Linear regression of interval-valued data based on complete information in hypercubes

Wang, Huiwen; Guan, Rong; Wu, Junjie

doi:10.1007/s11518-012-5203-4

Cited by 27 publications

(15 citation statements)

References 12 publications

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“…Standard regression has been extended in different ways to symbolic data. In the case of interval‐valued variables, see Refs . In the case of histogram‐valued variables, see Refs and .…”

Section: Some Methods For the Analysis Of Symbolic Datamentioning

confidence: 99%

Thinking by classes in data science: the symbolic data analysis paradigm

Diday

2016

WIREs Computational Stats

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Data Science, considered as a science by itself, is in general terms, the extraction of knowledge from data. Symbolic data analysis (SDA) gives a new way of thinking in Data Science by extending the standard input to a set of classes of individual entities. Hence, classes of a given population are considered to be units of a higher level population to be studied. Such classes often represent the real units of interest. In order to take variability between the members of each class into account, classes are described by intervals, distributions, set of categories or numbers sometimes weighted and the like. In that way, we obtain new kinds of data, called 'symbolic' as they cannot be reduced to numbers without losing much information. The first step in SDA is to build the symbolic data table where the rows are classes and the variables can take symbolic values. The second step is to study and extract new knowledge from these new kinds of data by at least an extension of Computer Statistics and Data Mining to symbolic data. SDA is a new paradigm which opens up a vast domain of research and applications by giving complementary results to classical methods applied to standard data. SDA also gives answers to big data and complex data challenges as big data can be reduced and summarized by classes and as complex data with multiple unstructured data tables and unpaired variables can be transformed into a structured data table with paired symbolic-valued variables.(X 0 , Y 0 ) describing a set of teams X 0 by a set of symbolic variables Y 0 .

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Section: Some Methods For the Analysis Of Symbolic Datamentioning

confidence: 99%

Thinking by classes in data science: the symbolic data analysis paradigm

Diday

2016

WIREs Computational Stats

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“…This is especially true when data is aggregated [22] or anonymized [8]. Consequently, several data analysis tools, including regression [23], [24], canonical analysis [25], and multi-dimensional scaling [26], have been developed for symbolic and interval-valued data. Given the popularity of PCA in data analysis, several interval-valued PCA algorithms have also been proposed [27], [28], [29], [30], most of which leverage the specific statistical and geometric meanings of principal components of a system of variables.…”

Section: Analysis Of Symbolic and Interval-valued Datamentioning

confidence: 99%

Matrix Factorization with Interval-Valued Data

Mauro

Candan

et al. 2021

IEEE Trans. Knowl. Data Eng.

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With many applications relying on multi-dimensional datasets for decision making, matrix factorization (or decomposition) is becoming the basis for many knowledge discoveries and machine learning tasks, from clustering, trend detection, anomaly detection, to correlation analysis. Unfortunately, a major shortcoming of matrix analysis operations is that, despite their effectiveness when the data is scalar, these operations become difficult to apply in the presence of non-scalar data, as they are not designed for data that include non-scalar observations, such as intervals. Yet, in many applications, the available data are inherently non-scalar for various reasons, including imprecision in data collection, conflicts in aggregated data, data summarization, or privacy issues, where one is provided with a reduced, clustered, or intentionally noisy and obfuscated version of the data to hide information. In this paper, we propose matrix decomposition techniques that consider the existence of interval-valued data. We show that naive ways to deal with such imperfect data may introduce errors in analysis and present factorization techniques that are especially effective when the amount of imprecise information is large.

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“…The more recent approaches [18]- [21] impose either positivity restrictions on parameters or design the regression model so as to ensure that the lower bound does not exceed the upper bound. For instance, Neto and Carvalho [18] apply the Lawson and Hanson's algorithm (LHA) [23], Wang et al [19] use Moore's linear combination [24], and Souza et al [21] the Box-Cox transformation [25] to guarantee mathematical coherence.…”

Section: Introductionmentioning

confidence: 99%

Towards Handling Uncertainty-at-Source in AI -- A Review and Next Steps for Interval Regression

Kabir¹,

Wagner²,

Ellerby³

2021

Preprint

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Most of statistics and AI draw insights through modelling discord or variance between sources of information (i.e., inter-source uncertainty). Increasingly, however, research is focusing upon uncertainty arising at the level of individual measurements (i.e., within-or intra-source), such as for a given sensor output or human response. Here, adopting intervals rather than numbers as the fundamental data-type provides an efficient, powerful, yet challenging way forward -offering systematic capture of uncertainty-at-source, increasing informational capacity, and ultimately potential for insight. Following recent progress in the capture of interval-valued data, including from human participants, conducting machine learning directly upon intervals is a crucial next step. This paper focuses on linear regression for interval-valued data as a recent growth area, providing an essential foundation for broader use of intervals in AI. We conduct an in-depth analysis of state-ofthe-art methods, elucidating their behaviour, advantages, and pitfalls when applied to datasets with different properties. Specific emphasis is given to the challenge of preserving mathematical coherence -i.e., ensuring that models maintain fundamental mathematical properties of intervals throughout -and the paper puts forward extensions to an existing approach to guarantee this. Carefully designed experiments, using both synthetic and real-world data, are conducted -with findings presented alongside novel visualizations for interval-valued regression outputs, designed to maximise model interpretability. Finally, the paper makes recommendations concerning method suitability for data sets with specific properties and highlights remaining challenges and important next steps for developing AI with the capacity to handle uncertainty-at-source. 1 Impact Statement-Capturing information as intervals, rather than numbers, provides a powerful means for handling uncertainty and inherent range in data. This paper focuses on a basic building block of statistics and AI: linear regression. In recent years, regression for intervals has become a topic of interest in AI, and has been applied to domains ranging from marketing to cyber-security. It allows direct modelling of relationships not only between variables per-se, but also concerning their associated uncertainty. For example, we can infer not only how a snack's nutritional benefits impact consumer purchase intention, but also how uncertainty about these benefits impacts purchase intention and its own associated uncertainty. Nonetheless, while there are considerable upsides to interval-valued regression and AI, substantial challenges remain. This paper reviews, analyses and extends state-of-the-art interval regression methods, presenting in-depth experimental results and introducing novel visualisations, to enhance model interpretability and provide context and data-specific recommendations for algorithm selection.

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Linear regression of interval-valued data based on complete information in hypercubes

Cited by 27 publications

References 12 publications

Thinking by classes in data science: the symbolic data analysis paradigm

Thinking by classes in data science: the symbolic data analysis paradigm

Matrix Factorization with Interval-Valued Data

Towards Handling Uncertainty-at-Source in AI -- A Review and Next Steps for Interval Regression

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