MATLAB Statistics and Machine Learning Toolbox – User’s Guide

MATLAB Statistics and Machine Learning Toolbox – User’s Guide 2022
Getting Started
1
Statistics and Machine Learning Toolbox Product Description . 1-2
Supported Data Types 1-3
Organizing Data
2
Other MATLAB Functions Supporting Nominal and Ordinal Arrays . 2-2
Create Nominal and Ordinal Arrays . 2-3
Create Nominal Arrays . 2-3
Create Ordinal Arrays 2-4
Change Category Labels 2-7
Change Category Labels 2-7
Reorder Category Levels 2-9
Reorder Category Levels in Ordinal Arrays 2-9
Reorder Category Levels in Nominal Arrays 2-10
Categorize Numeric Data 2-13
Categorize Numeric Data 2-13
Merge Category Levels 2-16
Merge Category Levels 2-16
Add and Drop Category Levels 2-18
Plot Data Grouped by Category . 2-21
Plot Data Grouped by Category 2-21
Test Differences Between Category Means 2-25
Summary Statistics Grouped by Category . 2-33
Summary Statistics Grouped by Category 2-33
Sort Ordinal Arrays . 2-35
Sort Ordinal Arrays 2-35
v
ContentsNominal and Ordinal Arrays 2-37
What Are Nominal and Ordinal Arrays? . 2-37
Nominal and Ordinal Array Conversion 2-37
Advantages of Using Nominal and Ordinal Arrays 2-39
Manipulate Category Levels 2-39
Analysis Using Nominal and Ordinal Arrays 2-39
Reduce Memory Requirements 2-40
Index and Search Using Nominal and Ordinal Arrays . 2-42
Index By Category . 2-42
Common Indexing and Searching Methods . 2-42
Grouping Variables . 2-46
What Are Grouping Variables? 2-46
Group Definition . 2-46
Analysis Using Grouping Variables . 2-47
Missing Group Values . 2-47
Dummy Variables . 2-49
What Are Dummy Variables? 2-49
Creating Dummy Variables . 2-50
Linear Regression with Categorical Covariates 2-53
Create a Dataset Array from Workspace Variables 2-58
Create a Dataset Array from a Numeric Array . 2-58
Create Dataset Array from Heterogeneous Workspace Variables . 2-60
Create a Dataset Array from a File . 2-63
Create a Dataset Array from a Tab-Delimited Text File 2-63
Create a Dataset Array from a Comma-Separated Text File . 2-65
Create a Dataset Array from an Excel File . 2-67
Add and Delete Observations . 2-69
Add and Delete Variables 2-72
Access Data in Dataset Array Variables . 2-75
Select Subsets of Observations . 2-80
Sort Observations in Dataset Arrays . 2-83
Merge Dataset Arrays . 2-86
Stack or Unstack Dataset Arrays 2-89
Calculations on Dataset Arrays . 2-93
Export Dataset Arrays . 2-96
Clean Messy and Missing Data 2-98
vi ContentsDataset Arrays in the Variables Editor 2-102
Open Dataset Arrays in the Variables Editor . 2-102
Modify Variable and Observation Names 2-103
Reorder or Delete Variables . 2-104
Add New Data . 2-106
Sort Observations . 2-107
Select a Subset of Data . 2-108
Create Plots . 2-110
Dataset Arrays 2-113
What Are Dataset Arrays? . 2-113
Dataset Array Conversion . 2-113
Dataset Array Properties . 2-114
Index and Search Dataset Arrays . 2-115
Ways To Index and Search 2-115
Examples . 2-115
Descriptive Statistics
3
Measures of Central Tendency . 3-2
Measures of Central Tendency . 3-2
Measures of Dispersion . 3-4
Compare Measures of Dispersion . 3-4
Exploratory Analysis of Data . 3-6
Resampling Statistics . 3-10
Bootstrap Resampling . 3-10
Jackknife Resampling . 3-12
Parallel Computing Support for Resampling Methods . 3-13
Statistical Visualization
4
Create Scatter Plots Using Grouped Data 4-2
Compare Grouped Data Using Box Plots . 4-4
Distribution Plots . 4-7
Normal Probability Plots 4-7
Probability Plots 4-9
Quantile-Quantile Plots 4-11
Cumulative Distribution Plots . 4-13
Visualizing Multivariate Data . 4-17
viiProbability Distributions
5
Working with Probability Distributions 5-3
Probability Distribution Objects 5-3
Apps and Interactive User Interfaces 5-6
Distribution-Specific Functions and Generic Distribution Functions 5-10
Supported Distributions . 5-16
Continuous Distributions (Data) . 5-16
Continuous Distributions (Statistics) 5-19
Discrete Distributions . 5-20
Multivariate Distributions 5-21
Nonparametric Distributions . 5-22
Flexible Distribution Families . 5-22
Maximum Likelihood Estimation 5-23
Negative Loglikelihood Functions . 5-25
Find MLEs Using Negative Loglikelihood Function . 5-25
Random Number Generation . 5-28
Nonparametric and Empirical Probability Distributions . 5-31
Overview 5-31
Kernel Distribution . 5-31
Empirical Cumulative Distribution Function 5-32
Piecewise Linear Distribution . 5-33
Pareto Tails 5-34
Triangular Distribution 5-35
Fit Kernel Distribution Object to Data . 5-37
Fit Kernel Distribution Using ksdensity 5-40
Fit Distributions to Grouped Data Using ksdensity . 5-42
Fit a Nonparametric Distribution with Pareto Tails . 5-44
Generate Random Numbers Using the Triangular Distribution . 5-48
Model Data Using the Distribution Fitter App . 5-52
Explore Probability Distributions Interactively 5-52
Create and Manage Data Sets . 5-53
Create a New Fit 5-56
Display Results 5-60
Manage Fits 5-61
Evaluate Fits . 5-63
Exclude Data . 5-65
Save and Load Sessions . 5-69
Generate a File to Fit and Plot Distributions 5-69
Fit a Distribution Using the Distribution Fitter App 5-72
Step 1: Load Sample Data 5-72
viii ContentsStep 2: Import Data 5-72
Step 3: Create a New Fit 5-74
Step 4: Create and Manage Additional Fits . 5-77
Define Custom Distributions Using the Distribution Fitter App . 5-82
Open the Distribution Fitter App . 5-82
Define Custom Distribution . 5-83
Import Custom Distribution 5-84
Explore the Random Number Generation UI 5-86
Compare Multiple Distribution Fits 5-88
Fit Probability Distribution Objects to Grouped Data . 5-93
Three-Parameter Weibull Distribution . 5-96
Multinomial Probability Distribution Objects . 5-103
Multinomial Probability Distribution Functions . 5-106
Generate Random Numbers Using Uniform Distribution Inversion . 5-109
Represent Cauchy Distribution Using t Location-Scale . 5-112
Generate Cauchy Random Numbers Using Student’s t . 5-115
Generate Correlated Data Using Rank Correlation 5-116
Create Gaussian Mixture Model 5-120
Fit Gaussian Mixture Model to Data 5-123
Simulate Data from Gaussian Mixture Model . 5-127
Copulas: Generate Correlated Samples 5-129
Determining Dependence Between Simulation Inputs 5-129
