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Independent component analysis : What is independent component analysis? by Aapo Hyvärinen Independent Component Analysis: A Tutorial by Aapo Hyvärinen A Tutorial on Independent Component Analysis FastICA as a package for Matlab, in R language, C++ ICALAB Toolboxes for Matlab, developed at RIKEN High Performance Signal... |
Locality-sensitive hashing : In computer science, locality-sensitive hashing (LSH) is a fuzzy hashing technique that hashes similar input items into the same "buckets" with high probability. (The number of buckets is much smaller than the universe of possible input items.) Since similar items end up in the same buckets... |
Locality-sensitive hashing : A finite family F of functions h : M → S is defined to be an LSH family for a metric space M = ( M , d ) =(M,d) , a threshold r > 0 , an approximation factor c > 1 , and probabilities p 1 > p 2 >p_ if it satisfies the following condition. For any two points a , b ∈ M and a hash functio... |
Locality-sensitive hashing : LSH has been applied to several problem domains, including: Near-duplicate detection Hierarchical clustering Genome-wide association study Image similarity identification VisualRank Gene expression similarity identification Audio similarity identification Nearest neighbor search Audio finge... |
Locality-sensitive hashing : One of the main applications of LSH is to provide a method for efficient approximate nearest neighbor search algorithms. Consider an LSH family F . The algorithm has two main parameters: the width parameter k and the number of hash tables L. In the first step, we define a new family G of ... |
Locality-sensitive hashing : Bloom filter – Data structure for approximate set membership Curse of dimensionality – Difficulties arising when analyzing data with many aspects ("dimensions") Feature hashing – Vectorizing features using a hash function Fourier-related transforms Geohash – Public domain geocoding invented... |
Locality-sensitive hashing : Samet, H. (2006) Foundations of Multidimensional and Metric Data Structures. Morgan Kaufmann. ISBN 0-12-369446-9 Indyk, Piotr; Motwani, Rajeev; Raghavan, Prabhakar; Vempala, Santosh (1997). "Locality-preserving hashing in multidimensional spaces". Proceedings of the twenty-ninth annual ACM ... |
Locality-sensitive hashing : Alex Andoni's LSH homepage LSHKIT: A C++ Locality Sensitive Hashing Library A Python Locality Sensitive Hashing library that optionally supports persistence via redis Caltech Large Scale Image Search Toolbox: a Matlab toolbox implementing several LSH hash functions, in addition to Kd-Trees,... |
Low-rank approximation : In mathematics, low-rank approximation refers to the process of approximating a given matrix by a matrix of lower rank. More precisely, it is a minimization problem, in which the cost function measures the fit between a given matrix (the data) and an approximating matrix (the optimization varia... |
Low-rank approximation : Given structure specification S : R n p → R m × n :\mathbb ^\to \mathbb ^ , vector of structure parameters p ∈ R n p ^ , norm ‖ ⋅ ‖ , and desired rank r , minimize over p ^ ‖ p − p ^ ‖ subject to rank ( S ( p ^ ) ) ≤ r . \quad \quad \|p-\|\quad \quad \operatorname ()\leq r. |
Low-rank approximation : Linear system identification, in which case the approximating matrix is Hankel structured. Machine learning, in which case the approximating matrix is nonlinearly structured. Recommender systems, in which cases the data matrix has missing values and the approximation is categorical. Distance ma... |
Low-rank approximation : The unstructured problem with fit measured by the Frobenius norm, i.e., minimize over D ^ ‖ D − D ^ ‖ F subject to rank ( D ^ ) ≤ r \quad \quad \|D-\|_\quad \quad \operatorname \leq r has an analytic solution in terms of the singular value decomposition of the data matrix. The result is refe... |
Low-rank approximation : Let A ∈ R m × n ^ be a real (possibly rectangular) matrix with m ≤ n . Suppose that A = U Σ V ⊤ is the singular value decomposition of A . Recall that U and V are orthogonal matrices, and Σ is an m × n diagonal matrix with entries ( σ 1 , σ 2 , ⋯ , σ m ) ,\sigma _,\cdots ,\sigma _) such... |
Low-rank approximation : Let A ∈ R m × n ^ be a real (possibly rectangular) matrix with m ≤ n . Suppose that A = U Σ V ⊤ is the singular value decomposition of A . We claim that the best rank k approximation to A in the Frobenius norm, denoted by ‖ ⋅ ‖ F , is given by A k = ∑ i = 1 k σ i u i v i ⊤ =\sum _^\sigma... |
Low-rank approximation : The Frobenius norm weights uniformly all elements of the approximation error D − D ^ . Prior knowledge about distribution of the errors can be taken into account by considering the weighted low-rank approximation problem minimize over D ^ vec ( D − D ^ ) ⊤ W vec ( D − D ^ ) subject to rank... |
