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Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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Eigendecomposition proof [migrated]

Let $X$ be an $(n \times n)$ matrix. Let $V$ be the $(n \times n-k)$ be the matrix of eigenvectors of $X$ which correspond to non-zero eigenvalues of $X$. Let $E$ be the $(n-k \times n-k)$ diagonal ...
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31 views

Eigenvalue decomposition of a covariance matrix using a fast Cholesky decomposition

Let $\mathbf{C}$ be a $n \times n$ covariance matrix and assume that the LDL' Cholesky decomposition can be obtained efficiently. Can we take advantage of this to obtain a fast eigenvalue ...
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7 views

clustering applied to squared symmetric matrix [closed]

Edit as per @Peter Flom guidelines to write question: https://www.statisticalanalysisconsulting.com/how-to-ask-a-statistics-question/ problem: modelling topological properties of economics networks. ...
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1answer
32 views

Does $\text{cov}(a_1' X, a_2' X) = 0$ imply $a_1 \cdot a_2 = 0$?

Let $X$ be a $p$-dimensional random vector with $p$ principal components $y_1, y_2, \dots, y_p$. By definition, a restriction put on the second principal component $y_2 = a_2'X$ is $$ \text{cov}(y_1, ...
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174 views

Why do PCA and PCoA give the same components but different explained variances?

I'm quite familiar with Principal Component Analysisis, as I use it to study genetic structure. Lately, I was revisiting some of the functions I was using in R (...
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1answer
37 views

Calculate first principal component direction and scores

Given that x1 = (9, 9, −18)^T and x2 = (18, 9, 9)^T with eigendecomposition of its sample covariance matrix Σ = cov(X) How do I calculate the first two principal component direction and the ...
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163 views

Failure to replicate calculation of PCA residuals in linear regression with heteroscedasticity

In their preprint, Rocha et al. suggest a new type of residual for linear regression models with heteroscedasticity. They call their new residual PCA residuals. I have tried to replicate some of their ...
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24 views

Is it right to use PCA in this scenario?

Physicist here. I have a dataset. The data is the emission from a molecule that has two dipoles. Molecules can only emit along these dipoles. As I rotate the molecule, I will selectively excite the ...
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10 views

Is eigenspace based classification possible

Imagine I would like to classify an image (e.g. into healthy and sick) and have a lot of labeled data. Could I classify any image by comparing it to the eigenspaces of the two sets? It sounds simple, ...
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21 views

Factor analysis

I have a couple of questions on factor analysis using Stata. How to decide whether to use pf (principal factors, default), pcf (principal-components factor), ipf (iterated principal factor) or mle (...
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When the elements of first basis are always positive for PCA?

I am computing the PCA projection matrix of some data. I notice that the elements of first basis vector (corresponding to the highest eigenvalue) are always positive. My data is real and contain both ...
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32 views

Is it better to interpret PCA components using the eigenvectors or the rescaled loadings?

I have a dataset to which I am applying PCA, and looking to each PCA component. Initially I was using the eigenvectors as a way to understand what each component "means". When using the eigenvectors ...
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35 views

eigen function in R

I want to ask about the eigen function of R. I am currently doing a project of NBA team analysis. I am trying to figure out correlation effect of two players lineup ...
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15 views

Basis vectors for categorical images

I have a sequence of categorical images. For a two category image, each image pixel can have one of two values. I would like to analyze these images using a technique like eigen images. The goal is to ...
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1answer
90 views

How to interpret eigenvectors in PCA analysis?

I'm trying to apply the output from PCA analysis I've run on some yield curve history and am getting a bit confused. I have followed the steps below, From a history of the yield curve ($m \times n$ ...
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24 views

Kaiser Criterion after Rotation

Within my PCA, I rotated the solution, so the output would make more sense given the complex structure using a Varimax rotation. However, when I am confirming the number of factors to extract based on ...
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1answer
204 views

Drawing 95% ellipse over scatter plot

The context is regression analysis using Eviews, but first I wanted to create a few scatter plots and overlay error ellipses on them. Eviews doesn't support that kind of graph ornamentation so I am ...
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55 views

Significance of eigenvector components in PCA

Long time reader, first time poster. Hopefully I won't screw this up... In the context of Principal Component Analysis, I have the sense that the components of an eigenvector are a measure of the ...
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2answers
188 views

adding a small constant to the diagonals of a matrix to stabilize

I have a large correlation matrix (110x110) with some small eigenvalues (about 20 < 0.1). It has been suggested that adding a constant (about 0.1) to the diagonals will help to stabilize the matrix....
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1answer
31 views

