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STATISTICAL
ANALYSIS OF ECOLOGICAL DIVERSITY
Defining and Measuring Ecological diversity
Shannon index
Alternative Diversity Indexes
Average Rarity Diversity Indexes
Species-Abundance curves
Related issues
Abundance Estimation
Simple Random Sampling.
Replicated Encounter Sampling.
Alternative Sampling Strategies.
Statistical Inference on Diversity
Design-Based Inference
Estimation of Species Richness
Model-based inference
Ecological Diversity Ordering
Intrinsic Diversity Ordering
Profile estimation.
Assessing diversity ordering
Field Studies
DESCRIPTIVE
MEASURES OF ECOLOGICAL DIVERSITY
Diversity, richness, evenness
Introduction to Essential Properties
A Comprehensive List of Diversity Indices
Richness
Evenness and Normalization of Diversity Indices
General properties of diversity indices
Statistical Proposals of General Properties
Diversity and Inequality
Majorization and Lorenz Curve
Special indices and families of indices
Gini-Simpson Indices and Generalizations
Derivation of the Gini-Simpson Index
A family of Indices Depending on the Euclidean Distance
Rao’s Family based on Dissimilarity Coefficients
Diversity Indices based on Distances between Distributions
Shannon and Entropy Measures
Diversity Indices Derived from Averages
SAMPLING
DESIGNS FOR MONITORING ECOLOGICAL DIVERSITY
Unit sampling
Sampling by plots
Sampling by points
Sampling by lines
Area sampling
Simple random sampling of areas
Sampling of areas
Further developments: two-stage sampling
INFERENCE
ON ECOLOGICAL DIVERSITY
Diversity index estimation
General Results
Inference for some specific Diversity Indexes under SRS
Jackknife for Diversity Index Estimation
Bayesian Approach to Diversity Index Estimation
Species-abundance curve models
Negative Binomial Distribution
Logarithmic Series Distribution
Geometric Distribution and Broken-Stick Distribution
Lognormal series distribution
THE
INVENTORY AND ESTIMATION OF PLANT SPECIES RICHNESS
Species Inventorying
Traditional collection of species data
Methods for perfecting species lists
Estimating and comparing species richness through samples
Fitting species abundance distributions
Estimation derived from species accumulation curves and
species area curves
Nonparametric estimators
Taxon surrogacy
SPATIAL
STATISTICS
Models
Geostatistical Models
Lattice Models
Spatial Point Processes
Exploring Spatial Structure
Geostatistical Data
Lattice Data
Spatial Point Patterns
Estimation
Prediction
Future Directions
GEOSTATISTICS:
PAST, PRESENT, AND FUTURE
Distribution-Free Methodology
Likelihood-Based Modeling
Model Based Prediction
Discussion and Future Directions
SPATIAL
DESIGN
A statistical framework
Single purpose spatial designs
Optimum Designs for Trend Estimation (Uncorrelated Processes)
Exploratory Designs
Optimum Designs for Trend Estimation (Correlated Processes)
Optimum Designs for Spatial Prediction
Optimum Designs for Covariogram Estimation
Multipurpose spatial designs
Constrained and Compound Designs
Entropy Sampling
Relationships among design criteria
STATISTICAL
ANALYSIS OF SPATIAL COUNT DATA
Random Spatial Indices
Non-Random Spatial Indices
Models with Continuous Spatial Index
Models with Discrete Spatial Index
Markov Random Field Models
Models with Exponential Family Conditionals
Estimation and Prediction
Application to Spatial Counts
Spatial Epidemiology and Disease Mapping
Spatial Epidemiology
Disease Mapping
SPATIAL
DISEASE MAPPING
Reasons for spatial pattern in disease data
Types of spatial disease data
Point Data
Regional Data
Geostatistical Data
Analytic methods by data type
Analytic Methods for Point Data
Analytic Methods for Regional Data
Analytic Methods for Geostatistical Data
Future directions
MULTIVARIATE
DATA ANALYSIS
Multivariate Distributions
The Multivariate Normal Distribution
Elliptically Contoured Distributions
Parameter Estimation for a Multivariate Normal
Population
Tests of Hypotheses for Mean Vectors and Covariance Matrices
The General Linear Hypothesis Model
Multiple Regression and Correlation
Multivariate Analysis of Variance
Discriminant Analysis
Principal Components
Factor Analysis
THE
ANALYSIS OF PUTATIVE SOURCES OF HEALTH HAZARD
Study Design
Retrospective and Prospective Studies
Study Region Design
Region Size
Region Shape
Replication and Control
Problems of Inference
Exploratory Techniques
Modeling the Hazard Exposure Risk
The Specification of in the Case Intensity
Models for Case Event Data
Estimation
Hypothesis Tests
Models for Count Data
Estimation
Hypothesis Tests
SPATIO-TEMPORAL
METHODS IN CLIMATOLOGY
Descriptive Statistical Methods
Empirical Orthogonal Function (EOF) Analysis
Continuous K-L Formulation
Discrete EOF Analysis
Estimation of EOFs
Complex EOF Analysis
Multivariate EOF Analysis
Extended EOF Analysis
Principal Oscillation Pattern (POP) Analysis
Formulation of POPs
Physical Implication of POPs
Estimation of POPs
Diagnostic Applications of POPs
Prognostic Application of POPs
POPs in Continuous Time
Complex POPs
Cyclostationary POPs
Space-Time Canonical Correlation Analysis (CCA)
Two-Field SpatialTemporal CCA
Estimation of CCA
Time Lagged CCA
Space-Time Spectral Analysis
RANK
TESTS FOR INDEPENDENCE
AND RANDOMNESS
Rank Tests for Independence
Optimal Tests
Treatment of Ties
Test for Randomness against trend
Contingency Tables
AREA
PRECIPITATION MEASUREMENT
The area precipitation measurement problem
The Kalman filter approach
A State Space Representation of the Problem.
The Kalman Filter Algorithm
Estimating the Model Parameters
Final Adjustments and Assessment of the Models
The Cokriging approach
The Kriging Estimator
The Cokriging Estimator
WATER-QUALITY
MONITORING OF RIVERS
Design Considerations in Water-Quality Monitoring Networks
Monitoring Objectives
Monitoring Approaches
Time Period for Measurement
Methods of Site Selection
Chemical vs. Biological Monitoring
Interpretive Context
Case Studies from the United States
Revised NASQAN
Maryland and Delaware Statewide Biomonitoring
Assessments
Massachusetts Comprehensive Monitoring Design
The Future of Water-Quality Monitoring Networks
STOCHASTIC
MODELLING IN LIFE SUPPORT SYSTEMS
The Concept of Stochastic Modelling
SM Metaphors and Reality Levels
Spatiotemporal Random Field Models
Towards a SM Program
Mathematical Forms of Natural Laws Considered in SM
Applications
SM in Genetic Research, Carcinogenesis and Toxicokinetics
applications
The Importance of Physical Geometry and Space/Time Scales
Knowledge Integration and the Epistemic Approach to
Space/time
Decision Making, Geographical Information Systems, and
Sampling Design
Physical Indicator Functions
Population Indicator Functions
Risk Assessment and Environmental Exposure-Health Effect
Associations
ECONOMIC
ASPECTS OF MONITORING ENVIRONMENTAL FACTORS: A COST-BENEFIT APPROACH
Setting environmental standards
Quantifying indicators
Emissions
Conservation
Bio-diversity
Economic implications of adopting environmental standards
Difficulties with neo-classical economic approach toward
environmental -valuation
Missing and incomplete markets
Failure of price mechanism
Environmental valuation
Direct valuation techniques
Contingent valuation method (CVM)
costs method (TCM)
Hedonic price analysis (HPA)
Value of life
Indirect valuation methods
Impact of pollution on health, material corrosion and
vegetation damage
Problems with environmental benefits estimates
The Rate of Discount
Irreversibility
Environmental policy regulation
Command and control policies
Economic instruments (EI)
TREND ANALYSIS
FOR ENVIRONMENTAL FACTORS: TIME EFFECTS ON NITROUS OXIDE (N2O) LEVELS AT
MACE HEAD, IRELAND
The Global Atmospheric Gases Experiment
Nitrous Oxide Levels at Mace Head
Identifying Trends
Trend Analysis for Variance Change
Problem Formulation
Detection of the Unknown Change-Point
Estimating the Unknown Change-Point
Change-Point Analysis of Nitrous Oxide Levels
MODERN
BIOMETRICS
History
Biometric Data Collection and Analysis
Experimental Design
Sample Surveys
Graphical Displays
Multivariate and Multidimensional Analysis
Linear Models and Generalized Linear Models
Categorical Data Analysis
Survival Analysis and Risk
Meta-Analysis
Bayes and Empirical Bayes
Computer-Intensive Biometrical Methods
Nonparametric Methods
Time Series
Longitudinal Studies
Spatial Analysis
Image Analysis
Biometry in Action
Agriculture
Forestry
Statistical Ecology and Biodiversity
Morphometrics and Stereology
Bioassay and Toxicology
Infectious Disease Epidemiology
Genetics
Bioinformatics and Genomics
Public Health and Biomedicine
The Case-Control Study
The Cohort Study
Clinical Trials
Mathematics in Biometry
Future
DATA
COLLECTION AND ANALYSIS IN BIOMETRICS
Experimental Design
Sample Surveys
Clinical Trials and Case Control Studies
Clinical Trials
Case Control Studies
Longitudinal Studies and Time Series
Longitudinal Studies
Time Series
Species Abundance
Data Collection
THE
DESIGN OF EXPERIMENTS
Standard Factorial Designs
Split-Plot Designs
Repeated Measures Designs
Importance of Correct Design and Analysis
SAMPLE SURVEYS
What is a Survey?
