Questions tagged [eigenvalues]
For questions involving calculation or interpretation of eigenvalues or eigenvectors.
402
questions
0
votes
0
answers
16
views
Does invariance of PCA under orthogonal transformation hold for data that is not centered?
I read the proof in the top answer to this question, but that page assumes that $\overline{A} = 0$. If the data instead has some nonzero mean $\mu$, I'm not sure if the same logic applies: ...
0
votes
0
answers
10
views
What is the variable v in this case? Is it standard deviation or variance?
The code below is from an example on the scikit-learn website for plotting the Gaussians of a GMM as ellipses. However, I don't know whether the final definition of the variable ...
- 111
3
votes
0
answers
44
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 ...
- 31
1
vote
1
answer
43
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}...
- 23
1
vote
1
answer
32
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 ...
- 513
1
vote
0
answers
39
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 ...
0
votes
0
answers
45
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) ...
- 1
0
votes
0
answers
25
views
The relationship between eigenvalues of a covariance matrix and the variances of the same data matrix after using eigenvectors as bases
Suppose we have a data matrix $\mathbf{X}\in \mathbb{R}^{M\times N}$ with $M$ features, $N$ samples and zero means ($M \lt N$). The covariance matrix of $\mathbf{X}$ is $\mathbf{C_x}=\frac{1}{N}\...
- 1
0
votes
0
answers
16
views
Are "weights", "loading scores", and "singular values" all synonyms?
I'm currently learning to use "eigenfaces" for facial image classification. Unfortunately, I've encountered some confusion with the following lines of code:
...
- 211
1
vote
0
answers
36
views
Generate a random covariance matrix with specified eigenspectra and diagonal elements and first off-diagonal?
I want to generate a random covariance matrix ($c \in \mathcal{R}^{n \times n} $) whose eigenspectra, i.e., $n$ eigenvalues $e_0 \in \mathcal{R}^{n\times 1}$ and diagonal elements $c_{ii} \,\, i=1 \,\,...
- 31
1
vote
0
answers
21
views
Eigenvalues of Johansen Trace Test
I'm currently taking a course in time series and have been struggling with understanding the Johansen trace test. Specifically, the calculation of the eigenvalues for the Likelihood ratio statistic. ...
0
votes
0
answers
36
views
EFA: Eigenvalues or Loadings after extraction (SPSS)
I've been doing a EFA with ML extraction and Promax rotation, whereby three factors were extracted. For reporting the results, I was wondering whether to use the 'Initial Eigenvalues' or the '...
0
votes
0
answers
31
views
Is it possible to derive the expected sampled eigenvalues given a population correlation matrix?
My question is : Is it possible to derive the expected sampled eigenvalues given a population correlation matrix and a sample size $n$.
I am pondering about how behaves eigenvalues of correlation ...
- 463
0
votes
0
answers
34
views
Popular symmetric positive definite matrices with explicitly known eigenvalues (at least largest eigenvalue)
I am looking for popular symmetric positive definite matrices with explicitly known eigenvalues (at least largest eigenvalue) arising as autocovariance matrices in time series (for example). In fact, ...
- 1,295
4
votes
2
answers
119
views
Why is sum of squares equal to eigenvalue in PCA?
We fit a line or a hyperplane to a set of points. We project the points onto the hyperplane. The sum of squared distances of the projected points to origin is equal to the eigenvalue. Why is that?
- 255
1
vote
0
answers
65
views
Positive semi definite matrix with negative eigenvalues?
From what I know, for any square real matrix A, a matrix generated with the following should be a positive semidefinite (PSD) matrix:
...
- 11
0
votes
0
answers
89
views
Why in PCA the sum of the distances of from the principal component is the explained variance?
I am trying to understand why, when I calculate the principal components, summing the distances of the original points to the best fit line (eigenvector) equals the variance this PC explains?
- 101
0
votes
1
answer
43
views
Eigenvector centrality comparison
Let's say we calculate the eigenvector centrality of the same set of nodes in different years (we have a network each year). Note that the eigenvector centrality is normalized in such a way that the ...
- 561
1
vote
0
answers
26
views
Finding the eigenvalues of covariance matrix [duplicate]
Came across this as an interview question that I saw online: given a covariance matrix with diagonals being all 1 and the off-diagonals being c, what are its eigenvalues?
Going by the definition $Av = ...
- 161
0
votes
0
answers
56
views
Show that the autocovariance matrix is positive definite
I've been working through the textbook Time Series Analysis and its Applications (R. H. Shumway & D. S. Stoffer 2ed). The topic I'm looking at is forecasting using ARMA models. The below assumes ...