Low-rank approximation : Let ‖ A ‖ p = ( ∑ i , j | A i , j p | ) 1 / p =\left(\sum _|A_^|\right)^ . For p = 2 , the fastest algorithm runs in n n z ( A ) + n ⋅ p o l y ( k / ϵ ) time. One of the important ideas been used is called Oblivious Subspace Embedding (OSE), it is first proposed by Sarlos. For p = 1 , it is ... |
Low-rank approximation : Let P = ,\ldots ,p_\ and Q = ,\ldots ,q_\ be two point sets in an arbitrary metric space. Let A represent the m × n matrix where A i , j = d i s t ( p i , q i ) =dist(p_,q_) . Such distances matrices are commonly computed in software packages and have applications to learning image manifold... |
Low-rank approximation : The low-rank approximation problems in the distributed and streaming setting has been considered in. |
Low-rank approximation : Using the equivalences rank ( D ^ ) ≤ r ⟺ there are P ∈ R m × r and L ∈ R r × n such that D ^ = P L ()\leq r\quad \iff \quad P\in \mathbb ^L\in \mathbb ^=PL and rank ( D ^ ) ≤ r ⟺ there is full row rank R ∈ R m − r × m such that R D ^ = 0 ()\leq r\quad \iff \quad R\in \mathbb ^R=0 the ... |
Low-rank approximation : The image representation of the rank constraint suggests a parameter optimization method in which the cost function is minimized alternatively over one of the variables ( P or L ) with the other one fixed. Although simultaneous minimization over both P and L is a difficult biconvex optimiza... |
Low-rank approximation : The alternating projections algorithm exploits the fact that the low rank approximation problem, parameterized in the image form, is bilinear in the variables P or L . The bilinear nature of the problem is effectively used in an alternative approach, called variable projections. Consider agai... |
Low-rank approximation : Usually, we want our new solution not only to be of low rank, but also satisfy other convex constraints due to application requirements. Our interested problem would be as follows, minimize over p ^ ‖ p − p ^ ‖ subject to rank ( S ( p ^ ) ) ≤ r and g ( p ^ ) ≤ 0 \quad \quad \|p-\|\quad \quad ... |
Low-rank approximation : CUR matrix approximation is made from the rows and columns of the original matrix |
Low-rank approximation : M. T. Chu, R. E. Funderlic, R. J. Plemmons, Structured low-rank approximation, Linear Algebra and its Applications, Volume 366, 1 June 2003, Pages 157–172 doi:10.1016/S0024-3795(02)00505-0 |
Low-rank approximation : C++ package for structured-low rank approximation |
Modes of variation : In statistics, modes of variation are a continuously indexed set of vectors or functions that are centered at a mean and are used to depict the variation in a population or sample. Typically, variation patterns in the data can be decomposed in descending order of eigenvalues with the directions rep... |
Modes of variation : Modes of variation are a natural extension of PCA and FPCA. |
Modes of variation : The formulation above is derived from properties of the population. Estimation is needed in real-world applications. The key idea is to estimate mean and covariance. |
Modes of variation : Modes of variation are useful to visualize and describe the variation patterns in the data sorted by the eigenvalues. In real-world applications, modes of variation associated with eigencomponents allow to interpret complex data, such as the evolution of function traits and other infinite-dimension... |
Multidimensional scaling : Multidimensional scaling (MDS) is a means of visualizing the level of similarity of individual cases of a data set. MDS is used to translate distances between each pair of n objects in a set into a configuration of n points mapped into an abstract Cartesian space. More technically, MDS refe... |
Multidimensional scaling : MDS algorithms fall into a taxonomy, depending on the meaning of the input matrix: |
Multidimensional scaling : The data to be analyzed is a collection of M objects (colors, faces, stocks, . . .) on which a distance function is defined, d i , j := := distance between i -th and j -th objects. These distances are the entries of the dissimilarity matrix D := ( d 1 , 1 d 1 , 2 ⋯ d 1 , M d 2 , 1 d 2 , 2 ... |
Multidimensional scaling : There are several steps in conducting MDS research: Formulating the problem – What variables do you want to compare? How many variables do you want to compare? What purpose is the study to be used for? Obtaining input data – For example, :- Respondents are asked a series of questions. For eac... |
Multidimensional scaling : ELKI includes two MDS implementations. MATLAB includes two MDS implementations (for classical (cmdscale) and non-classical (mdscale) MDS respectively). The R programming language offers several MDS implementations, e.g. base cmdscale function, packages smacof (mMDS and nMDS), and vegan (weigh... |
