Related papers: Dimensional Analysis: A Centenary Update
We present an innovative approach to dimensional analysis, based on a general representation theorem for complete quantity functions admitting a covariant scalar representation; this theorem is in turn grounded in a purely algebraic theory…
In one-dimensional case the search for presence of the anomalous phenomena in multiplicity distributions is usually performed in frame of the horizontal, vertical and mixed types of the analysis. We show that if the data involve a…
Fluctuations in the return time statistics of a dynamical system can be described by a new spectrum of dimensions. Comparison with the usual multifractal analysis of measures is presented, and difference between the two corresponding sets…
In this paper, we develop a systematic theory for high dimensional analysis of variance in multivariate linear regression, where the dimension and the number of coefficients can both grow with the sample size. We propose a new \emph{U}~type…
This chapter explores the notion of "dimension" of a set. Various power laws by which an Euclidean space can be characterized are used to define dimensions, which then explore different aspects of the set. Also discussed are the…
Both a general and a diagonal u-invariant for forms of higher degree are defined, generalizing the u-invariant of quadratic forms. Both old and new results on these invariants are collected.
Generalized dimensions of multifractal measures are usually seen as static objects, related to the scaling properties of suitable partition functions, or moments of measures of cells. When these measures are invariant for the flow of a…
The work relates to a new way for analysis of one-dimensional stochastic systems, based on consideration of its higher order difference structure. From this point of view, the deterministic and random processes are analyzed. A new numerical…
A novel method for common and individual feature analysis from exceedingly large-scale data is proposed, in order to ensure the tractability of both the computation and storage and thus mitigate the curse of dimensionality, a major…
We introduce a new method of performing high dimensional discriminant analysis, which we call multiDA. We achieve this by constructing a hybrid model that seamlessly integrates a multiclass diagonal discriminant analysis model and feature…
In the rapidly growing literature on explanation algorithms, it often remains unclear what precisely these algorithms are for and how they should be used. In this position paper, we argue for a novel and pragmatic perspective: Explainable…
Dimensionality reduction is a common method for analyzing and visualizing high-dimensional data across domains. Dimensionality-reduction algorithms involve complex optimizations and the reduced dimensions computed by these algorithms…
In current research, machine and deep learning solutions for the classification of temporal data are shifting from single-channel datasets (univariate) to problems with multiple channels of information (multivariate). The majority of these…
Based on the matrix expression of general nonlinear numerical analogues presented by the present author, this paper proposes a novel philosophy of nonlinear computation and analysis. The nonlinear problems are considered an ill-posed linear…
The paper investigates the problem of performing correlation analysis when the number of observations is very large. In such a case, it is often necessary to combine the random observations to achieve dimensionality reduction of the…
A hierarchical scheme for clustering data is presented which applies to spaces with a high number of dimension ($N_{_{D}}>3$). The data set is first reduced to a smaller set of partitions (multi-dimensional bins). Multiple clustering…
Archetype and archetypoid analysis can be extended to functional data. Each function is represented as a mixture of actual observations (functional archetypoids) or functional archetypes, which are a mixture of observations in the data set.…
Dimensionality reduction is a popular preprocessing and a widely used tool in data mining. Transparency, which is usually achieved by means of explanations, is nowadays a widely accepted and crucial requirement of machine learning based…
Interactive model analysis, the process of understanding, diagnosing, and refining a machine learning model with the help of interactive visualization, is very important for users to efficiently solve real-world artificial intelligence and…
We consider an enlarged dimension reduction space in functional inverse regression. Our operator and functional analysis based approach facilitates a compact and rigorous formulation of the functional inverse regression problem. It also…