Structure-preserving non-linear PCA for matrices

We propose MNPCA, a novel non-linear generalization of (2D)$^2${PCA}, a classical linear method for the simultaneous dimension reduction of both rows and columns of a set of matrix-valued data. MNPCA is based on optimizing over separate non-linear mappings on the left and right singular spaces of the observations, essentially amounting to the decoupling of the two sides of the matrices. We develop a comprehensive theoretical framework for MNPCA by viewing it as an eigenproblem in reproducing kernel Hilbert spaces. We study the resulting estimators on both population and sample levels, deriving their convergence rates and formulating a coordinate representation to allow the method to be used in practice. Simulations and a real data example demonstrate MNPCA's good performance over its competitors.Comment: 23 pages, 4 figure

Sufficient dimension reduction via principal Lq support vector machine

Principal support vector machine was proposed recently by Li, Artemiou and Li (2011) to combine L$1$ support vector machine and sufficient dimension reduction. We introduce the principal L$q$ support vector machine as a unified framework for linear and nonlinear sufficient dimension reduction. By noticing that the solution of L$1$ support vector machine may not be unique, we set $q>1$ to ensure the uniqueness of the solution. The asymptotic distribution of the proposed estimators are derived for $q> 1$. We demonstrate through numerical studies that the proposed L$2$ support vector machine estimators improve existing methods in accuracy, and are less sensitive to the tuning parameter selection

Penalized principal logistic regression for sparse sufficient dimension reduction

Sufficient dimension reduction (SDR) is a successive tool for reducing the dimensionality of predictors by finding the central subspace, a minimal subspace of predictors that preserves all the regression information. When predictor dimension is large, it is often assumed that only a small number of predictors is informative. In this regard, sparse SDR is desired to achieve variable selection and dimension reduction simultaneously. We propose a principal logistic regression (PLR) as a new SDR tool and extend it to a penalized version for sparse SDR. Asymptotic analysis shows that the penalized PLR enjoys the oracle property. Numerical investigation supports the advantageous performance of the proposed methods

Sliced inverse median difference regression

In this paper we propose a suffcient dimension reduction algorithm based on the difference of inverse medians. The classic methodology based on inverse means in each slice was recently extended, by using inverse medians, to robustify existing methodology at the presence of outliers. Our effort is focused on using differences between inverse medians in pairs of slices. We demonstrate that our method outperforms existing methods at the presence of outliers. We also propose a second algorithm which is not affected by the ordering of slices when the response variable is categorical with no underlying ordering of its value

Dominant vs non-dominant shoulder morphology in volleyball players and associations with shoulder pain and spike speed

Background: The aims of our study were to compare the dominant (DOM) and non-dominant (NDOM) shoulders of high-level volleyball athletes and identify possible associations of shoulder adaptations with spike speed (SS) and shoulder pathology. Materials-Methods: A total of 22 male volleyball players from two teams participating in the first division of the Cypriot championship underwent clinical shoulder tests and simple measurements around their shoulder girdle joints bilaterally. SS was measured with the use of a sports speed radar. Results: Compared with the NDOM side, the DOM scapula was more lateralised, the DOM dorsal capsule demonstrated greater laxity, the DOM dorsal muscles stretching ability was compromised, and the DOM pectoralis muscle was more lengthened. Players with present or past DOM shoulder pain demonstrated greater laxity in their DOM dorsal capsule, tightening of their DOM inferior capsule, and lower SS compared with those without shoulder pain. Dorsal capsule measurements bilaterally were significant predictors of SS. None of the shoulder measurements was associated with team roles or infraspinatus atrophy, while scapular lateralisation was more pronounced with increasing years of experience, and scapular antetilting was greater with increasing age. Conclusions: Adaptations of the DOM shoulder may be linked to pathology and performance. We describe simple shoulder measurements that may have the potential to predict chronic shoulder injury and become part of injury prevention programmes. Detailed biomechanical and large prospective studies are warranted to assess the validity of our findings and reach more definitive conclusions

