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Questions tagged [eigenvalues]

For questions involving calculation or interpretation of eigenvalues or eigenvectors.

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Error term in SGD with momentum

I am reading the article "How Momentum really works" (https://distill.pub/2017/momentum/), and i am confused in one point: I am trying to derive the convergence rate for momentum from the ...
Patricio's user avatar
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Why do I need Loadings when I can reconstruct data with eigenvectors in PCA?

I have read the responses to this question here, here and here, but I am still confused on the application of loadings and eigenvectors. Principally this statement (from the first link), It is ...
insomniac's user avatar
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Linear algebra properties of a confusion matrix (eigenvalues, eigenvectors, and determinants)

This answer to a question on Math Stack Exchange got me thinking about a confusion matrix as more than just a rectangular array of numbers. We don’t talk about a confusion matrix as a linear ...
Dave's user avatar
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Compare Eigenvector Centrality of two networks

We have two networks (G1, G2), one with 4 times more nodes than the other, and we want to compare the eigenvector centrality of their three most central nodes (e.g. top_1 node of G1 vs top_1 node of ...
s223's user avatar
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Are squared loadings and squared distances the same in Principal Component Analysis?

I read online that: Eigenvalue: Represents the variance explained by the principal component. It equals both the sum of squared loadings for that component and the sum of squared projections of data ...
Bennett K.N.'s user avatar
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1 answer
85 views

Expected value of largest eigen value of sample correlation matrix

Suppose $X$ follows some multivariate distribution with zero mean and Identity covariance matrix. Suppose $X$ is N dimensional. Suppose $R$ is the sample correlation matrix, calculated based on n ...
deb's user avatar
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Distribution of Maximum Eigen Value

Suppose I have X, k*n, where $M=X'X$. Suppose $n>>k$, and $rank(M) =k-1$. Suppose $\lambda_1, \cdots, \lambda_{k-1}$ are the eigen values of M. Under the assumption that the columns of X are ...
deb's user avatar
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Interpreting eigenvalues of non-normalized covariance matrix of physical system

Cross-posted from physics stackexchange Summary: Eigenvalues of a "non-normalized" covariance matrix of time-series measurements from a linear system have units of Action (energy * time). ...
user3716267's user avatar
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The eigen values of Johansen's cointegration procedure

Assume a K dimension VECM model for cointegration analysis $$\Delta y_t=\Pi y_{t-1}+\Gamma_1\Delta y_{t-1}+...+\Gamma_{p-1}\Delta y_{t-p+1}+u_t$$ The Johansen approach for maximum eigenvalue test or ...
qiu's user avatar
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Using sample covariance to estimate eigenvalues of population covariance?

Suppose $X$ is an $m\times n$ data matrix with $m\approx \infty$ and $X_k$ consists of random $k$ rows of $X$. How do I estimate the spectrum of $X$ from $X_k$? I can only reliably estimate $K$ ...
Yaroslav Bulatov's user avatar
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How to balance PCA and LDA in subspace learning?

PCA is a generative model, by which input images or data can be reconstructed. LDA (Linear Discriminant Analysis) is a discriminative model, which extracts better features for classification. How to ...
chickensoup's user avatar
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Solving an exercise about admissible coefficient values for a MA(1) process

I'm studying "Principles of system identification : Theory and Pratice" by Arun K. Tangirala and well... I've just entered the part about moving averages and I'm confused. I don't understand ...
NokiYola's user avatar
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1 answer
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Distribution of Data Within Hypersphere That Maximizes the Nuclear Norm?

Suppose I have $N$ points in $\mathbb{R}^D$ such that for each point $x_n$, its L2 norm is at most 1: $$||x_n||_2^2 = x_n^T x_n \leq 1$$ Assuming $N > D >> 0$, if I construct a matrix $X \in \...
Rylan Schaeffer's user avatar
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Using cross correlation and an uncorrelated eigenvector basis to minimize exposure

I have the following problem: We are looking at stock market data and have a historic price dataset $X$ with two stock types: \begin{equation} X = \begin{bmatrix} x_1 & x_2 \end{bmatrix} = \begin{...
tbolind's user avatar
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What is the Intution behind eigen vector in context of Google Page rank Algorithm?

I am trying to understand the intuition behind Eigen Values and vectors. Theoretically, I understand that a linear transformation can be understood as scaling for special vectors and those special ...
Gupta's user avatar
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PCA with gram matrix produces different results from PCA done using covariance matrix?

I was trying PCA on a dataset (#samples=24, #dims=42) via eigendecomposition using numpy. I read that for matrices where the number of features exceeds the number of samples, we should use the gram ...
Ranjan's user avatar
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Difference between conducting PCA on $XX^\top$ vs $X^\top X$?

PCA: For a given set of centered data $\mathscr D =\{x_i\}_{i=1}^N \subset \mathbb R^d$, i.e. the data has $N$ examples with dimension $d$. Then the principal directions of PCA can be obtained from ...
Fong Lam's user avatar
2 votes
1 answer
134 views

Functional Principal Component Analysis - Explaining Functional Principal Component Scores

I was wondering if someone can help with explaining Functional Principal Component Scores? I am working with a dataset which reflects participants in a weight loss management trial (longitudinal data)....
Data_Science_Mick's user avatar
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Why do we expect the eigenvalues of the Gramian matrix found by Maximum Variance Unfolding to capture the number of degrees of freedom in the data?

