Questions tagged [eigenvalues]
For questions involving calculation or interpretation of eigenvalues or eigenvectors.
369
questions
-3
votes
0answers
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
...
2
votes
0answers
34 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{...
1
vote
0answers
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 ...
1
vote
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 ...
1
vote
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 ...
1
vote
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 ...
2
votes
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 ...
0
votes
0answers
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:
<...
1
vote
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& ...
2
votes
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 ...
1
vote
0answers
17 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 ...
1
vote
0answers
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 ...
0
votes
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. ...
1
vote
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 ...
2
votes
0answers
24 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 ...
3
votes
0answers
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 ...
0
votes
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 ...
1
vote
0answers
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 ...
0
votes
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?
1
vote
0answers
41 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 ...
0
votes
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 $\...
5
votes
0answers
66 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 $\lambda$ ...
0
votes
0answers
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 $v\in \mathbb{R}^{np}$ ...
1
vote
0answers
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 ...
0
votes
0answers
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 $N$ ...
1
vote
0answers
21 views
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 ...
5
votes
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-...
0
votes
0answers
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 ...
2
votes
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 ...
0
votes
0answers
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 ...
4
votes
0answers
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, ...
0
votes
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 ...
4
votes
0answers
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 ...
4
votes
0answers
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) ...
1
vote
0answers
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 ...
3
votes
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 ...
2
votes
0answers
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 ...
1
vote
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 = \...
0
votes
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 ...
1
vote
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 ...
1
vote
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\...
1
vote
0answers
30 views
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 ...
7
votes
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? ...
1
vote
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) ...
0
votes
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 ...
0
votes
0answers
12 views
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 ...
4
votes
1answer
71 views
0
votes
0answers
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 ...
0
votes
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 ...
0
votes
1answer
135 views
