Publications by Aaron Schlegel
Image Compression with Principal Component Analysis
Image compression with principal component analysis is a frequently occurring application of the dimension reduction technique. Recall from a previous post that employed singular value decomposition to compress an image, that an image is a matrix of pixels represented by RGB color values. Thus, principal component analysis can be used to reduce t...
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Factor Analysis Introduction with the Principal Component Method and R
Factor analysis is a controversial technique that represents the variables of a dataset \(y_1, y_2, \cdots, y_p\) as linearly related to random, unobservable variables called factors, denoted \(f_1, f_2, \cdots, f_m\) where \(m \lt p\). The factors are representative of ‘latent variables’ underlying the original variables. The existence of th...
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Factor Analysis with the Principal Component Method Part Two
In the first post on factor analysis, we examined computing the estimated covariance matrix \(S\) of the rootstock data and proceeded to find two factors that fit most of the variance of the data using the principal component method. However, the variables in the data are not on the same scale of measurement, which can cause variables with compar...
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Factor Analysis with the Principal Factor Method and R
As discussed in a previous post on the principal component method of factor analysis, the \(\hat{\Psi}\) term in the estimated covariance matrix \(S\), \(S = \hat{\Lambda} \hat{\Lambda}’ + \hat{\Psi}\), was excluded and we proceeded directly to factoring \(S\) and \(R\). The principal factor method of factor analysis (also called the principal ...
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Iterated Principal Factor Method of Factor Analysis with R
The iterated principal factor method is an extension of the principal factor method that seeks improved estimates of the communality. As seen in the previous post on the principal factor method, initial estimates of \(R – \hat{\Psi}\) or \(S – \hat{\Psi}\) are found to obtain \(\hat{\Lambda}\) from which the factors are computed. In the itera...
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Hierarchical Clustering Nearest Neighbors Algorithm in R
Hierarchical clustering is a widely used and popular tool in statistics and data mining for grouping data into ‘clusters’ that exposes similarities or dissimilarities in the data. There are many approaches to hierarchical clustering as it is not possible to investigate all clustering possibilities. One set of approaches to hierarchical cluste...
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QR Decomposition with the Gram-Schmidt Algorithm
QR decomposition is another technique for decomposing a matrix into a form that is easier to work with in further applications. The QR decomposition technique decomposes a square or rectangular matrix, which we will denote as \(A\), into two components, \(Q\), and \(R\). \( A = QR \) Where \(Q\) is an orthogonal matrix, and \(R\) is an upper tria...
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QR Decomposition with Householder Reflections
The more common approach to QR decomposition is employing Householder reflections rather than utilizing Gram-Schmidt. In practice, the Gram-Schmidt procedure is not recommended as it can lead to cancellation that causes inaccuracy of the computation of \(q_j\), which may result in a non-orthogonal \(Q\) matrix. Householder reflections are another...
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Introduction to Set Theory and Sets with R
Sets define a ‘collection’ of objects, or things typically referred to as ‘elements’ or ‘members.’ The concept of sets arises naturally when dealing with any collection of objects, whether it be a group of numbers or anything else. Conceptually, the following examples can be defined as a ‘set’: {1, 2, 3, 4} {Red, Green, Blue} {Ca...
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Set Operations Unions and Intersections in R
Part 2 of 2 in the series Set TheoryThe set operations of unions and intersections should ring a bell for those who’ve worked with relational databases and Venn Diagrams. The ‘union’ of two of sets \(A\) and \(B\) represents a set that comprises all members of \(A\) and \(B\) (or both). One of the most natural ways to visualize set unions a...
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