Maximum Variance Unfolding (MVU) is a manifold learning method which, like other forms of dimensionality reduction, makes the assumption that whatever (high-dimensional) data we're dealing with "...
Saucy Goat's user avatar
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Is there an eigenvector based decomposition scheme that will find the longest axis of variation collinear with a named variable?

So far as I understand, algorithms like principal component analysis, spectral decomposition, etc. are similar algorithms that identify orthogonal vectors in a dataset using a different set of ...
Vincent Laufer's user avatar
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I decomposed only the first four components from PCA/EOF. Can I get the percent variance of each component against the entire dataset?

I am using NIPALS (non-linear iterative partial least squares) to get the loadings and scores from a PCA analysis. The reason I am using NIPALS rather than singular value decomposition is because ...
Fat Salami's user avatar
1 vote
1 answer
76 views

Which features corresponds to which eigenvalues when use SVD in PCA?

Today, after learning about performing $PCA$ using $SVD$, I know $PCA$ will choose $K$ components that have the highest eigenvalues. I have a question which feature will correspond to which ...
kenj's user avatar
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How do I check if the spectral radius of a matrix is greater than 1? [closed]

I have a real matrix whose two largest eigenvalues are two conjugate complex numbers. I need to check if their absolute value is greater than 1. Since the largest by absolute value eigenvalue is ...
seed's user avatar
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1 answer
416 views

Singular value and Eigen value for Square Matrix [duplicate]

Is it true that for a square symmetric matrix such as the covariance matrix, the singular values are equal to the eigenvalues? The eigen decomposition for covariance is the same as singular value ...
user3153824's user avatar
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24 views

Do similar PCA feature importance in first few top PCs mean these variables are nearly same in the original space?

I am using PCA to do the data inspection. First 3 PCs explain nearly 82% of the total variance. Suppose the number of features is $n$. And I found 4 of the features have similar PCA feature ...
Xu Shan's user avatar
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2 votes
1 answer
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Eigenvalues of VAR(1) coefficient matrix

Suppose we have a VAR(1) model: $$ \mathbf{X}_t = \boldsymbol{\Phi} \mathbf{X}_{t-1} + \mathbf{Z}_t, \hspace{10mm} \mathbf{Z}_t \sim WN(0,\Sigma) $$ If we can keep plugging that equation into itself, ...
Taylor's user avatar
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Using Dudley Integral to estimate maximum singular value of Gaussian random matrices [duplicate]

On Exercise 5.14 of Wainwright, it provides a way to estimate maximum singular value of Gaussian random matrices using the one-step discretization bound and Gaussian comparison inequality. Can we use ...
dc3506's user avatar
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Johansen test: why testing for the algebraic multiplicity of 0 and not for the nullity?

From what I already know about the Johansen test, it tests the rank of the VAR matrix (in error correction form) through steps testing whether every eigenvalue is signifincantly different from 0 (...
Mauro's user avatar
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Johansen test accepts first null hypothesis but would reject last one

Suppose that we perfrom a Johansen test over three I(1) variables that give us these results through the maximum eigenvalues statistic: as you can see, we accept the null hypothesis in the first step ...
Mauro's user avatar
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Rotation Matrix from Covariance of 3D point-cloud

I am trying to retreive rotation matrix from a rotated 3D point cloud covariance matrix, using SVD decomposition (as done in SimNet and MVTrans). Here how I computed the covariance matrix from 3D ...
Samuele's user avatar
14 votes
3 answers
2k views

Eigenvalues/Eigenvectors of Correlation and Covariance matrices

Suppose $\Sigma$ is a covariance matrix $P$ is its corresponding correlation matrix. Let $\lambda_1, \dots, \lambda_p$ and $\tau_1, \dots, \tau_p$ denote the ordered eigenvalues of $\Sigma$ and $P$, ...
Greenparker's user avatar
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4 votes
0 answers
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PCA: Why can’t we say loadings are the coefficients used to back predict the components by the variables?

In this answer @ttnphns writes Both eigenvectors and loadings are similar in respect that they serve regressional coefficients in predicting the variables by the components (not vice versa!) , and ...
user1205901 - Слава Україні's user avatar
4 votes
1 answer
169 views

Why is the Scaling Matrix in LDA unnormalized?

I was carrying out LDA (linear Discriminant Analysis) and noticed that the Scaling matrix produced by R is not normalized. Here is an example: ...
Onyambu's user avatar
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4 votes
2 answers
360 views

Does the average eigenvalue equals 1 in PCA applied to standardised data?

From what I understood when we are doing PCA, we can work both with raw or standardised data, depending on the situation we're in. Is it true that the average of the eigenvalues is equal to 1 when we ...
Paolo Totaro's user avatar
1 vote
1 answer
56 views

Can I apply Kaiser Rule without knowing the eigenvalues?

