Quadratic discriminant analysis r

Linear & Quadratic Discriminant Analysis. In the previous tutorial you learned that logistic regression is a classification algorithm traditionally limited to only two-class classification problems (i.e. default = Yes or No).However, if you have more than two classes then Linear (and its cousin Quadratic) Discriminant Analysis (LDA & QDA) is an often-preferred classification technique In this post, we will look at linear discriminant analysis (LDA) and quadratic discriminant analysis (QDA). Discriminant analysis is used when the dependent variable is categorical. Another commonly used option is logistic regression but there are differences between logistic regression and discriminant analysis. Both LDA and QDA are used in situations in which there i Quadratic discriminant analysis - QDA. QDA is little bit more flexible than LDA, in the sense that it does not assumes the equality of variance/covariance. In other words, for QDA the covariance matrix can be different for each class. LDA tends to be a better than QDA when you have a small training set. In contrast, QDA is recommended if the training set is very large, so that the variance of.

Linear & Quadratic Discriminant Analysis - UC R Programmin

  1. ant Function Analysis . The MASS package contains functions for perfor
  2. ant analysis (QDA) is a variant of LDA that allows for non-linear separation of data. Finally, regularized discri
  3. ant Analysis (QDA) QDA is not really that much different from LDA except that you assume that the covariance matrix can be different for each class and so, we will estimate the covariance matrix \(\Sigma_k\).
  4. ant analysis (LDA) and the related Fisher's linear discri
  5. ant Analysis (QDA) which does not assume equal covariance across the classes. Both LDA and QDA require the number of independent variables to be less than the sample size and both assume multivariate normality among the independent variables. That is, the independent variables come from a normal (or Gaussian) distribution. Data and Required Packages. We will use Fisher's.

Linear VS Quadratic Discriminant Analysis in R

Everything is not linear - quadratic discriminant analysis. MASS package also contains the qda() function which stands for quadratic discriminant analysis. The idea is simple - if the data can be discriminated using a quadratic function, we can use qda() instead of lda(). The rest of the nuances are the same for qda() as were in lda() #QDA qda_iris=qda(Species~.,data=dataset) qda_iris Call. R Pubs by RStudio. Sign in Register Quadratic Discriminant Analysis; by Aaron Schlegel; Last updated over 3 years ago; Hide Comments (-) Share Hide Toolbars × Post on: Twitter Facebook Google+ Or copy & paste this link into an email or IM:. Machine Learning Classification in R | Quadratic Discriminant Analysis | Data Science for Beginners:  If you care about SETScholars, please donate to support us. We will try our best to bring end-to-end Python & R examples in the field of Machine Learning and Data Science. Latest end-to-end Learn by Coding Recipes in Project-Based Learning: All Notebooks in One Bundle: Data Science Recipes. i don't know the appropriate way to post data as this is my first post on the site. i have 63 rows and 35 columns 14 of which are categorical and 19 numerical (continuous) and i didn't use any categorical variables in the model. so far i've been able to use only 10 variables - oussama_ag Mar 13 '15 at 18:2

R Pubs by RStudio. Sign in Register Discriminant Analysis with R; by Gabriel Martos; Last updated over 5 years ago; Hide Comments (-) Share Hide Toolbars × Post on: Twitter Facebook Google+ Or copy & paste this link into an email or IM:. Linear Discriminant Analysis (LinearDiscriminantAnalysis) and Quadratic Discriminant Analysis (QuadraticDiscriminantAnalysis) are two classic classifiers, with, as their names suggest, a linear and a quadratic decision surface, respectively. These classifiers are attractive because they have closed-form solutions that can be easily computed, are inherently multiclass, have proven to work well.

Quadratic discriminant analysis is not available using SPSS. However, you can choose to classify cases based upon separate covariance matrices (as opposed to the default use of the pooled covariance matrix). Using separate covariance matrices is one way to get around the problem of inequality of covariance matrices. When the number of predictors does not exceed the number of groups minus 1. 4.7.1 Quadratic Discriminant Analysis (QDA) Like LDA, the QDA classifier results from assuming that the observations from each class are drawn from a Gaussian distribution, and plugging estimates for the parameters into Bayes' theorem in order to perform prediction. However, unlike LDA, QDA assumes that each class has its own covariance matrix. Under this assumption, the Bayes classifier.

