Analysis of Distributional Data 1st Edition by Paula Brito, Sónia Dias ISBN 9781498725453 1498725457

I Data Representation and Exploratory Analysis

1 Fundamental Concepts about Distributional Data

1.1 Introduction

1.2 The framework of distributional data

1.2.1 Definition and classification of symbolic variables

1.2.2 Histogram-valued variables

1.3 Operations with distributions

1.3.1 Histogram Arithmetic

1.3.2 Operations with Quantile Functions

1.4 Distances between distributions

1.5 Conclusion

Bibliography

2 Descriptive Statistics based on Frequency Distribution

2.1 Introduction

2.2 Univariate statistics

2.2.1 Frequency of ranges of values in distributional data

2.2.2 Location and dispersion symbolic measures

2.3 Bivariate descriptive statistics

2.3.1 Empirical joint distribution and density functions

2.3.2 Empirical symbolic covariances

2.4 Illustrative Example

2.5 Conclusion

Bibliography

3 Descriptive Statistics for Numeric Distributional Data

3.1 Introduction

3.2 Modal-numeric data and numeric distributional data

3.3 Univariate statistics

3.3.1 Fréchet and Chisini mean for numeric distributional variables

3.3.1.1 Euclidean distance-based mean

3.3.1.2 L2 Wasserstein distance-based mean

3.3.2 The variance of a distributional variable

3.3.2.1 Euclidean distance-based variance

3.3.2.2 L2 Wasserstein distance based variance

3.4 Bivariate descriptive indices for numeric distributional variables

3.4.1 L2 Wasserstein distance-based covariance

3.4.2 The correlation coefficient

3.5 Example

3.6 Conclusion

Bibliography

4 The Quantile Methods to Analyze Distributional Data

4.1 Introduction

4.2 Common representation of distributional data by the quantiles

4.2.1 Histogram-valued data

4.2.2 Other types of variables

4.2.2.1 Interval-valued data

4.2.2.2 Categorical modal data

4.2.3 Quantile vectors and monotone property

4.2.4 Hardwood data

4.3 Visualization: The data accumulation graph (DAG)

4.3.1 Quantile vectors data accumulation graph (QVDAG)

4.3.2 Variable-wise data accumulation graph (VWDAG)

4.3.3 Total data accumulation graph (TDAG)

4.4 The quantile method Principal Component Analysis (PCA)

4.4.1 Quantile method PCA

4.4.2 Application to the Hardwood data

4.5 The quantile method of hierarchical conceptual clustering

4.5.1 Compactness as the measures of similarity and cluster quality

4.5.2 Algorithm of hierarchical conceptual clustering

4.5.3 A sufficient condition for mutually disjoint concepts, and examples

4.6 Summary of the quantile methods

Bibliography

II Clustering and Classification

5 Partitive and Hierarchical Clustering of Distributional Data using the Wasserstein Distance

5.1 Introduction

5.2 Dynamic Clustering Algorithm for distributional data

5.3 Agglomerative hierarchical clustering using Wasserstein distance

5.4 Example

5.4.1 USA temperature dataset: Dynamic Clustering

5.4.2 USA temperature dataset: Hierarchical clustering

5.5 Conclusions

5.6 Appendix: decomposition of the TSS for grouped distributional data

Bibliography

6 Divisive Clustering of Histogram Data

6.1 Introduction

6.2 Divisive clustering

6.2.1 The criterion

6.2.2 Binary Questions and Assignment

6.2.3 Algorithm

6.3 Application: Crime dataset

6.4 Conclusion

Bibliography

7 Clustering of Modal Valued Data

7.1 Distributional symbolic data

7.1.1 Example: Tourism

7.1.2 Example: Structure of population

7.1.3 Example: Ego-centered networks and TIMSS

7.2 Clustering modal-valued symbolic data

7.2.1 Overview of the current literature on clustering modal-valued symbolic data

7.2.2 Clustering problem and algorithms

7.2.3 Leaders’ method

7.2.4 Agglomerative hierarchical method

7.2.5 Huygens Theorem for δ1

7.3 Interpretation of results

7.4 Application

7.5 Conclusions

Bibliography

8 Mixture Models for Distributional Data

8.1 Distributional Data

8.1.1 Random Distributions

8.1.2 Sample of Distributional Data

8.1.3 Examples of Distributional Data

8.2 The Mixture Problem

8.2.1 Finite Mixtures

8.2.2 Identifiability

8.2.3 Mixture and Classification

8.2.4 Examples

8.3 Mixtures of Dirichlet Distributions

8.3.1 EM algorithm for a Mixture of Dirichlet Distributions

8.3.2 Clustering quality and Consistency

8.4 Kernel Finite Mixtures

8.5 Dirichlet Process Mixture (DPM)

8.6 Dependent random distributions

8.6.1 Dependent Dirichlet Distributions, Model 1

8.6.2 Dependent Dirichlet Distributions, Model 2

8.6.3 Dependent Dirichlet Distributions, Model 3

8.7 Mixture models for general histograms

