Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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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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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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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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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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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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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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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-...
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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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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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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 ...
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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, ...
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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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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 ...
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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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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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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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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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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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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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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 ...
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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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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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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: ...
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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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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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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 ...
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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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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 \...
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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 ...
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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 ...