Multidimensional scaling : Data clustering t-distributed stochastic neighbor embedding Factor analysis Discriminant analysis Dimensionality reduction Distance geometry Cayley–Menger determinant Sammon mapping Iconography of correlations |
Multifactor dimensionality reduction : Multifactor dimensionality reduction (MDR) is a statistical approach, also used in machine learning automatic approaches, for detecting and characterizing combinations of attributes or independent variables that interact to influence a dependent or class variable. MDR was designed... |
Multifactor dimensionality reduction : Consider the following simple example using the exclusive OR (XOR) function. XOR is a logical operator that is commonly used in data mining and machine learning as an example of a function that is not linearly separable. The table below represents a simple dataset where the relati... |
Multifactor dimensionality reduction : As illustrated above, the basic constructive induction algorithm in MDR is very simple. However, its implementation for mining patterns from real data can be computationally complex. As with any machine learning algorithm there is always concern about overfitting. That is, machine... |
Multifactor dimensionality reduction : Numerous extensions to MDR have been introduced. These include family-based methods, fuzzy methods, covariate adjustment, odds ratios, risk scores, survival methods, robust methods, methods for quantitative traits, and many others. |
Multifactor dimensionality reduction : MDR has mostly been applied to detecting gene-gene interactions or epistasis in genetic studies of common human diseases such as atrial fibrillation, autism, bladder cancer, breast cancer, cardiovascular disease, hypertension, obesity, pancreatic cancer, prostate cancer and tuberc... |
Multifactor dimensionality reduction : www.epistasis.org provides an open-source and freely-available MDR software package. An R package for MDR. An sklearn-compatible Python implementation. An R package for Model-Based MDR. MDR in Weka. Generalized MDR. |
Multifactor dimensionality reduction : Data mining Dimensionality reduction Epistasis Feature Engineering Machine learning Multilinear subspace learning |
Multifactor dimensionality reduction : Michalski, R. S., "Pattern Recognition as Knowledge-Guided Computer Induction," Department of Computer Science Reports, No. 927, University of Illinois, Urbana, June 1978. |
Multilinear principal component analysis : Multilinear principal component analysis (MPCA) is a multilinear extension of principal component analysis (PCA) that is used to analyze M-way arrays, also informally referred to as "data tensors". M-way arrays may be modeled by linear tensor models, such as CANDECOMP/Parafac,... |
Multilinear principal component analysis : The MPCA solution follows the alternating least square (ALS) approach. It is iterative in nature. As in PCA, MPCA works on centered data. Centering is a little more complicated for tensors, and it is problem dependent. |
Multilinear principal component analysis : MPCA features: Supervised MPCA is employed in causal factor analysis that facilitates object recognition while a semi-supervised MPCA feature selection is employed in visualization tasks. |
Multilinear principal component analysis : Various extension of MPCA: Robust MPCA (RMPCA) Multi-Tensor Factorization, that also finds the number of components automatically (MTF) |
Multilinear principal component analysis : Matlab code: MPCA. Matlab code: UMPCA (including data). R code: MTF |
Multilinear subspace learning : Multilinear subspace learning is an approach for disentangling the causal factor of data formation and performing dimensionality reduction. The Dimensionality reduction can be performed on a data tensor that contains a collection of observations have been vectorized, or observations that... |
Multilinear subspace learning : Multilinear methods may be causal in nature and perform causal inference, or they may be simple regression methods from which no causal conclusion are drawn. Linear subspace learning algorithms are traditional dimensionality reduction techniques that are well suited for datasets that are... |
Multilinear subspace learning : MATLAB Tensor Toolbox by Sandia National Laboratories. The MPCA algorithm written in Matlab (MPCA+LDA included). The UMPCA algorithm written in Matlab (data included). The UMLDA algorithm written in Matlab (data included). |
Multilinear subspace learning : 3D gait data (third-order tensors): 128x88x20(21.2M); 64x44x20(9.9M); 32x22x10(3.2M); |
