Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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Sparsity regularization for eigenvectors

One way to think about finding the eigenvectors of a matrix $A$ is that they are the critical points of the functional $\vec x^\top A \vec x$ subject to $\|\vec x\|_2=1$. To regularize this problem, ...
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1answer
4k views

Analysis of compounds using PCA - selecting the right PCA “type” for the data…?

I'm completing scientific analysis of chemical compounds in consumer products. As a non-statistician, I would really appreciate any thoughts from the experts here. My data is non-normal so I've used ...
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1answer
267 views

Physical significance: multiplying matrix by outer product of its eigenvector

I stumbled around this piece of code: v1 <- eigen(X.center %*% t(X.center))$vectors[,1] X.0 <- v1 %*% t(v1) %*% X.center while v1 is the eigenvector ...
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1answer
2k views

Why do R function `eigen()' and Armadillo's `eig_sym()` give different results

I am trying to compute eigenvalues in C++ using the Armadillo function eig_sym via RcppArmadillo. The results are not entirely ...
2
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1answer
71 views

Regression Estimation difficulties

My regression problem is properly formulated, but is encountering serious computational difficulties. Dependent: $Y$ = multinomial Independent: $X_1, \dots, X_{90}$ = linearly independent set of ...
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0answers
551 views

Eigenvectors of a shuffled correlation matrix

A question about a shuffled vs. unshuffled correlation matrix I took the correlation matrix and shuffled between its values symmetrically. (shuffled only the left lower triangle of the matrix and ...
8
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1answer
261 views

Distribution of eigenvalues given one is known

I'm familiar with using insights from Random Matrix Theory to determine the number of principal components from the PCA of a covariance/correlation matrix to use to form factors. If the eigenvalue ...
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1answer
1k views

Exploratory factor analysis and eigenvalues

So, I ran an EFA on 60 items. Analysis resulted in 19 components with an eigenvalue of a score greater than 1. The only factors that theoretically make sense and that include more then 3 items have ...
0
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0answers
1k views

How do I convert the variances explained by a principal component into an eigenvalue?

I'm using SigmaPlot to run PCA on various measurements that should all correspond to size of an animal. Running PCA using a covariance matrix (instead of a correlation matrix, since all of the ...
6
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1answer
1k views

Adversarial noise in PCA

PCA is known to be quite sensitive to outlier noise (and this is why several Robust PCA techniques exists.) However, I am looking for a concrete example of sensitivity of PCA to adversarial noise that ...
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1answer
2k views

Dimensionality Reduction using PCA, with SVD of correlation matrix

I have computed a correlation matrix from certain data set of dimension 6 The correlation matrix is: ...
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1answer
3k views

PCA eigenvectors with dimensionality reduction

I want to understand how I can compute the eigenvectors and the eigenvalues of a matrix using dimensional reduction.I have a Matrix $M$ of dimensions $n$ x $d$ using dimension reduction I can compute ...
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1answer
2k views

Determining eigenvalues and eigenvectors in R

x<-model.matrix(d) [,-1] e<-eigen(t(x) %*% x) e values [1] 1174600.548 21261.741 16133.842 6206.181 1856.894 First my $\alpha_1=1174600.548$, but ...
9
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2answers
553 views

Why is the amount of variance explained by my 1st PC so close to the average pairwise correlation?

What is the relationship between the first principal component(s) and the average correlation in the correlation matrix? For example, in an empirical application I observe that the average ...
0
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1answer
73 views

Show that all the characteristic roots of a dispersion matrix of a random variable are non-negative

Show that all the characteristic roots of a dispersion matrix of a random variable are non-negative. $$\begin{vmatrix} \sigma_{11}-\lambda & \sigma_{12} & \cdots & \sigma_{1p}\\ \sigma_{...
4
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0answers
3k views

What is the correct way to calculate the explained variance of each EOF as calculated from a gappy data set?

I am trying to determine the correct amount of variance explained by each mode of an Empirical Orthogonal Function (EOF) analysis (similar to "PCA") as applied to a gappy data set. (i.e., containing ...
12
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1answer
3k views

Why are eigen and svd decompositions of a covariance matrix based on sparse data yielding different results?

I am trying to decompose a covariance matrix based on a sparse / gappy data set. I'm noticing that the sum of lambda (explained variance), as calculated with svd, ...
4
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1answer
756 views

Eigenvalues nearly all 1 in canonical correlation analysis

I ran a Canonical Correlation Analysis on about 845 cases with 1000 variables each. (It originally started with 1000 cases and 400 variables but by using a kernel I got a 1000x1000 matrix) As a ...
9
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1answer
533 views

Estimated distribution of eigenvalues for i.i.d. (uniform or normal) data

Assuming I have a data set with $d$ dimensions (e.g. $d=20$) so that each dimension is i.i.d. $X_i \sim U[0;1]$ (alternatively, each dimension $X_i \sim \mathcal N[0;1]$) and independent of each other....
1158
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28answers
721k views

Making sense of principal component analysis, eigenvectors & eigenvalues

In today's pattern recognition class my professor talked about PCA, eigenvectors and eigenvalues. I understood the mathematics of it. If I'm asked to find eigenvalues etc. I'll do it correctly like ...

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