0
votes
0
answers
19
views
Different eigenvalues output running EFA in SPSS VS R
I am new to R. I ran exploratory factor analysis (EFA) in SPSS and in R. The SPSS output suggests 3 factors of eigenvalue>1 (8.78;1.5;1.05), while R indicates there is just 1 factor with eigenvalue&...
0
votes
0
answers
96
views
The discrepancy of results of PCA via Eigendecomposition vs via SVD in Python with scipy.linalg [duplicate]
I recently learned about different methods of PCA. I decided to manually implement PCA in Python with Eigendecomposition of cov(X) and the Singular Value ...
- 165
2
votes
0
answers
43
views
Is a bivariate copula relevant in this physics setting manifesting uniform univariate marginals--and, if so, how can it be constructed?
To quickly place our probabilistic (copula) question in its subject matter setting, we note that a fundamental concept in quantum theory is that of entanglement QuantumEntanglement.
The states of ...
- 161
0
votes
0
answers
269
views
All eigenvalues less than 1 in factor analysis
Is it possible to have all eigenvalues less than 1 during factor analysis? then what can be concluded here? since a factor is considered a factor when the eigenvalue of the factor is greater than 1. ...
- 1
1
vote
0
answers
67
views
How do I get the stationary distribution of a Markov chain matrix from SVD?
I have a matrix that represents a Markov chain.
...
- 1,327
0
votes
0
answers
7
views
data projection on k - dime hyperplane (for eigenvector with largest eigenvalue) provides corrected representation
This is in reference to the text, outlier analysis by Charu C. Aggarwal:
The author mentions, the projection of data points on the k - dimensional hyperplane corresponding to the largest eigenvalues (...
- 207
0
votes
0
answers
47
views
Eigenvalue decomposition
Why is the eigenvalue decomposition of the covariance matrix formula, $UΛU^T$ (assuming T means transpose) and not $UΛU^{-1}$? Where $U$ is eigenvectors and $Λ$ is a diagonal matrix consisting of ...
1
vote
1
answer
110
views
Find first Principal Component (and loading) using a fast iterative algorithm without covariance matrix
I have a matrix $X$ and I would like to find its first principal component and the corresponding loadings. I would like to do this without computing the covariance matrix of $X$. How can I do so?
This ...
- 499
3
votes
2
answers
154
views
Inequality on norm in terms of eigenvalue
I came across this inequality but not sure if it's true:
$|\lambda_{\text{min}}|^2\|\hat{\beta}-\beta\|^2_2\leq (\hat{\beta}-\beta)'\hat{\Sigma}(\hat{\beta}-\beta)$
where $\lambda_{\text{min}}$ is the ...
- 353
1
vote
0
answers
94
views
Eigenvalues of time lagged covariance matrix - should be always real and positive?
Consider two random variables $X_t$ and $X_{t+\tau}$; where both come from random variable $X$ but are lagged by $\tau$. Assume that both are mean-free. If we want calculate covariance matrix of them, ...
- 547
0
votes
0
answers
34
views
How to calculate eigen-decomposition for a positive kernel in R^2 using R
Suppose $K$ is a continuous positive semi-definite kernel on $\mathcal{T} = [0,1]\times[0,1] \subseteq \mathbb{R}^2$. By Mercer Theorem, there is a eigen-decomposition of $K$ such that
$$
K(x,y) = \...
- 1
2
votes
0
answers
332
views
Negative Eigenvalues in EFA
Assuming a set of data meet all assumptions for EFA and we are doing a factor analysis (with the SMC used to define the shared variance), how do negative eigenvalues appear and what does that say ...
- 375
1
vote
1
answer
72
views
Why is the first canonical direction equal to the left singular vector i.e. why is $w_1 = a = u_1$ in CCA (Canonical Correlation Analysis)?
I want to understand why the canonical direction $a$ is equal to the left singular value of $M = \Sigma^{-1/2}_X \Sigma_{X, Y} \Sigma^{-1/2}_Y$ and not $a = \sigma_1 u_1$.
My calculation tell me that ...
- 6,186
1
vote
0
answers
43
views
Relation between PCA eigen values and data visualization
We have matrix data $X$ which is $n\times d$. We use the covariance matrix/ design matrix/ gram matrix $X^T X$ to perform least-squares/ PCA. I compute the eigen basis representation of said matrix
$$...