An example of doing quadratic discriminant analysis in R. Thanks for watching!! ️ //Discriminant analysis code used in the video https://rpubs.com/mathetal/.. Value. an object of class qda containing the following components:. prior. the prior probabilities used. means. the group means. scaling. for each group i, scaling[i] is an array which transforms observations so that within-groups covariance matrix is spherical.. ldet. a vector of half log determinants of the dispersion matrix

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Discriminant Analysis Essentials in R - Articles - STHD

Quick-R: Discriminant Function Analysis

Details. This function is a method for the generic function predict() for class qda.It can be invoked by calling predict(x) for an object x of the appropriate class, or directly by calling predict.qda(x) regardless of the class of the object.. Missing values in newdata are handled by returning NA if the quadratic discriminants cannot be evaluated. If newdata is omitted and the na.action of. Discriminant Analysis (Part 1) - Duration: 21:00. Galit Shmueli 15,871 views. 21:00. John Conway: Surreal Numbers - How playing games led to more numbers than anybody ever thought of.

PROC DISCRIM: Univariate Density Estimates and Posterior

Linear and Quadratic Discriminant Analysis: Tutorial 4 which is in the quadratic form x>Ax+ b>x+ c= 0. Therefore, if we consider Gaussian distributions for the two classes, the decision boundary of classification is quadratic. Because of quadratic decision boundary which discrimi-nates the two classes, this method is named quadratic dis- criminant analysis. For obtaining Eq. (25), we brought. Quadratic discriminant analysis (qda) extends lda by allowing the intraclass covariance ma-trices to difier between classes, so that discrimination is based on quadratic rather than linear functions of X. With qda, however, there are no natural canonical variates and no general meth-ods for displaying the analysis graphically. When there are just two feature variables, it is possible to. Linear Discriminant Analysis is frequently used as a dimensionality reduction technique for pattern recognition or classification and machine learning. If you want to quickly do your own linear discriminant analysis, use this handy template! The intuition behind Linear Discriminant Analysis. Linear Discriminant Analysis takes a data set of cases (also known as observations) as input. For each. Linear Discriminant Analysis Quadratic Discriminant Analysis Naíve Bayes Logistic Regression Evaluation Methodology. Given training data with K classes, assume a parametric form for f k(x), where for each class X|Y = k ∼ (µ k, Σ k), i.e. instead of assuming that every class has a different mean µ k with the same covariance matrix Σ, we now allow each class to have its own covariance. The use of quadratic discriminant analysis (QDA) or its regularized version (R-QDA) for classification is often not recommended, due to its well-acknowledged high sensitivity to the estimation noise of the covariance matrix

Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that the measurements from each class are normally distributed. Unlike LDA however, in QDA there is no assumption that the covariance of each of the classes is identical This tutorial explains Linear Discriminant Anal- ysis (LDA) and Quadratic Discriminant Analysis (QDA) as two fundamental classification meth- ods in statistical and probabilistic learning. We start with the optimization of decision boundary on which the posteriors are equal. Then, LDA and QDA are derived for binary and multiple classes Of course, we could group terms to obtain an expression of the form const+linear+quadratic const + linear + quadratic. The important thing is that we obtain a quadratic function of x x; this leads us to Quadratic Discriminant Analysis (QDA). Having a quadratic discriminant function causes the decision boundaries in QDA to be quadratic surfaces A closely related generative classifier is Quadratic Discriminant Analysis (QDA). It is based on all the same assumptions of LDA, except that the class variances are different. Let us continue with Linear Discriminant Analysis article and se Quadratic discriminant analysis allows for the classifier to assess non -linear relationships. This of course something that linear discriminant analysis is not able to do. This post will go through the steps necessary to complete a qda analysis using Python. The steps that will be conducted are as follow

Linear, Quadratic, and Regularized Discriminant Analysis Quadratic Discriminant Analysis Model The univariate statistical method of QDA is used for constructing a model based on groups that consider the observed effective factors (Hong et al., 2017). In the QDA model, it is assumed that the measurements in each class have normal dispensation Therefore, when the groups do not have equal covariance matrices, observations are frequently assigned to groups with large variances on the diagonal of its corresponding covariance matrix (Rencher, n.d., pp. 321). Quadratic discriminant analysis is a modification of LDA that does not assume equal covariance matrices amongst the groups Linear and Quadratic Discriminant Analysis with covariance ellipsoid¶ This example plots the covariance ellipsoids of each class and decision boundary learned by LDA and QDA. The ellipsoids display the double standard deviation for each class. With LDA, the standard deviation is the same for all the classes, while each class has its own.

