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Kernel PCA
In the field of multivariate statistics, kernel principal component analysis (kernel PCA) is an extension of principal component analysis (PCA) using techniques of kernel methods. Using a kernel, the originally linear operations of PCA are performed in a reproducing kernel Hilbert space. Background: Linear PCA Recall that conventional PCA operates on zero-centered data; that is, :\frac\sum_^N \mathbf_i = \mathbf, where \mathbf_i is one of the N multivariate observations. It operates by diagonalizing the covariance matrix, :C=\frac\sum_^N \mathbf_i\mathbf_i^\top in other words, it gives an eigendecomposition of a matrix, eigendecomposition of the covariance matrix: :\lambda \mathbf=C\mathbf which can be rewritten as :\lambda \mathbf_i^\top \mathbf=\mathbf_i^\top C\mathbf \quad \textrm~i=1,\ldots,N. (See also: Covariance matrix#Covariance matrix as a linear operator, Covariance matrix as a linear operator) Introduction of the Kernel to PCA To understand the utility of kernel PCA, pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Multivariate Statistics
Multivariate statistics is a subdivision of statistics encompassing the simultaneous observation and analysis of more than one outcome variable. Multivariate statistics concerns understanding the different aims and background of each of the different forms of multivariate analysis, and how they relate to each other. The practical application of multivariate statistics to a particular problem may involve several types of univariate and multivariate analyses in order to understand the relationships between variables and their relevance to the problem being studied. In addition, multivariate statistics is concerned with multivariate probability distributions, in terms of both :*how these can be used to represent the distributions of observed data; :*how they can be used as part of statistical inference, particularly where several different quantities are of interest to the same analysis. Certain types of problems involving multivariate data, for example simple linear regression an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centering Matrix
In mathematics and multivariate statistics, the centering matrixJohn I. Marden, ''Analyzing and Modeling Rank Data'', Chapman & Hall, 1995, , page 59. is a symmetric and idempotent matrix, which when multiplied with a vector has the same effect as subtracting the mean of the components of the vector from every component of that vector. Definition The centering matrix of size ''n'' is defined as the ''n''-by-''n'' matrix :C_n = I_n - \tfracJ_n where I_n\, is the identity matrix of size ''n'' and J_n is an ''n''-by-''n'' matrix of all 1's. For example :C_1 = \begin 0 \end , :C_2= \left \begin 1 & 0 \\ 0 & 1 \end \right- \frac\left \begin 1 & 1 \\ 1 & 1 \end \right = \left \begin \frac & -\frac \\ -\frac & \frac \end \right , :C_3 = \left \begin 1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 1 \end \right- \frac\left \begin 1 & 1 & 1 \\ 1 & 1 & 1 \\ 1 & 1 & 1 \end \right = \left \begin \frac & -\frac & -\frac \\ -\frac & \frac & -\frac \\ -\frac & -\frac & \frac \end \right ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Machine Learning Algorithms
The following outline is provided as an overview of and topical guide to machine learning. Machine learning is a subfield of soft computing within computer science that evolved from the study of pattern recognition and computational learning theory in artificial intelligence.http://www.britannica.com/EBchecked/topic/1116194/machine-learning In 1959, Arthur Samuel defined machine learning as a "field of study that gives computers the ability to learn without being explicitly programmed". Machine learning explores the study and construction of algorithms that can learn from and make predictions on data. Such algorithms operate by building a model from an example training set of input observations in order to make data-driven predictions or decisions expressed as outputs, rather than following strictly static program instructions. What ''type'' of thing is machine learning? * An academic discipline * A branch of science ** An applied science *** A subfield of computer science ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Signal Processing
Signal processing is an electrical engineering subfield that focuses on analyzing, modifying and synthesizing ''signals'', such as audio signal processing, sound, image processing, images, and scientific measurements. Signal processing techniques are used to optimize transmissions, Data storage, digital storage efficiency, correcting distorted signals, subjective video quality and to also detect or pinpoint components of interest in a measured signal. History According to Alan V. Oppenheim and Ronald W. Schafer, the principles of signal processing can be found in the classical numerical analysis techniques of the 17th century. They further state that the digital refinement of these techniques can be found in the digital control systems of the 1940s and 1950s. In 1948, Claude Shannon wrote the influential paper "A Mathematical Theory of Communication" which was published in the Bell System Technical Journal. The paper laid the groundwork for later development of information c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimension Reduction
Dimensionality reduction, or dimension reduction, is the transformation of data from a high-dimensional space into a low-dimensional space so that the low-dimensional representation retains some meaningful properties of the original data, ideally close to its intrinsic dimension. Working in high-dimensional spaces can be undesirable for many reasons; raw data are often sparse as a consequence of the curse of dimensionality, and analyzing the data is usually computationally intractable (hard to control or deal with). Dimensionality reduction is common in fields that deal with large numbers of observations and/or large numbers of variables, such as signal processing, speech recognition, neuroinformatics, and bioinformatics. Methods are commonly divided into linear and nonlinear approaches. Approaches can also be divided into feature selection and feature extraction. Dimensionality reduction can be used for noise reduction, data visualization, cluster analysis, or as an intermediate ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