Probability sampling
Common probability sampling designs
Simple random sampling
Stratified sampling
Cluster sampling
Unequal probability sampling
Systematic sampling
Stratified multistage sampling
Survey estimates and standard errors
Nonsampling errors
Sampling rare populations
Issues in Survey Design
RESPONSE
ADAPTIVE RANDOMIZATION IN CLINICAL TRIALS
The Design
Likelihood Based Inference
Nonparametric Inference
Regression Models
TIME
SERIES MODELS
Standard Linear ARMA Models
Bilinear Models
Spatial Models
Spatial Bilinear Models
Exponential Models
ESTIMATING
SPECIES ABUNDANCE
Quadrat Sampling
Design-Based Quadrat sampling
Model-based quadrat sampling
Quadrat Size
Adaptive Cluster Sampling
Line and Point Transect Sampling
The classical approach
Methodological issues
The calibration approach
Nearest-Neighbour Distance Methods
Homogeneous Poisson Processes
Alternatives to homogeneous Poisson processes
Nonparametric methods
Edge Correction
Capture-Recapture Methods
A single recapture
Multiple recaptures
Related methods
STATISTICAL
METHODOLOGY IN BIOMETRY
Linear Regression, Generalized Linear Models, Exponential
Family and Logistic Regression
Gaussian Outcomes
Non-Gaussian Outcomes
Regression Models for Ordinal Data
Hierarchical Data
Multivariate Analysis
Longitudinal and Other Hierarchical Data
The Linear Mixed Model
From Gaussian to Non-Gaussian Longitudinal Data
Survival Analysis
LINEAR
REGRESSION MODELS
Simple Linear Regression model
The Model
Estimation
Inference
Inferences About the Regression Coefficients
Diagnostics and Remedial Measures
Multiple Linear Regression Model
Estimation of Regression Coefficients
Inferences About Regression Coefficients
Model Adequacy and Diagnostic
Comments on Interpreting Regression Analysis
GENERALIZED
LINEAR MODELLING
A Corner Stone: the Exponential Family of Distributions
Generalized Linear Modelling
Estimation for Generalized Linear Models
Quasi-likelihood and Generalized Estimating Equations (GEE)
GEE1
GEE2
CATEGORICAL
DATA ANALYSIS
Inference for a Single Proportion
Analysis of 2 × 2 Contingency Tables
Analysis of R x C Contingency Tables
Analysis of Sets of 2 × 2 Contingency Tables
Log-linear Models
Logistic Regression
Multinomial Regression Models
Poisson Regression
Clustered Categorical Data
SURVIVAL
ANALYSIS
Basic concepts of survival analysis
Censoring
Terminology and notation
Goals of survival analysis
Basic analysis layout for survival analysis
Descriptive measures of survival experience
The Kaplan Meier method and the log-rank test
Kaplan-Meier curves
Analysis layout for Kaplan-Meier curves
Calculation of estimated survival probabilities
The Log-rank test
The log-rank test for two groups
The log-rank test for several groups
The Wilcoxon test
The Cox proportional hazards model
Properties of the Cox PH model
Testing the significance of interaction
Computing and interpreting the hazard ratio from the Cox PH
model
Calculating a confidence interval for the hazard ratio
Adjusted survival curves using the Cox PH model
Evaluating the proportional hazards assumption
The proportional hazards assumption
A graphical method for evaluating the PH assumption: log-log
survival curves
Using time-dependent variables
Goodness-of-fit (GOF) tests
The stratified Cox model
Properties of the stratified Cox model
Testing the no-interaction assumption in the SC model
Extension of the Cox PH model for time-dependent variables
Time-dependent variables
Using time-dependent variables to test the proportional
hazards assumption
The extended Cox model for time-dependent variables
The hazard ratio formula for the extended Cox model
Use of the extended Cox model versus the stratified Cox model
MULTIVARIATE
AND MULTIDIMENSIONAL ANALYSIS
Continuous Outcomes
Multivariate Linear Regression
Multivariate Analysis of Variance and Covariance
Canonical Correlation and Redundancy Analysis
Structural Equation Modelling and Path Analysis
Discriminant and Cluster Analysis
Linear Projection Methods
Principal Components and Factor Analysis
Projection Pursuit
Non-linear Projection Methods
Multidimensional Scaling
Non-continuous Outcomes
Graphical Analysis
Pre- and Post-Modelling
Graphical Modelling
A Magician at Work?
REPEATED
MEASURES AND MULTILEVEL MODELLING
General Model
Some Models for Continuous Data
Multivariate Linear Regression Models
Linear Mixed Models
Non-linear Mixed Models
Models for Discrete Data
Conditional Model
Marginal Models
The Bahadur Model
The Dale and Probit Models
Random-effects Models
The Beta-binomial Model
The Generalized Linear Mixed Model
Generalized Estimating Equations
Discussion
META-ANALYSIS
Types of meta-analyses
Statistical principles of meta-analysis
Estimation
Stratification
Heterogeneity
Interaction
Statistical models for meta-analysis
Fixed effects models
Test for treatment effect
Estimate of overall treatment effect
Test for heterogeneity
Test for interaction
Test for trend
Random effects models
Bayesian models
Example of a meta-analysis
Tests for treatment effect and heterogeneity
Graphical display
Tests for trend and interaction
Further topics in meta-analysis
Weighted, cumulative, and prospective meta-analysis
Further uses of meta-analysis
Meta-regression
COMPUTATION
AND BIOMETRY
Computer Language and Systems Past, Present and Future
The Beginnings of Scientific Computing
High Level versus Low Level Languages
Unfulfilled Promises
Constraints on Future Development
Changing Views of Statistical Computing
Numerical Statistical Computation
Changes in Methodology
Connections into Other Software
Document Preparation and Display Systems
Project Management Systems
Human Interface Systems
Networking and Internet Connection Systems.
Computational Biology
Bioconductor
Statistical Computing in the Larger Context of Scientific
Computing
Computing Requirements for Scientific Projects
Interdependence Between Statistical Computing and Other
Scientific Computing Tasks
Limitations of Coverage
Articles Included Under This Theme
Numerical Statistical Computation
The Design of Data Collection
Directions for Future Development
Progress to Date
Incremental Development
Areas Where Improvements Can Be Expected
Articles Included Under This Theme
STATISTICAL
GRAPHICS
Graphs for models involving two or more variables
Two-dimensional graphics
Plots based on residuals
Graphs for models involving several covariates
Dynamic Displays
Outliers and Influential Points
Graphics for model building
Graphs for modelling data developing in time or space
Dependence
Graphs for modelling survival data
Graphs for multivariate data
Principal Components Analysis
Ordination Methods
Cluster Analysis
COMPUTER-INTENSIVE
STATISTICAL METHODS
Resampling and Monte
Carlo methods
The Bootstrap
Monte Carlo Methods
Numerical optimization and integration
Density estimation and smoothing
Scatterplot Smoothers
Relaxing least-squares and linearity
Non-linearity
Neural Networks
Support Vector Machines
Classification and regression trees
Selecting and combining models
STATISTICAL
COMPUTING
Advances in Routines Used for Statistical Computation
Numerical versus Non-numerical Routines
The Structuring of Numerical Routines
Non-numerical Routines
Numerical Statistical Computing
Calculations that Challenge Current Programs
Languages and Systems for Statistical Computing
Communication between Human and Machine
Different Modes of Communication for Different Users
Systems for professional use
Systems that are aimed at novices
Key Ideas for Statistical Systems
Automation
Connectivity - Interfaces between Systems
Unifying and Enabling Ideas
Different Types of Unifying and Enabling Ideas
Do as the Object Requires
Unifying Theoretical and Computational Ideas
Desirable Unifications
Computing on Language Objects
Computable Documents
Desiderata for Statistical Systems
General Requirements
Results that Can be Trusted
Analysis and Interpretation faults
Faults in Software
Faulty Tolerance Settings
Wrestling with New Questions - the Analysis of
Microarray Data
Large Data Bases - Data Mining
What is Data Mining?
