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Regularized Canonical Correlation Analysis
Regularized canonical correlation analysis is a way of using ridge regression to solve the singularity problem in the cross-covariance matrices of canonical correlation analysis. By converting \operatorname(X, X) and \operatorname(Y, Y) into \operatorname(X, X) + \lambda I_X and \operatorname(Y, Y) + \lambda I_Y, it ensures that the above matrices will have reliable inverses. The idea probably dates back to Hrishikesh D. Vinod's publication in 1976 where he called it "Canonical ridge". It has been suggested for use in the analysis of functional neuroimaging data as such data are often singular. It is possible to compute the regularized canonical vectors in the lower-dimensional space. Section 3.18.5 References * {{cite journal, last1=Leurgans, first1=S.E., author1-link=Sue Leurgans, last2=Moyeed, first2=R.A., last3=Silverman, first3=B.W., title=Canonical correlation analysis when the data are curves, journal=Journal of the Royal Statistical Society The ''Journal o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ridge Regression
Ridge regression is a method of estimating the coefficients of multiple-regression models in scenarios where the independent variables are highly correlated. It has been used in many fields including econometrics, chemistry, and engineering. Also known as Tikhonov regularization, named for Andrey Tikhonov, it is a method of regularization of ill-posed problems. It is particularly useful to mitigate the problem of multicollinearity in linear regression, which commonly occurs in models with large numbers of parameters. In general, the method provides improved efficiency in parameter estimation problems in exchange for a tolerable amount of bias (see bias–variance tradeoff). The theory was first introduced by Hoerl and Kennard in 1970 in their ''Technometrics'' papers “RIDGE regressions: biased estimation of nonorthogonal problems” and “RIDGE regressions: applications in nonorthogonal problems”. This was the result of ten years of research into the field of ridge analysis. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singularity Theory
In mathematics, singularity theory studies spaces that are almost manifolds, but not quite. A string can serve as an example of a one-dimensional manifold, if one neglects its thickness. A singularity can be made by balling it up, dropping it on the floor, and flattening it. In some places the flat string will cross itself in an approximate "X" shape. The points on the floor where it does this are one kind of singularity, the double point: one bit of the floor corresponds to more than one bit of string. Perhaps the string will also touch itself without crossing, like an underlined "U". This is another kind of singularity. Unlike the double point, it is not ''stable'', in the sense that a small push will lift the bottom of the "U" away from the "underline". Vladimir Arnold defines the main goal of singularity theory as describing how objects depend on parameters, particularly in cases where the properties undergo sudden change under a small variation of the parameters. These ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cross-covariance Matrix
In probability theory and statistics, a cross-covariance matrix is a matrix whose element in the ''i'', ''j'' position is the covariance between the ''i''-th element of a random vector and ''j''-th element of another random vector. A random vector is a random variable with multiple dimensions. Each element of the vector is a scalar random variable. Each element has either a finite number of ''observed'' empirical values or a finite or infinite number of ''potential'' values. The potential values are specified by a theoretical joint probability distribution. Intuitively, the cross-covariance matrix generalizes the notion of covariance to multiple dimensions. The cross-covariance matrix of two random vectors \mathbf and \mathbf is typically denoted by \operatorname_ or \Sigma_. Definition For random vectors \mathbf and \mathbf, each containing random elements whose expected value and variance exist, the cross-covariance matrix of \mathbf and \mathbf is defined by where \math ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Canonical Correlation Analysis
In statistics, canonical-correlation analysis (CCA), also called canonical variates analysis, is a way of inferring information from cross-covariance matrices. If we have two vectors ''X'' = (''X''1, ..., ''X''''n'') and ''Y'' = (''Y''1, ..., ''Y''''m'') of random variables, and there are correlations among the variables, then canonical-correlation analysis will find linear combinations of ''X'' and ''Y'' which have maximum correlation with each other. T. R. Knapp notes that "virtually all of the commonly encountered parametric tests of significance can be treated as special cases of canonical-correlation analysis, which is the general procedure for investigating the relationships between two sets of variables." The method was first introduced by Harold Hotelling in 1936, although in the context of angles between flats the mathematical concept was published by Jordan in 1875. Definition Given two column vectors X = (x_1, \dots, x_n)^T ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Inverse
In linear algebra, an -by- square matrix is called invertible (also nonsingular or nondegenerate), if there exists an -by- square matrix such that :\mathbf = \mathbf = \mathbf_n \ where denotes the -by- identity matrix and the multiplication used is ordinary matrix multiplication. If this is the case, then the matrix is uniquely determined by , and is called the (multiplicative) ''inverse'' of , denoted by . Matrix inversion is the process of finding the matrix that satisfies the prior equation for a given invertible matrix . A square matrix that is ''not'' invertible is called singular or degenerate. A square matrix is singular if and only if its determinant is zero. Singular matrices are rare in the sense that if a square matrix's entries are randomly selected from any finite region on the number line or complex plane, the probability that the matrix is singular is 0, that is, it will "almost never" be singular. Non-square matrices (-by- matrices for which ) do not hav ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hrishikesh D