Constructing Dependent Bivariate Distributions 5-132
Using Rank Correlation Coefficients . 5-136
Using Bivariate Copulas 5-138
Higher Dimension Copulas 5-145
Archimedean Copulas 5-146
Simulating Dependent Multivariate Data Using Copulas 5-147
Fitting Copulas to Data . 5-151
Simulating Dependent Random Variables Using Copulas . 5-155
Fit Custom Distributions . 5-173
Avoid Numerical Issues When Fitting Custom Distributions 5-186
Nonparametric Estimates of Cumulative Distribution Functions and Their
Inverses . 5-192
ixModelling Tail Data with the Generalized Pareto Distribution . 5-207
Modelling Data with the Generalized Extreme Value Distribution 5-215
Curve Fitting and Distribution Fitting . 5-226
Fitting a Univariate Distribution Using Cumulative Probabilities 5-234
Gaussian Processes
6
Gaussian Process Regression Models . 6-2
Compare Prediction Intervals of GPR Models 6-3
Kernel (Covariance) Function Options 6-6
Exact GPR Method 6-10
Parameter Estimation . 6-10
Prediction 6-11
Computational Complexity of Exact Parameter Estimation and Prediction
. 6-13
Subset of Data Approximation for GPR Models 6-14
Subset of Regressors Approximation for GPR Models . 6-15
Approximating the Kernel Function 6-15
Parameter Estimation . 6-16
Prediction 6-16
Predictive Variance Problem 6-17
Fully Independent Conditional Approximation for GPR Models . 6-19
Approximating the Kernel Function 6-19
Parameter Estimation . 6-19
Prediction 6-20
Block Coordinate Descent Approximation for GPR Models . 6-22
Fit GPR Models Using BCD Approximation . 6-22
Predict Battery State of Charge Using Machine Learning 6-27
Random Number Generation
7
Generating Pseudorandom Numbers 7-2
Common Pseudorandom Number Generation Methods . 7-2
Representing Sampling Distributions Using Markov Chain Samplers . 7-9
Using the Metropolis-Hastings Algorithm . 7-9
Using Slice Sampling 7-9
x ContentsUsing Hamiltonian Monte Carlo . 7-10
Generating Quasi-Random Numbers . 7-12
Quasi-Random Sequences 7-12
Quasi-Random Point Sets 7-13
Quasi-Random Streams . 7-18
Generating Data Using Flexible Families of Distributions 7-20
Bayesian Linear Regression Using Hamiltonian Monte Carlo . 7-26
Bayesian Analysis for a Logistic Regression Model . 7-35
Hypothesis Tests
8
Hypothesis Test Terminology 8-2
Hypothesis Test Assumptions 8-4
Hypothesis Testing 8-5
Available Hypothesis Tests . 8-10
Selecting a Sample Size . 8-12
Analysis of Variance
9
One-Way ANOVA . 9-2
Introduction to One-Way ANOVA 9-2
Prepare Data for One-Way ANOVA 9-3
Perform One-Way ANOVA . 9-4
Mathematical Details 9-8
Two-Way ANOVA 9-11
Introduction to Two-Way ANOVA . 9-11
Prepare Data for Balanced Two-Way ANOVA 9-12
Perform Two-Way ANOVA 9-13
Mathematical Details . 9-15
Multiple Comparisons . 9-18
Multiple Comparisons Using One-Way ANOVA 9-18
Multiple Comparisons for Three-Way ANOVA . 9-20
Multiple Comparison Procedures 9-22
N-Way ANOVA 9-26
Introduction to N-Way ANOVA 9-26
Prepare Data for N-Way ANOVA . 9-28
xiPerform N-Way ANOVA 9-28
ANOVA with Random Effects . 9-33
Other ANOVA Models . 9-38
Analysis of Covariance 9-39
Introduction to Analysis of Covariance 9-39
Analysis of Covariance Tool 9-39
Confidence Bounds . 9-43
Multiple Comparisons . 9-45
Nonparametric Methods . 9-47
Introduction to Nonparametric Methods . 9-47
Kruskal-Wallis Test . 9-47
Friedman’s Test . 9-47
MANOVA 9-49
Introduction to MANOVA 9-49
ANOVA with Multiple Responses . 9-49
Model Specification for Repeated Measures Models 9-54
Wilkinson Notation . 9-54
Compound Symmetry Assumption and Epsilon Corrections 9-55
Mauchly’s Test of Sphericity 9-57
Multivariate Analysis of Variance for Repeated Measures 9-59
Bayesian Optimization
10
Bayesian Optimization Algorithm . 10-2
Algorithm Outline 10-2
Gaussian Process Regression for Fitting the Model . 10-3
Acquisition Function Types . 10-3
Acquisition Function Maximization . 10-5
Parallel Bayesian Optimization . 10-7
Optimize in Parallel 10-7
Parallel Bayesian Algorithm 10-7
Settings for Best Parallel Performance 10-8
Differences in Parallel Bayesian Optimization Output . 10-9
Bayesian Optimization Plot Functions . 10-11
Built-In Plot Functions . 10-11
Custom Plot Function Syntax 10-12
Create a Custom Plot Function . 10-12
Bayesian Optimization Output Functions 10-19
What Is a Bayesian Optimization Output Function? 10-19
xii ContentsBuilt-In Output Functions . 10-19
Custom Output Functions . 10-19
Bayesian Optimization Output Function 10-20
Bayesian Optimization Workflow . 10-25
What Is Bayesian Optimization? 10-25
Ways to Perform Bayesian Optimization 10-25
Bayesian Optimization Using a Fit Function . 10-26
Bayesian Optimization Using bayesopt . 10-26
Bayesian Optimization Characteristics . 10-27
Parameters Available for Fit Functions . 10-28
Hyperparameter Optimization Options for Fit Functions 10-30
Variables for a Bayesian Optimization . 10-34
Syntax for Creating Optimization Variables 10-34
Variables for Optimization Examples . 10-35
Bayesian Optimization Objective Functions 10-37
Objective Function Syntax 10-37
Objective Function Example . 10-37
Objective Function Errors . 10-37
Constraints in Bayesian Optimization . 10-39
Bounds . 10-39
Deterministic Constraints — XConstraintFcn 10-39
Conditional Constraints — ConditionalVariableFcn 10-40
Coupled Constraints . 10-41
Bayesian Optimization with Coupled Constraints . 10-42
Optimize Cross-Validated Classifier Using bayesopt 10-46
Optimize Classifier Fit Using Bayesian Optimization . 10-56
Optimize a Boosted Regression Ensemble . 10-67
Parametric Regression Analysis
11
Choose a Regression Function 11-2
Update Legacy Code with New Fitting Methods . 11-2
What Is a Linear Regression Model? . 11-6
Linear Regression 11-9
Prepare Data . 11-9
Choose a Fitting Method . 11-10
Choose a Model or Range of Models . 11-11
Fit Model to Data . 11-13
Examine Quality and Adjust Fitted Model . 11-14
Predict or Simulate Responses to New Data . 11-31
Share Fitted Models . 11-33
xiiiLinear Regression Workflow . 11-35
Regression Using Dataset Arrays . 11-40
Linear Regression Using Tables 11-43
Linear Regression with Interaction Effects . 11-46
Interpret Linear Regression Results 11-52
Cook’s Distance . 11-57
Purpose 11-57
Definition . 11-57
How To . 11-57