Data Must Support the Intended Use
Connectivity
Connections between Different Computing Tasks
Text Formatting and Document Preparation Systems
Internet Connectivity
The Future of Statistical Computing
SPATIAL
STATISTICAL MODELING IN BIOLOGY
Gaussian Random Process Models
Linear Mixed Model Framework
Covariance models
Estimation and
Prediction
Bayesian Estimation
Markov Random Field Spatial Models
Gaussian Markov Random Field Model
Non-Gaussian Random Process Models
Generalized Mixed Model Framework
Spatial GLMM Estimation and Prediction
Spatial GLMM Example: Mapping Bird Counts
Multivariate Spatial Models
Cokriging
Hierarchical Models
Spatiotemporal Models
Computation
High-Level or Low-Level Language
Bayesian Computation
Geographic Information Systems
Technology
Future Directions
Nonstationary Spatial Processes
Multivariate Non-Gaussian Formulations
BIOSTATISTICAL
METHODS AND RESEARCH DESIGNS
Biostatistical Research Strategies
Understanding Scientific Disciplines
Study Design and Data Collection
Statistical Data Analysis
Dissemination of Results
Study Designs
Observational Studies
Cross-Sectional Studies
Cohort/Follow-up Studies
Case-control Studies
Randomized Studies
Statistical Models and Methods
Linear Models and Generalized Linear Models
Statistical Models for Survival Data
Statistical Models for Longitudinal Data
Other Statistical Models
Statistical Inference
Estimation
Hypothesis Testing
EPIDEMIOLOGY
METHODS
Types of Investigation
Measures of Association
Common Designs
Cohort Design
Case-Control Design
Other Designs
Discussion
COMMUNICABLE
DISEASES AND DATA ANALYSIS
Transmission probability
The binomial model of transmission
Contacts with persons of unknown infection status
The secondary attack rate
Transmission probability ratio
Augmented study designs
Validation sets
Basic reproductive number
Characteristics of Ro
Estimation of Ro
The dependent happening relation
Population-level effects of intervention
Challenges for the future
NUTRITIONAL
EPIDEMIOLOGY
Research Designs and Methods
Hypothesis Development
Hypothesis Testing
Analytic Epidemiology Studies
Dietary Intervention and Nutritional Supplementation Trials
Example of Dietary Fat and Post-Menopausal Breast Cancer
Hypothesis Generation
Association Studies
Ongoing Intervention Trials
Future Directions, Research Needs and Opportunities
Hypothesis Development
Hypothesis Testing
STATISTICAL
METHODS IN LABORATORY AND BASIC SCIENCE RESEARCH
Theory: Universal Distributions
The Role of Statistics
Exceptional Cases
Endemic Methods
Statistical Strategies
Descriptive Statistics
Randomization
Modeling
Case Studies
Microbial Biodiversity and Conditional Inference
Comparative Genomic Hybridization and Mixture Modeling
Mouse Mutagenesis, Randomization Testing and Modeling
Gene Expression Data Analysis: Hierarchical Modeling
Closing Remarks
STATISTICAL
METHODS FOR TOXICOLOGY
Applications of Biostatistics to Toxicology
Carcinogenicity Studies
Developmental Toxicity Studies
Reproductive Toxicity Studies
General Methods in Dose-Response Modeling
Models for Quantal Responses
Models for Continuous and Ordinal Responses
Adjustment for Litter Effects
Biologically Based Models for Carcinogenesis
Quantitative Risk Assessment
Cancer Risk Assessment
Non-Cancer Risk Assessment
SELECTED
TOPICS IN BIOMETRY
Inference
Hypothesis testing
Confidence intervals
Model selection
Design and analysis of experiments
Spatial analysis
Point patterns - complete enumeration
Point patterns - sparse sampling
Random fields
Multivariate methods
Inference from multivariate data
Classification
Ordination
Variation over time
Simulation
Statistical genetics
Qualitative variation
Quantitative variation
Bioinformatics
STATISTICAL
METHODOLOGY IN AGRICULTURE AND HORTICULTURE
Current methodology
Experimental Design
Analysis of Variance
Regression Analysis
Linear Regression
Non-linear Regression
Generalised Linear Models (GLMs)
Residual or Restricted Maximum Likelihood (REML)
Future developments
Analysis of Spatial Data
Precision Agriculture
On-farm Experimentation
STATISTICAL
METHODOLOGY IN FORESTRY
Forest Inventory
Modeling Characteristics of Individual Trees
Tree Height Models
Bole Taper Models
Bole Volume and Biomass Models
Models for Aboveground Biomass
Modeling Crowns and Roots
Quantitative Characteristics of Forest
Stands
Stand Density
Site Quality
Growth and Yield Models
Statistically Designed Experiments in Forestry
Greenhouse Experiments
In Situ Experiments
Spacing Trials
STATISTICAL
ECOLOGY AND ENVIRONMENTAL STATISTICS
Simple Stories but Challenging Concerns
Life and Death with Averages and Variability
Innovative Statistical Mind Sets
Comprehensive vs. Comprehensible
Space Age/Stone Age
Cycle of No Information, New Information, and Non Information
Mechanization/Computerization
Normality, Lognormality and Beyond Lognormality
Triad
Follow-up
Ecological Sampling and Statistical Inference
Encounter Sampling
Adaptive Sampling
Distance Sampling
Capture-Recapture Sampling
Biodiversity Measurement and Comparison
Biodiversity with Presence/Absence Data
Biodiversity with Relative Abundance Data
Am I a Specialist or a Generalist?
Resource Apportionment
Diversity as Average Species Rarity
Diversity Profiles
Environmental Data and Cost-Effective Acquisition
Observational Economy
Design and Analysis with Composite Samples
Ranked Set Samples
Sampling Heterogeneous Media
Combining Environmental Information
Landscape Ecology and Multi-Scale Assessment
Hierarchical Markov Transition Matrix Models
Spatial Dependence, Auto-Association, and Adjacency Matrix:
Hierarchical Classified Map Simulation Model
Fragmentation Profiles
Echelon Analysis for Multispectral Environmental Change
Detection
Introduction and Background
Echelons of Spatial Variation
Echelon Characteristics
Echelon Trees
Echelon Profiles
Echelon Research
Environmental Applications
Statistics as an Instrument to Deal with Environmental and
Ecological Crisis
Increasing Use of Statistical Language in the Regulation of
Environment and Natural Resources
Conflict Resolution and Sustainable Development
How Many of Them are Out There
Long-Term Ecological Research
Design, Analysis, and Nature of Our Observations
Information Age and Sustainable Development
Synthesis and Analysis with Integrated Satellite Data, Site
Data, and Survey Data
Future Areas of Concern and Challenge
Environmental Monitoring and Assessment
Environmental Sampling and Observational Economy
Geo-Spatial Statistics and Spatio-Temporal Statistics
Ecological Assessment and Multi-Scale Analysis
Environmental Data Synthesis and Statistical Meta-Analysis
Statistics in Environmental Toxicology and Epidemiology
Environmental Risk Assessment and Reduction
Computational Ecometrics and Environmetrics
Looking Ahead
POPULATION
GENETICS
Basic Principles
Genetic Variation
Hardy Weinberg Principle
Non-random Mating
Mutation
Migration and Population Structure
Genetic Drift
Selection
Explanations for Genetic Variation
Mutation-Selection Balance
Balancing Selection
Mutation-Drift Balance: The Neutral Theory
Mutation, Drift and Selection: The Nearly-neutral Theory
STATISTICAL
GENETICS
Basic Principles
Allele and Genotype Frequencies
Hypothesis Tests
Hardy-Weinberg Disequilibrium
Linkage Equilibrium
Segregation
Relatedness
Inbreeding
Kinship
Estimating Relatedness
Testing Relationships
Exclusion
Hypothesis Testing
Plant and Animal Breeding
Infinitesimal Model
Genetic Parameters
Estimating Genetic Parameters
Heritability
Repeatability
Maternal Effects and Dominance
Selection
Locus Mapping
Two-Point Linkage
Multi-Point Linkage
Ordering Loci
Genetic Maps
Physical Mapping
Quantitative Trait Locus Mapping
Segregation Analysis
Single Marker Analysis
Interval Mapping
Multi-Marker Methods
Allele Sharing Methods
Type I and II Errors
Significance Thresholds
Confidence Intervals for Location
Power
Designs to Increase Power
Fine Mapping
BIOINFORMATICS:
PAST, PRESENT AND FUTURE
Biological sequence analysis
Background
Scoring systems
Sequence alignment
Assessment of Significance
Overview of basic theory
Complications and Developments
Applications of hidden Markov models in bioinformatics
Evolutionary models and phylogenetic reconstruction
Gene expression analysis
Background
Issues concerning outcome measures
Experimental Design
Analysis of microarray data.
Statistical methods in proteomics
Systems biology
Federated data integration and bio-grids
Discussion
A VIEW OF MATHEMATICS
The Unity of Mathematics
The concept of Space
Projective Geometry
The Angel of Geometry and the Devil
of Algebra
Non-Euclidean Geometry
Symmetries
Line element and Riemannian geometry
Noncommutative Geometry
Grothendieck’s Motives
Topos theory
Fundamental Tools
Positivity
Cohomology
Calculus
Trace and Index Formulas
Abelian Categories
Symmetries
The input from Quantum Field Theory
The Standard Model
Renormalization
Symmetries
DIFFERENTIAL
AND INTEGRAL CALCULUS
Historical survey
Convergence of Sequences
Definition of Convergence
The Basic Property of Real Numbers
Real Line
Continuous Functions
Continuous Functions and Their Limits
Properties of Continuous Functions.
The Intermediate Value Theorem
Maxima and Minima of Continuous Functions
The graph of a function
Differential Calculus
Derivative
Linear Approximations
The Mean Value Theorem
Higher Order Derivatives
Higher Order Derivatives
Leibnitz Rule
Taylor’s
Formula
Integral Calculus
Motivation for a definite integral.
Riemann Integral
Fundamental Theorem of Calculus
Basic Properties of Integrals
Explicitly Integrable Functions
Integration of Rational Functions
Differential Calculus of Functions of Many Variables
Partial Derivatives
Total Differential
Derivatives of Composite Functions.