Quadratic discriminant function: This quadratic discriminant function is very much like the linear discriminant function except that because Σ k, the covariance matrix, is not identical, you cannot throw away the quadratic terms. This discriminant function is a quadratic function and will contain second order terms For quadratic discriminant analysis, there is nothing much that is different from the linear discriminant analysis in terms of code. The quadratic discriminant analysis algorithm yields the best classification rate. This might be due to the fact that the covariances matrices differ or because the true decision boundary is not linear. We play again with the ROC curve and determine a threshold.

On Discriminant Analysis Overview. An academical project (for SY09 at the University of Technology of Compiègne). This project was meant to compare different discriminant analysis methods, mainly: Linear Discriminant Analysis (LDA) Quadratic Discriminant Analysis (QDA) Logistic Regression (LR) (Normal) Naive Bayse Classifier (NBC) Decision. Quadratic Discriminant Analysis . In this type of analysis, your observation will be classified in the forms of the group that has the least squared distance. However, in this, the squared distance will never be reduced to the linear functions. #4. Canonical Discriminant Analysis. In this type of analysis, dimension reduction occurs through the canonical correlation and Principal Component. 4.6.4 Quadratic Discriminant Analysis ¶ We will now fit a QDA model to the Smarket data. QDA is implemented in R using the qda () function, which is also part of the MASS library. The syntax is identical to that of lda () Linear discriminant analysis is also known as canonical discriminant analysis, or simply discriminant analysis. If we want to separate the wines by cultivar, the wines come from three different cultivars, so the number of groups (G) is 3, and the number of variables is 13 (13 chemicals' concentrations; p = 13). The maximum number of useful discriminant functions that can separate.

Robust Generalised Quadratic Discriminant Analysis. 04/11/2020 ∙ by Abhik Ghosh, et al. ∙ 0 ∙ share . Quadratic discriminant analysis (QDA) is a widely used statistical tool to classify observations from different multivariate Normal populations. The generalized quadratic discriminant analysis (GQDA) classification rule/classifier, which generalizes the QDA and the minimum Mahalanobis. Regularization for Quadratic Discriminant Analysis Strategy 1 : If QDA is ill- or poorly-posed Replacing the individual class sample covariance matrices by their average (pooled covariance matrix) =^ PK k = 1 Sk PK k = 1 N k regularization by reducing the number of parameters to be estimated this can result in superior performance, especially in small-sample settings leads to LDA) the choice. Shrinkage-mean-based Diagonal Quadratic Discriminant Analysis (SmDQDA) from Tong, Chen, and Zhao (2012) print.mdeb: Outputs the summary for a MDEB classifier object. quadform: Quadratic form of a matrix and a vector: quadform_inv: Quadratic Form of the inverse of a matrix and a vector: solve_chol : Computes the inverse of a symmetric, positive-definite matrix using the Cholesky decomposition. QDA is implemented in sklearn using the QuadraticDiscriminantAnalysis () function, which is again part of the discriminant_analysis module. The syntax is identical to that of LinearDiscriminantAnalysis (). qda = QuadraticDiscriminantAnalysis() model2 = qda.fit(X_train, y_train) print(model2.priors_) print(model2.means_ Linear discriminant function analysis (i.e., discriminant analysis) performs a multivariate test of differences between groups. In addition, discriminant analysis is used to determine the minimum number of dimensions needed to describe these differences. A distinction is sometimes made between descriptive discriminant analysis and predictive discriminant analysis. We will be illustrating.