Determine Outliers Using Cook’s Distance 11-57
Coefficient Standard Errors and Confidence Intervals 11-60
Coefficient Covariance and Standard Errors . 11-60
Coefficient Confidence Intervals 11-61
Coefficient of Determination (R-Squared) . 11-63
Purpose 11-63
Definition . 11-63
How To . 11-63
Display Coefficient of Determination . 11-63
Delete-1 Statistics . 11-65
Delete-1 Change in Covariance (CovRatio) 11-65
Delete-1 Scaled Difference in Coefficient Estimates (Dfbetas) 11-67
Delete-1 Scaled Change in Fitted Values (Dffits) 11-68
Delete-1 Variance (S2_i) 11-70
Durbin-Watson Test 11-72
Purpose 11-72
Definition . 11-72
How To . 11-72
Test for Autocorrelation Among Residuals . 11-72
F-statistic and t-statistic 11-74
F-statistic . 11-74
Assess Fit of Model Using F-statistic . 11-74
t-statistic . 11-76
Assess Significance of Regression Coefficients Using t-statistic . 11-77
Hat Matrix and Leverage . 11-79
Hat Matrix 11-79
Leverage . 11-80
Determine High Leverage Observations 11-80
Residuals 11-82
Purpose 11-82
Definition . 11-82
How To . 11-83
Assess Model Assumptions Using Residuals . 11-83
xiv ContentsSummary of Output and Diagnostic Statistics 11-91
Wilkinson Notation 11-93
Overview . 11-93
Formula Specification 11-93
Linear Model Examples 11-96
Linear Mixed-Effects Model Examples . 11-97
Generalized Linear Model Examples . 11-98
Generalized Linear Mixed-Effects Model Examples 11-99
Repeated Measures Model Examples . 11-100
Stepwise Regression 11-101
Stepwise Regression to Select Appropriate Models . 11-101
Compare large and small stepwise models . 11-101
Reduce Outlier Effects Using Robust Regression . 11-106
Why Use Robust Regression? . 11-106
Iteratively Reweighted Least Squares . 11-106
Compare Results of Standard and Robust Least-Squares Fit 11-107
Steps for Iteratively Reweighted Least Squares . 11-109
Ridge Regression . 11-111
Introduction to Ridge Regression 11-111
Ridge Regression 11-111
Lasso and Elastic Net . 11-114
What Are Lasso and Elastic Net? 11-114
Lasso and Elastic Net Details . 11-114
References . 11-115
Wide Data via Lasso and Parallel Computing 11-117
Lasso Regularization 11-122
Lasso and Elastic Net with Cross Validation . 11-125
Partial Least Squares . 11-128
Introduction to Partial Least Squares . 11-128
Perform Partial Least-Squares Regression . 11-128
Linear Mixed-Effects Models . 11-133
Prepare Data for Linear Mixed-Effects Models . 11-136
Tables and Dataset Arrays . 11-136
Design Matrices . 11-137
Relation of Matrix Form to Tables and Dataset Arrays . 11-139
Relationship Between Formula and Design Matrices 11-140
Formula . 11-140
Design Matrices for Fixed and Random Effects 11-141
Grouping Variables . 11-143
Estimating Parameters in Linear Mixed-Effects Models . 11-145
Maximum Likelihood (ML) . 11-145
Restricted Maximum Likelihood (REML) . 11-146
xvLinear Mixed-Effects Model Workflow 11-148
Fit Mixed-Effects Spline Regression . 11-160
Train Linear Regression Model . 11-163
Analyze Time Series Data 11-181
Partial Least Squares Regression and Principal Components Regression
. 11-190
Generalized Linear Models
12
Multinomial Models for Nominal Responses 12-2
Multinomial Models for Ordinal Responses . 12-4
Hierarchical Multinomial Models . 12-7
Generalized Linear Models . 12-9
What Are Generalized Linear Models? 12-9
Prepare Data . 12-9
Choose Generalized Linear Model and Link Function 12-11
Choose Fitting Method and Model 12-13
Fit Model to Data . 12-15
Examine Quality and Adjust the Fitted Model 12-16
Predict or Simulate Responses to New Data . 12-23
Share Fitted Models . 12-26
Generalized Linear Model Workflow 12-28
Lasso Regularization of Generalized Linear Models . 12-32
What is Generalized Linear Model Lasso Regularization? . 12-32
Generalized Linear Model Lasso and Elastic Net 12-32
References 12-33
Regularize Poisson Regression 12-34
Regularize Logistic Regression 12-36
Regularize Wide Data in Parallel . 12-43
Generalized Linear Mixed-Effects Models 12-48
What Are Generalized Linear Mixed-Effects Models? 12-48
GLME Model Equations 12-48
Prepare Data for Model Fitting . 12-49
Choose a Distribution Type for the Model . 12-50
Choose a Link Function for the Model 12-50
Specify the Model Formula 12-51
Display the Model . 12-53
Work with the Model 12-55
xvi ContentsFit a Generalized Linear Mixed-Effects Model 12-57
Fitting Data with Generalized Linear Models . 12-65
Train Generalized Additive Model for Binary Classification . 12-77
Train Generalized Additive Model for Regression . 12-86
Nonlinear Regression
13
Nonlinear Regression . 13-2
What Are Parametric Nonlinear Regression Models? 13-2
Prepare Data . 13-2
Represent the Nonlinear Model . 13-3
Choose Initial Vector beta0 . 13-5
Fit Nonlinear Model to Data 13-6
Examine Quality and Adjust the Fitted Nonlinear Model . 13-6
Predict or Simulate Responses Using a Nonlinear Model 13-9
Nonlinear Regression Workflow 13-13
Mixed-Effects Models 13-18
Introduction to Mixed-Effects Models 13-18
Mixed-Effects Model Hierarchy 13-18
Specifying Mixed-Effects Models . 13-19
Specifying Covariate Models 13-21
Choosing nlmefit or nlmefitsa 13-22
Using Output Functions with Mixed-Effects Models . 13-24
Examining Residuals for Model Verification 13-28
Mixed-Effects Models Using nlmefit and nlmefitsa 13-33
Weighted Nonlinear Regression 13-45
Pitfalls in Fitting Nonlinear Models by Transforming to Linearity 13-53
Nonlinear Logistic Regression . 13-59
Time Series Forecasting
14
Time Series Forecasting Using Ensemble of Boosted Regression Trees
. 14-2
xviiSurvival Analysis
15
What Is Survival Analysis? . 15-2
Introduction 15-2
Censoring 15-2
Data 15-2
Survivor Function 15-4
Hazard Function . 15-6
Kaplan-Meier Method 15-10
Hazard and Survivor Functions for Different Groups . 15-16
Survivor Functions for Two Groups . 15-22
Cox Proportional Hazards Model . 15-26
Introduction . 15-26
Hazard Ratio 15-26
Extension of Cox Proportional Hazards Model . 15-27
Partial Likelihood Function 15-27
Partial Likelihood Function for Tied Events 15-28
Frequency or Weights of Observations . 15-29
Cox Proportional Hazards Model for Censored Data . 15-31
Cox Proportional Hazards Model with Time-Dependent Covariates . 15-35
Cox Proportional Hazards Model Object . 15-39