Taylor’s
Formula for Functions of Several Variables
Extrema of Functions of Several Variables
Multiple Integrals
Riemann Integrals
The Iterated Integral
Change of Variables in Multiple Integrals
CONTINUOUS
FUNCTIONS
Complex number
Holomorphic functions
Conditions for Holomorphic Functions
Examples of Holomorphic Functions
Zero and an Isolated Singularity
Analytic Functions and Analytic Continuation
Residue and residue calculus
Analytic functions of several complex variables
Brief history
MEASURE
AND PROBABILITY
Measure
Fields of Sets
Lebesgue Measure
Measures
Measurable Functions
Integral
Product Measures
Relations between two Measures
Signed Measures
Radon Measures
Haar Measures
Probability
Basic Definitions and Results
Sum of Independent Random Variables Infinite Divisible Distributions
Conditional Expectation and Martingale.
Conditional Expectation.
Martingale
Stationary Process Ergodic Theory
Discrete Parameter
Continuous Parameter
Stationary Gaussian Processes
Markov Processes
Heat Equation and Corresponding Markov Processes.
Markov Chains
Stochastic Dynamical System Itô Calculus
FUNCTIONAL
ANALYSIS AND FUNCTION SPACES
Function Spaces and Some Examples
Basic Concepts in Functional Analysis
Normed spaces and Banach Spaces
Hilbert Spaces
Bounded Linear operators
Applications of Bair’s Category Theorem
The Dual Space of a Banach Space
The Duality of Hilbert Spaces
Some Advanced Concept in Functional Analysis
Topological Vector Spaces
The Weak Topology and the Weak * Topology.
Locally Convex Spaces
Banach Algebras
Miscellaneous Function Spaces
Spaces of Continuous Functions
Spaces of Measurable Functions.
Spaces of Differentiable Fucntions.
Spaces of Holomorphic Fucntions
NUMERICAL
ANALYSIS AND COMPUTATION
Linear Systems of Equations
An Example
Condition Number
Norms and Vector Spaces
Application to Error Analysis
Stable Algorithms and Stable Problems
Application to Numerical Solution of Linear Systems
Iterative Methods
Eigenvalue Problems
The Singular Value Decomposition
Software and References
INFINITE
ANALYSIS
Ising Model and Monodromy Preserving Deformation
Two-dimensional Ising Model and Onsager’s Result
Transfer Matrix
Harmonic Oscillator
Clifford Algebra and Clifford Group
Free Fermions and Creation/Annihilation Operators
Magnetization and Scaled Two-point Correlation Function
Ising Field Theory and Monodromy Preserving Deformation
Soliton Equations and Vertex Operators
Bosonic Fock spaces and Vertex Operators.
Conformal Coinvariants and Vertex Operators
Sugawara Construction and the Level Vertex Operators
Conformal Blocks and Coinvariants
XXZ Model and Quantum Vertex Operators
Quantum Hamiltonian and Commuting Transfer Matrix
Level One Modules and Space of States
Method of Corner Transfer Matrix and Vacuum States
Quantum Vertex Operator and Diagonalization of Transfer Matrix
Form Factor Bootstrap Approach in Sine-Gordon Model
S-Matrix and Form Factor Axioms
Level Zero Action Revisited
FOURIER
ANALYSIS AND INTEGRAL TRANSFORMS
Fourier series
Definition
Convolution and Fourier Series
Pointwise Convergence of Fourier Series
Norm Convergence of Fourier Series.
Analytic Functions in the Unit Disk.
Orthogonal Function Expansions.
Orthogonal Systems
Examples of Orthogonal Systems
Wavelet expansion
Multiresolution Analysis.
Examples of Wavelets.
Fourier transforms
Fourier Transform in One Variable.
Definition and Inversion Formula
Examples
Convergence of Fourier Integrals
Poisson Summation Formula.
Fourier Transform and Analytic Functions
Hardy Space.
Real Method in Hardy Spaces.
Fourier Transform in Several Variables
Definition and Examples.
Some Fundamental Properties.
Fourier analysis on locally compact Abelian groups
Finite Fourier transforms
Finite Fourier Transform
Fast Fourier Transform.
Integral transforms
Mellin Transform.
Hankel Transform.
Laplace Transform.
Wavelet Transform.
OPERATOR
THEORY AND OPERATOR ALGEBRA
Hilbert space
Bounded linear operator
Compact Operator
Miscellaneous Operators.
Polar Decomposition and Spectral Decomposition
Spectrum
Operator theory
Dilation Theory
Generalization of Normality
Toeplitz Operator
Operator Inequalities
Operator algebra
von Neumann Algebra
Basic Theory
Modular Theory and Structure of Type III Factors
Classification of AFD Factors
Index Theory
Free Probability Theory
PROOF
THEORY AND CONSTRUCTIVE MATHEMATICS
Introduction
Constructivism
Proof Theory
Intuitionistic Logic, I
The BHK-interpretation
Natural Deduction; Formulas as Types
The Hilbert-type Systems Hi and Hc
Metamathematics of I and its Relation
to classical logic C
Semantics of Intuitionistic logic
I-completeness
Kripke semantics
Topological and Algebraic Semantics
Intuitionistic (Heyting) arithmetic,
HA
Realizability
Characterization of Realizability
Constructive Mathematics
Bishop's Constructive Mathematics
(BCM)
Constructive Recursive Mathematics
(CRM)
Intuitionism (INT)
Lawless Sequences and I-validity
Comparison of BCM, CRM and INT
Proof Theory of first-order logic
The Gentzen systems Gc and Gi
Cut Elimination
Natural Deduction and Normalization
The Tait-Calculus
Proof Theory of mathematical theories
The language
Order types
Truth-Complexity
The Proof-theoretic Ordinal of a
Theory
The Method
COMPUTABILITY
AND COMPLEXITY
Recursive and Recursively Enumerable
Sets
m-Complete Sets, Creative Sets, and
Simple Sets
Algorithmic View of Gödel
Incompleteness
Unsolvable Problems
The Word Problem for Semigroups
The Word Problem for Groups
Hilbert's 10th Problem
Applications and Extensions of MRDP.
Classifying Unsolvable Problems
Degrees of Unsolvability.
The Arithmetical Hierarchy.
R.e. Degrees.
Complexity
Abstract Complexity Theory
Polynomial-time Computability.
SET THEORY
Some elementary tools
Ordinals
The Wellordering Theorem
The Cumulative Hierarchy; Proper
Classes
Cardinals
Cofinality, Inaccessibility, and
König’s Theorem
Club and Stationary Sets
Trees
Transitive Models and the Levy
Hierarchy
Large Cardinals and the
Consistency-Strength Hierarchy
Constructible sets
Gödel’s Work on L
Suslin Trees
Canonical Inner Models Larger Than L
Forcing
The Basics of Forcing
CH via Adding Cohen Reals
Easton’s theorem
The Singular Cardinals Problem
A model where the Axiom of Choice
fails
Cardinal Collapsing and Solovay’s
Model
Suslin’s Hypothesis and Martin’s
Axiom
Martin’s Maximum
Descriptive set theory
Gödel’s Program
Classical Descriptive Set Theory
Determinacy
Large Cardinals and Determinacy
Generic Absoluteness and CH
Other topics
LOGIC
AND COMPUTER SCIENCE
Complexity Classes and the P = NP problem
Propositional Logic and Complexity Classes
The Complexity of First-Order Logic and Richer Logics
The Complexity of First-Order Logic
The Complexity of Existential Second-Order Logic
Fagin’s Theorem and Descriptive Complexity
Least Fixed-Point Logic and Polynomial-Time
Partial Fixed-Point Logic and Polynomial Space
Finite Model Theory
Classical Model Theory in the Finite
Ehrenfeucht-Fraїssé Games and First-Order Logic
Pebble Games and Fixed-Point Logics
0-1 Laws in Finite Model Theory
Logic and Databases
Database Query Languages
Constraints in Databases
A BASIC
EXAMPLE OF NONLINEAR EQUATIONS: THE NAVIER-STOKES EQUATIONS
Scaling, Hierarchies and formal
Derivations
Stabilities and instabilities of
macroscopic solutions
Turbulence, weak convergence and
Wigner measures
Some special properties of the
dimension 2
CALCULUS OF VARIATIONS, PARTIAL DIFFERENTIAL EQUATIONS, AND
GEOMETRY
Generalities
Parameterization
of Geometrical Problems
An
example: minimal surfaces
Graphs
Conformal
Parameterization
Bubbles
Phase
transitions and interfaces
Ginzburg-Landau
Functionals
The
Scalar Case
The
Case N = 1
The
Higher Dimensional Case
LINEAR
DIFFERENTIAL EQUATIONS
Linearity and Continuity
Continuity
Linearity
Perturbation Theory and Linearity
Axiomatically Linear Equations
Fields: Maxwell Equations
Densities on Phase Space in Classical Physics
Quantum Mechanics and Schrödinger Equation
Examples
Ordinary Differential Equations
The Laplace Equation
The Wave Equation
The Heat Equation and Schrödinger Equation
Equations of Complex Analysis
The Cauchy-Riemann Equation
The Hans Lewy Equation
The Mizohata Equation
Methods
Well posed Problems
Initial Value Problem, Cauchy-Kowalewsky Theorem
Other Boundary Conditions
Distributions
Distributions
Weak Solutions
Elementary Solutions
Fourier Analysis
Fourier Transform
Equations with Constant Coefficients
Asymptotic Analysis, Microanalysis
DIFFERENTIAL
EQUATIONS AND SYMPLECTIC GEOMETRY
Lagrangian Mechanics
Hamiltonian Systems and Symplectic Geometry
Nonlinear First order Partial Differential Equations
Oscillatory Integrals
Fourier Integral Operators
FROM THE
ATOMIC HYPOTHESIS TO MICROLOCAL ANALYSIS
The Schrödinger Equation And Semiclassical Analysis
Schrödinger equation
WKB Asymptotics
Newtonian and Hamiltonian Mechanics
Caustic
Feynman Integral
Stationary Phase.