Multilinear subspace learning : CP decomposition Dimension reduction Multilinear algebra Multilinear Principal Component Analysis Tensor Tensor decomposition Tensor software Tucker decomposition == References == |
Multiple correspondence analysis : In statistics, multiple correspondence analysis (MCA) is a data analysis technique for nominal categorical data, used to detect and represent underlying structures in a data set. It does this by representing data as points in a low-dimensional Euclidean space. The procedure thus appea... |
Multiple correspondence analysis : MCA is performed by applying the CA algorithm to either an indicator matrix (also called complete disjunctive table – CDT) or a Burt table formed from these variables. An indicator matrix is an individuals × variables matrix, where the rows represent individuals and the columns are du... |
Multiple correspondence analysis : Since MCA is adapted to draw statistical conclusions from categorical variables (such as multiple choice questions), the first thing one needs to do is to transform quantitative data (such as age, size, weight, day time, etc) into categories (using for instance statistical quantiles).... |
Multiple correspondence analysis : In recent years, several students of Jean-Paul Benzécri have refined MCA and incorporated it into a more general framework of data analysis known as geometric data analysis. This involves the development of direct connections between simple correspondence analysis, principal component... |
Multiple correspondence analysis : In the social sciences, MCA is arguably best known for its application by Pierre Bourdieu, notably in his books La Distinction, Homo Academicus and The State Nobility. Bourdieu argued that there was an internal link between his vision of the social as spatial and relational --– captur... |
Multiple correspondence analysis : MCA can also be viewed as a PCA applied to the complete disjunctive table. To do this, the CDT must be transformed as follows. Let y i k denote the general term of the CDT. y i k is equal to 1 if individual i possesses the category k and 0 if not. Let denote p k , the proportion ... |
Multiple correspondence analysis : There are numerous software of data analysis that include MCA, such as STATA and SPSS. The R package FactoMineR also features MCA. This software is related to a book describing the basic methods for performing MCA . There is also a Python package for [1] which works with numpy array m... |
Multiple correspondence analysis : Le Roux, B. and H. Rouanet (2004), Geometric Data Analysis, From Correspondence Analysis to Structured Data Analysis at Google Books: [2] FactoMineR A R software devoted to exploratory data analysis. |
NOMINATE (scaling method) : NOMINATE (an acronym for Nominal Three-Step Estimation) is a multidimensional scaling application developed by US political scientists Keith T. Poole and Howard Rosenthal in the early 1980s to analyze preferential and choice data, such as legislative roll-call voting behavior. In its most we... |
NOMINATE (scaling method) : The main procedure is an application of multidimensional scaling techniques to political choice data. Though there are important technical differences between these types of NOMINATE scaling procedures; all operate under the same fundamental assumptions. First, that alternative choices can b... |
NOMINATE (scaling method) : Poole and Rosenthal demonstrate that—despite the many complexities of congressional representation and politics—roll call voting in both the House and the Senate can be organized and explained by no more than two dimensions throughout the sweep of American history. The first dimension (horiz... |
NOMINATE (scaling method) : For illustrative purposes, consider the following plots which use W-NOMINATE scores to scale members of Congress and uses the probabilistic voting model (in which legislators farther from the "cutting line" between "yea" and "nay" outcomes become more likely to vote in the predicted manner) ... |
NOMINATE (scaling method) : DW-NOMINATE scores have been used widely to describe the political ideology of political actors, political parties and political institutions. For instance, a score in the first dimension that is close to either pole means that such score is located at one of the extremes in the liberal-cons... |
NOMINATE (scaling method) : Poole and Rosenthal (beginning with their 1984 article "The Polarization of American Politics") have also used NOMINATE data to show that, since the 1970s, party delegations in Congress have become ideologically homogeneous and distant from one another (a phenomenon known as "polarization").... |
NOMINATE (scaling method) : NOMINATE has been used to test, refine, and/or develop wide-ranging theories and models of the United States Congress. In Ideology and Congress (pp. 270–271), Poole and Rosenthal agree that their findings are consistent with the "party cartel" model that Cox and McCubbins present in their 19... |
