Publications by Aaron Schlegel
Image Compression with Singular Value Decomposition
As mentioned in a previous post, image compression with singular value decomposition is a frequently occurring application of the method. The image is treated as a matrix of pixels with corresponding color values and is decomposed into smaller ranks that retain only the essential information that comprises the image. In this example, we are inter...
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Discriminant Analysis for Group Separation in R
The term ‘discriminant analysis’ is often used interchangeably to represent two different objectives. These objectives of discriminant analysis are: Description of group separation. Linear combinations of variables, known as discriminant functions, of the dependent variables that maximize the separation between the groups are used to identif...
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Multiple Analysis of Variance (MANOVA)
MANOVA, or Multiple Analysis of Variance, is an extension of Analysis of Variance (ANOVA) to several dependent variables. The approach to MANOVA is similar to ANOVA in many regards and requires the same assumptions (normally distributed dependent variables with equal covariance matrices). This post will explore how MANOVA is performed and interpr...
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MANOVA Test Statistics with R
Multiple tests of significance can be employed when performing MANOVA. The most well known and widely used MANOVA test statistics are Wilk’s \(\Lambda\), Pillai, Lawley-Hotelling, and Roy’s test. Unlike ANOVA in which only one dependent variable is examined, several tests are often utilized in MANOVA due to its multidimensional nature. Each M...
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Discriminant Analysis of Several Groups
Discriminant analysis is also applicable in the case of more than two groups. In the first post on discriminant analysis, there was only one linear discriminant function as the number of linear discriminant functions is \(s = min(p, k – 1)\), where \(p\) is the number of dependent variables and \(k\) is the number of groups. In the case of more...
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Classification with Linear Discriminant Analysis
Classification with linear discriminant analysis is a common approach to predicting class membership of observations. A previous post explored the descriptive aspect of linear discriminant analysis with data collected on two groups of beetles. In this post, we will use the discriminant functions found in the first post to classify the observation...
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Quadratic Discriminant Analysis of Two Groups
As mentioned in the post on classification with linear discriminant analysis, LDA assumes the groups in question have equal covariance matrices \((\Sigma_1 = \Sigma_2 = \cdots = \Sigma_k)\). Therefore, often when the groups do not have equal covariance matrices, observations are frequently assigned to groups with large variances on the diagonal o...
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LDA for Classification into Several Groups
Similar to the two-group linear discriminant analysis for classification case, LDA for classification into several groups seeks to find the mean vector that the new observation \(y\) is closest to and assign \(y\) accordingly using a distance function. The several group case also assumes equal covariance matrices amongst the groups \((\Sigma_1 = ...
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Quadratic Discriminant Analysis of Several Groups
Quadratic discriminant analysis for classification is a modification of linear discriminant analysis that does not assume equal covariance matrices amongst the groups \((\Sigma_1, \Sigma_2, \cdots, \Sigma_k)\). Similar to LDA for several groups, quadratic discriminant analysis of several groups classification seeks to find the group that maximize...
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Principal Component Analysis
Often, it is not helpful or informative to only look at all the variables in a dataset for correlations or covariances. A preferable approach is to derive new variables from the original variables that preserve most of the information given by their variances. Principal component analysis is a widely used and popular statistical method for reduci...
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