Analyzing Survival or Reliability Data . 15-47
Multivariate Methods
16
Multivariate Linear Regression . 16-2
Introduction to Multivariate Methods . 16-2
Multivariate Linear Regression Model 16-2
Solving Multivariate Regression Problems . 16-3
Estimation of Multivariate Regression Models . 16-5
Least Squares Estimation 16-5
Maximum Likelihood Estimation . 16-7
Missing Response Data 16-9
Set Up Multivariate Regression Problems . 16-11
Response Matrix 16-11
Design Matrices 16-14
Common Multivariate Regression Problems . 16-14
Multivariate General Linear Model . 16-20
xviii ContentsFixed Effects Panel Model with Concurrent Correlation 16-24
Longitudinal Analysis 16-30
Multidimensional Scaling . 16-35
Nonclassical and Nonmetric Multidimensional Scaling 16-36
Nonclassical Multidimensional Scaling . 16-36
Nonmetric Multidimensional Scaling 16-37
Classical Multidimensional Scaling . 16-40
Compare Handwritten Shapes Using Procrustes Analysis . 16-42
Introduction to Feature Selection 16-47
Feature Selection Algorithms 16-47
Feature Selection Functions . 16-48
Sequential Feature Selection 16-59
Introduction to Sequential Feature Selection 16-59
Select Subset of Features with Comparative Predictive Power . 16-59
Nonnegative Matrix Factorization 16-63
Perform Nonnegative Matrix Factorization . 16-64
Principal Component Analysis (PCA) 16-66
Analyze Quality of Life in U.S. Cities Using PCA . 16-67
Factor Analysis 16-76
Analyze Stock Prices Using Factor Analysis 16-77
Robust Feature Selection Using NCA for Regression . 16-83
Neighborhood Component Analysis (NCA) Feature Selection 16-97
NCA Feature Selection for Classification 16-97
NCA Feature Selection for Regression . 16-99
Impact of Standardization . 16-100
Choosing the Regularization Parameter Value . 16-100
t-SNE 16-102
What Is t-SNE? 16-102
t-SNE Algorithm . 16-102
Barnes-Hut Variation of t-SNE 16-105
Characteristics of t-SNE . 16-105
t-SNE Output Function 16-108
t-SNE Output Function Description 16-108
tsne optimValues Structure . 16-108
t-SNE Custom Output Function . 16-109
Visualize High-Dimensional Data Using t-SNE . 16-111
xixtsne Settings 16-115
Feature Extraction 16-127
What Is Feature Extraction? 16-127
Sparse Filtering Algorithm . 16-127
Reconstruction ICA Algorithm 16-129
Feature Extraction Workflow . 16-132
Extract Mixed Signals . 16-161
Select Features for Classifying High-Dimensional Data . 16-168
Perform Factor Analysis on Exam Grades . 16-177
Classical Multidimensional Scaling Applied to Nonspatial Distances 16-186
Nonclassical Multidimensional Scaling 16-194
Fitting an Orthogonal Regression Using Principal Components Analysis
. 16-202
Tune Regularization Parameter to Detect Features Using NCA for
Classification 16-207
Cluster Analysis
17
Choose Cluster Analysis Method 17-2
Clustering Methods 17-2
Comparison of Clustering Methods . 17-4
Hierarchical Clustering . 17-6
Introduction to Hierarchical Clustering . 17-6
Algorithm Description 17-6
Similarity Measures 17-7
Linkages . 17-8
Dendrograms . 17-9
Verify the Cluster Tree . 17-10
Create Clusters 17-15
DBSCAN . 17-19
Introduction to DBSCAN 17-19
Algorithm Description . 17-19
Determine Values for DBSCAN Parameters 17-20
Partition Data Using Spectral Clustering 17-26
Introduction to Spectral Clustering 17-26
Algorithm Description . 17-26
Estimate Number of Clusters and Perform Spectral Clustering . 17-27
xx Contentsk-Means Clustering 17-33
Introduction to k-Means Clustering . 17-33
Compare k-Means Clustering Solutions 17-33
Cluster Using Gaussian Mixture Model 17-39
How Gaussian Mixture Models Cluster Data . 17-39
Fit GMM with Different Covariance Options and Initial Conditions 17-39
When to Regularize . 17-44
Model Fit Statistics . 17-45
Cluster Gaussian Mixture Data Using Hard Clustering . 17-46
Cluster Gaussian Mixture Data Using Soft Clustering 17-52
Tune Gaussian Mixture Models 17-57
Cluster Evaluation . 17-63
Cluster Analysis . 17-66
Anomaly Detection with Isolation Forest 17-81
Introduction to Isolation Forest 17-81
Parameters for Isolation Forests 17-81
Anomaly Scores 17-81
Anomaly Indicators 17-82
Detect Outliers and Plot Contours of Anomaly Scores 17-82
Examine NumObservationsPerLearner for Small Data . 17-85
Unsupervised Anomaly Detection 17-91
Outlier Detection . 17-91
Novelty Detection . 17-99
Model-Specific Anomaly Detection 17-107
Detect Outliers After Training Random Forest 17-107
Detect Outliers After Training Discriminant Analysis Classifier 17-110
Parametric Classification
18
Parametric Classification 18-2
ROC Curve and Performance Metrics 18-3
Introduction to ROC Curve . 18-3
Performance Curve with MATLAB 18-4
ROC Curve for Multiclass Classification . 18-9
Performance Metrics 18-11
Classification Scores and Thresholds 18-13
Pointwise Confidence Intervals . 18-17
Performance Curves by perfcurve 18-19
Input Scores and Labels for perfcurve . 18-19
Computation of Performance Metrics 18-20
xxiMulticlass Classification Problems 18-22
Confidence Intervals . 18-22
Observation Weights . 18-22
Classification . 18-24
Nonparametric Supervised Learning
19
Supervised Learning Workflow and Algorithms 19-2
What Is Supervised Learning? 19-2
Steps in Supervised Learning . 19-3
Characteristics of Classification Algorithms 19-6
Misclassification Cost Matrix, Prior Probabilities, and Observation Weights
. 19-8
Visualize Decision Surfaces of Different Classifiers 19-11
Classification Using Nearest Neighbors . 19-14
Pairwise Distance Metrics 19-14
k-Nearest Neighbor Search and Radius Search . 19-16
Classify Query Data . 19-20
Find Nearest Neighbors Using a Custom Distance Metric . 19-26
K-Nearest Neighbor Classification for Supervised Learning . 19-29
Construct KNN Classifier . 19-30
Examine Quality of KNN Classifier 19-30
Predict Classification Using KNN Classifier . 19-31
Modify KNN Classifier . 19-31
Framework for Ensemble Learning . 19-33
Prepare the Predictor Data 19-34
Prepare the Response Data 19-34
Choose an Applicable Ensemble Aggregation Method 19-34
Set the Number of Ensemble Members . 19-37
Prepare the Weak Learners . 19-37
Call fitcensemble or fitrensemble . 19-39
Ensemble Algorithms 19-41
Bootstrap Aggregation (Bagging) and Random Forest 19-44
