Questions tagged [eigenvalues]
For questions involving calculation or interpretation of eigenvalues or eigenvectors.
290
questions
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12 views
How to quantify the similarity of EOF loading in multiple matrices?
I have five 3-D matrices (time, latitude, and longitude) representing the same variable but from different sources, denoted as A, B, C, D, and E. I calculated the first five EOF loadings for each of ...
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40 views
Derive Confidence Intervals for Eigenvalues of a Covariance Matrix
I am working with a dataset of n = 273 observations, p = 9 variables for which I have generated principal components. The task I am faced with is:
Assume the eigenvalues of a covariance matrix ${cov(...
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1answer
42 views
Upper bound trace of inverse of covariance matrix
Let C be the covariance matrix from any normal distribution. If the trace of C is upper-bounded by a constant k (i.e., tr(C)<=k), can I find an upper bound for the trace of the inverse of C (i.e., ...
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9 views
What is the correct explanation for the definition of Eigen vectors of covariance matrix, Principal components and Eigenfaces?
We have an input matrix X consisting of n images. We need to do PCA on this matrix.
We compute covariance matrix of X, and find the Eigen values. The Eigen vector corresponding to highest Eigen value ...
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24 views
Relation between eigenvalues of original and transformed matrices
Let the matrix $X$ be some data arranged in rows. Consider the following eigenvalue decomposition $X^\top X = Q \Theta Q^\top=\sum_{i=1}^n \theta_iq_iq_i^\top$ where $q_i$ are the eigenvectors and $\...
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11 views
Matrix GLM estimation Example Explanation
**Below is my question:
It is an example from the book " A Primer on linear modes" by Monahan**
I have 3 questions
Question 1:
is how did he calculated the matrix λ
Question 2:
Why $λ $ ...
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41 views
Getting sense from loadings plot of scaled eigenvectors
I wanted to confirm my intution about the meaning of "loadings" I have made out of eigenvalue/eigenvector decomposition, but I still fail to do that.
Note that I made 3 pairs of highly correlated ...
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1answer
57 views
Scatter Plot - Basics [closed]
I am stuck in understanding a basic scatter plot. I am working in two dimensions i.e. there are two variables X & Y.
So, the question is that in the scatter plot, what do the two axes mean?
...
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1answer
29 views
How can I adjust for negative eigenvalues?
I wish to run a path analysis from a pooled correlation matrix that I have imputed using the maximum-likelihood procedure. There was considerable missing data. The resulting correlation matrix is:
<...
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66 views
Using NbClust on datasets that produce some negative eigenvalues. When to exclude data, when to force to positive, when to exclude test index?
Background on why I am using clustering:
I am analyzing data from a multistep biological experiment, where each step is done in batches of varying sizes. I want to account for any biases that might ...
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38 views
Why are the eigenvalues of $X'X$ equal to that of $XX'$ when $X$ is a design matrix? [duplicate]
The title says it all. If $X$ is a design matrix (columns containing variables, rows containing observations), I have observed that eigs($X'X$)=eigs($XX'$). I actually found this by accident when I ...
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2answers
72 views
How to apply Principal components (Eigenvectors) in PCA?
I struggle to understand how to get further in my PCA analysis after computing the eigenvectors of the covariance matrix.
I choose the number of principal components which contain 95% of the ...
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1answer
73 views
Is the first principal component is the one with the largest eigenvalue and how to convert it to explained variance?
In PCA, after we calculate the eigenvalues of each variable, we need to get the explained variance, I read an article which suggests:
...
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0answers
38 views
Procedure to quickly find the near zero eigenvalues (and corresponding eigenvectors) of a positive semidefinite square matrix?
I have an ill-conditioned positive semidefinite $n \times n$ square matrix (Hessian), with unknown rank $r$, that I need to compute the inverse for. I'd prefer not to compute the SVD for performance ...
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26 views
eigen value decomposition of co-variance a series generated by factor model
Let's assume $N\times T$ series $Y_t$ is generated by the following equation.
$$ Y_t = \begin{bmatrix}A_x & A_m\end{bmatrix}\begin{bmatrix}x_t \\ m_t \end{bmatrix}$$
Where $A_x$ and $A_m$ are $N\...
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8 views
How to measure changes in condition indices over time
I am trying to understand how adding data, one observation at a time, affects the condition indices of a model. A similar question is how adding individual observations affects the principal ...
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1answer
24 views
Does the first principle component maximize variance on the two variables with greatest covariance or on all variables simultaneously?
Suppose I am performing PCA on 3 standardized variables: height, weight, and income. I understand that each principle component maximizes variance along a new line, but there are two ways I can see ...
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33 views
How to show that VIF's are the main diagonal elements of $\mathbf{T\Lambda^{-1}T'}$?
I'm stucked in the Exercise 9.29 of Introduction to Linear Regression Analysis (5th edition), by Montgomery:
9.29) Show that if $\mathbf{X'X}$ is in correlation form, $\mathbf{\Lambda}$ is the ...
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0answers
14 views
Parallel analysis for principle components analysis and Multi-dimensional scaling
Does the same method for conducting a parallel analysis for principal component analyses apply to find the cut-off point for multidimensional scaling?
Under the pretense that PCA and linear MDS are ...
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1answer
46 views
Interpretation of Eigenvalue vs. Singular Value plot
I'm doing some preliminary analysis on the feature matrix for a certain dataset (rows are observations, columns are feature dimensions).
I have computed the SVD and PCA decompositions for this matrix ...
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21 views
Eigenvectors - principle lines of force
In the article about PCA and coavariance matrix I've read the following:
Finding the eigenvectors and eigenvalues of the covariance matrix is the equivalent of fitting those straight, principal-...
2
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1answer
18 views
Determine missing eigenvalues given only correlations between variables and components
Given that I just have a correlation matrix ($X$ Variables vs. $Y$ Principal Components), and that I am trying to find 2 missing eigenvalues (e.g., missing $\lambda_1$ and $\lambda_5$) from the total $...
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1answer
33 views
Gradient descent derivation in Eigenspace [duplicate]
I am trying to decode article on https://distill.pub/2017/momentum/
I was able to follow everything until the part with a change of basis x$^k=Q^T(w^k−w^⋆)$ to eigenspace... I conceptually understand ...
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0answers
98 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 ...
2
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1answer
47 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, ...
5
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0answers
267 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
396 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 ...
3
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0answers
190 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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0answers
11 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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36 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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19 views
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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0answers
48 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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0answers
68 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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0answers
17 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
208 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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0answers
43 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
455 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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2answers
59 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 ...
3
votes
2answers
234 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
36 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
56 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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0answers
41 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:
...
0
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0answers
61 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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0answers
65 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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0answers
58 views
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 ...
0
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1answer
44 views
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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0answers
52 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 \...
0
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1answer
333 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 ...
0
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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 ...