8.8 Conclusion

Bibliography

9 Classification of Continuous Distributional Data Using the Logratio Approach

9.1 Introduction

9.2 Bayes spaces

9.3 Functional linear discriminant analysis for density functions

9.3.1 The FLDA model

9.3.2 Classification using the FLDA model

9.3.3 Spline representation of PDFs

9.4 Classification of particle size distributions

9.4.1 Preprocessing of the raw data

9.4.2 Spline representation

9.4.3 Running the FLDA procedure

9.4.4 Quality of the classification

9.5 Conclusions

Bibliography

III Dimension Reduction

10 Principal Component Analysis of Distributional Data

10.1 Introduction

10.2 Approach based on mean of histograms

10.2.1 Parametric coding

10.2.2 Non parametric codings

10.2.3 Ridit scores

10.2.4 Metabin coding

10.2.5 PCA of means and representation of dispersion of concepts on individual map

10.2.5.1 The use of a numerical coding

10.2.5.2 Use of interval metabins as supplementary elements

10.3 Approach based on metabin coding

10.3.1 Determination of optimal metabins

10.3.2 Numerical and graphical tools provided by metabin coding

10.4 The use of quantiles as an alternative to the coding

10.4.1 Representation of histograms by common number of quantiles

10.5 Framework Extension for three-way data

10.5.1 Average three-way Histogram PCA

10.5.1.1 Transform three-way Histogram into classical three-way data

10.5.1.2 Determination of a compromise table

10.5.2 PCA of the compromise and projection of supplementary elements

10.5.3 Repeated Histogram PCA

10.6 Application of Histogram PCA

10.6.1 Histogram PCA on a macroeconomical dataset

10.6.1.1 Histogram PCA based on barycenters

10.6.1.2 PCA based on the average correlation matrix of quantiles tables

10.6.2 Histogram PCA using OECDGrowth data

10.7 Conclusion

Bibliography

11 Principal Component Analysis of Numeric Distributional Data

11.1 Introduction

11.2 Numeric Distributional Data

11.3 Linear Combination of Numeric Distributional Data

11.4 Numerical Characteristics of Numeric Distributional Data

11.5 The PCA Algorithm on Numeric Distributional Data

11.6 Properties of DPCA

11.7 Experimental Results of Synthetic Data Set

11.8 Real-life Applications

11.8.1 JOURNALS Data

11.8.2 STOCK Data

11.9 Conclusions

Bibliography

12 Multidimensional Scaling of Distributional Data

12.1 Introduction

12.2 Interval MDS

12.2.1 Circle Model

12.2.2 Rectangle Model

12.3 Histogram MDS

12.3.1 Example: Timbre Data

12.3.2 Histogram MDS and Fuzzy Theory

12.4 Local Minima

12.5 Conclusions and Discussion

Bibliography

IV Regression and Forecasting

13 Regression Analysis with the Distribution and Symmetric Distribution Model

13.1 Introduction

13.2 The Distribution and Symmetric Distribution Model

13.2.1 Model definition

13.2.2 Interpretation of the model

13.2.3 Estimation of the parameters

13.3 Properties and results deduced from the model

13.3.1 Kuhn Tucker conditions and consequences

13.3.2 Goodness-of-fit measure

13.4 An extension of the model

13.5 Illustrative example

13.6 Conclusion

Bibliography

14 Regression Analysis of Distributional Data Based on a Two-Component Model

14.1 Introduction

14.2 OLS linear regression for distributional data

14.2.1 A decomposition of the L2 Wasserstein distance

14.2.2 The two-component linear regression model

14.2.2.1 Interpretation of the parameters

14.2.2.2 Goodness-of-fit indices

14.3 Application

14.3.1 Data description

14.3.2 The Two-component model estimation

14.4 Conclusions

Bibliography

15 Forecasting Distributional Time Series

15.1 Introduction

15.2 Fundamentals of distributional time series

15.3 A theoretical approximation to distributional time series

15.4 Error measurement for distributional time series

15.4.1 Distance-based error measures for distributional time series

15.4.2 Quantile error measures for distributional time series

15.5 Tools for the analysis of distributional time series

15.5.1 Components in the distributional time series

15.5.1.1 Trend

15.5.1.2 Seasonal component

15.5.2 Autocorrelation in the distributional time series

15.6 Forecasting methods for distributional time series

15.6.1 Exponential smoothing methods

15.6.1.1 Simple exponential smoothing

15.6.1.2 Exponential smoothing with additive trend in the location

15.6.1.3 Exponential smoothing with seasonality

15.6.2 The k Nearest Neighbors method

15.6.3 Autoregressive approach

15.7 Applications

15.7.1 Cairns rainfall

15.7.2 IBM intra-daily returns

15.8 Concluding remarks

Bibliography