Cost-based reweighting for Principal Lq SVM for sufficient dimension reduction

In this work we try to address the imbalance of the number of points which naturally occurs when slicing the response in Sufficient Dimension Reduction methods (SDR). Specifically, some recently proposed support vector machine based (SVM-based) methodology suffers a lot more due to the properties of the SVM algorithm. We target a recently proposed algorithm called Principal LqSVM and we propose the reweighting based on a different cost. We demonstrate that our reweighted proposal works better than the original algorithm in simulated and real data

Using adaptively weighted large margin classifiers for robust sufficient dimension reduction

In this paper we combine adaptively weighted large margin classifiers with Support Vector Machine (SVM)-based dimension reduction methods to create dimension reduction methods robust to the presence of extreme outliers. We discuss estimation and asymptotic properties of the algorithm. The good performance of the new algorithm is demonstrated through simulations and real data analysis

Comunidades zooplanctónicas estivales en relación con parámetros ambientales en el golfo de Kavala, norte del mar Egeo

Shallow coastal areas are ecosystems with high productivity. Although the eastern part of the Mediterranean Sea is oligotrophic, the shallow coastal waters of the northern Aegean, such as Kavala Gulf, are productive due to the influence of the Black Sea water and the presence of freshwater input from three rivers. The aim of this work was to determine the structure of zooplankton communities in Kavala Gulf in the summer of 2002 and 2003 and to investigate their relation to environmental variables. Zooplankton communities were characterized by the presence of common coastal Cladocera, such as Penilia avirostris, small pelagic Copepoda, such as the calanoida Acartia clausi and the cyclopoida Oithona plumifera, and Tunicata, such as Oikopleura, Fritillaria and Doliolidae. The abundances corresponded to the peak of the warm period and were significantly greater in 2002 because of a P. avirostris bloom, which seemed to have better exploited the environmental sources favouring its dominance in the area. Overall, the structure of summer mesozooplankton communities in Kavala Gulf follows the pattern exhibited by mesozooplankton communities in other Greek coastal areas of the northern Aegean Sea.Las aguas costeras poco profundas son ecosistemas con alta productividad. Aunque el mar Mediterráneo oriental es oligotrófico, las aguas costeras poco profundas al norte del mar Egeo, como el golfo de Kavala, son productivas debido a la influencia de aguas provenientes del mar Negro y a los aportes de agua dulce procedentes de tres ríos. El objetivo de este trabajo fue determinar la estructura de las comunidades de zooplancton en el golfo de Kavala durante los veranos de 2002 y 2003, e investigar su relación con variables ambientales. Las comunidades de zooplancton se caracterizaron por la presencia de cladóceros costeros comunes, como Penilia avirostris, pequeños copépodos pelágicos, tales como el calanoide Acartia clausi y el ciclopoide Oithona plumifera, y tunicados como Oikopleura, Fritillaria y Doliolidae. Las abundancias correspondieron al pico del periodo cálido y fueron significativamente mayores en 2002 debido a una proliferación de P. avirostris, que parece que supo explotar mejor las condiciones ambientales favoreciendo su dominio en la zona. En general, la estructura de las comunidades de mesozooplancton de verano en el golfo de Kavala siguen el patrón exhibido por las comunidades de mesozooplancton en otras areas costeras griegas al norte del mar Egeo

On principal components regression with hilbertian predictors

We demonstrate that, in a regression setting with a Hilbertian predictor, a response variable is more likely to be more highly correlated with the leading principal components of the predictor than with trailing ones. This is despite the extraction procedure being unsupervised. Our results are established under the conditional independence model, which includes linear regression and single-index models as special cases, with some assumptions on the regression vector. These results are a generalisation of earlier work which showed that this phenomenon holds for predictors which are real random vectors. A simulation study is used to quantify the phenomenon

High-dimensional sufficient dimension reduction through principal projections

We develop in this work a new dimension reduction method for high-dimensional settings. The proposed procedure is based on a principal support vector machine framework where principal projections are used in order to overcome the non-invertibility of the covariance matrix. Using a series of equivalences we show that one can accurately recover the central subspace using a projection on a lower dimensional subspace and then applying an ℓ1 penalization strategy to obtain sparse estimators of the sufficient directions. Based next on a desparsified estimator, we provide an inferential procedure for high-dimensional models that allows testing for the importance of variables in determining the sufficient direction. Theoretical properties of the methodology are illustrated and computational advantages are demonstrated with simulated and real data experiments