Graphical understanding of PCA

I learned about PCA and how to find the principal components via eigenvectors/values. Now for the following problems my professor says that "Feature 2 is constant and can hence be ignored, so you can ...
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1answer
25 views

importance of correlation between data for a PROC CLUSTER

i'm working on a clustering analysis on SAS. I need to improve an actual code : ...
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37 views

Determining the Direction of Eigenvectors in PCA [duplicate]

I'm using R to get the principal components for several datasets. An example result, using prcomp yields: ...
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33 views

Idea behind change of basis and how it relates to projecting your points onto principal components

I would like to clarify if my understanding is correct. In the traditional X-Y coordinate system, our choice of basis vectors are $\vec{i} = (1, 0)$ and $\vec{j} = (0, 1)$ and when you I have a point $...
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55 views

Change in eigenvalues due to perturbation to a correlation matrix

Let $A$ be a $m \times n$ matrix defined as $ A = \Big[\frac{a_1}{\|a_1\|} \cdots \frac{a_n}{\|a_n\|}\Big]$ and $a_k \in \mathbb{R}^{m\times 1}$ where $k \in [1,\dots,n]$. Now, we define a ...
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Maximum likelihood: Why is the number of non-zero eigenvalues equal to $x^T \hat{\Sigma}^{-1} x$

I've been reading this code (based on this R package) and I found that the number of non-zero eigenvalues of the estimated covariance is roughly equal to $x_i^T \hat{\Sigma}^{-1} x_i$. I want to know ...
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The miracle of the Lanczos/conjugate gradient algorithm

Lanczos/Arnoldi/Rietz/CG-like algorithm share the same core strategy... In each, a little miracle appears, most of the Gram-Schmidt inner products are zeroes ! In others words, new direction need only ...
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51 views

Limiting results for non-unique eigenvalues and eigenvectors for a sample covariance matrix

I am working on the limiting behavior for the eigenvalue and the corresponding eigenvectors, especially the minimum eigenvalues. For instance, let $S_X=\frac{1} {T} \sum_t X_t X_t ^\prime$ be a $p \...
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1answer
217 views

Are eigenfaces same as eigenvectors?

I'm trying to understand the difference between eigenvectors and eigenfaces, are they different names for same concepts? I ask this because I got confused when I am trying to compute eigenvectors for ...
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1answer
20 views

PCA- creating a model with values obtained

Hoping somebody can help me. I cannot find an example that 'finishes' a problem. I run a proc princomp in SAS. I have hundreds of variables but used four for the purpose of an example. I ...
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1answer
124 views

Compute the $k$ largest eigenvector in spectral clustering

In Spectral Clustering, we need to compute the top $k$ largest eigenvector of normalized $L$. $$L = D^{-\frac{1}{2}}SD^{-\frac{1}{2}}$$ In Andrew NG's paper, L is not positive definite (unless ...
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36 views

Eigenvalue decomposition/SVD and the filtering perspective

I have been studying the SVD algorithm recently and I can understand how it might be used for compression but I am trying to figure out if there is a perspective of SVD where it can be seen as a low ...
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83 views

Intuition behind finding number of clusters in spectral clustering according to Zelnik-Manor and Perona

I have a quick question on the intuition behind the estimation strategy for the number of clusters presented in the paper "Self-Tuning Spectral Clustering" by Lihi Zelnik-Manor and Pietro Perona. See ...
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1answer
257 views

PCA in psych package with more columns than rows

Why is it impossible to do a PCA in R using principal from psych package without warnings with a matrix, which has more columns ...
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32 views

PCA for three-dimensional linear fit on time-resolved trajectory

I study the behavior of organisms that are able of self-locomotion and that show directed motion toward one another. This directed motion occurs through the detection of chemical trails released by ...
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3answers
700 views

PCA basic: Must eigenvalues converge to zero at high dimension?

Recently, I obtained several PCA plots, and because I am unable to produce eigenvalues for higher dimensions, I tried to extrapolate them based on the available data. The reason why I want to do this ...
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1answer
107 views

PCA Basic: Do eigenvalues remain constant with increase in dimensions?