Quadratic discriminant analysis : c'est une généralisation de la LDA, sauf qu'on ne fait pas l'hypothèse que la matrice de covariance est indépendante de la classe. Les frontières entre classes prédites ne sont alors plus nécessairement linéaires. On peut utiliser la fonction qda du package MASS : model <- qda(y ~ x1 + x2, fr): calcule le modèle en utilisant y comme variable à. Quadratic discriminant analysis performed exactly as in linear discriminant analysis except that we use the following functions based on the covariance matrices for each category: Example 1: We want to classify five types metals based on four properties (A, B, C and D) based on the training data shown in Figure 1. We also want to see how good this classification is for the mean vectors for. In sparsediscrim: Sparse and Regularized Discriminant Analysis. Description Usage Arguments Details Value References Examples. View source: R/dqda.r. Description. Given a set of training data, this function builds the Diagonal Quadratic Discriminant Analysis (DQDA) classifier, which is often attributed to Dudoit et al. (2002) Linear discriminant analysis (LDA) and the related Fisher's linear discriminant are used in machine learning to find the linear combination of features which best separate two or more classes of object or event. The resulting combinations may be used as a linear classifier, or more commonly in dimensionality reduction before later classification.. LDA is closely related to ANOVA and regression. r linear-regression statistical-learning r-markdown logistic-regression regularization knn quadratic-discriminant-analysis linear-discriminant-analysis generalized-additive-models Updated Jul 31, 201

A new theory of discriminant analysis the Theory after R. Fisher is explained. There are five serious problems with discriminant analysis. I completely solve these problems through five mathematical programming-based linear discriminant functions (MP-based LDFs). First, I develop an optimal linear discriminant function using integer programming (IP-OLDF) based on a minimum number of. DiscriMiner: Tools of the Trade for Discriminant Analysis. Functions for Discriminant Analysis and Classification purposes covering various methods such as descriptive, geometric, linear, quadratic, PLS, as well as qualitative discriminant analyse Discriminant Analysis for Matrix Variate Distributions Geoffrey Thompson 2019-11-14. library (MixMatrix) In the MixMatrix package, there are two functions for training a linear or quadratic classifier. The usage is fairly similar to the function MASS::lda() or MASS::qda(), however it requires the input as an array and the group variable provided as a vector (that is, it cannot handle data.

Quadratic Discriminant Analysis is linked closely with the Linear Discriminant Analysis in which the assumption is made that the calculations are distributed normally. In Quadratic Discriminant Analysis, unlike Linear Discriminant Analysis, it is not assumed that the covariance of every class is same. To calculate the parameters needed in quadratic discrimination further data and computation. QDA(Quadratic discriminant Analysis)의 개념 앞선 LDA에서는 클래스 집단 별 공분산 행렬 $\Sigma_k$가 모두 동일하다고 했습니다. 이걸 조금 더 일반화시켜서 클래스 집단 별 공분산 행렬이 같다는 가정이 없다고 하겠습니다 I've seen posts for discriminant analysis in R using linear and quadratic discriminant analysis Quadratic discriminant analysis (QDA) is closely related to linear discriminant analysis (LDA), where it is assumed that the measurements from each class are normally distributed. My understanding for GDA was that the joint distribution of the variables was supposed to be roughly normally. In the case of heteroscedasticity of the classes, Quadratic Discriminant Analysis (QDA) can be used to determine an appropriate classification rule, but QDA does not serve for dimensionality reduction. Sliced Average Variance Estimation (SAVE) has been shown to be adequate in such situations as implemented in R in the package dr

线性判别分析(Linear Discriminant Analysis) Duanxx 2016-07-11 16:34:37 56890 收藏 52 分类专栏: 监督学 Pattern Recognition and Scene Analysis, R. E. Duda and P. E. Hart, Wiley, 1973. Bayesien Discriminant functions Séance 16 16-2 Notation x a variable X a random variable (unpredictable value) N The number of possible values for X (Can be infinite). →! ! x A vector of D variables. X ! ! A vector of D random variables. D The number of dimensions for the vector X ! ! x or ! ! E An observation. I hope you have enjoyed the Linear vs. Quadratic Discriminant Analysis tutorial. If you have any questions, let me know in the comments below. Categories Machine Learning Tags assumption checking, bayes classifier, linear discriminant analysis, quadratic discriminant analysis. Leave a Reply Cancel reply. Your email address will not be published. Required fields are marked * Comment. Name. Discriminant analysis is a classification method. In discriminant analysis, the idea is to: model the distribution of X in each of the classes separately. use what's known as Bayes theorem to flip things around to get the probability of Y given X. Th Christopher J. Fonnesbeck Ph.D. Student Georgia Cooperative Wildlife Unit University of Georgia Athens, GA 30602 Email: cjf at fonnesbeck.net Yahoo: fonnesbeck_chri

