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Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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36 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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0answers
27 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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44 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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54 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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16 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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48 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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1answer
89 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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1answer
19 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 ...
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1answer
30 views

Compute the $k$ largest eigenvector in spectral clustering

In Spectral Clustering, we need to compute the top $k$ largest eigenvector of normalized $L$. $$L = D^{-\frac{1}{2}}SD^{-\frac{1}{2}}$$ In Andrew NG's paper, L is not positive definite (unless ...
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13 views

Approximation error on standard and sparse PCA

I am trying to understand the approximation error of pca explained in this tutorial. $\sum_{n=1}^{N}||x_n-\tilde{x}_n||=N\sum_{i=M+1}^{D}\lambda_i$ where $||x_n-\tilde{x}_n||=\sum_{i=M+1}^{D}\left(\...
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16 views

Eigenvalue decomposition/SVD and the filtering perspective

I have been studying the SVD algorithm recently and I can understand how it might be used for compression but I am trying to figure out if there is a perspective of SVD where it can be seen as a low ...
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23 views

Does it make sense to analyse an autocorrelation matrix with Random Matrix Theory?

I wonder whether you can gain valuable information about a time series by analysing its autocorrelation matrix using RMT. I know that RMT can help to extract information about the collective ...
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13 views

Signs in SPSS's PCA with rotations with the FACTOR algorithm

I am trying to reproduce the results of the PCA with rotations from SPSS in python. But there is some information I didn't find in their documentation. I am trying to do the PCA like in the FACTOR ...
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0answers
42 views

Intuition behind finding number of clusters in spectral clustering according to Zelnik-Manor and Perona

I have a quick question on the intuition behind the estimation strategy for the number of clusters presented in the paper "Self-Tuning Spectral Clustering" by Lihi Zelnik-Manor and Pietro Perona. See ...
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1answer
146 views

PCA in psych package with more columns than rows

Why is it impossible to do a PCA in R using principal from psych package without warnings with a matrix, which has more columns ...
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26 views

PCA for three-dimensional linear fit on time-resolved trajectory

I study the behavior of organisms that are able of self-locomotion and that show directed motion toward one another. This directed motion occurs through the detection of chemical trails released by ...
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3answers
560 views

PCA basic: Must eigenvalues converge to zero at high dimension?

Recently, I obtained several PCA plots, and because I am unable to produce eigenvalues for higher dimensions, I tried to extrapolate them based on the available data. The reason why I want to do this ...
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1answer
47 views

PCA Basic: Do eigenvalues remain constant with increase in dimensions?

Say I have a set of genomic data to be analysed for population structure. I ran two different analyses using two different maximum principal components. For the first analysis, I modeled the data ...
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0answers
35 views

Weighting a Covariance Matrix via Eigendecomposition

Given a covariance matrix $\mathbf{\Sigma} \in \mathbb{R}^{N \times N}$ between $N$ variables of interest, generated by some kernel function $\mathbf{\kappa}(\mathbf{x}, \mathbf{y})$, and some vector $...
2
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3answers
142 views

PCA to choose variables based on its loadings on PC1 [duplicate]

I have a dataset of cave dimensions (and other variables related to their features). The problem is that 3 of these variables are: Length, Area, and Volume. These 3 are highly correlated as they ...
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1answer
108 views

Negative eigenvalues when computing MultiDimensional Scaling given nonnegative distance matrix

I am using the Smile MDS https://github.com/haifengl/smile/blob/master/core/src/main/java/smile/mds/MDS.java and occasionally running into: ...
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60 views

Interpreting low Cronbach's alpha value in CATPCA

I ran categorical principal component analysis (CATPCA) on the data that I collected through a questionnaire. The purpose of the questionnaire was to understand pedestrians intentions and attitudes ...
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55 views

Why are the discriminant axes in linear discriminant analysis (LDA) not orthogonal?

This may be a quite silly question and please correct me if I'm wrong. The discriminants (discriminant axes) are essentially eigenvectors of $\mathrm{Cov}_\mathrm{within}^{-1} \mathrm{Cov}_\mathrm{...
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0answers
33 views

Is max. Eigenvalue of k-sparse PCA always $\leq$ max. Eigenvalue of normal PCA on same dataset?

Is max. Eigenvalue of k-sparse PCA always less than or equal to the max. Eigenvalue of normal PCA on same dataset? K refers to the number of non zero eigenvalues when the dataset is of dimension n <...
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1answer
31 views

How to infer the eigenvalue distribution from matrix where each entry has a known Gaussian distribution?

Problem Given $X \in \mathbb{R}^{n \times n}$ where $X_{ij} \sim \mathcal{N}(\mu_{ij}, \sigma_{ij}^2 I)$ Find the eigenvalue distribution using whatever you can. Background In my field, I have a ...
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1answer
81 views

Usage of the term “feature vector” in Lindsay I Smith's PCA tutorial

I'm currently following Lindsay Smith's (in large parts very well written) PCA tutorial. I'm a bit confused about the usage of the term "feature vector" in this paper though. A quote from this paper ...
2
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1answer
38 views

Can I use combination of eigenvectors as a single vector to explain most of variance?

I have a problem trying to find a combination (or weighted average) of variables (statistics) that best explains the sample statistics. A – n x p matrix (n: observations p: variables, here are ...
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1answer
160 views

Must a matrix of sample pairwise covariances be PSD?