Random Subspace 19-47
Boosting Algorithms . 19-48
Train Classification Ensemble . 19-56
Train Regression Ensemble . 19-59
Select Predictors for Random Forests . 19-62
Test Ensemble Quality 19-68
Ensemble Regularization . 19-72
Regularize a Regression Ensemble 19-72
xxii ContentsClassification with Imbalanced Data 19-81
Handle Imbalanced Data or Unequal Misclassification Costs in
Classification Ensembles . 19-86
Train Ensemble With Unequal Classification Costs 19-87
Surrogate Splits . 19-92
LPBoost and TotalBoost for Small Ensembles 19-97
Tune RobustBoost 19-102
Random Subspace Classification 19-105
Train Classification Ensemble in Parallel . 19-110
Bootstrap Aggregation (Bagging) of Regression Trees Using TreeBagger
. 19-114
Bootstrap Aggregation (Bagging) of Classification Trees Using
TreeBagger 19-125
Detect Outliers Using Quantile Regression . 19-138
Conditional Quantile Estimation Using Kernel Smoothing . 19-143
Tune Random Forest Using Quantile Error and Bayesian Optimization
. 19-146
Support Vector Machines for Binary Classification . 19-151
Understanding Support Vector Machines 19-151
Using Support Vector Machines . 19-155
Train SVM Classifiers Using a Gaussian Kernel 19-157
Train SVM Classifier Using Custom Kernel . 19-160
Optimize Classifier Fit Using Bayesian Optimization 19-164
Plot Posterior Probability Regions for SVM Classification Models 19-174
Analyze Images Using Linear Support Vector Machines . 19-176
Assess Neural Network Classifier Performance 19-181
Assess Regression Neural Network Performance . 19-188
Automated Feature Engineering for Classification . 19-194
Interpret Linear Model with Generated Features 19-194
Generate New Features to Improve Bagged Ensemble Accuracy . 19-197
Automated Feature Engineering for Regression . 19-201
Interpret Linear Model with Generated Features 19-201
Generate New Features to Improve Bagged Ensemble Performance 19-204
Moving Towards Automating Model Selection Using Bayesian
Optimization 19-208
xxiiiAutomated Classifier Selection with Bayesian and ASHA Optimization
. 19-216
Automated Regression Model Selection with Bayesian and ASHA
Optimization 19-235
Credit Rating by Bagging Decision Trees . 19-256
Combine Heterogeneous Models into Stacked Ensemble 19-272
Label Data Using Semi-Supervised Learning Techniques 19-279
Bibliography 19-285
Decision Trees
20
Decision Trees . 20-2
Train Classification Tree . 20-2
Train Regression Tree 20-2
View Decision Tree . 20-4
Growing Decision Trees . 20-7
Prediction Using Classification and Regression Trees . 20-9
Predict Out-of-Sample Responses of Subtrees 20-10
Improving Classification Trees and Regression Trees 20-13
Examining Resubstitution Error 20-13
Cross Validation 20-13
Choose Split Predictor Selection Technique . 20-14
Control Depth or “Leafiness” 20-15
Pruning 20-19
Splitting Categorical Predictors in Classification Trees 20-25
Challenges in Splitting Multilevel Predictors 20-25
Algorithms for Categorical Predictor Split 20-25
Inspect Data with Multilevel Categorical Predictors . 20-26
Discriminant Analysis
21
Discriminant Analysis Classification . 21-2
Create Discriminant Analysis Classifiers . 21-2
xxiv ContentsCreating Discriminant Analysis Model . 21-4
Weighted Observations 21-4
Prediction Using Discriminant Analysis Models 21-6
Posterior Probability 21-6
Prior Probability . 21-6
Cost 21-7
Create and Visualize Discriminant Analysis Classifier . 21-9
Improving Discriminant Analysis Models 21-15
Deal with Singular Data 21-15
Choose a Discriminant Type . 21-15
Examine the Resubstitution Error and Confusion Matrix 21-16
Cross Validation 21-17
Change Costs and Priors . 21-18
Regularize Discriminant Analysis Classifier 21-21
Examine the Gaussian Mixture Assumption 21-27
Bartlett Test of Equal Covariance Matrices for Linear Discriminant Analysis
21-27
Q-Q Plot 21-29
Mardia Kurtosis Test of Multivariate Normality . 21-31
Naive Bayes
22
Naive Bayes Classification . 22-2
Supported Distributions . 22-2
Plot Posterior Classification Probabilities . 22-5
Classification Learner
23
Machine Learning in MATLAB 23-2
What Is Machine Learning? 23-2
Selecting the Right Algorithm . 23-3
Train Classification Models in Classification Learner App 23-6
Train Regression Models in Regression Learner App . 23-7
Train Neural Networks for Deep Learning . 23-8
Train Classification Models in Classification Learner App . 23-10
Automated Classifier Training . 23-10
Manual Classifier Training 23-13
Parallel Classifier Training 23-14
Compare and Improve Classification Models . 23-14
xxvSelect Data for Classification or Open Saved App Session . 23-18
Select Data from Workspace . 23-18
Import Data from File 23-19
Example Data for Classification 23-19
Choose Validation Scheme 23-20
(optional) Reserve Data for Testing 23-22
Save and Open App Session . 23-22
Choose Classifier Options . 23-23
Choose Classifier Type . 23-23
Decision Trees . 23-27
Discriminant Analysis 23-29
Logistic Regression . 23-30
Naive Bayes Classifiers . 23-30
Support Vector Machines . 23-31
Nearest Neighbor Classifiers 23-34
Kernel Approximation Classifiers . 23-36
Ensemble Classifiers 23-37
Neural Network Classifiers 23-40
Feature Selection and Feature Transformation Using Classification
Learner App 23-42
Investigate Features in the Scatter Plot 23-42
Select Features to Include 23-44
Transform Features with PCA in Classification Learner . 23-46
Investigate Features in the Parallel Coordinates Plot 23-46
Misclassification Costs in Classification Learner App 23-49
Specify Misclassification Costs . 23-49
Assess Model Performance 23-52
Misclassification Costs in Exported Model and Generated Code 23-53
Hyperparameter Optimization in Classification Learner App 23-54
Select Hyperparameters to Optimize 23-54
Optimization Options 23-59
Minimum Classification Error Plot 23-61
Optimization Results 23-63
Visualize and Assess Classifier Performance in Classification Learner
23-66
Check Performance in the Models Pane 23-66
View Model Metrics in Summary Tab and Models Pane . 23-67
Compare Model Information and Results in Table View . 23-68
Plot Classifier Results 23-69
Check Performance Per Class in the Confusion Matrix . 23-70
Check ROC Curve . 23-72
