Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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36 views

Making sure that the design matrix is positive (-semi) definite

In bayesian linear regression, how to make sure that the design matrix produced by a neural network $ \Phi$ is positive definite? Because to computing the covariance matrix on the weight requires ...
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11 views

How does eigendecomposition form principal components? [duplicate]

I have a few questions regarding how specifically principal components are formed: What is the relevance of the magnitudes of the covariance when it comes to eigendecomposition? How does ...
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3answers
187 views

Get accurate eigenvectors, when eigenvalues are minuscule

I have a symmetric matrix A. I'm not able to compute all the eigenvectors accurately, and I believe it is due to the last few eigenvalues for ...
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50 views

What are the metrics to assess the quality of a multiple correspondence analysis (MCA) model?

We are trying to implement a multiple correspondence analysis (MCA) model. I was looking for metrics to assess the quality of an MCA to evaluate our model. Sadly, I didn’t find much literature about ...
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0answers
25 views

How to interpret a PCA Biplot? [duplicate]

I constructed a PCA plot from a very high-dimensional dataset that contains features relating to fraud. After creating the PCA plot, I created a biplot with the features to see how they interact. The ...
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47 views

Why absolute value of eigenvalues are used in PCA or LDA?

In PCA and LDA techniques, eigenvectors with the $k$ largest eigenvalues give principal components. However, when selecting these eigenvalues, are they to be sorted by the absolute value (regardless ...
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0answers
15 views

What does it mean for Katz Centralities to "diverge"

In Mark Newman's Networks book, 2010 edition, page 173, he explains some mathematical details behind the Katz Centrality measure: In matrix terms, Eq. (7.8) can be written x = αAx + β1, (7.9) where 1 ...
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16 views

why do we need orthogonal basis for PCA? [duplicate]

In PCA, why do we map to the orthogonal basis? What is the important point that they should be orthogonal with each other?
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0answers
28 views

Are eigenvectors of PCA guaranteed to be orthonormal?

Are eigenvectors (principal components) of PCA orthonormal or only orthogonal ? Or only some of them are orthonormal or they are orthonormal if data were normalized before doing PCA ?
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62 views

Does the Cholesky decomposition of a covariance matrix lead to a lower triangular matrix with positive diagonals?

We know that an $N\times N$ covariance matrix $\Sigma$ is symmetric positive definite, and can be factorized using Cholesky decomposition as follows \begin{equation} \Sigma=LL' \end{equation} where $L$...
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6 views

Normalizing a custom Weight Shifted or Spiked Gaussian distribution

I have a custom weight shifted bivariate gaussian distribution that I wish to normalize. W is the weighted symmetric matrix that shifts the entire distribution and the λ below is the diagonal matrix ...
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1answer
40 views

Very low values of explained variance for the first Axes from PCoA

I am comparing different sites based on their floristic composition in R. Therefore, I have created a huge community datamatrix (presence/absence data) from 53 sites including over 1000 species. To ...
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29 views

In the rotational cost algorithm, are eigenvectors sorted according to eigenvalues, or according to absolute value of eigenvalues?

I am trying compute Rotational Cost, as defined in the 2004 paper "Self-Tuning Spectral Clustering" by Zelnik-Manor and P. Perona (http://www.vision.caltech.edu/lihi/Publications/...
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13 views

Disambiguation of spectral clustering

I've read about spectral clustering, and feel like there are two related but different ways people use the words "spectral clustering". Way 1 Spectral clustering as a pre-processing step ...
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10 views

How to check if my matrices are 'random' using Marchenko-Pastur?

For each $n=1,2,\dots,100$, I have an $r \times s$ integer-valued matrix $X(n)$ of increasing dimension as $n$ increases. That is, the number of rows $r$ and columns $s$ increases as $n$ increases. ...
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53 views

Is it possible that Fisher information matrix be indefinite? [duplicate]

I`m using the Newton-Raphson method for obtaining MLE for parameters for maximizing my objective function. At each iteration, I want to check that is the Hessian matrix negative definite or not and I ...
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18 views

Why must the non-zero eigenvalues in the Johansen test be between 0 and 1?