Say I have a set of genomic data to be analysed for population structure. I ran two different analyses using two different maximum principal components. For the first analysis, I modeled the data ...
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44 views

Weighting a Covariance Matrix via Eigendecomposition

Given a covariance matrix $\mathbf{\Sigma} \in \mathbb{R}^{N \times N}$ between $N$ variables of interest, generated by some kernel function $\mathbf{\kappa}(\mathbf{x}, \mathbf{y})$, and some vector $...
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3answers
387 views

PCA to choose variables based on its loadings on PC1 [duplicate]

I have a dataset of cave dimensions (and other variables related to their features). The problem is that 3 of these variables are: Length, Area, and Volume. These 3 are highly correlated as they ...
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1answer
206 views

Negative eigenvalues when computing MultiDimensional Scaling given nonnegative distance matrix

I am using the Smile MDS https://github.com/haifengl/smile/blob/master/core/src/main/java/smile/mds/MDS.java and occasionally running into: ...
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122 views

Interpreting low Cronbach's alpha value in CATPCA

I ran categorical principal component analysis (CATPCA) on the data that I collected through a questionnaire. The purpose of the questionnaire was to understand pedestrians intentions and attitudes ...
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Why are the discriminant axes in linear discriminant analysis (LDA) not orthogonal?

This may be a quite silly question and please correct me if I'm wrong. The discriminants (discriminant axes) are essentially eigenvectors of $\mathrm{Cov}_\mathrm{within}^{-1} \mathrm{Cov}_\mathrm{...
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38 views

Is max. Eigenvalue of k-sparse PCA always $\leq$ max. Eigenvalue of normal PCA on same dataset?

Is max. Eigenvalue of k-sparse PCA always less than or equal to the max. Eigenvalue of normal PCA on same dataset? K refers to the number of non zero eigenvalues when the dataset is of dimension n <...
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1answer
38 views

How to infer the eigenvalue distribution from matrix where each entry has a known Gaussian distribution?

Problem Given $X \in \mathbb{R}^{n \times n}$ where $X_{ij} \sim \mathcal{N}(\mu_{ij}, \sigma_{ij}^2 I)$ Find the eigenvalue distribution using whatever you can. Background In my field, I have a ...
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1answer
142 views

Usage of the term “feature vector” in Lindsay I Smith's PCA tutorial

I'm currently following Lindsay Smith's (in large parts very well written) PCA tutorial. I'm a bit confused about the usage of the term "feature vector" in this paper though. A quote from this paper ...
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1answer
51 views

Can I use combination of eigenvectors as a single vector to explain most of variance?

I have a problem trying to find a combination (or weighted average) of variables (statistics) that best explains the sample statistics. A – n x p matrix (n: observations p: variables, here are ...
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187 views

Must a matrix of sample pairwise covariances be PSD?

Consider a random vector $\mathbf{X}=(X)_{i=1}^n$. Then the covariance matrix $$C=\mathbb{E}[(\mathbf{X}-\mu(\mathbf{X}))(\mathbf{X}-\mu(\mathbf{X}))^\top]$$ is by definition positive-semidefinite. (...
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1answer
256 views

Orthogonality as found by the Gram-Schmidt process vs. uncorrelated basis vectors

I have a data matrix $Y$ of size $n \times p$, a basis vector in $\mathbb{R}^p$ $v_1$, and a potential basis vector in $\mathbb{R}^p$ $v_2'$. Then, if I use the Gram-Schmidt process on $[v_1, v_2']$ ...
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1answer
38 views

Exploratory factor analysis with 5 positively loaded and 1 negatively loaded factor. How to interpret?

I conducted an EFA with Maximum Likelihood and Direct Oblimin Rotation. Following the Kaiser Eigenvalue 1 rule, I identified 6 latent factors. 5 of them are positively loaded, while 1 is not. Here ...
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1answer
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Is it possible to have a basis for a covariance matrix such that the greatest variance is greater than the variance of the first eigenvector?

Suppose we have a covariance matrix $C$. Define the eigendecomposition, $C = Q^{-1} \Lambda Q$ and some other arbitrary basis $C = B^{-1} D B$. Define $V_\text{PCA} = \text{diag}(QCQ^{-1})$ and $V_B =...
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4k views

How does eigenvalues measure variance along the principal components in PCA? [duplicate]

I understand that eigenvalues measure variance along the principal components. Questions How are eigenvalues and variance same for PCA? What is the intuition behind this being the same? What is ...