Interpret Model Using Partial Dependence Plots 23-74
Compare Model Plots by Changing Layout 23-76
Evaluate Test Set Model Performance 23-76
Export Plots in Classification Learner App . 23-78
Export Classification Model to Predict New Data 23-83
Export the Model to the Workspace to Make Predictions for New Data
23-83
xxvi ContentsMake Predictions for New Data 23-83
Generate MATLAB Code to Train the Model with New Data . 23-84
Generate C Code for Prediction 23-85
Deploy Predictions Using MATLAB Compiler 23-87
Export Model for Deployment to MATLAB Production Server 23-88
Train Decision Trees Using Classification Learner App . 23-89
Train Discriminant Analysis Classifiers Using Classification Learner App
23-99
Train Logistic Regression Classifiers Using Classification Learner App
. 23-103
Train Support Vector Machines Using Classification Learner App . 23-107
Train Nearest Neighbor Classifiers Using Classification Learner App 23-111
Train Kernel Approximation Classifiers Using Classification Learner App
. 23-115
Train Ensemble Classifiers Using Classification Learner App . 23-120
Train Naive Bayes Classifiers Using Classification Learner App . 23-124
Train Neural Network Classifiers Using Classification Learner App 23-133
Train and Compare Classifiers Using Misclassification Costs in
Classification Learner App . 23-137
Train Classifier Using Hyperparameter Optimization in Classification
Learner App . 23-145
Check Classifier Performance Using Test Set in Classification Learner App
. 23-152
Interpret Classifiers Trained in Classification Learner App . 23-157
Deploy Model Trained in Classification Learner to MATLAB Production
Server 23-167
Choose Trained Model to Deploy 23-167
Export Model for Deployment . 23-168
(Optional) Simulate Model Deployment 23-169
Package Code . 23-170
Build Condition Model for Industrial Machinery and Manufacturing
Processes . 23-171
Load Data . 23-171
Import Data into App and Partition Data . 23-172
Train Models Using All Features . 23-173
Assess Model Performance . 23-174
Export Model to the Workspace and Save App Session 23-177
Check Model Size 23-178
Resume App Session . 23-178
Select Features Using Feature Ranking . 23-178
xxviiInvestigate Important Features in Scatter Plot 23-180
Further Experimentation 23-181
Assess Model Accuracy on Test Set . 23-184
Export Final Model . 23-186
Regression Learner
24
Train Regression Models in Regression Learner App 24-2
Automated Regression Model Training 24-2
Manual Regression Model Training . 24-4
Parallel Regression Model Training 24-5
Compare and Improve Regression Models . 24-6
Select Data for Regression or Open Saved App Session 24-9
Select Data from Workspace 24-9
Import Data from File 24-10
Example Data for Regression 24-10
Choose Validation Scheme 24-11
(optional) Reserve Data for Testing 24-12
Save and Open App Session . 24-12
Choose Regression Model Options 24-14
Choose Regression Model Type 24-14
Linear Regression Models 24-16
Regression Trees . 24-18
Support Vector Machines . 24-20
Gaussian Process Regression Models 24-22
Kernel Approximation Models 24-24
Ensembles of Trees . 24-26
Neural Networks . 24-27
Feature Selection and Feature Transformation Using Regression Learner
App . 24-30
Investigate Features in the Response Plot . 24-30
Select Features to Include 24-31
Transform Features with PCA in Regression Learner 24-33
Hyperparameter Optimization in Regression Learner App 24-35
Select Hyperparameters to Optimize 24-35
Optimization Options 24-41
Minimum MSE Plot . 24-43
Optimization Results 24-45
Visualize and Assess Model Performance in Regression Learner . 24-48
Check Performance in Models Pane . 24-48
View Model Statistics in Summary Tab and Models Pane . 24-49
Compare Model Information and Results in Table View . 24-50
Explore Data and Results in Response Plot 24-52
Plot Predicted vs. Actual Response 24-54
Evaluate Model Using Residuals Plot 24-55
Interpret Model Using Partial Dependence Plots 24-56
xxviii ContentsCompare Model Plots by Changing Layout 24-58
Evaluate Test Set Model Performance 24-59
Export Plots in Regression Learner App . 24-61
Export Regression Model to Predict New Data 24-65
Export Model to Workspace . 24-65
Make Predictions for New Data 24-65
Generate MATLAB Code to Train Model with New Data 24-66
Generate C Code for Prediction 24-67
Deploy Predictions Using MATLAB Compiler 24-69
Export Model for Deployment to MATLAB Production Server 24-69
Train Regression Trees Using Regression Learner App . 24-71
Train Regression Neural Networks Using Regression Learner App . 24-82
Train Kernel Approximation Model Using Regression Learner App . 24-89
Train Regression Model Using Hyperparameter Optimization in
Regression Learner App 24-97
Check Model Performance Using Test Set in Regression Learner App
. 24-103
Interpret Regression Models Trained in Regression Learner App . 24-108
Deploy Model Trained in Regression Learner to MATLAB Production
Server 24-119
Choose Trained Model to Deploy 24-119
Export Model for Deployment . 24-120
(Optional) Simulate Model Deployment 24-120
Package Code . 24-121
Support Vector Machines
25
Understanding Support Vector Machine Regression 25-2
Mathematical Formulation of SVM Regression 25-2
Solving the SVM Regression Optimization Problem . 25-5
Fairness
26
Introduction to Fairness in Binary Classification . 26-2
Reduce Statistical Parity Difference Using Fairness Weights 26-2
Reduce Disparate Impact of Predictions . 26-5
xxixInterpretability
27
Interpret Machine Learning Models . 27-2
Features for Model Interpretation 27-2
Interpret Classification Model . 27-3
Interpret Regression Model . 27-10
Shapley Values for Machine Learning Model . 27-18
What Is a Shapley Value? . 27-18
Shapley Value with MATLAB . 27-18
Algorithms 27-18
Specify Computation Algorithm 27-20
Computational Cost . 27-23
Reduce Computational Cost . 27-23
Incremental Learning
28
Incremental Learning Overview . 28-2
What Is Incremental Learning? 28-2
Incremental Learning with MATLAB 28-3
Configure Incremental Learning Model 28-9
Call Object Directly . 28-11
Convert Traditionally Trained Model 28-15
Implement Incremental Learning for Regression Using Succinct Workflow
28-19
Implement Incremental Learning for Classification Using Succinct