9.2.8 - Quadratic Discriminant Analysis (QDA) STAT 50

Discriminant analysis encompasses a wide variety of techniques used for classification purposes. These techniques, commonly recognized among the class of model-based methods in the field of machine learning (Devijver and Kittler, 1982), rely merely on the fact that we assume a parametric model in which the outcome is described by a set of explanatory variables that follow a certain distribution Quadratic Discriminant Analysis (QDA) Suppose only 2 classes C, D. Then r ⇤(x) = (C if Q C(x) Q D(x) > 0, D otherwise. [Pick the class with the biggest posterior probability] Decision fn is quadratic in x. Bayes decision boundary is Q C(x) Q D(x) = 0. - In 1D, B.d.b. may have 1 or 2 points. [Solutions to a quadratic equation] - In d-D, B.d.b. is a quadric. [In 2D, that's a conic. Retrouvez Discriminant Analysis et des millions de livres en stock sur Amazon.fr. Achetez neuf ou d'occasion Amazon.fr - Discriminant Analysis - Klecka, W.R. - Livres Passer au contenu principa Shannon Zavorka, Jamis J. Perrett, Minimum Sample Size Considerations for Two-Group Linear and Quadratic Discriminant Analysis with Rare Populations, Communications in Statistics - Simulation and Computation, 10.1080/03610918.2012.744041, 43, 7, (1726-1739), (2014). Crossref. Paul R. Yarnold, Robert C. Soltysik, Gary J. Martin, Heart rate variability and susceptibility for sudden cardiac death.

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Quadratic Discriminant Analysis(QDA), an extension of LDA is little bit more flexible than the former, in the sense that it does not assumes the equality of variance/covariance. In other words. Discriminant analysis is used to determine which variables discriminate between two or more naturally occurring groups, it may have a descriptive or a predictive objective. Description. The regularized discriminant analysis (RDA) is a generalization of the linear discriminant analysis (LDA) and the quadratic discreminant analysis (QDA). Both. Quadratic discriminant analysis (QDA) with qualitative predictors in R. 2. What is the correct formula for covariance matrix in quadratic discriminant analysis (QDA)? 8. Gaussian process and Correlation. 1. Discriminant function for diagonal LDA. Hot Network Questions Does increases in lift always cause an increase in lift induced drag? What was this Giant, Metal Death Monster I remember.

Discriminant Analysis in R - Amazon Web Service

Quadratic Discriminant Analysis Classification Learner Source: R/LearnerClassifQDA.R. mlr_learners_classif.qda.Rd. Quadratic discriminant analysis. Calls MASS::qda() from package MASS. Details. Parameters method and prior exist for training and prediction but accept different values for each. Therefore, arguments for the predict stage have been renamed to predict.method and predict.prior. Análisis discriminante lineal (LDA) y análisis.

covariance matrix of group i for quadratic discriminant analysis : m t: column vector of length p containing the means of the predictors calculated from the data in group t : S t: covariance matrix of group t |S t | determinant of S t : Linear discriminant function. The linear discriminant function corresponds to the regression coefficients in multiple regression and is calculated as follows. prove the classi cation performance of the quadratic discriminant analysis in a high dimension context. Key-words: Discriminant, redundant or independent variables, Variable se-lection, Gaussian classi cation models, Linear regression, BIC Institut de Math ematiques de Toulouse, INSA de Toulouse, Universit e de Toulouse yINRIA Saclay - ^Ile-de-France, Projet select, Universit e Paris-Sud 11.