ħ Fourier Integral Operator
High Frequency Asymptotics and Microlocal Analysis
Differential Operators
Microsupport
Pseudo-differential Operators
Symbolic calculus, principal Symbol
Symplectic geometry
Fourier Integral Distributions, Fourier Integral Operators
Models, propagation of singularities
Eigenvalues of elliptic operators
Miscellaneous
Microlocal regularity
Weyl Calculus
Analytic Pseudo-differential Operators and Analytic Wave
Front set
Gevrey Classes
Uncertainty principle
Carleman estimates
DISCRETE
MATHEMATICS
Bipartite Matchings
Matchings and Covers
Dulmage-Mendelsohn Decomposition
Discrete Convex Functions
Definitions
Convexity
Matroids and Submodular Functions
Graphs and Networks
Polyhedra and Conjugacy
Optimization and Duality
Algorithms and Computational Complexity
GRAPH
THEORY
Degrees and Distances
Connectivity
Operations
Trees
Factor Theory
Eulerian Circuits and Hamiltonian Cycles
Coloring
Planar Graph
COMBINATORICS
Selected Topics in Combinatorics
Hypergraphs
Graphs and Hypergraphs
Transversal, Matching, and Covering
Pigeonhole Principle and Ramsey Theory
Block Design
Codes and Hypergraphs
Partially Ordered Set
Chain and Antichain of a Poset
Möbius Inversion Formula
Matroids
Axiom and Examples of Matroids
Matroids and Combinatorial Optimization
Combinatorial Geometry
Arrangements and Convex Polytopes
Semispaces and K-Sets
Helly Type Theorems
Theory of Partitions
Partitions and Young Diagrams
Number of Partitions and Partition Functions
Young Tableaux
COMPUTATIONAL
COMPLEXITY
Machine Models and Complexity Measures
Complexity Classes
Fundamental Results and Questions
Inclusion by Simulation
Important Open Questions
Comparison of Hardness by Reducibility
Selected Topics
OPTIMIZATION
Integer Programming
Definitions
Total Unimodularity
Integer Polyhedron
Enumerative Algorithms for Integer Programming
Branch-and-Bound Algorithms
Branch-and-Cut Algorithms
Solvable Cases of Integer Programming
Approximation Algorithms
Approximation Algorithms with Performance Guarantees
Polynomial Time Approximation Schemes
Metaheuristics
Local Search
Fundamentals of Metaheuristic Algorithms
MATHEMATICS
THROUGH MILLENIA
The dawn of mathematics
Egyptian Mathematics
Mesopotamian Mathematics
Mayan Mathematics
The Greek heritage in mathematics
Geometry
Number Theory
The golden period of the Hindus and the Arabs in mathematics
Hindu Mathematics
Islamic Mathematics
Mathematics in Europe
in the Middle Ages
Mathematics in China
Ancient Chinese Mathematics
The
“Nine Chapters on the Mathematical Art”
In
the Shadows of the Great Masters
A
Golden Century for Mathematics in China
European mathematics in the Renaissance
The Solution of Cubic Equations
Mathematics inspired by Applications
Mathematics and the scientific revolution
Analytic Geometry
Calculus gets off the Ground
Other Mathematical Discoveries from
the Seventeenth Century
The tools of calculus are developed and consolidated
The Birth of Mathematical Analysis
Further Remarks on Mathematics in the Eighteenth Century
Abstract mathematical structures emerges
New Algebraic Structures
Ground-breaking New Discoveries in Geometry
Rigor in Analysis
Further Developments in the Nineteenth Century
Mathematics in the twentieth century
Problems in the Foundations of Set Theory
Tendencies in Twentieth Century Mathematics
Highlights from Twentieth Century Mathematics
Mathematics forever
MATHEMATICS
ALIVE AND IN ACTION
Pure Mathematics at the Turn of
Millennium
Stochastic Modeling
Mathematics in the Biological
Sciences
ALGEBRA
Equivalence
Relations
Grobner
Bases
Homological
Algebra
The
case of modules
Categories
and functors
Abelian
categories
Derived
functors
Derived
categories
MATRICES,
VECTORS, DETERMINANTS,
AND LINEAR ALGEBRA
Matrices, Vectors and their Basic Operations
Matrices
Vectors
Addition and Scalar Multiplication of Matrices
Multiplication of Matrices
Determinants
Square Matrices
Determinants
Cofactors and the Inverse Matrix
Systems of Linear Equations
Linear Equations
Cramer’s Rule
Eigenvalues of a Complex Square Matrix
Jordan Canonical Form
Symmetric Matrices and Quadratic Forms
Real Symmetric Matrices and Orthogonal Matrices
Hermitian Symmetric Matrices and Unitary Matrices
Vector Spaces and Linear Algebra
Vector spaces
Subspaces
Direct Sum of Vector Spaces
Linear Maps
Change of Bases
Properties of Linear Maps
A System of Linear Equations Revisited
Quotient Vector Spaces
Dual Spaces
Tensor Product of Vector Spaces
Symmetric Product of a Vector Space
Exterior Product of a Vector Space
GROUPS
AND APPLICATIONS
Groups
Commutative Groups
Examples
Subgroups
Homomorphism
Quotient Groups
Homomorphism and Isomorphism Theorems
Cyclic Groups
Direct Products
Finitely Generated Abelian Groups
Group Actions and Symmetry
Solvable Groups
Representations of Finite Groups
RINGS
AND MODULES
Definition of Rings
Basic Properties and Examples
Noetherian Rings
Completion
Localization and Local Rings
Modules
Integral Extensions
FIELDS
AND ALGEBRAIC EQUATIONS
Basic Properties and Examples of Fields
Algebraic Equations
Algebraic Extensions
Separability
Galois Theory
Finite Fields
Cyclotomic Extensions
Kummer Extensions
Solvability
Ruler and Compass Constructions
NUMBER
THEORY AND APPLICATIONS
The Additive Structure of Natural Numbers
The Well-Ordered Structure and the Principle of Mathematical
Induction
Triangular Numbers and Square Numbers
The Multiplicative Structure of Natural Numbers
Prime Numbers
Infinitude of Prime Numbers and Euler Product
Euclidean Algorithm and the Greatest Common Divisors
Dirichlet’s Prime Number Theorem on Arithmetic Progressions
The Ring of Integers
The Ring of Integers
Linear Equations in Integers and Divisibility
Multiplicative Structure of the Integral Solutions of Pell’s
Equations
Multiplicative Structure on Binary Quadratic Equations
Congruence
Congruence Relation and Residue Rings
Euler’s Phi Function
Chinese Remainder Theorem
Linear Congruence Equations
Quadratic Congruence Equations and Quadratic Residues
The Reciprocity Law of Quadratic Residues
The Multiplicative Group of a Finite Field and Primitive
Roots modulo
Caesar’s Cipher in Cryptography and Congruence
Public Key Cryptology
Analytic Methods in Number Theory
Counting Prime Numbers
Densities of some Sets of Prime Numbers
The Riemann Zeta Function and the Riemann Hypothesis
Dirichlet Characters and Dirichlet’s L-functions
Arithmetic of Quadratic Fields
Quadratic Fields and the Rings of Integers
Ideals and the Fundamental Theorem of Arithmetic in a
Quadratic Field
Units of Quadratic Fields and Pell’s Equations
Ideal Class Groups and Class Numbers
Cyclotomic Fields
Algebraic Bases of Cyclotomic Fields
Arithmetic Bases of Cyclotomic Fields
Kronecker-Weber Theorem on Abelian Polynomials over Q
Comments on Kronecker’s Dream in his Youth and Class Field
Theory
Kronecker’s Dream in his Youth
The Ideal Class Group of an imaginary Quadratic Field and Automorphism
Classes of Elliptic Function Fields with Complex Multiplication
ALGEBRAIC
GEOMETRY AND APPLICATIONS
Affine Algebraic Varieties
Projective Algebraic Varieties
Sheaves and General Algebraic Varieties
Properties of Algebraic Varieties
Divisors
Algebraic Geometry over Algebraically Closed Fields
Schemes
Applications
BASIC
NOTIONS OF GEOMETRY AND EUCLIDEAN GEOMETRY
Basic Notions
Metric Space
Transformation Group
Lie Group
Euclidean Space
Euclidean Vector Space
Euclidean Space
Equations of Plane and Sphere
Triangle and Plane
Trigonometry
Euclidean Group
Translation and Rotation