Workflow . 28-22
Implement Incremental Learning for Regression Using Flexible Workflow
28-25
Implement Incremental Learning for Classification Using Flexible
Workflow . 28-29
Initialize Incremental Learning Model from SVM Regression Model
Trained in Regression Learner . 28-33
Initialize Incremental Learning Model from Logistic Regression Model
Trained in Classification Learner . 28-40
Perform Conditional Training During Incremental Learning 28-45
Perform Text Classification Incrementally . 28-49
Incremental Learning with Naive Bayes and Heterogeneous Data 28-52
xxx ContentsMarkov Models
29
Markov Chains . 29-2
Hidden Markov Models (HMM) . 29-4
Introduction to Hidden Markov Models (HMM) 29-4
Analyzing Hidden Markov Models 29-5
Design of Experiments
30
Design of Experiments 30-2
Full Factorial Designs . 30-3
Multilevel Designs . 30-3
Two-Level Designs . 30-3
Fractional Factorial Designs 30-5
Introduction to Fractional Factorial Designs 30-5
Plackett-Burman Designs 30-5
General Fractional Designs . 30-5
Response Surface Designs . 30-8
Introduction to Response Surface Designs . 30-8
Central Composite Designs . 30-8
Box-Behnken Designs 30-10
D-Optimal Designs 30-12
Introduction to D-Optimal Designs 30-12
Generate D-Optimal Designs . 30-13
Augment D-Optimal Designs . 30-14
Specify Fixed Covariate Factors 30-15
Specify Categorical Factors . 30-16
Specify Candidate Sets . 30-16
Improve an Engine Cooling Fan Using Design for Six Sigma Techniques
30-19
Statistical Process Control
31
Control Charts . 31-2
Capability Studies 31-4
xxxiTall Arrays
32
Logistic Regression with Tall Arrays . 32-2
Bayesian Optimization with Tall Arrays . 32-9
Statistics and Machine Learning with Big Data Using Tall Arrays 32-24
Parallel Statistics
33
Quick Start Parallel Computing for Statistics and Machine Learning
Toolbox . 33-2
Parallel Statistics and Machine Learning Toolbox Functionality 33-2
How to Compute in Parallel 33-2
Use Parallel Processing for Regression TreeBagger Workflow 33-4
Concepts of Parallel Computing in Statistics and Machine Learning
Toolbox . 33-6
Subtleties in Parallel Computing . 33-6
Vocabulary for Parallel Computation 33-6
When to Run Statistical Functions in Parallel . 33-7
Why Run in Parallel? 33-7
Factors Affecting Speed . 33-7
Factors Affecting Results 33-7
Analyze and Model Data on GPU 33-9
Working with parfor . 33-14
How Statistical Functions Use parfor 33-14
Characteristics of parfor 33-14
Reproducibility in Parallel Statistical Computations . 33-16
Issues and Considerations in Reproducing Parallel Computations . 33-16
Running Reproducible Parallel Computations 33-16
Parallel Statistical Computation Using Random Numbers . 33-17
Implement Jackknife Using Parallel Computing 33-20
Implement Cross-Validation Using Parallel Computing . 33-21
Simple Parallel Cross Validation 33-21
Reproducible Parallel Cross Validation . 33-21
Implement Bootstrap Using Parallel Computing 33-23
Bootstrap in Serial and Parallel 33-23
Reproducible Parallel Bootstrap 33-24
xxxii ContentsCode Generation
34
Introduction to Code Generation 34-2
Code Generation Workflows 34-2
Code Generation Applications . 34-4
General Code Generation Workflow 34-5
Define Entry-Point Function 34-5
Generate Code 34-5
Verify Generated Code 34-7
Code Generation for Prediction of Machine Learning Model at Command
Line 34-9
Code Generation for Incremental Learning 34-13
Code Generation for Nearest Neighbor Searcher 34-20
Code Generation for Prediction of Machine Learning Model Using
MATLAB Coder App 34-23
Code Generation and Classification Learner App 34-32
Load Sample Data 34-32
Enable PCA . 34-33
Train Models 34-34
Export Model to Workspace . 34-36
Generate C Code for Prediction 34-37
Deploy Neural Network Regression Model to FPGA/ASIC Platform . 34-40
Predict Class Labels Using MATLAB Function Block . 34-51
Specify Variable-Size Arguments for Code Generation . 34-56
Create Dummy Variables for Categorical Predictors and Generate C/C++
Code 34-61
System Objects for Classification and Code Generation 34-65
Predict Class Labels Using Stateflow 34-73
Human Activity Recognition Simulink Model for Smartphone Deployment
34-77
Human Activity Recognition Simulink Model for Fixed-Point Deployment
34-86
Code Generation for Prediction and Update Using Coder Configurer . 34-92
Code Generation for Probability Distribution Objects 34-94
Fixed-Point Code Generation for Prediction of SVM . 34-99
xxxiiiGenerate Code to Classify Data in Table 34-112
Code Generation for Image Classification . 34-115
Predict Class Labels Using ClassificationSVM Predict Block . 34-123
Predict Responses Using RegressionSVM Predict Block . 34-127
Predict Class Labels Using ClassificationTree Predict Block . 34-133
Predict Responses Using RegressionTree Predict Block . 34-139
Predict Class Labels Using ClassificationEnsemble Predict Block . 34-142
Predict Responses Using RegressionEnsemble Predict Block 34-149
Predict Class Labels Using ClassificationNeuralNetwork Predict Block
. 34-156
Predict Responses Using RegressionNeuralNetwork Predict Block 34-160
Predict Responses Using RegressionGP Predict Block 34-164
Predict Class Labels Using ClassificationKNN Predict Block . 34-170
Code Generation for Logistic Regression Model Trained in Classification
Learner . 34-176
Code Generation for Anomaly Detection 34-179
Compress Machine Learning Model for Memory-Limited Hardware . 34-185
Functions
35
Sample Data Sets
A
Sample Data Sets A-2
Probability Distributions
B
Bernoulli Distribution . B-2
Overview . B-2
xxxiv ContentsParameters . B-2
Probability Density Function B-2
Cumulative Distribution Function . B-2
Descriptive Statistics B-2
Examples . B-3
Related Distributions B-4
Beta Distribution B-6
Overview . B-6
Parameters . B-6
Probability Density Function B-6
Cumulative Distribution Function . B-7
Examples . B-7
Related Distributions B-9
Binomial Distribution . B-10
Overview B-10
Parameters B-10
Probability Density Function . B-10
Cumulative Distribution Function B-11
Descriptive Statistics . B-11
Example . B-11
Related Distributions . B-16
Birnbaum-Saunders Distribution . B-18
Definition B-18
Background B-18
Parameters B-18
Burr Type XII Distribution . B-19
