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Master advanced topics in the analysis of large, dynamically dependent datasets with this insightful resource

Statistical Learning with Big Dependent Data delivers a comprehensive presentation of the statistical and machine learning methods useful for analyzing and forecasting large and dynamically dependent data sets. The book presents automatic procedures for modelling and forecasting large sets of time series data. Beginning with some visualization tools, the book discusses procedures and methods for finding outliers, clusters, and other types of heterogeneity in big dependent data. It then introduces various dimension reduction methods, including regularization and factor models such as regularized Lasso in the presence of dynamical dependence and dynamic factor models. The book also covers other forecasting procedures, including index models, partial least squares, boosting, and now-casting. It further presents machine-learning methods, including neural network, deep learning, classification and regression trees and random forests. Finally, procedures for modelling and forecasting spatio-temporal dependent data are also presented.

Throughout the book, the advantages and disadvantages of the methods discussed are given. The book uses real-world examples to demonstrate applications, including use of many R packages. Finally, an R package associated with the book is available to assist readers in reproducing the analyses of examples and to facilitate real applications.

Analysis of Big Dependent Data includes a wide variety of topics for modeling and understanding big dependent data, like:

New ways to plot large sets of time series
An automatic procedure to build univariate ARMA models for individual components of a large data set
Powerful outlier detection procedures for large sets of related time series
New methods for finding the number of clusters of time series and discrimination methods , including vector support machines, for time series
Broad coverage of dynamic factor models including new representations and estimation methods for generalized dynamic factor models
Discussion on the usefulness of lasso with time series and an evaluation of several machine learning procedure for forecasting large sets of time series
Forecasting large sets of time series with exogenous variables, including discussions of index models, partial least squares, and boosting.
Introduction of modern procedures for modeling and forecasting spatio-temporal data

Perfect for PhD students and researchers in business, economics, engineering, and science: Statistical Learning with Big Dependent Data also belongs to the bookshelves of practitioners in these fields who hope to improve their understanding of statistical and machine learning methods for analyzing and forecasting big dependent data.




Table of contents

1 Introduction to Big Dependent Data 1

1.1 Examples of Dependent Data 2

1.2 Stochastic Processes 11

1.2.1 Scalar Processes 11

1.2.1.1 Stationarity 12

1.2.1.2 White Noise Process 14

1.2.1.3 Conditional Distribution 14

1.2.2 Vector Processes 14

1.2.2.1 Vector White Noises 17

1.2.2.2 Invertibility 17

1.3 Sample Moments of Stationary Vector Process 18

1.3.1 Sample Mean 18

1.3.2 Sample Covariance and Correlation Matrices 19

1.3.3 Example 1.1 20

1.3.4 Example 1.2 23

1.4 Nonstationary Processes 23

1.5 Principal Component Analysis 26

1.5.1 Discussion 30

1.5.2 Properties of the Principal Components 30

1.5.3 Example 1.3 31

1.6 Effects of Serial Dependence 35

1.6.1 Example 1.4 37

1.7 Appendix 1: Some Matrix Theory 38

2 Linear Univariate Time Series 43

2.1 Visualizing a Large Set of Time Series 45

2.1.1 Dynamic Plots 45

2.1.2 Static Plots 51

2.1.3 Example 2.1 55

2.2 Stationary ARMA Models 56

2.2.1 The Autoregressive Process 58

2.2.1.1 Autocorrelation Functions 59

2.2.2 The Moving Average Process 60

2.2.3 The ARMA Process 62

2.2.4 Linear Combinations of ARMA Processes 63

2.2.5 Example 2.2 64

2.3 Spectral Analysis of Stationary Processes 65

2.3.1 Fitting Harmonic Functions to a Time Series 65

2.3.2 The Periodogram 67

2.3.3 The Spectral Density Function and its Estimation 70

2.3.4 Example 2.3 71

2.4 Integrated Processes 72

2.4.1 The Random Walk Process 72

2.4.2 ARIMA Models 74

2.4.3 Seasonal ARIMA Models 75

2.4.3.1 The Airline Model 77

2.4.4 Example 2.4 78

2.5 Structural and State Space Models 80

2.5.1 Structural Time Series Models 80

2.5.2 State-Space Models 81

2.5.3 The Kalman Filter 85

2.6 Forecasting with Linear Models 88

2.6.1 Computing Optimal Predictors 88

2.6.2 Variances of the Predictions 90

2.6.3 Measuring Predictability 91

2.7 Modeling a Set of Time Series 92

2.7.1 Data Transformation 93

2.7.2 Testing for White Noise 95

2.7.3 Determination of the Difference Order 95

2.7.4 Example 2.5 96

2.7.5 Model Identification 97

2.8 Estimation and Information Criteria 97

2.8.1 Conditional Likelihood 97

2.8.2 On-line Estimation 99

2.8.3 Maximum Likelihood Estimation 100

2.8.4 Model Selection 101

2.8.4.1 The Akaike Information Criterion 102

2.8.4.2 The BIC Criterion 103

2.8.4.3 Other Criteria 103

2.8.4.4 Cross-Validation 104

2.8.5 Example 2.6 104

2.9 Diagnostic Checking 107

2.9.1 Residual Plot 107

2.9.2 Portmanteau Test for Residual Serial Correlations 107

2.9.3 Homoscedastic Tests 109

2.9.4 Normality Tests 109

2.9.5 Checking for Deterministic Components 109

2.9.6 Example 2.7 110

2.10 Forecasting 111

2.10.1 Out-of-Sample Forecasts 111

2.10.2 Forecasting with Model Averaging 112

2.10.3 Forecasting with Shrinkage Estimators 113

2.10.4 Example 2.8 114

2.11 Appendix 2: Difference Equations 115

3 Analysis of Multivariate Time Series 125

3.1 Transfer Function Models 126

3.1.1 Single Input and Single Output 126

3.1.2 Example 3.1 131

3.1.3 Multiple Inputs and Multiple Outputs 132

3.2 Vector AR Models 133

3.2.1 Impulse Response Function 135

3.2.2 Some Special Cases 136

3.2.3 Estimation 137

3.2.4 Model Building 139

3.2.5 Prediction 141

3.2.6 Forecast Error Variance Decomposition 143

3.2.7 Example 3.2 144

3.3 Vector Moving-Average Models 152

3.3.1 Properties of VMA Models 153

3.3.2 VMA Modeling 153

3.4 Stationary VARMA Models 157

3.4.1 Are VAR Models Sufficient? 157

3.4.2 Properties of VARMA Models 158

3.4.3 Modeling VARMA Process 159

3.4.4 Use of VARMA Models 159

3.4.5 Example 3.4 160

3.5 Unit Roots and Co-integration 165

3.5.1 Spurious Regression 165

3.5.2 Linear Combinations of a Vector Process 166

3.5.3 Co-integration 167

3.5.4 Over-Differencing 167

3.6 Error-Correction Models 169

3.6.1 Co-integration Test 170

3.6.2 Example 3.5 171

4 Handling Heterogeneity in Many Time Series 179

4.1 Intervention Ana