- 181
2
votes
1
answer
51
views
number and size of eigenvectors in PCA
As I understand, the size of eigenvector produced in PCA should be min{n,N}, where N=number of samples and n=dimension of each sample (Right?). However, I have seen in couple of cases that this size ...
- 43
1
vote
1
answer
255
views
Making sure that the design matrix is positive (semi-) definite
In Bayesian linear regression, how do I make sure that the design matrix produced by a neural network $ \Phi$ is positive definite?
Computing the covariance matrix on the weight requires inverting --- ...
- 11
8
votes
3
answers
644
views
Get accurate eigenvectors, when eigenvalues are minuscule
I have a symmetric matrix A. I'm not able to compute all the eigenvectors accurately, and I believe it is due to the last few eigenvalues for ...
- 117
1
vote
0
answers
191
views
What are the metrics to assess the quality of a multiple correspondence analysis (MCA) model?
We are trying to implement a multiple correspondence analysis (MCA) model. I was looking for metrics to assess the quality of an MCA to evaluate our model. Sadly, I didn’t find much literature about ...
- 23
1
vote
0
answers
70
views
How to interpret a PCA Biplot? [duplicate]
I constructed a PCA plot from a very high-dimensional dataset that contains features relating to fraud. After creating the PCA plot, I created a biplot with the features to see how they interact. The ...
- 153
0
votes
0
answers
275
views
Why absolute value of eigenvalues are used in PCA or LDA?
In PCA and LDA techniques, eigenvectors with the $k$ largest eigenvalues give principal components. However, when selecting these eigenvalues, are they to be sorted by the absolute value (regardless ...
- 1
2
votes
2
answers
84
views
What does it mean for Katz Centralities to "diverge"
In Mark Newman's Networks book, 2010 edition, page 173, he explains some mathematical details behind the Katz Centrality measure:
In matrix terms, Eq. (7.8) can be written x = αAx + β1, (7.9) where 1
...
0
votes
0
answers
43
views
Are eigenvectors of PCA guaranteed to be orthonormal?
Are eigenvectors (principal components) of PCA orthonormal or only orthogonal ? Or only some of them are orthonormal or they are orthonormal if data were normalized before doing PCA ?
- 547
2
votes
0
answers
213
views
Does the Cholesky decomposition of a covariance matrix lead to a lower triangular matrix with positive diagonals?
We know that an $N\times N$ covariance matrix $\Sigma$ is symmetric positive definite, and can be factorized using Cholesky decomposition as follows
\begin{equation}
\Sigma=LL'
\end{equation}
where $L$...
- 1,072
0
votes
1
answer
272
views
Very low values of explained variance for the first Axes from PCoA
I am comparing different sites based on their floristic composition in R. Therefore, I have created a huge community datamatrix (presence/absence data) from 53 sites including over 1000 species. To ...
1
vote
0
answers
53
views
Is it possible that Fisher information matrix be indefinite? [duplicate]
I`m using the Newton-Raphson method for obtaining MLE for parameters for maximizing my objective function.
At each iteration, I want to check that is the Hessian matrix negative definite or not and I ...
- 71
1
vote
1
answer
65
views
Is the QR Algorithm guaranteed to compute eigenvectors?
I'm writing some C++ matrix library for hobby.
For computing eigenvalues and eigenvectors, I referred the following "Francis double step QR algorithm":
In particular, page 82 of https://...
- 131
1
vote
1
answer
269
views
Mercer's theorem and eigenfunctions
Consider a kernel $K$ which satisfies the conditions of Mercer's theorem. We know that $K(x,y) = \sum\limits_{i=1}^{+\infty}\lambda_ie_i(x)e_i(y)$ where $\lambda_i$ and $e_i$ are the eigenvalues/...
- 161
2
votes
1
answer
639
views
How to translate eigenvectors and eigenvalues to the number of clusters in spectral clustering?
I have generated this output, where L is the Laplacian Matrix, D is the degree and A is the adjacency matrix:
I can see the eigenvalues and eigenvectors are returned. I am unsure how to interpret ...
- 73
2
votes
0
answers
47
views
How to calculate the eigenvector?
I've been struggling to solve this math problem for two days...
So I calculated the mean of all samples (0,0).
Put it into the equation and got V as
\begin{array}
{rrr}
4 & 2 \\
2 & 2
\end{...
- 21
1
vote
0
answers
17
views
Why is sum of squared PCA column loadings equal to corresponding Eigenvalue?
I'm wondering why or how is the sum of squared PCA loadings in the column equal to Eigenvalue. I understood that the sum of squared PCA loadings in the row is equal to 1 or 100% because original ...
- 11