Discriminant Analysis: Statistics All The Way R-blogger

We present here an approach based on quadratic discriminant analysis (QDA). This approach is evaluated on antimeric pairs of humeri and femora from the openly available Goldman Data Set and compared with two classical and previously published methods for osteometric pair‐matching, based respectively on linear regressions and t tests Quadratic Discriminant Analysis (QDA) does not assume that the groups of matrices have equal covariances. However, the assumptiong that the variables are normally distributed still exists. Mixture Discriminant Anlaysis (MDA) assumes that each class is a Gaussian mixture of subclasses Quadratic Discriminant Analysis (QDA) using Principal Component Analysis (PCA) Description. The principal components (PCs) for predictor variables provided as input data are estimated and then the individual coordinates in the selected PCs are used as predictors in the qda Predict using a PCA-LDA model built with function 'pcaLDA' Usage pcaQDA(formula = NULL, data = NULL, grouping = NULL, n.pc. Gaussian Discriminant Analysis (GDA)is the name for a family of classifiers that includes the well-known linearand quadraticclassifiers. These classifiers use class-conditional normal distributions as the data model for their observed features: \[(X \mid C = c) \sim Normal(\mu_c, \Sigma_c) \ The study further examined the sensitivity of the Quadratic Discriminant Function in predicting pregnancy outcomes with variations in the training and test samples of deliveries recorded in a hospital in Accra, Ghana. The study considered the scenarios; 50:50, 60:40, 70:30 and 75:25 ratios of training sets to testing sets. Predictor variables on both maternal factors (maternal age, parity and.

Role of dimensionality reduction in segment-based

# /* END OF QUADRATIC DISCRIMINANT ANALYSIS */ Posted by Renegade at 10:28 AM. Email This BlogThis! Share to Twitter Share to Facebook Share to Pinterest. Labels: quardatic discriminanat analysis, R. No comments: Post a Comment. Newer Post Older Post Home. Subscribe to: Post Comments (Atom) Followers. Quadratic Discriminant Analysis is another machine learning classification technique. Like, LDA, it seeks to estimate some coefficients, plug those coefficients into an equation as means of making predictions. LDA and QDA are actually quite similar. Both assume that the k classes can be drawn from Gaussian Distributions

When different groups of observations have different covariance structures, the canonical transformation is not optimal for group separation. For normally distributed data, the theoretical equivalent of Linear discriminant analysis (LDA) in the presence of different group covariance matrices is Quadratic discriminant analysis (QDA) Can't plot the result of a quadratic discriminant analysis using partimat in the klaR package in R. Ask Question Asked 5 years, 7 months ago. Active 5 years, 7 months ago. Viewed 839 times 0. I am working with the classic Pima indians and diabetes data set found here. I have read this data set in and called it Pima. With the klaR package, I am attempting to use the partimat function to plot. $\begingroup$ Could you please indicate which R package you are using for rank methods and M-estimators or MCD-estimators. The discriminant analysis function LDA in MASS package has only options for 3 methods (moment for standard estimators of the mean and variance, mle for MLEs, mve to use cov.mve, or t for robust estimates based on a t distribution)

Linear vs. quadratic discriminant analysis classifier: a tutorial. Article (PDF Available) · January 2016 with 15,775 Reads How we measure 'reads' A 'read' is counted each time someone views a. 9.2.8 - Quadratic Discriminant Analysis (QDA) 9.2.9 - Connection between LDA and logistic regression; 9.3 - Nearest-Neighbor Methods; Lesson 10: Support Vector Machines. 10.2 - Support Vector Classifier; 10.1 - When Data is Linearly Separable; 10.3 - When Data is NOT Linearly Separable; 10.4 - Kernel Functions ; 10.5 - Multiclass SVM; Lesson 11: Tree-based Methods. 11.1 - Construct the Tree.

How To Implement Linear Discriminant Analysis in R?Dimensionality Reduction for Visualization and PredictionIris data analysis example in RLinear models and multiclass classificationКлассификация, регрессия и другие алгоритмы Data Mining сArtificial Intelligence in High Content Screening andNo Title

And also, by the way, quadratic discriminant analysis. But let's start with linear discriminant analysis. And to illustrate that connection, let's start with a very simple mixture model. So let's start with a mixture model of the form, f(x) = the sum from 1 to 2. So I'm going to be working only with two components of omega k, 1 over the square root 2pi to the p determinant of sigma to the -1. Linear Discriminant Analysis with only one variable (p = 1). For a generalization, see Statistics - Fisher (Multiple Linear Discriminant Analysis|multi-variant Gaussian) Articles Related Assumption The variance from the distribution of the value when is the same in each of the classes k T1 - Graphical tools for quadratic discriminant analysis. AU - Pardoe, Iain. AU - Cook, R. Dennis. AU - Yin, Xiangrong. PY - 2007/5/1. Y1 - 2007/5/1. N2 - Sufficient dimension-reduction methods provide effective ways to visualize discriminant analysis problems. For example, Cook and Yin showed that the dimension-reduction method of sliced average variance estimation (SAVE) identifies variates.

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