Consider a random vector $\mathbf{X}=(X)_{i=1}^n$. Then the covariance matrix $$C=\mathbb{E}[(\mathbf{X}-\mu(\mathbf{X}))(\mathbf{X}-\mu(\mathbf{X}))^\top]$$ is by definition positive-semidefinite. (...
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1answer
110 views

Orthogonality as found by the Gram-Schmidt process vs. uncorrelated basis vectors

I have a data matrix $Y$ of size $n \times p$, a basis vector in $\mathbb{R}^p$ $v_1$, and a potential basis vector in $\mathbb{R}^p$ $v_2'$. Then, if I use the Gram-Schmidt process on $[v_1, v_2']$ ...
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1answer
35 views

Exploratory factor analysis with 5 positively loaded and 1 negatively loaded factor. How to interpret?

I conducted an EFA with Maximum Likelihood and Direct Oblimin Rotation. Following the Kaiser Eigenvalue 1 rule, I identified 6 latent factors. 5 of them are positively loaded, while 1 is not. Here ...
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1answer
61 views

Is it possible to have a basis for a covariance matrix such that the greatest variance is greater than the variance of the first eigenvector?

Suppose we have a covariance matrix $C$. Define the eigendecomposition, $C = Q^{-1} \Lambda Q$ and some other arbitrary basis $C = B^{-1} D B$. Define $V_\text{PCA} = \text{diag}(QCQ^{-1})$ and $V_B =...
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2answers
2k views

How does eigenvalues measure variance along the principal components in PCA? [duplicate]

I understand that eigenvalues measure variance along the principal components. Questions How are eigenvalues and variance same for PCA? What is the intuition behind this being the same? What is ...
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0answers
46 views

Find the missing eigenvector values in PCA

A principal component analysis is carried out using the correlation matrix R of a data set with n = 25 observations and p = 4 variables. The ordered eigenvalues of R are given by $$\lambda_1 = 2.25, \...
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0answers
67 views

Interpretation of Square of Covariance Matrix

I have a random variable, distributed as a sum of independent chi-squared random variables each with one degree of freedom. $$ X = \sum_{i=1}^n \lambda_i \chi^2_{i(1)}$$ where $ \lambda_i$ are ...
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0answers
70 views

Expected eigenvalues of a Wishart Matrix

I consider a $n\times n$ Wishart Matrix with expected value $p \cdot I_n$, i.e. a matrix of the form $$W = XX'$$ with $X$ a $n\times p$ matrix with independent standard normal entries. It is easy ...
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0answers
40 views

Associated eigenvalue problem to spline penalization

In Natural Spline functions, their associated eigenvalue problem by F. Utreras they prove a result. I am interested to know if a more generalized form exists and where to find it. Consider the space ...
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2answers
199 views

Most important variable in a principal component

How do I tell what is the most important variable for a particular principal component? Do I calculate the absolute value for all the eigenvectors associated with that particular principal component ...
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0answers
147 views

General PCA optimization problem

I was looking at the PCA optimization problem, which is finding a matrix $U \in \mathbb{R}^{d\times n}$, $n \le d$, that solves the problem $$\max{tr(U^TCU)},\ \ \ s.t. U^TU = I, $$ where $C$ is the ...
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0answers
14 views

What is the point of applying PCA to correlation matrix? [duplicate]

I have always applying PCA to covariance matrix . This is what most of text books and web based tutorials do : $X^{T}X$ Is the covariance matrix when X is centered. PCA tries to capture the big ...
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0answers
22 views

What significance can be attached to the eigenmodes of a transition matrix?

I am studying a continuous dynamical system and am categorising preferred areas of the system's phase space using a 3 state hidden Markov model. As the resulting transition matrix is row stochastic, ...
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1answer
240 views

Covariance matrix of image data is not positive definite matrix

I've really hit the wall here and need help with direction :). I am trying to use mvnpdf as part of basic EM algorithm but the covariance matrix of data seems to be not positive definite. There are ...
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1answer
19 views

Rank of N x D vs D x N matrices

If $X$ is a random $N \times D$ matrix where $N > D$, then why is the rank of X - mean(X, 1) $D$ while the rank of ...
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2answers
260 views

If the eigenvalues of a covariance matrix have very low variance, what does it mean?

If we have a covariance matrix $A$ $AP=PD$ where $D$ is a diagonal matrix which contains all the eigenvalues then if the variance of the eigenvalues is very small, what does this tell us about $A$? ...
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0answers
46 views

What should I check after I do PCA?

I have X in which each row is a sample and each column is a predictor. I de-mean X first and then construct the co-variance matrix $A=X^{T}X$ after that I do PCA $AP=PD$ while each column in P is ...
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1answer
46 views

Why is the result a set of factors instead of just only one?

I am trying to get into SEM and factor analysis. I understand a factor is a latent construct, say e.g. $intelligence$, user-defined by the (weighted) average of a set of indicators $x_1, x_2\dots x_n$....
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0answers
55 views

Interpreting cointegration test

I am very new to interpreting cointegration tests and eigenvalues, so any help would be appreciated.
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1answer
72 views

For an eigenvector v, is it always true that v^tv =1?

I am reading this link PCA which is a very insightful tutorial however, in this tutorial, the author mentioned a constraint on PCA: $C^{T}C=1$ when we look at eigenvalue/eigenvector definition $...
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0answers
109 views

Why does pnorm() change the correlation in my multivariate normal sample?

I came across 2 problems that puzzle me while simulating variables for a Monte Carlo simulation, using the rnormalcopula command from the rCopula example. The first one is the one from the title, the ...
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1answer
52 views

Adding/removing variables to PCA

If I have a PCA that I ran on some set of variables, how (if at all) will it relate to the PCA results if I add or remove one variable? Will the PCA components change in some well-defined way, or is ...
3
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1answer
777 views

Maximum number of principal components in PCA. Is sklearn wrong?

Recently I've been interested in applying PCA to a dataset I have and I wanted to develop a deep understanding of what I would actually be doing when I implement it. Today I encountered two ...