Why are the non-zero eigenvalues in the matrix $\Pi$ in the Johansen test between 0 and 1? Why can't they be greater than 1 or less than zero? My lecturer just dropped in that's its because "the ...
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16 views

PCA face recognition (problem) faces are not recognized

I need advice regarding PCA face recognition. I have a dataset of 86 faces, which I stretch into a column vector. (so my matrix is 3000 by 86, where columns are faces). I calculate the average face ...
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0answers
31 views

LDA (linear discriminant analysis) for images: strange eigenvalues

I have the following dataset: I represent each image as a $(67 \times 67, 1)$ vector and add it to the dataframe. df.head() My goal is to determine the ...
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1answer
30 views

Is the QR Algorithm guaranteed to compute eigenvectors?

I'm writing some C++ matrix library for hobby. For computing eigenvalues and eigenvectors, I referred the following "Francis double step QR algorithm": In particular, page 82 of https://...
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0answers
10 views

Is there any relationship between the mode value of a data set and the main component in PCA for this dataset?

I'm trying to intuitively understand if there is any direct relationship between the mode value of a dataset, which represents the value that repeats the most, and the main PCA component for this same ...
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0answers
80 views

MANOVA: How to identify the most relevant dependent variables

I am facing a problem with a big number of dependent variables and relatively small sample size. Not all dependent variables might be relevant, though. As multivariate analysis of variance (MANOVA) ...
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0answers
18 views

Alternative of the Nyström method

Say you want to obtain the eigenvalues/vectors of the integral operator associated to a kernel $K$. I know there is the Nyström method to obtain an approximation of these eigenvalues/vectors. What are ...
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1answer
90 views

Mercer's theorem and eigenfunctions

Consider a kernel $K$ which satisfies the conditions of Mercer's theorem. We know that $K(x,y) = \sum\limits_{i=1}^{+\infty}\lambda_ie_i(x)e_i(y)$ where $\lambda_i$ and $e_i$ are the eigenvalues/...
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1answer
148 views

How to translate eigenvectors and eigenvalues to the number of clusters in spectral clustering?

I have generated this output, where L is the Laplacian Matrix, D is the degree and A is the adjacency matrix: I can see the eigenvalues and eigenvectors are returned. I am unsure how to interpret ...
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0answers
38 views

How to calculate the eigenvector?

I've been struggling to solve this math problem for two days... So I calculated the mean of all samples (0,0). Put it into the equation and got V as \begin{array} {rrr} 4 & 2 \\ 2 & 2 \end{...
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12 views

Why is sum of squared PCA column loadings equal to corresponding Eigenvalue?

I'm wondering why or how is the sum of squared PCA loadings in the column equal to Eigenvalue. I understood that the sum of squared PCA loadings in the row is equal to 1 or 100% because original ...
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1answer
289 views

How to use principal components as inputs in hierarchical clustering analysis in R

For my statistical analysis I want to follow the steps of a paper I read. I have a dataset in which each row corresponds to a dive carried out by a whale ('id' in table below) and the columns to the ...
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1answer
51 views

Why is there a discrepancy between the eigenvalues of the covariance matrix (PCA) and the eigenvalues of the kernel matrix (kernel PCA)?

I've done PCA on my data matrix $ \mathbf{X} $ which gives me i.a. the eigenvalues $ \lambda $ and eigenvectors $ v $ of the data covariance matrix $ C=\mathbf{X}^T \mathbf{X} $. I'm now extending my ...
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1answer
94 views

Can you sum principal components?

I am currently reviewing principal components from my recent output. For each principal component I know you get an eigenvalue which represents how much of the variance is explained. If I was to take ...
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1answer
46 views

Why 'eigen()' and 'fa.parallel()' give different eigenvalues? [closed]

I ran an EFA on 10 items in Rstudio. I did parallel analysis (fa.parallel) using psych package and also eigenvalues using these ...
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1answer
52 views

Doubt in the range of variance of First Principal Component

This is my first problem on this forum. Here is the problem: The covariance matrix of a four dimensional random vector $\boldsymbol X$ is of the form $$\begin{bmatrix}1&\rho&\rho&\rho\\ \...
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22 views

How to recover true eigenvectors after creating data matrix from synthetic PCA results?

