On the Concept of Depth for Functional Data
The statistical analysis of functional data is a growing need in many research areas. In particular, a robust methodology is important to study curves, which are the output of many experiments in applied statistics. As a starting point for this robust analysis, we propose, analyze, and apply a new d...
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| Veröffentlicht in: | Journal of the American Statistical Association Jg. 104; H. 486; S. 718 - 734 |
|---|---|
| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
| Veröffentlicht: |
Alexandria, VA
Taylor & Francis
01.06.2009
American Statistical Association Taylor & Francis Ltd |
| Schlagworte: | |
| ISSN: | 0162-1459, 1537-274X |
| Online-Zugang: | Volltext |
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| Abstract | The statistical analysis of functional data is a growing need in many research areas. In particular, a robust methodology is important to study curves, which are the output of many experiments in applied statistics. As a starting point for this robust analysis, we propose, analyze, and apply a new definition of depth for functional observations based on the graphic representation of the curves. Given a collection of functions, it establishes the "centrality" of an observation and provides a natural center-outward ordering of the sample curves. Robust statistics, such as the median function or a trimmed mean function, can be defined from this depth definition. Its finite-dimensional version provides a new depth for multivariate data that is computationally feasible and useful for studying high-dimensional observations. Thus, this new depth is also suitable for complex observations such as microarray data, images, and those arising in some recent marketing and financial studies. Natural properties of these new concepts are established and the uniform consistency of the sample depth is proved. Simulation results show that the corresponding depth based trimmed mean presents better performance than other possible location estimators proposed in the literature for some contaminated models. Data depth can be also used to screen for outliers. The ability of the new notions of depth to detect "shape" outliers is presented. Several real datasets are considered to illustrate this new concept of depth, including applications to microarray observations, weather data, and growth curves. Finally, through this depth, we generalize to functions the Wilcoxon rank sum test. It allows testing whether two groups of curves come from the same population. This functional rank test when applied to children growth curves shows different growth patterns for boys and girls. |
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| AbstractList | The statistical analysis of functional data is a growing need in many research areas. In particular, a robust methodology is important to study curves, which are the output of many experiments in applied statistics. As a starting point for this robust analysis, we propose, analyze, and apply a new definition of depth for functional observations based on the graphic representation of the curves. Given a collection of functions. it establishes the "centrality" of an observation and provides a natural center-outward ordering of the sample curves. Robust statistics, such as the median function or a trimmed mean function, can be defined from this depth definition. Its finite-dimensional version provides a new depth for multivariate data that is computationally feasible and useful for studying high-dimensional observations. Thus, this new depth is also suitable for complex observations such as microarray data, images. and those arising in some recent marketing and financial studies. Natural properties of these new concepts are established and the uniform consistency of the sample depth is proved. Simulation results show that the corresponding depth based trimmed mean presents better performance than other possible location estimators proposed in the literature for some contaminated models. Data depth can be also used to screen for outliers. The ability of the new notions of depth to detect "shape" outliers is presented. Several real datasets are considered to illustrate this new concept of depth, including applications to microarray observations, weather data, and growth curves. Finally, through this depth. we generalize to functions the Wilcoxon rank sum test. It allows testing whether two groups of curves come from the same population. This functional rank test when applied to children growth curves shows different growth patterns for boys and girls. [PUBLICATION ABSTRACT] The statistical analysis of functional data is a growing need in many research areas. In particular, a robust methodology is important to study curves, which are the output of many experiments in applied statistics. As a starting point for this robust analysis, we propose, analyze, and apply a new definition of depth for functional observations based on the graphic representation of the curves. Given a collection of functions, it establishes the "centrality" of an observation and provides a natural center-outward ordering of the sample curves. Robust statistics, such as the median function or a trimmed mean function, can be defined from this depth definition. Its finite-dimensional version provides a new depth for multivariate data that is computationally feasible and useful for studying high-dimensional observations. Thus, this new depth is also suitable for complex observations such as microarray data, images, and those arising in some recent marketing and financial studies. Natural properties of these new concepts are established and the uniform consistency of the sample depth is proved. Simulation results show that the corresponding depth based trimmed mean presents better performance than other possible location estimators proposed in the literature for some contaminated models. Data depth can be also used to screen for outliers. The ability of the new notions of depth to detect "shape" outliers is presented. Several real datasets are considered to illustrate this new concept of depth, including applications to microarray observations, weather data, and growth curves. Finally, through this depth, we generalize to functions the Wilcoxon rank sum test. It allows testing whether two groups of curves come from the same population. This functional rank test when applied to children growth curves shows different growth patterns for boys and girls. |
| Author | López-Pintado, Sara Romo, Juan |
| Author_xml | – sequence: 1 givenname: Sara surname: López-Pintado fullname: López-Pintado, Sara – sequence: 2 givenname: Juan surname: Romo fullname: Romo, Juan |
| BackLink | http://pascal-francis.inist.fr/vibad/index.php?action=getRecordDetail&idt=21515002$$DView record in Pascal Francis |
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| Copyright | American Statistical Association and the American Society for Quality 2009 2009 American Statistical Association 2009 INIST-CNRS Copyright American Statistical Association Jun 2009 |
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| Keywords | Functional data Prediction theory Median Multivariate analysis Stochastic process Microarray Marketing Statistical data Outlier Statistical test Depth function Ordering Rank test Child Human Wilcoxon test Discriminant analysis Data analysis Statistical analysis Finance Functional analysis Statistical estimation Mean estimation Data depth Rank test for functions Growth curve Trimmed mean Functional Statistical method Graphics Simulation Observation data Filtering theory Application |
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| References | p_29 Liu R. (p_18) 1993; 88 p_25 p_26 Brown B. (p_2) 1989; 51 p_22 Mahalanobis P. C. (p_20) 1936; 12 Hettmansperger T. (p_8) 1994; 56 p_16 p_1 p_4 p_34 p_3 p_13 p_35 p_6 p_14 p_5 p_15 p_7 p_30 Liu R. (p_17) 1999; 27 p_31 p_10 p_32 p_11 p_33 López-Pintado S. (p_19) 2005; 72 |
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| SubjectTerms | Applications Boys Child development Children Coordinate systems Data Data analysis Data depth Datasets Estimation Exact sciences and technology Experiments Functional analysis Functional data General topics Girls Insurance, economics, finance Marketing Mathematical analysis Mathematical functions Mathematics Measurement Median Modeling Multivariate analysis Observation Outliers Probability and statistics Property Quantitative analysis Rank test for functions Rank tests Research methodology Sampling Sciences and techniques of general use Simulation Simulations Statistical analysis Statistical median Statistical methods Statistics Studies Tests Theory and Methods Weather |
| Title | On the Concept of Depth for Functional Data |
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