Definition B-19
Background B-19
Parameters B-20
Fit a Burr Distribution and Draw the cdf B-21
Compare Lognormal and Burr Distribution pdfs . B-23
Burr pdf for Various Parameters . B-24
Survival and Hazard Functions of Burr Distribution B-26
Divergence of Parameter Estimates B-27
Chi-Square Distribution . B-29
Overview B-29
Parameters B-29
Probability Density Function . B-29
Cumulative Distribution Function B-30
Inverse Cumulative Distribution Function . B-30
Descriptive Statistics . B-30
Examples B-30
Related Distributions . B-32
Exponential Distribution B-34
Overview B-34
Parameters B-34
Probability Density Function . B-35
Cumulative Distribution Function B-35
Inverse Cumulative Distribution Function . B-35
xxxvHazard Function B-35
Examples B-36
Related Distributions . B-39
Extreme Value Distribution B-41
Definition B-41
Background B-41
Parameters B-43
Examples B-44
F Distribution . B-46
Definition B-46
Background B-46
Examples B-46
Gamma Distribution B-48
Overview B-48
Parameters B-48
Probability Density Function . B-49
Cumulative Distribution Function B-49
Inverse Cumulative Distribution Function . B-50
Descriptive Statistics . B-50
Examples B-50
Related Distributions . B-54
Generalized Extreme Value Distribution B-56
Definition B-56
Background B-56
Parameters B-57
Examples B-58
Generalized Pareto Distribution B-60
Definition B-60
Background B-60
Parameters B-61
Examples B-62
Geometric Distribution B-64
Overview B-64
Parameters B-64
Probability Density Function . B-64
Cumulative Distribution Function B-65
Descriptive Statistics . B-65
Hazard Function B-65
Examples B-65
Related Distributions . B-67
Half-Normal Distribution B-69
Overview B-69
Parameters B-69
Probability Density Function . B-69
Cumulative Distribution Function B-71
Descriptive Statistics . B-73
Relationship to Other Distributions B-73
xxxvi ContentsHypergeometric Distribution . B-74
Definition B-74
Background B-74
Examples B-74
Inverse Gaussian Distribution B-76
Definition B-76
Background B-76
Parameters B-76
Inverse Wishart Distribution . B-77
Definition B-77
Background B-77
Example . B-77
Kernel Distribution . B-79
Overview B-79
Kernel Density Estimator B-79
Kernel Smoothing Function B-79
Bandwidth . B-83
Logistic Distribution B-86
Overview B-86
Parameters B-86
Probability Density Function . B-86
Relationship to Other Distributions B-86
Loglogistic Distribution . B-87
Overview B-87
Parameters B-87
Probability Density Function . B-87
Relationship to Other Distributions B-87
Lognormal Distribution . B-89
Overview B-89
Parameters B-89
Probability Density Function . B-90
Cumulative Distribution Function B-90
Examples B-90
Related Distributions . B-95
Loguniform Distribution B-97
Overview B-97
Parameters B-97
Probability Density Function . B-97
Cumulative Distribution Function B-97
Descriptive Statistics . B-98
Examples B-98
Related Distributions B-101
Multinomial Distribution . B-102
Overview . B-102
Parameter B-102
Probability Density Function B-102
Descriptive Statistics B-102
xxxviiRelationship to Other Distributions . B-103
Multivariate Normal Distribution B-104
Overview . B-104
Parameters . B-104
Probability Density Function B-104
Cumulative Distribution Function . B-105
Examples . B-105
Multivariate t Distribution B-110
Definition . B-110
Background . B-110
Example . B-110
Nakagami Distribution . B-114
Definition . B-114
Background . B-114
Parameters . B-114
Negative Binomial Distribution B-115
Definition . B-115
Background . B-115
Parameters . B-115
Example . B-117
Noncentral Chi-Square Distribution B-119
Definition . B-119
Background . B-119
Examples . B-119
Noncentral F Distribution B-121
Definition . B-121
Background . B-121
Examples . B-121
Noncentral t Distribution . B-123
Definition . B-123
Background . B-123
Examples . B-123
Normal Distribution . B-125
Overview . B-125
Parameters . B-125
Probability Density Function B-126
Cumulative Distribution Function . B-126
Examples . B-127
Related Distributions B-133
Piecewise Linear Distribution . B-136
Overview . B-136
Parameters . B-136
Cumulative Distribution Function . B-136
Relationship to Other Distributions . B-136
xxxviii ContentsPoisson Distribution . B-137
Overview . B-137
Parameters . B-137
Probability Density Function B-137
Cumulative Distribution Function . B-138
Examples . B-138
Related Distributions B-141
Rayleigh Distribution B-143
Definition . B-143
Background . B-143
Parameters . B-143
Examples . B-143
Rician Distribution B-145
Definition . B-145
Background . B-145
Parameters . B-145
Stable Distribution B-147
Overview . B-147
Parameters . B-147
Probability Density Function B-148
Cumulative Distribution Function . B-150
Descriptive Statistics B-152
Relationship to Other Distributions . B-153
Student’s t Distribution B-156
Overview . B-156
Parameters . B-156
Probability Density Function B-156
Cumulative Distribution Function . B-157
Inverse Cumulative Distribution Function B-157
Descriptive Statistics B-157
Examples . B-157
Related Distributions B-161
t Location-Scale Distribution B-163
Overview . B-163
Parameters . B-163
Probability Density Function B-163
Cumulative Distribution Function . B-164
Descriptive Statistics B-164
Relationship to Other Distributions . B-164
Triangular Distribution B-165
Overview . B-165
Parameters . B-165
Probability Density Function B-165
Cumulative Distribution Function . B-166
Examples . B-166
Uniform Distribution (Continuous) . B-170
Overview . B-170
Parameters . B-170
xxxixProbability Density Function B-171
Cumulative Distribution Function . B-171
Descriptive Statistics B-171
Random Number Generation B-171
Examples . B-171
Related Distributions B-174
Uniform Distribution (Discrete) . B-175
Definition . B-175
Background . B-175
Examples . B-175
Weibull Distribution . B-177
Overview . B-177
Parameters . B-177
Probability Density Function B-178
Cumulative Distribution Function . B-178
Inverse Cumulative Distribution Function B-178
Hazard Function . B-179
Examples . B-179
Related Distributions B-182
Wishart Distribution . B-184
Overview . B-184
Parameters . B-184
Probability Density Function B-184
Example . B-184
Bibliography
C
Bibliography . C-2
xl Contents

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