I try to create a raw data matrix given a matrix of eigenvectors and some synthetic principle components and want to discover the true eigenvectors. Assume I have the following random data matrix: <...
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1answer
31 views

Computing the principal components using the given variance covariance matrix

I have been given the following variance covariance matrix and I am required to compute the principal components of this matrix. $\Sigma = \begin{pmatrix} 1 & 0.5& 0.5& 0.5\\ 0.5& ...
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1answer
174 views

Interpreting variables "weights" and "loadings" from PCA parallel coordinates plot

I am presented with the following parallel coordinate plots in PCA: The following is then said: PCA of the Raw Breast Cancer Data Variables 24 and 4 dominate the parallel coordinate plot of the raw ...
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0answers
18 views

In PCA, why does maximising projections maximise variance?

My understanding could be wrong as the lecture series from my University isn't clear and there are no links to share. But having rewatched a few times, I think the point being made is: . We want the ...
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0answers
50 views

Random matrix theory impact on covariance matrix analysis

Framework: From RMT, eigenvalues have a semicircle distribution for symmetric matrices each with i.i.d normally distributed entries as the size of the matrix grows. The restrictions on i.i.d have ...
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1answer
24 views

Why is Eigenvalue different from Variance.percent?

I am conducting PCA analysis on results from a Likert-scale (10-point base) survey on user preferences. When using the code below, I obtain the list of variables with their respective eigenvalues. ...
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1answer
42 views

Explanation on Google’s PageRank is Webpages as Eigenvectors

Help understand what is the matrix A and the vector x discussed below. Mathematics for Machine Learning Example 4.9 Google uses the eigenvector corresponding to the maximal eigenvalue of a matrix A ...
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0answers
26 views

Training dataset from analytical solution

I am currently redesigning an inverse problem on an experimental technique, but I am having doubts about how to create a training dataset. Here is the problem I am trying to solve: I have already ...
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0answers
71 views

Decorrelation, PCA and rotation

I am not a PCA expert, nor do I have a good knowledge of linear algebra, so bear with me and my ignorance. I am trying to understand how the authors of some papers I have been reading decorrelate two ...
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0answers
11 views

Why eigen vector of a covariance matrix is the largest principle components? [duplicate]

I am self studying principle component analysis using this tutorial, I got most of the reasoning behind PCA but I don't get the intuitive reason why eigen vectors of a covariance matrix is also its ...
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0answers
39 views

Order of eigenvalues when using different methods

I'm doing PCA in a covariance matrix where each column and row represents tenors of the yield curve. I have coded the Jacobi rotation method and I also have a QR algorithm based on numpy.linalg.qr in ...
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1answer
120 views

Eigenvectors of covariance matrix and inertia tensor

The moment of inertia tensor from physics looks very similar to the covariance matrix, used for PCA. How are their eigenvectors and eigenvalues related?
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58 views

Are there problems associated with choosing the smallest eigenvalue/eigenvector when performing PCA?

In relation to PCA which is usually used in a setting where one wishes to maximize the variance as well as a reduction in dimensionality thereby choosing the eigenvalues with corresponding ...
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1answer
57 views

Eigenvalues of idempotent matrix of rank $r$

In the proof for the following theorem in Linear Models in Statistics, Render & Schaalje $\textbf{Theorem 5.5}$ Let y be distributed as $N_p\left({\mathbf{\mu}, \mathbf{\Sigma}}\right)$, let $\...
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0answers
75 views

What is the precise relation between the eigenvalues of a covariance *function* and the eigenvalues of a covariance *matrix*?

Assume we have a temporal Gaussian Process $\mathcal{GP}(t;\ m,k)$ (GP) with mean $m$ and covariance function (aka. kernel) $k$ on some compact time interval $[0,T]$. Then, the eigenvalues $\lambda$ ...
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64 views

Multivariate normal distribution - diagonalizable matrix

Let $X_{1},..,X_{n}$ be independent random vectors drawn from the $N_{p}(\mu ,\Sigma )$ distribution and let $\bar{X}$ be their mean. Find a $p\times np$ matrix $A$ and vector $v\in \mathbb{R}^{np}$ ...
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0answers
114 views

OLS estimator is consistent if the smallest eigenvalue of $X^TX$ goes to infinity as $n\to\infty$

I want to show that if $\lambda_{min}(X^T X)$ (i.e., the smallest eigenvalue of $X^TX$) goes to infinity as $n\to\infty$, then $\hat{\beta}$ is a consistent estimator of $\beta$. My approach is the ...
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0answers
32 views

Variables' weights using PCA

I have recently read a paper where the authors applied PCA to determine the weights of the variables used to calculate a composite index. In the methodology, they mentioned that for a set of $N$ ...

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