial Object Name ...... 156 Global Name Resolution in Views, Synonyms, and Procedures ............ 157 What Happens When Global Names Change ...................................... 157 Scenarios for Global Name Changes ................................................... 158 Global Database Names and Global Object Names ............................. 159 Types of Database Links .................................................................... 160 Connection Qualifiers .......................................................................... 164 Views, Synonyms, Procedures and Global Name Resolution................ 166 Techniques for Location Transparency................................................ 168 Database Administration and Automation ............................................ 169 DBA Responsibilities .......................................................................... 170 Types of Database Administration ....................................................... 170 Nature of Database Administration ..................................................... 171 Database Administration Tools ............................................................ 171 The Impact of it Automation on Database Administration .................... 172 Distributed Database Management System ......................................... 172 Relational Databases .......................................................................... 172 Types of DBMS: Object-Oriented Databases (OODBMS) ................. 173 Hierarchical Databases ....................................................................... 174 The Hierarchical Structure .................................................................. 175 Types of DBMS: Network Databases ................................................. 177 Distributed Transactions and Database ............................................... 178 DML and DDL Transactions ............................................................... 179 Session Trees for Distributed Transactions .......................................... 179 Two-Phase Commit Mechanism.......................................................... 184 Manual Resolution of in-Doubt Transactions ....................................... 188 Relational Data Model ........................................................................ 189 Relational Database ............................................................................ 191 Terminology ........................................................................................ 191 Relational Operations .......................................................................... 194 Relational Database Management Systems ......................................... 195 Constraints in Relational Data ............................................................. 195 Relational Database ............................................................................ 198 Mapping ER Designs Relational Scheme............................................. 200 Relational Database Design ................................................................ 205 Chapter 4 Linear Models ................................................................................. 217 Linear Algebra .................................................................................... 217 History ................................................................................................ 218 x Scope of Study.................................................................................... 219 Some Main useful Theorems ............................................................... 224 Applications ........................................................................................ 224 Geometric Introduction........................................................................ 227 Introduction to Linear Transformations ................................................ 228 Generalizations and Related Topics ..................................................... 230 Linear Programming Formulation ........................................................ 231 Linear Programming ........................................................................... 232 Application of Linear Programming ..................................................... 233 Standard form of Linear Programming Problem .................................. 233 Augmented Form (Slack Form) ........................................................... 233 Duality ................................................................................................ 234 A Prototype LP Problem ..................................................................... 234 The General LP Formulation ............................................................... 237 Graphical Solution of Two Variable Linear Programming ..................... 239 Feasible Regions of Two Variable Linear Programming....................... 239 The Solution Space of A Single Equality Constraint ............................. 239 The Solution Space of A Single Inequality Constraint ........................... 240 Representing the Objective Function in the LP Solution Space............. 241 Graphical Solution of the Prototype Example: A 2-Var LP with a Unique Optimal Solution ...................................................................... 242 2-Var LP’s with Many Optimal Solutions............................................. 243 Infeasible 2-Var LP’s .......................................................................... 243 Unbounded 2-Var LP’s ....................................................................... 244 Non-Linear Programming ................................................................... 256 Applicability ........................................................................................ 256 Methods for Solving the Problem......................................................... 257 Examples ............................................................................................ 257 Sample Applications ............................................................................ 258 The Product-Mix Problem with Price Elasticity ................................... 258 The Transportation Problem with Volume Discounts on Shipping Costs .... 259 Portfolio Selection with Risky Securities.............................................. 260 Graphical Illustration of Nonlinear Programming Problems .................. 262 Types of Nonlinear Programming Problems ........................................ 266 Unconstrained Optimization................................................................. 266 Linearly Constrained Optimization ....................................................... 267 Quadratic Programming ...................................................................... 267 Convex Programming ......................................................................... 268 Separable Programming ...................................................................... 268 Non-convex Programming .................................................................. 269 Geometric Programming ..................................................................... 269 Fractional Programming ...................................................................... 270 The Complementarity Problem ............................................................ 271 xi One-Variable Unconstrained Optimization ........................................... 272 One-Dimensional Search Procedure.................................................... 272 Summary of the One-Dimensional Search Procedure .......................... 273 Multivariable Unconstrained Optimization ........................................... 275 The Gradient Search Procedure .......................................................... 276 Summary of the Gradient Search Procedure ....................................... 277 Karush-Kuhn-Tucker (KKT) Conditions for Constrained Optimization... 280 Quadratic Programming ...................................................................... 285 The KKT Conditions for Quadratic Programming ............................... 287 The Modified Simplex Method ............................................................ 288 Separable Programming ...................................................................... 291 Reformulation as a Linear Programming Problem ............................... 293 Extensions ........................................................................................... 297 Convex Programming ......................................................................... 298 A Sequential Linear Approximation Algorithm (Sang Wan) .................. 299 Summary of the Sangwan Algorithm ................................................... 299 Non-Convex Programming.................................................................. 303 Sequential Unconstrained Minimization Technique (SUMT) ................ 303 Summary of Sumt ............................................................................... 305 Problems ............................................................................................. 306 Bibliography ..................................................................................... 310