Categorization of Isometries
Conic
Sections
Binary Quadratic Equation
Focus, Eccentricity and Directrix
Confocal Conic Sections
Inverse Square Central Force
Discrete Groups of Isometries
Finite
Subgroups of Isom (En) and Polyhedra
Space Group
AFFINE GEOMETRY, PROJECTIVE GEOMETRY, AND
NON-EUCLIDEAN GEOMETRY
Affine Geometry
Affine Space
Affine Lines
Affine transformations
Affine Collinearity
Conic Sections
Projective Geometry
Perspective
Projective Plane
Projective Transformations
Projective Collinearity
Conics
Geometries and Groups
Transformation Groups
Erlangen Program
Non-euclidean Geometry
Elliptic Geometry
Hyperbolic Geometry
Poincaré Model
Riemannian Geometry
DIFFERENTIAL GEOMETRY
Curves
in Euclidean Plane and Euclidean Space
Surfaces
in Euclidean Space
Differentiable
Manifolds
Tensor
Fields and Differential Forms
Riemannian Manifolds
Geometric
Structures on Manifolds
Variational
Methods and PDE
TOPOLOGY
Metric spaces and the convergence of
sequences
Connectedness and homotopy
Homotopy groups
Homotopy exact sequence
Simplicial complexes
Poincaré duality theorem
Morse theory
Intersection numbers and
linking numbers
Knot theory
COMPLEX ANALYTIC GEOMETRY
Analytic Functions of One
Complex Variable
Analytic Functions of
Several Complex Variables
Germs
of Holomorphic Functions
Complex
Manifolds and Analytic Varieties
Germs of
Varieties
Vector
bundles
Vector Fields
and Differential Forms
Chern Classes
of Complex Vector Bundles
Divisors
Complete
Intersections and Local Complete Intersections
Grothendieck
Residues
Residues at
an Isolated Zero
Examples
Sheaves and
Cohomology
de Rham and
Dolbeault Theorems
Poincare and
Kodaira-Serre dualities
Riemann-Roch
theorem
PROBABILITY
AND STATISTICS
Origin and History
Probability
Continuous Probability Distributions
Discrete Probability Distributions
Mixed Probability Distributions
Mixture Distributions
Descriptive Statistics
Empirical Distributions
Empirical Distribution Function
Multivariate Data
Regression
Indices
Time Series
Stochastic Models
Stochastic Vectors
Discrete Stochastic Vectors
Continuous Stochastic Vectors
Correlation and Independence
of Stochastic Quantities
Conditional Distributions
Sequences of Stochastic Quantities
The Law of Large Numbers
Central Limit Theorem
Stochastic Processes
From Stochastic Models to Statistical Inference
Classical Statistical Inference
Point Estimates for Parameters and Other Characteristic
Values of Probability Distributions
The Fundamental Theorem of Statistics
Further Methods in Classical Statistical Inference
Bayesian Statistical Inference
Information and Decision
Types of Uncertainty and Data Quality
Outlook
PROBABILITY
THEORY
Early Concepts of Probability
Ancient Times
Renaissance
The Problem of Points
Other Problems from Gambling
The First Steps Towards a Theory of Probability
New Contributions
Still More Gambling Problems
Theoretical Contributions
Earliest Applications
Insurance
Demography
The Theory of Errors
Social Sciences
Geometry
The Axiomatization of Probability Theory
Doubts about Probability Theory
The Need for Axiomatization
The Russian School
Probability and Statistics in Life Support Systems
Spatial Statistics
Time Series
Image Analysis
Simulation
Water
Rainfall
Water Storage
Floods
Environmental Issues
Seawater
Energy
Stochastic Hydrology
Wind
Reliability Theory
Environment
Wildlife Sampling
Abundance
Forestry
System Analysis
Exposure Data
Environment and Health
Food
Agriculture
Design and Analysis of Experiments
Weather Modification
MATHEMATICAL
FOUNDATIONS AND INTERPRETATIONS OF PROBABILITY
Finite Probability Spaces
Conditional Probability
Discrete Probability Spaces
Kolmogorov Triplets
RANDOM VARIABLES
AND THEIR DISTRIBUTION
The distribution function of a random variable.
Classification of random variables.
Some special discrete probability distributions.
Some special continuous probability distributions.
Location characteristics of a real-valued random variable.
Dispersion characteristics of a real-valued random variable.
Joint distribution functions.
Independence
of random variables.
Random variables in statistics.
The moments and the characteristic function of a random variable.
Conditional probability distributions,
Probability distributions presented as Borel measures.
LIMIT
THEOREMS OF PROBABILITY THEORY
Laws of Large Numbers
Weak Laws of Large Numbers
Strong Laws of Large Numbers
An Application, the Glivenko-Cantelli Theorem
The Law of Iterated Logarithm
Central Limit Theorem
Moivre-Laplace Central Limit Theorem
Lindeberg-Lévy and Lindeberg-Feller Central Limit Theorems
Error Bounds and Asymptotic Expansions in Central Limit Theorem
Multivariate Central Limit Theorem
Limit Theorems of Large Deviations
Classical Summation Theory
Local Limit Theorems
Approximation by the Density of the Normal
Law
Approximation by Poisson Probabilities
Limit Theorems for Extreme Values
ALTERNATIVE
PROBABILISTIC SYSTEMS
Early developments
Capacities
The 1970s and 80s
From the 1990s on
STOCHASTIC
PROCESSES AND RANDOM FIELDS
Important Concepts and Methods
Dependency
Correlation Theory
Convergence
Types of Stochastic Process
Gaussian Process:
Independent Increment Processes and Martingales
Markov Processes and Stochastic Differential Equations
Stationary Processes
Self-similar (Fractal) Process
Point Process
Empirical Processes
Processes in Random Environment
Random Fields
Basic Facts
Gaussian Random Fields
Gibbs Random Fields
CONSTRUCTION
OF RANDOM FUNCTIONS AND PATH PROPERTIES
Examples
Poisson Process
Markov Chain
Martingale
Brownian Motion
Definition of the Stochastic Process
Poisson Process
The Distribution
A Construction of the Poisson Process
Generalizations
Brownian Motion
Random Walk
Distributions
Path Properties
Local Time
Fourier Series Representation
MARKOV
PROCESSES
Discrete Markov Chains
Classification of the State of a Markov Chain
Recurrence
Transient States
Continuous Time Markov Chains
Examples of Markov Chains
Simple Random Walk
Birth and Death Processes
Galton-Watson Processes
Stopping Times and the Strong Markov Property
Path Properties and Continuity
Transition Operators
Definition and Basic Properties
The Infinitesimal Operator
The Resolvent
Path Properties and the Infinitesimal Operator
Examples of Markov Processes
STOCHASTIC
CALCULUS
Stochastic Integral
Ito Formula
Tanaka Formula
Differential of the Brownian Motion
STOCHASTIC
DIFFERENTIAL EQUATIONS
Existence and Unicity
A Stochastic Chain Rule
A Property of the Solution of a Stochastic Differential Equation
STATIONARY
PROCESSES
Spaces and operators related to stationary processes
Spaces of square-integrable functions
Shift operators
The correlation function
Spectral representations
Prediction
Basic facts about prediction
Singularity and regularity
The best linear prediction for weakly stationary sequences
ERGODIC
PROPERTIES OF STATIONARY, MARKOV, AND REGENERATIVE PROCESSES
Ergodic theory for stationary processes
The Mean Square Ergodic Theorem
The Strong Ergodic Theorem
Ergodic properties of Markov processes
Irreducible Markov Chains
Regenerative processes
Definition
Examples of Regenerative Processes
Ergodic Theorems for Regenerative Processes
Applications of ergodic theorems
Statistical Inference for Markov Chains
The Range of a Random Walk
Entropy
HOMOGENEOUS
RANDOM FIELDS AND THEIR EVALUATION
Homogenous random fields and their spectral representation
Meteorological applications.
Approximation and positive definiteness of correlation functions.