NOMINATE (scaling method) : Multidimensional scaling (MDS) Scale analysis (statistics) |
NOMINATE (scaling method) : Special issue of Studies in American Political Development on NOMINATE, 2016. |
NOMINATE (scaling method) : Archive of K7MOA.com ("Permanent Archive of the Research of Poole and Rosenthal and Colleagues"). Includes "NOMINATE and American Political History: A Primer." A more extensive introduction to NOMINATE "NOMINATE: A Short Intellectual History" (by Keith T. Poole) W-NOMINATE in R: Software and... |
Nonlinear dimensionality reduction : Nonlinear dimensionality reduction, also known as manifold learning, is any of various related techniques that aim to project high-dimensional data, potentially existing across non-linear manifolds which cannot be adequately captured by linear decomposition methods, onto lower-dimen... |
Nonlinear dimensionality reduction : High dimensional data can be hard for machines to work with, requiring significant time and space for analysis. It also presents a challenge for humans, since it's hard to visualize or understand data in more than three dimensions. Reducing the dimensionality of a data set, while ke... |
Nonlinear dimensionality reduction : A method based on proximity matrices is one where the data is presented to the algorithm in the form of a similarity matrix or a distance matrix. These methods all fall under the broader class of metric multidimensional scaling. The variations tend to be differences in how the proxi... |
Nonlinear dimensionality reduction : Waffles is an open source C++ library containing implementations of LLE, Manifold Sculpting, and some other manifold learning algorithms. UMAP.jl implements the method for the programming language Julia. The method has also been implemented in Python (code available on GitHub) |
Nonlinear dimensionality reduction : Manifold hypothesis Spectral submanifold Taken's theorem Whitney embedding theorem Discriminant analysis Elastic map Feature learning Growing self-organizing map (GSOM) Self-organizing map (SOM) |
Nonlinear dimensionality reduction : Murphy, Kevin P. (2022). "Manifold Learning". Probabilistic Machine Learning. MIT Press. pp. 682–699. ISBN 978-0-262-04682-4. |
Nonlinear dimensionality reduction : Isomap Generative Topographic Mapping Mike Tipping's Thesis Gaussian Process Latent Variable Model Locally Linear Embedding Relational Perspective Map DD-HDS homepage RankVisu homepage Short review of Diffusion Maps Nonlinear PCA by autoencoder neural networks |
Ordination (statistics) : Ordination or gradient analysis, in multivariate analysis, is a method complementary to data clustering, and used mainly in exploratory data analysis (rather than in hypothesis testing). In contrast to cluster analysis, ordination orders quantities in a (usually lower-dimensional) latent space... |
Ordination (statistics) : Ordination methods can broadly be categorized in eigenvector-, algorithm-, or model-based methods. Many classical ordination techniques, including principal components analysis, correspondence analysis (CA) and its derivatives (detrended correspondence analysis, canonical correspondence analys... |
Ordination (statistics) : Ordination can be used on the analysis of any set of multivariate objects. It is frequently used in several environmental or ecological sciences, particularly plant community ecology. It is also used in genetics and systems biology for microarray data analysis and in psychometrics. |
Ordination (statistics) : Multivariate statistics Principal components analysis Correspondence analysis Multiple correspondence analysis Detrended correspondence analysis Intrinsic dimension Latent space Latent variable model |
Ordination (statistics) : Birks, H.J.B., 1998. An Annotated Bibliography Of Canonical Correspondence Analysis And Related Constrained Ordination Methods 1986–1996. Botanical Institute, University of Bergen. World Wide Web: http://www.bio.umontreal.ca/Casgrain/cca_bib/index.html Braak, C.J.F. ter & I.C. Prentice 1988 A ... |
Ordination (statistics) : General http://ordination.okstate.edu/ The Ordination Web Page - Ordination Methods for Ecologists https://www.davidzeleny.net/anadat-r/doku.php/en:start https://link.springer.com/article/10.1007/s11192-015-1744-x Specific Techniques http://www.statsoft.com/textbook/stcoran.html http://www.sta... |
Preference regression : Preference regression is a statistical technique used by marketers to determine consumers’ preferred core benefits. It usually supplements product positioning techniques like multi dimensional scaling or factor analysis and is used to create ideal vectors on perceptual maps. |
