Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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15 views

numpy eigenvector and eigenvalues not making sense [closed]

sorry, maybe the subject is too strong? but its driving my little brain crazy ..please look at the code below ...
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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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10 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
67 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
25 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
34 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
24 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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20 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
26 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
121 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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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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32 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
20 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
35 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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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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69 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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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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29 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
37 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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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
39 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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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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34 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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67 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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26 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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Why are the directions of eigenvectors in SVD and Eigen-Decomposition for PCA opposite? [duplicate]

As you may know, scikit-learn library utilizes singular value decomposition (SVD) of data matrix X to produce eigenvectors for PCA. I decided to code PCA by using ...
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1answer
119 views

How to prove positive (semi-)definitness in matrix notation without numbers

I'd like to show that $$V[\hat\beta_{OLS}]-V[\hat\beta_{GLS}]=\sigma^2(X'X)^{-1}(X'\Omega X)(X'X)^{-1}-\sigma^2(X'\Omega X)^{-1}\geq 0$$ is positive (semi-) definite. $\Omega$ and $X$ are square-...
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35 views

Fisher's formalism : How to find a complementary matrix to respect the Maximum Likelihood Estimator (MLE)?

I make following a previous post : Bad attempt to do cross-correlations between 2 matrices Indeed, I say "Bad attempt" since the beginning of this study, I did a major error. By wanting to ...
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1answer
84 views

How to get the eigenvalue expansion of the covariance matrix?

Working through Bishops’s Pattern Recognition and Machine Learning and have the following question regarding the Eigenvalue expansion of a covariance matrix: “ Assume we have a symmetric real-valued ...
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39 views

Eigenvectors and eigenvalues of covariance matrix or its inverse in drawing ellipsoid

I'm trying to draw an ellipsoid of the $3 \times 3$ covariance matrix. Usually, I see the sentence an ellipsoid corresponding to the eigenvectors and eigenvalues of covariance matrix. But from the ...
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464 views

How to draw an ellipsoid corresponding to the eigenvectors and eigenvalues of covariance matrix?

I'm doing PCA with Python with dataset decathlon in which I'm interested in 3 variables 100m, ...
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1answer
410 views

How to Compute the Reconstruction error in Principal Component Analysis at lower dimensions

I have m examples and d features where m<<d. So I managed to compute the eigen value and corresponding its eigen vector ... I want to compute the reconstruction error for various value of ...
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72 views

Physical interpretation of $U$ and $V$ matrices in SVD

I have a question about the physical interpretation of $U$ and $V$ matrices in SVD. I collect measurements at multiple devices across time are collected into an $m$ × $T$ matrix $M$, where m is the ...
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101 views

PCA: inference on the proportion of explained variance, in a large p setting

I am interested in doing inference on the proportion of total variance explained by the first principal component, for a PCA based on the correlation matrix R. I want to know the (asymptotic) ...
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22 views

Kernel matrix decomposition

I had a look at the sklearn.kernel_approxiamtion.Nystroem implementation, which is also described in this post: Nystroem Method for Kernel Approximation Here, a ...
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1answer
102 views

Eigenvalues in Ridge regression [duplicate]

The ridge regression estimate is given by $$\beta^{*}=(X'X+kI)^{-1}X'y, k≥0,$$ where $X$ is the feature matrix. The original paper, Hoerl and Kennard's Ridge Regression: Biased Estimation for ...
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40 views

Dimensionality reduction of a large covariance matrix

I have a large covariance matrix $\Sigma$ and I am reducing its dimensionality by using a truncated eigendecomposition. $\Sigma \approx VDV^T$. I remember somewhere that you could also decompose it as ...
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1answer
46 views

Question about the Proof of PCA in “Learning from Data” by Shwartz and Ben-David, P. 280-281

Does anyone know how to justify the red and blue line in the attached proof of PCA? Red line: $B \in \mathbb{R}^{ d \times n}$, arrange $B = [B_{j,1} | B_{j,2} | \cdots | B_{j,n}]$, then $B^\top B = \...
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1answer
91 views

how to get total Fisher matrix that makes cross synthesis of 2 Fisher matrix

I have initially posted on physics.stackexchange but I think my issue is more adapted on Cross-Validated (so I am going to delete the initial post on physics.stackexchange). I have 2 Fisher matrixes ...
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1answer
30 views

What do eigenvectors of a data matrix consisting of house features/prices tell us? [duplicate]

I know this is one of the most repetitive question but bear with me please. I am trying to gain an intuitive understanding of eigenvectors. I had this example in my mind where there is a matrix A, the ...
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0answers
30 views

Checking that $ 𝐸[𝑥𝑥′⊗𝑥𝑥′] \prec \Sigma\otimes I+I\otimes \Sigma$

I have a random variable with mean 0 and covariance $\Sigma$, and I need to check that the following condition is satisfied $$2\Sigma\otimes \Sigma+\text{vec} \Sigma(\text{vec}\Sigma)'\prec \Sigma\...
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How to find eigenvalues and eigenvectors of the cokurtosis matrix?

Kurtosis is the fourth statistical moment of a random variable's distribution. Unlike the variance-covariance matrix $\Sigma$, which had a shape of $p\times p$, the kurtosis-cokurtosis matrix is ...
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1answer
221 views

Multiplying vectors by the covariance matrix?

I thought I knew covariance but I'm starting to think that there's more to it. For example, what happens when you multiply observations by their corresponding covariance matrix? ...
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0answers
157 views

What's the importance of parallel eigenvectors?

I'm studying eigenvectors. I read that if a matrix is symmetric and if the eigenvalues are real numbers, the eigenvectors will be perpendicular. However, I have no idea what it means (if anything) ...
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0answers
15 views

Interpreting SAS output - Roots of AR Characteristic Polynomial

I urgently need help on interpreting the numbers from a SAS output on Characteristic Roots: The VARMAX Procedure The VARMAX Procedure Roots of AR Characteristic Polynomial Index Real Imaginary Modulus ...
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Can trials have a differing number of samples when running PCA ? Why not?

Can somebody confirm that the number of "samples for each trial" doesn't matter(i guess that's right the language) for PCA. The case at hand is this: i have 5 sets of 3-dimensional ...
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1answer
71 views

PCA: Using R to generate and plot eigenvalues

I have the following data: ...
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29 views

Interpretation of eigenvalues from `decorana()` summary table

I just want to make sure I understand the example in the vegan package. Per the summary table on page 2 of this document, can I say that the first two axes of the ...
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0answers
44 views

Can Principal Component Analysis Be Used Here?

My professor has given us test prep in the form of a scenario essay question (For studying purposes/not graded) I want to see if my method of Principal Component Analysis would be applicable here. I ...
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1answer
135 views

Eigenvalues from `prcomp`

I used prcomp to calculate the follow PCA values: ...

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