Kaiser's rule suggests the number of principal components to be included in an analysis by looking at eigenvalues. If I'm given standard deviations only, instead of eigenvalues, can I still somehow ...
Paolo Totaro's user avatar
1 vote
0 answers
54 views

Are low-rank kernel approximations implementing implicit regularization?

Consider a kernel estimation problem as follows. We have functions $f^* \sim GP(0, C^*)$ drawn from a Gaussian process. We want to construct a kernel $K$ that does well in regressing functions drawn ...
Tanishq Kumar's user avatar
2 votes
0 answers
72 views

Eigenvalues of a block matrix

Let $\bar{\lambda}$ be the smallest eigenvalue of $$M=\Omega^{-1/2}Y'Z(Z'Z)^{-1}Z'Y\Omega^{-1/2}$$, where $\Omega$, $Y$, and $Z$ are $(l\times l)$, $(N\times l)$, and $(N \times k)$ matrices, ...
MinChul Park's user avatar
1 vote
0 answers
62 views

is it fair to use a subset of eigenvalues to evaluate the multidimensional variance

I want to find a single metric to assess how spread (or how much variance) a multidimensional dataset (a large number of features) is. I learned that the determinant (or pseudo-determinant) of the ...
qliang's user avatar
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1 vote
0 answers
73 views

How to reconstruct Cartesian coordinates from a Gram matrix?

I read this article and am wondering why we can reconstruct Cartesian coordinates from a Gram matrix generated by taking dot product of the distance from the origin. They had a Euclidean distance ...
delilah's user avatar
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0 answers
22 views

Finding Eigen Vector [duplicate]

I followed this tutorial from this question. The dataset and example used in the mentioned tutorial seems famous one and it gives following eigen vectors:- ...
jaykio77's user avatar
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0 answers
70 views

Why do we multiply the centered data with eigen vector instead of taking inverse of the eigen vector while performing PCA

The question may sound stupid but I really don't understand the logic behind this. Whenever we do a PCA, we take a covariance matrix on the centered data and do eigen decomposition. In order to ...
Jacob Simon Areickal's user avatar
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0 answers
45 views

PCA for 3D Lidar data

I'm trying to work with Lidar data to classify road. I'm wondering how to use pca (or something else?) to identify if region has connectivity > threshold then label the region to be road? Would ...
h612's user avatar
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7 votes
1 answer
265 views

Expectation of a function of the sample covariance matrix

Let $X \sim N(\mu, \Sigma)$ and let $\hat\Sigma$ be the sample covariance matrix calculated from an iid sample $X_1, \dots, X_N$. Also, let $\phi_i$ and $\phi_j$ be the orthonormal eigenvectors of $\...
BurlyPotatoMan's user avatar
1 vote
1 answer
46 views

Why do eigenvalues of $\mathbf\Phi^T\mathbf\Phi$ increase with the size of data set?

The question comes from a paragraph in page 171 of "Pattern Recognition and Machine Learning" by Christopher M. Bishop: Here $\mathbf\Phi$ is the design matrix for a data set of $N$ samples ...
zzzhhh's user avatar
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4 votes
2 answers
454 views

Why is $\mathbf\Phi^{\top}\mathbf\Phi$ a positive definite matrix?

I had this question when reading section 3.5.3 on page 170 of "Pattern Recognition and Machine Learning" written by Christopher M. Bishop: Here $\mathbf\Phi$ represents the design matrix ...
zzzhhh's user avatar
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3 votes
0 answers
104 views

Relation between generalization bounds of Kernel Ridge Regression and largest eigenvalue of the kernel Gram matrix

Consider a positive-definite, symmetric function $k(x_1, x_2)$ which is used, given the dataset $\{(x_i, y_i)\}_{i=1}^m$, to construct the Gram matrix $K = [k(x_i, x_j)]_{i,j \in 1, ..., m}$. What is ...
incud's user avatar
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1 vote
1 answer
96 views

How to derive the three matrices of SVD from eigenvalue decomposition in Kernel PCA?

Kernel PCA is usually done via eigenvalue decomposition of the Kernel Matrix $\mathbf{K}$ and standard PCA via SVD of the input $\mathbf{X}$. In standard PCA as far as I know we can derive $\mathbf{S}...
arod's user avatar
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2 votes
1 answer
239 views

Confusion about principal component and major axis of the ellipse corresponding to the covariance matrix

Based on my understanding, in PCA, we try to find a linear combination of axes such that the variance in that direction is maximized. If variables have the covariance matrix $\Sigma$, then, the first ...
Amin's user avatar
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1 vote
0 answers
55 views

The convergence of random variables to standard normal distribution

Let $V_s$ be $n\times s$ real matrix and consisting i.i.d $\mathcal{N}(0,1)$ random variables [*]. Suppose that $O_s^1$ is the orthogonal matrix, its first column being the normalization of the first ...
Atina Husnaqilati's user avatar
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47 views

Finding the vectors/modes "maximally responsible" for the fluctutation of a quantity?

I have a question which I fear might be simple but i am completely unable to figure out. Lets say we have a time-series of a vector of coordinates which define a molecule ( column vector of x,y,z) ...
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