Preference regression : Starting with raw data from surveys, researchers apply positioning techniques to determine important dimensions and plot the position of competing products on these dimensions. Next they regress the survey data against the dimensions. The independent variables are the data collected in the surve... |
Preference regression : Self-stated importance method is an alternative method in which direct survey data is used to determine the weightings rather than statistical imputations. A third method is conjoint analysis in which an additive method is used. |
Preference regression : Marketing Product management Positioning (marketing) Marketing research Perceptual mapping Multidimensional scaling Factor analysis Linear discriminant analysis#Marketing Preference-rank translation |
Preference regression : Park, S. T.; Chu, W. (2009). "Pairwise preference regression for cold-start recommendation". Proceedings of the third ACM conference on Recommender systems - RecSys '09. p. 21. doi:10.1145/1639714.1639720. ISBN 9781605584355. Jarboe, G.R.; McDaniel, C.D.; Gates, R.H. (1992). "Preference regressi... |
Principal component analysis : Principal component analysis (PCA) is a linear dimensionality reduction technique with applications in exploratory data analysis, visualization and data preprocessing. The data is linearly transformed onto a new coordinate system such that the directions (principal components) capturing t... |
Principal component analysis : When performing PCA, the first principal component of a set of p variables is the derived variable formed as a linear combination of the original variables that explains the most variance. The second principal component explains the most variance in what is left once the effect of the fi... |
Principal component analysis : PCA was invented in 1901 by Karl Pearson, as an analogue of the principal axis theorem in mechanics; it was later independently developed and named by Harold Hotelling in the 1930s. Depending on the field of application, it is also named the discrete Karhunen–Loève transform (KLT) in sign... |
Principal component analysis : PCA can be thought of as fitting a p-dimensional ellipsoid to the data, where each axis of the ellipsoid represents a principal component. If some axis of the ellipsoid is small, then the variance along that axis is also small. To find the axes of the ellipsoid, we must first center the v... |
Principal component analysis : PCA is defined as an orthogonal linear transformation on a real inner product space that transforms the data to a new coordinate system such that the greatest variance by some scalar projection of the data comes to lie on the first coordinate (called the first principal component), the se... |
Principal component analysis : The singular values (in Σ) are the square roots of the eigenvalues of the matrix XTX. Each eigenvalue is proportional to the portion of the "variance" (more correctly of the sum of the squared distances of the points from their multidimensional mean) that is associated with each eigenvect... |
Principal component analysis : The following is a detailed description of PCA using the covariance method as opposed to the correlation method. The goal is to transform a given data set X of dimension p to an alternative data set Y of smaller dimension L. Equivalently, we are seeking to find the matrix Y, where Y is th... |
Principal component analysis : Let X be a d-dimensional random vector expressed as column vector. Without loss of generality, assume X has zero mean. We want to find ( ∗ ) a d × d orthonormal transformation matrix P so that PX has a diagonal covariance matrix (that is, PX is a random vector with all its distinct compo... |
Principal component analysis : In practical implementations, especially with high dimensional data (large p), the naive covariance method is rarely used because it is not efficient due to high computational and memory costs of explicitly determining the covariance matrix. The covariance-free approach avoids the np2 ope... |
Principal component analysis : In PCA, it is common that we want to introduce qualitative variables as supplementary elements. For example, many quantitative variables have been measured on plants. For these plants, some qualitative variables are available as, for example, the species to which the plant belongs. These ... |
Principal component analysis : ALGLIB – a C++ and C# library that implements PCA and truncated PCA Analytica – The built-in EigenDecomp function computes principal components. ELKI – includes PCA for projection, including robust variants of PCA, as well as PCA-based clustering algorithms. Gretl – principal component an... |
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