Rishikesh, also spelt as Hrishikesh, is a city near Dehradun in Dehradun district of the Indian state Uttarakhand. It is situated on the right bank of the Ganges River and is a pilgrimage town for Hindus, with ancient sages and saints meditating here in search of higher knowledge. There are numerous temples and ashrams built along the banks of the river. It is known as the "''Gateway to the Garhwal Himalayas''" and "''Yoga Capital of the World''". The city has hosted the annual "International Yoga Festival" on the first week of March since 1999. Rishikesh is a vegetarian-only and alcohol-free city. The Tehri Dam is just away and Uttarkashi, a popular yoga destination, is uphill on the way to Gangotri. Rishikesh is the starting point for travelling to the four Chota Char Dham pilgrimage places: Badrinath, Kedarnath, Gangotri, and Yamunotri. It's also a starting point for the Himalayan tourist destinations such as Harsil, Chopta, Auli and famous summer and winter trekking des ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Econometrics
The ''Journal of Econometrics'' is a scholarly journal in econometrics. It was first published in 1973. Its current managing editors are Serena Ng and Elie Tamer, Torben Andersen and Xiaohong Chen serve as editors. The journal publishes work dealing with estimation and other methodological aspects of the application of statistical inference to economic data, as well as papers dealing with the application of econometric techniques to economics. The journal also publishes a supplement to the Journal of Econometrics which is called "Annals of Econometrics". Each issue of the Annals includes a collection of papers on a single topic selected by the editor of the issue. See also * ''Econometrics Journal'' References External links Homepage Econometrics, Journal of Econometrics journals Econometrics Econometrics is the application of Statistics, statistical methods to economic data in order to give Empirical evidence, empirical content to economic relationships.M. Hashem P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Functional Neuroimaging
Functional neuroimaging is the use of neuroimaging technology to measure an aspect of brain function, often with a view to understanding the relationship between activity in certain brain areas and specific mental functions. It is primarily used as a research tool in cognitive neuroscience, cognitive psychology, neuropsychology, and social neuroscience. Overview Common methods of functional neuroimaging include * Positron emission tomography (PET) * Functional magnetic resonance imaging (fMRI) * Electroencephalography (EEG) * Magnetoencephalography (MEG) * Functional near-infrared spectroscopy (fNIRS) * Single-photon emission computed tomography (SPECT) * Functional ultrasound imaging (fUS) PET, fMRI, fNIRS and fUS can measure localized changes in cerebral blood flow related to neural activity. These changes are referred to as ''activations''. Regions of the brain which are activated when a subject performs a particular task may play a role in the computational neuroscience, n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
NeuroImage
''NeuroImage'' is a peer-reviewed scientific journal covering research on neuroimaging, including functional neuroimaging and functional human brain mapping. The current Editor in Chief is Michael Breakspear. Abstracts from the annual meeting of the Organization for Human Brain Mapping have been published as supplements to the journal. Members of the Organization for Human Brain Mapping are eligible for reduced subscription rates. In 2012, Elsevier launched an online-only, open access sister journal to ''NeuroImage'', entitled '' NeuroImage: Clinical''. Among prolific authors publishing numerous articles in the journal are Karl J. Friston, Arthur W. Toga, Paul M. Thompson and Karl Zilles. Related journals are Human Brain Mapping, Frontiers in Human Neuroscience, Cerebral Cortex, Magnetic Resonance in Medicine, Journal of Neuroscience and Journal of Cognitive Neuroscience. Abstracting and indexing The journal is abstracted and indexed in Scopus, Science Citation Index, Curre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Technical University Of Denmark
The Technical University of Denmark ( da, Danmarks Tekniske Universitet), often simply referred to as DTU, is a polytechnic university and school of engineering. It was founded in 1829 at the initiative of Hans Christian Ørsted as Denmark's first polytechnic, and it is today ranked among Europe's leading engineering institutions. It is located in the town Kongens Lyngby, north of central Copenhagen, Denmark. Along with École Polytechnique in Paris, École Polytechnique Fédérale de Lausanne, Eindhoven University of Technology, Technical University of Munich and Technion – Israel Institute of Technology, DTU is a member of EuroTech Universities Alliance. History DTU was founded in 1829 as the "College of Advanced Technology" (Danish: Den Polytekniske Læreanstalt). The Physicist Hans Christian Ørsted, at that time a professor at the University of Copenhagen, was one of the driving forces behind this initiative. He was inspired by the École Polytechnique in Paris, Fran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The Royal Statistical Society
The ''Journal of the Royal Statistical Society'' is a peer-reviewed scientific journal of statistics. It comprises three series and is published by Wiley for the Royal Statistical Society. History The Statistical Society of London was founded in 1834, but would not begin producing a journal for four years. From 1834 to 1837, members of the society would read the results of their studies to the other members, and some details were recorded in the proceedings. The first study reported to the society in 1834 was a simple survey of the occupations of people in Manchester, England. Conducted by going door-to-door and inquiring, the study revealed that the most common profession was mill-hands, followed closely by weavers. When founded, the membership of the Statistical Society of London overlapped almost completely with the statistical section of the British Association for the Advancement of Science. In 1837 a volume of ''Transactions of the Statistical Society of London'' were wri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Analysis
Analysis is the branch of mathematics dealing with continuous functions, limit (mathematics), limits, and related theories, such as Derivative, differentiation, Integral, integration, measure (mathematics), measure, infinite sequences, series (mathematics), series, and analytic functions. These theories are usually studied in the context of Real number, real and Complex number, complex numbers and Function (mathematics), functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any Space (mathematics), space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space). History Ancient Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