Perturbation theory for improvement of positive definiteness
Computational algorithm
Errors of Sample Correlation Coefficients
Approximation of Correlations
Corrections of Matrix Coefficients
Deviations to Scale Multipliers along Horizontal Direction
Separation of the CF for “true” Meteorological Parameters and Observation
Errors
Results
PROBABILISTIC
MODELS AND METHODS
A Simple Probabilistic Model
Risk Management
Independence
Stochastic Processes
Processes with Independent Increments
Poisson Processes
Brownian Motion
Markov Processes
Markov Chains
Diffusion Processes
Stochastic Differential Equation
Martingale
STATISTICAL
SIMULATION AND NUMERICAL PROCEDURES
Random Number Generation
Linear Congruential Generators
Other Sources of Uniform Random Numbers
Non Uniform Random Variate Generation
General Methods
The Inverse Method
The Composition (Mixture) Method
The Acceptance-Rejection Method
Other Methods
Ratio-of-uniform Method
Simulation of some Particular Distributions
Simulation of Some Multivariate Distributions
The Use of Simulation in Statistics
Estimation of Parameters via Simulation
Use of Simulation in Hypotheses Testing
The Bootstrap Technique
Use of Simulation in Numerical Calculations
Generalities on the Monte Carlo Method
Evaluating an Integral
Crude Monte Carlo
Variance Reduction Techniques
Importance Sampling
Antithetic Variates
Control Variates
Solving Operatorial Equations
Solving Systems of Linear Equations
Solving Integral Equations
Monte Carlo Optimization
Markov Chain Monte Carlo
INSURANCE
MATHEMATICS
Non-life Insurance
Premium Principles
Credibility Theory
Bonus Systems
Collective Risk Theory
Reserves
Aggregate Claims Distributions
The Stop Loss Transform
Life Insurance
The Single Life Model
Spouse Pension
Second Order Basis
MATHEMATICAL
MODELS IN FINANCE
A Tutorial on Mathematical Finance without Formula
The Pricing of Financial Derivatives by Mathematical Means
The Approach by Black, Scholes, and Merton
Pricing by Change of Measure
Interest Rate Models
Financial Time Series Models
RELIABILITY
AND MAINTAINABILITY
Some Reliability Concepts
Commonly Used Reliability Measures
Some Lifetime Distribution Models
System Reliability Analysis
Coherent Systems
Methods for System Reliability Calculation
System Reliability Bounds and Approximations
Availability and Maintainability
Reliability Data Analysis
Graphical Techniques for the Analysis of Failure Data
Statistical Estimation for the Exponential Distribution
Repairable System Reliability Analysis
Towards the 21st Century
INVENTORIES,
WATER STORAGE AND QUEUES
Inventory Models
The (Q, r) Model and the EOQ Formula
The Newsvendor Problem
The (s, S) Inventory Model
A Periodic Review Base Stock Inventory Model
Models for Water Storage
Moran's Model for the Finite Dam
A Continuous Time Model for the Dam
The Queueing System GI /G /S
The Queueing System M/M/S; Erlang's Loss Formulas
The M/G/1 System; The Pollaczek-Khintchine Formula
Queueing Networks
Closed Networks
Open Networks: The Product Formula
Jackson
Networks
A Closed Network Model for Flexible Manufacturing Systems
An Open Network Model for Flexible Manufacturing Systems
INFORMATION
THEORY AND COMMUNICATION
Information source
Source coding
Uniquely Decodable Codes
Entropy
Source Coding with Small Decoding Error Probability
Universal Codes
Facsimile Coding
Electronic Correspondence
Measures of information
Transmission channel
Classification of Channels
The Noisy Channel Coding Problem
Error Detecting and Correcting Codes
The practice of classical telecommunication
Analog-to-Digital Conversion
Quantization
Modulation
Multiplexing
Multiple Access
Mobile communication
Cryptology
Classical Cryptography
Simple Substitution
Transposition
Polyalphabetic Methods
One Time Pad
DES: Data Encryption Standard. AES: Advanced Encryption Standard
Vocoders
Public Key Cryptography
Public Key Crypto-algorithms
Proving Integrity: Hashing
Cryptographic Protocols
Cryptanalysis
FOUNDATIONS
OF STATISTICS
Statistical data
Uncertainty
Probability and philosophical foundations
Classical Probabilities
Geometric Probabilities
Probabilities as Idealized Relative Frequencies
Probability Spaces
General Axiomatic Probability
Subjective Probabilities
Transition Probabilities
Fuzzy Probability Densities
Philosophical Questions
Statistical populations and samples
Statistics and Sample Moments
Sample Mean
Sampling from the normal distribution
The Chi-square Distribution
Gosset's t-distribution
F-distribution
Confidence statements and statistical tests
A-priori information
A-priori Knowledge
A-priori Distributions
Sensitivity and robustness
Model Robustness
Data Robustness
Bayesian Robustness
Information and decisions
PRELIMINARY
DATA ANALYSIS
Univariate Data Sets
Graphical Displays
Frequencies
Cumulative Frequencies
Measures of Location
Depths
Measures of Spread
Outliers
Seven-point summaries
Box-Plots
Data Transformation
Probability Plots
P-P-Plots
Q-Q-Plots
Bivariate Data Sets
Graphical Displays
Numerical Characteristics
Multivariate Data Sets
Data Matrix and Summary Statistics
Data Transformations
Graphical Displays
STATISTICAL
INFERENCE
Parametric and Nonparametric Inference
Parametric Inference
Nonparametric Inference
Semiparametric Inference
Sufficiency and Information
The Likelihood Function
Classical Statistical Inference
Point Estimators
Confidence Regions
Testing Statistical Hypotheses
Bayesian Inference
Sufficiency
Conjugate Families of Distributions
Data Quality and Statistical Inference
Statistical Inference and Decisions
Selection of Stochastic Models
Parameter Estimation as a Decision Process
Statistical Tests as Decisions
General Decisions Subject to Loss
Decisions in the Case of Fuzzy Data
STATISTICAL
PARAMETER ESTIMATION
Fundamental Concepts
Optimality Properties
Methods of Parameter Estimation
Classical Confidence Regions
STATISTICAL
TESTING OF HYPOTHESES
Statistical Hypothesis
Statistical Test
Errors of the First and the Second Kind
The Power Function, the Power and the Significance Level of the Test
Non-randomized Test
Randomized Test
Unbiased Test
Uniformly Most Powerful Test
Neyman-Pearson Lemma
Consistency
Neyman Structure
Likelihood Ratio Test for Composite Hypotheses
ROBUST
STATISTICS
Motivation and Introduction
The Meaning of Robust Statistics
Outliers
Aims of Robust Statistics
History
Basic Concepts
The Breakdown Value
Positive-Breakdown Regression
Multivariate Location and Scatter
Regression Diagnostics
Other Robust Methods
The Maxbias Curve
Perspective and Future Directions
BAYESIAN
STATISTICS
Foundations
Probability as a Measure of Conditional Uncertainty
Statistical Inference and Decision Theory
Exchangeability and Representation Theorem
The Bayesian Paradigm
The Learning Process
Predictive Distributions
Asymptotic Behavior
Inference Summaries
Estimation
Hypothesis Testing
Reference Analysis
Reference Distributions
Frequentist Properties
A Simplified Case Study
Objective Bayesian Analysis
Sensitivity analysis
Discussion and Further Issues
Coherence
Objectivity
Applicability
STATISTICAL
INFERENCE WITH IMPRECISE DATA
Imprecise data
Imprecise numbers and characterizing functions
Special Imprecise Numbers
Convex Hull
of a Non-convex Pseudo-characterizing Function
Construction of characterizing functions
Multivariate data, imprecise vectors, and combination of imprecise samples
Imprecise multivariate data and imprecise vectors
Combination of Imprecise Observations
Multivariate Data
Functions and imprecision
Functions of Imprecise Variables
Functions with Imprecise Values
Generalized inference procedures for imprecise samples
Classical statistical inference for imprecise data
Generalized Point Estimators for Parameters
Generalized Confidence Regions for Parameters
Statistical Tests based on Imprecise Data
Bayesian inference for imprecise data
Bayes’ Theorem for Imprecise Data
Bayes Estimators
Generalized HPD-regions
Fuzzy Predictive Distributions
Fuzzy a priori Distributions
APPLIED
STATISTICS
Foundations Probability theory is the root upon which statistical methods
are built
Exploratory Data Analysis Let the data show its relevant information
Models Transform relevant information into concise and usable forms
Statistical Inference From the sampled data to the relevant population and
actions
Design of Experiments Tools for efficient collection of most relevant data
The Future of Applied Statistics Modern technology opens new phase
CORRELATION
ANALYSIS
Correlation Between Two Random Variables (Simple Correlation)
Partial Correlation
Multiple Correlation
Canonical Correlation
REGRESSION
ANALYSIS
Simple Regression
Multiple Regression
Gauß-Markov Theorem
Unequal Variances
Quasi-linear Regression
Multivariate Regression
ANALYSIS
OF VARIANCE AND ANALYSIS OF COVARIANCE
Analysis of Variance (ANOVA)
Fixed Models
One-Way Classification
Complete Higher-Way Classification
Random (Effects) Models
One-way Classification (with Unequal Numbers of Observations)
Higher-way Classification
Mixed (Effects) Models
Analysis of Covariance
(1,1)-Classification of the analysis of covariance
Mathematical model
SAMPLE
METHOD AND QUALITY CONTROL
Concepts of Quality
Item Quality Indicators
Lot Quality Indicators
Process Variation
Process Quality Indicators
Lots of Segments of Processes
Definition of Quality Levels
Inspection and Prevention in Quality Control
Inspection and Prevention in Product Quality Control
Inspection and Prevention in Process Quality Control
Decision Making and its Statistical Tools in Quality Control
The Decision Problem of Reactive Control
Decision Making by Sampling
Random Sampling and Statistical Hypothesis Testing
Interpretations of Acceptance and Rejection in Industrial Practice
Acceptance and Rejection in SLI
Acceptance and Rejection in SPI
Design Components of Statistical Decision Techniques in Quality Control
