Some equivalence relationships of regularized regressions

Zhang, Yun; Thakar, Juilee; Topham, David J.; Falsey, Ann R.; Zeng, Delu; Qiu, Xing

doi:10.13164/ma.2018.01

Cited by 2 publications

(1 citation statement)

References 18 publications

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“…For the cases (i) and (ii), examples include Cardot et al (1991Cardot et al ( , 2003, James (2002), Hu et al (2004, Amato et al (2006), Horowitz (2007), Ferraty andVieu (2009), Cook et al (2010), Chen et al (2011), Dou et al (2012), Febrero-Bande and Gonzalez-Manteiga (2013, and Goia and Vieu (2015). For the case (iii), see Ramsay and Dalzell (1991), Fan and Zhang (1999), S ¸ent ürk and M üller (2005, 2008, Yao et al (2005), Harezlak et al (2007), Matsui et al (2009), Valderrama et al (2010, He et al (2005), Jiang and Wang (2011), Ivanescu et al (2015), Chiou et al (2016) and Zhang et al (2018). Also, M üller and Stadtmuller (2005), Horvath andKokoszka (2012), andCuevas (2014) present an excellent overview of research on functional regression models and their applications.…”

Section: Introductionmentioning

confidence: 99%

Functional linear models for interval-valued data

Beyaztaş

Shang

Abdel‐Salam

2020

Communications in Statistics - Simulation and Computation

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Aggregation of large databases in a specific format is a frequently used process to make the data easily manageable. Interval-valued data is one of the data types that is generated by such an aggregation process. Using traditional methods to analyze interval-valued data results in loss of information, and thus, several interval-valued data models have been proposed to gather reliable information from such data types. On the other hand, recent technological developments have led to high dimensional and complex data in many application areas, which may not be analyzed by traditional techniques. Functional data analysis is one of the most commonly used techniques to analyze such complex datasets. While the functional extensions of much traditional statistical techniques are available, the functional form of the interval-valued data has not been studied well. This paper introduces the functional forms of some well-known regression models that take interval-valued data. The proposed methods are based on the function-on-function regression model, where both the response and predictor/s are functional. Through several Monte Carlo simulations and empirical data analysis, the finite sample performance of the proposed methods is evaluated and compared with the state-of-the-art.Due to recent technological advances, the process of collecting data has become complicated, causing high dimensional and complex data structures. Symbolic data analysis is one of the commonly used methods in modeling such complex and large datasets, see Billard (2011) and Noirhomme-Fraiture and Brito (2011) for recent developments in symbolic data analysis. Contrary to single-valued observations in p-dimensional space where classical statistical methods work on, symbolic data may be in the form of hypercubes in p-dimensional space. There are many symbolic data types, for example, list, histogram, modal-valued, and interval-valued data. In this research, we restrict our attention to the interval-valued data only. The data expressed in an interval format (minimum and maximum values of the data) is called the interval-valued data. Such datasets are frequently encountered in daily life, for example, air and/or surface temperature, wind speed, energy production, blood pressure, and exchange rates. The main problem encountered during the modeling of the interval-valued data with classical statistical techniques is "how the variability of observations within the range is involved in modeling?". Traditional methods analyze interval-valued data using its summary (i.e., mid-points), and this approach results in loss of information. Therefore, interval-valued data analysis techniques are needed to obtain more accurate information.The early studies about the interval-valued data regression were conducted by Billard and Diday (2000), who extended traditional statistical techniques to the interval-valued data. Billard and Diday (2002) extended several classical regression models to interval-valued data and they proposed a regression equation for fitting histogram...

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Section: Introductionmentioning

confidence: 99%

Functional linear models for interval-valued data

Beyaztaş

Shang

Abdel‐Salam

2020

Communications in Statistics - Simulation and Computation

View full text Add to dashboard Cite

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A comparison of parameter estimation in function-on-function regression

Beyaztaş

Shang

2020

Communications in Statistics - Simulation and Computation

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Recent technological developments have enabled us to collect complex and high-dimensional data in many scientific fields, such as population health, meteorology, econometrics, geology, and psychology. It is common to encounter such datasets collected repeatedly over a continuum. Functional data, whose sample elements are functions in the graphical forms of curves, images, and shapes, characterize these data types. Functional data analysis techniques reduce the complex structure of these data and focus on the dependences within and (possibly) between the curves. A common research question is to investigate the relationships in regression models that involve at least one functional variable. However, the performance of functional regression models depends on several factors, such as the smoothing technique, the number of basis functions, and the estimation method. This paper provides a selective comparison for function-on-function regression models where both the response and predictor(s) are functions, to determine the optimal choice of basis function from a set of model evaluation criteria. We also propose a bootstrap method to construct a confidence interval for the response function. The numerical comparisons are implemented through Monte Carlo simulations and two real data examples.where Φ(t) = {φ 1 (t), · · · , φ K Y (t)} and Ψ(s) = {ψ m1 (s), · · · , ψ mK m,X (s)} denote the vectors of the basis functions, c i = {c i1 , · · · , c iK Y } and d im = {d im1 , · · · , d imK m,X } are the coefficient vectors, and K Y and K m,X are the number of basis functions used for approximating the functional response and functional predictors, respectively. Similarly, the bivariate coefficient function is defined as: β m (s, t) = ∑ j,k ψ mj (s)b mjk φ k (t) = Ψ m (s)B m Φ(t),where B m = (b mjk ) j,k denote a K m,X × K Y dimensional coefficient matrix. Accordingly, the regression

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Some equivalence relationships of regularized regressions

Cited by 2 publications

References 18 publications

Functional linear models for interval-valued data

Functional linear models for interval-valued data

A comparison of parameter estimation in function-on-function regression

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