Economic Evaluation of SQC Decision Procedures
The Cost of Sampling
The Cost of Intervention
The Profit from Continuing Business
Economic Evaluation Indices
The Expected Value Approach
The Worst Case Approach
The Restricted Worst Case Approach
Statistical Evaluation of SQC Decision Procedures
Methods of Designing SQC Decision Procedures
Economic Design of SQC Decision Procedures
Statistical Design of SQC Decision Procedures
Mixed Economic-Statistical Design of SQC Decision Procedures
Acceptance of Designs by Industry
Statistical Lot Inspection Schemes
Classification of Sampling Plans by Decision Algorithm
Single Sampling Plans
Double Sampling Plans
Multiple Sampling Plans
Sequential Sampling Plans
Comparison of Decision Algorithms
Classification of Sampling Plans by Quality Model
Attributes Sampling for Lot Proportion
Nonconforming
Attributes Sampling for Lot Average Number
of Defects
Variables Sampling for Lot Mean
Variables Sampling for Lot Proportion
Nonconforming
Special Acceptance Sampling Schemes
Dodge-Romig Plans
Prescription of Two Points of the OC Function
Military Standard
The α -Optimal Sampling Scheme
Statistical Process Inspection Schemes
The Time to Signal and the Run Length
Classification of Control Charts by Sampling Algorithm
Classification of Control Charts by Decision Algorithm
Shewhart Charts
Extended Shewhart Charts
Sequential Charts
Comparison of Decision Algorithms
Classification of Control Charts by Quality Model
Attributes Control Chart for Process Proportion Nonconforming
Attributes Control Charts for Process Average Number of Defects
Variables Control Charts for Process Mean
Variables Control Chart for Process Variance
Special Control Charts
Shewhart Charts with Prescribed Points of the OC
Function
Control Charts in ISO Standards
Economic Design: Duncan’s
Approach
Economic Design: Von Collani’s Approach
Recent Trends and Outlook
TIME
SERIES ANALYSIS
Finite-difference equations
Linear FDE
Boundary Value Problems
Non-linear FDE, that approximate, e.g., Lotka Volterra eq., Lorenz System
Lyapunov Exponents and their Evaluation
Interpolation, approximation, and checking
Polynomial and Trigonometric Interpolation
The Lebesgue Constant
Spline Interpolation
Wavelet Methods
Checking of Time Series
Chebyshev (uniform) Approximation by Polynomials and Rational Functions
Correlations
Correlation Coefficient and Regression Method
Empirical Orthogonal Functions
Operators on Random Processes
STATISTICAL
EXPERIMENTS AND OPTIMAL DESIGN
Linear models
How to measure the information obtained in an experiment modeled linearly
Information Matrices
Information Matrices for Parameter Subsets
Geometrical Presentation of the Information Contained in the Experiment
The Ellipsoid for Parameter Subsets
Optimality Criteria
The design of experiments with uncorrelated observations and non-restricted
replications
The Set-up
The Equivalence Theorem a Check of Optimality
The Elfving Set
A Quick Numerical Check of Information Properties of a Given Design
Numerical Computation of Φ-optimum Designs
Experiments with Constraints on Designs
Optimal design in linear models under a given covariance structure
Design of nonlinear regression experiments
The Model
The Optimal Design based on the Information Matrix
Other Measures of Information in Nonlinear Models
Perspectives and further developments
MATHEMATICAL
MODELS OF LIFE SUPPORT SYSTEMS
Basic Principles of Mathematical Modeling
Types of Modeling: Mathematical Modeling
Stages of Mathematical Modeling.
Requirements for Mathematical Models
Plurality and Unity of Models
Adequacy Requirement
Requirement of Sufficient Simplicity
Other Requirements of Mathematical Models
Determining Components and Relations
Determining Relations
Finite Equations
Equations for Functions of One Variable
Equations for Functions of Several Variables
Extremum Problems with Finite Degrees of Freedom: Mathematical Programming
Extremum Problems With a Sought-For Function
Classification of Mathematical Models
Structural and Functional Models
Discrete and Continuous Models
Linear and Nonlinear Models
Deterministic and Probabilistic Models; Other Types of Model
Classification of Mathematical Models of Earth’s Life Support Systems
General Methods of Analysis; Simplification and Specification of Models
Dimension Analysis
Similarity of Objects
Methods of Simplifying and Specifying Models
Mathematical Models in Water Sciences
Some classes of mathematical models in water sciences
Mathematical Models of Hydrodynamics
Incompressible Non-Viscous Fluids
Viscous Incompressible Fluid
Mathematical Models of Flows in Rivers, Lakes, and Coastal Waters
Rivers
Lakes
Coastal Waters and Estuaries
Mathematical Models of Circulation in Oceans and Seas
Mathematical Models for Water Resources Management
Mathematical Models of Atmosphere and Climate
General Information on Atmosphere and Climate: Classes of Mathematical
Model
Basic Equations of Atmospheric Processes of Hydrothermodynamics
Derivation of Simplified Models and Weather Forecast Models
Models of Solar Radiation
The Use of Climate Models for Estimating Anthropogenic Impact
Mathematical Models in Energy Sciences
Classes of Mathematical Model in Energy Sciences
Electrodynamic Models
Maxwell’s Equations
Main Elements of Electrodynamics
Kirchhoff’s Equations
Mathematical Models of Electric Systems and Networks
Mathematical Models for Nuclear Reactors
Nuclear Reactors
The Four Co-Multiplier Formula: A Simple Mathematical Model of a
Nuclear Reactor
Mathematical Modeling of the Critical Size of a Nuclear Reactor
Mathematical Model of the Nonstationary Process of Diffusion
Mathematical Models of Electric Machines
Mathematical Models of Plasma
Mathematical Models in Food and Agricultural Sciences
Classes of Mathematical Models
Important Factors in Modern Food and Agricultural Modeling
EconomicMathematical Models in Agriculture
Optimizing the Structure of Herds in Animal Husbandry
Optimizing the Balance Between Branches of the Industry
Network Models
Mathematical Models in Biological, Health, and Medical Sciences
Classes of Mathematical Model
Population Growth Models
The Model of Exponential Growth (Malthus)
The Model of the Dynamics of Population Size Subject to
Competitive Coexistence (Verhulst)
The “Prey Predator” Model (Volterra)
Pharmacological Kinetic Models
Mathematical Models in Immunology
Models of Epidemic Spread
Deterministic Model
Stochastic Model
Mathematical Models in Human Social Relations and Global Biosphere
Processes
Classes of Mathematical Model
Global Modeling and Global Models
BASIC
PRINCIPLES OF MATHEMATICAL MODELLING
Physical and mathematical models
Mathematical modeling
Fundamental and applied models
Using computers in mathematical modeling
Using Computers
Mathematical methods in experimental studies
Experimental data treatment
Mathematical Model of a Device
Computational experiment
Main Stages of Numerical Experiment
Main Peculiarities of New Technology of Scientific Researches
Computational experiment in science and technology
Areas of Application of Computational Experiment
Classification
Types of computational experiment: an example
Search Computational Experiment
Optimization computational experiment
Diagnostic computational experiment
Constructing mathematical models
System Analysis
Aims of modeling
Mathematical Model
Hierarchy of Mathematical Models
Closing Mathematical Models
Previous study of mathematical models
Qualitative Analysis
Dimensionless Analysis of Problems
Approximate solutions
Exact Solutions
Numerical algorithms
Systems of Equations
Systems of Ordinary Differential Equations
Problems of Mathematical Physics
Inverse Problems
Optimization Problems
Numerical Algorithms and Parallel
MATHEMATICAL
MODELING OF LIFE SUPPORT SYSTEMS: CLASSIFICATION OF MODELS
Mathematical models
Some classes of mathematical models
Linear and nonlinear models
Well-and ill-posed problems
Point models
Distributed models
Discrete models
Imitation modeling
MATHEMATICAL
MODELS IN WATER SCIENCES
Mathematical Models in Hydrodynamics
Incompressible Inviscid Fluid
Compressible Inviscid Fluid
Viscous Incompressible Fluid
Mathematical Models of Flows in Rivers, Lakes, and Coastal Waters
Rivers
Lakes
Coastal Waters and Estuaries
Mathematical Models of Circulation in Oceans and Seas
General Circulation of Seas and Oceans
Equations of a General Circulation of Seas and Oceans.
Peculiarities of Large-scale Dynamics of Seas and Oceans
Data Analysis
Mathematical Models of Water Waves
Tidal waves
Wind waves
Internal waves
Tsunami
Mathematical Models for Water Resources Management
Modeling of Water Quality and Ecosystems
Structure of Water-Ecosystem Models
Simplified Ecosystem Model
Adjoint Equation Analysis
MATHEMATICAL
MODELS OF CIRCULATION IN OCEANS AND SEAS
Mathematical Modeling of Oceanic and Marine General Circulation
Equations of the General Circulation in Oceans and Seas
Boundary Conditions
Initial Conditions
Total Energy Conservation Law.
Parameterization of Sub-Scale Physical Processes
Solvability of Problems of the Ocean and Sea Dynamics
Linear Problems
Nonlinear Problems.
Alternative and Generalized Models of the General Circulation in Oceans and
Seas
Model Based on Nonlinear Shallow-Water Equations
Nonhydrostatic Model of the Sea Dynamics
Numerical Methods
The Choice of Differential Formulation of the Problem
Methods of Spatial Approximation
Methods for Solving the Ocean Problems with Respect to Time
Forward and Adjoint Models
Statement of the Data Assimilation Problem
Initialization Problem
MATHEMATICAL
MODELS FOR WATER RESOURCES MANAGEMENT
Mathematical modeling in water resources planning
Total state
Classes of models of water resources planning and management
Optimization models
Deterministic river basin modeling
Water f
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