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2
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1answer
140 views

Scaling a Covariance Matrix by a factor?

I have a $n\times n$ covariance matrix representing the covariance between n random processes. I want to increase all of the covariances by a certain factor, say $1.5x$. Multiplying each covariance ...
0
votes
0answers
5 views

“Tucker decomposition often assumes Gaussian noises”. Are the noises here input or output?

I saw in some papers saying " Tucker decomposition methods have implicitly or explicitly assumed the noises are Gaussian" in ...
0
votes
2answers
43 views

optimizing over a set of symmetric matrices

I need to minimize a complicated loss function, $f\left(\Lambda\right)$ over a set of symmetric matrices, $S_{p}$ of dimension p, such that all the eigenvalues of $\Lambda \in \left[0,1\right]$. I ...
1
vote
1answer
54 views

Calculating Cosine Similarity with Matrix Decomposition (matrix multiplication with normalized columns)

To calculate the column cosine similarity of $\mathbf{R} \in \mathbb{R}^{m \times n}$, $\mathbf{R}$ is normalized by Norm2 of their columns, then the cosine similarity is calculated as ...
1
vote
0answers
19 views

Shrinkage of Schafer and Strimmer

As we all know that the sample covariance matrix $(S = (s_{ij}))$ is postive definite when the number of observations is smaller than the number of samples, that is n>p. But, the sample covariance ...
0
votes
0answers
33 views

Interpretation of matrix factorization results

Matrix factorization methods are known to give good results pertaining to problems like movie recommendation. The method reduces the feature space, which is then used for recommendations. For example ...
0
votes
1answer
50 views

How to find unknown correlation coefficients in a correlation matrix from known correlation coefficients? [duplicate]

I have a correlation matrix A given below. Here A should be a positive-definite matrix so that we can perform Cholesky decomposition of A. ...
0
votes
0answers
35 views

Non-negative matrix factorization in recommender systems

As i understand, in NMF we should have our three matrices elements non-negative. But i can't understand how to do it so far. Shouldn't we just initialize our factor matrices at the start with random ...
2
votes
2answers
53 views

Singular value and eigen-decomposition of a square symmetric matrix should be identical, but differ in sign

As far as I know, singular value decomposition (SVD) and eigendecomposition give the same result for symmetric square matrices. But when I check the results in R, that's not what I see. Please see ...
1
vote
1answer
45 views

What is the relationship between SVD and UV decomposition?

I am aware that both are Matrix decomposition techniques. SVD decomposes the Matrix as $\mathbf C = \mathbf U_1 \mathbf L \mathbf V_1^\top$. UV decomposes the Matrix as $\mathbf C = \mathbf ...
2
votes
0answers
58 views

orthogonalized impulse response's contradictory forms in a VAR(p) model

I have so far discovered three different ways of utilizing the Cholesky decomposition for calculating the OIRFs of a VAR(k). The different methods seem contradictory so I would like some input on ...
1
vote
0answers
19 views

Why “nonnegative” matrix factorisation?

Factorising a nonnegative data matrix $X \approx WH$ is very popular in machine learning and statistics. It has a number of applications on image processing, audio processing etc. I wonder why there ...
2
votes
1answer
57 views

Decomposition of inverse covariance matrix

Let $\Sigma$ be a covariance matrix and let $$x^T \Sigma^{-1} x = \|Ax\|_2^2.$$ What is the interpretation of matrix $A$? I tried solving for $A$ with an eigenvalue decomposition of $\Sigma$ as ...
2
votes
1answer
104 views

Derivative of $x^T A^Ty$ with respect to $\Sigma$ where $A$ is (an upper triangle matrix and ) Cholesky decomposition of $\Sigma$

I would like to evaluate: $$ \frac{ \partial x^T A^Ty}{\partial \Sigma} $$ where $A$ is a Cholesky decomposition of $\Sigma$ and an upper triangle matrix such that $\Sigma = A^T A$, $x$ and $y$ are a ...
0
votes
1answer
86 views

Relationship between Poisson generation and generalized Kullback-Leibler divergence

I have read that, in the context of matrix factorization, performing maximum likelihood estimation under the assumption that the entries are Poisson generated is equivalent to minimizing the ...
1
vote
0answers
47 views

What is the advantage of non-negativity in matrix factorization?

I am wondering why matrix factorization techniques in the machine learning domain almost always expect the provided matrix to be non-negative. What is the advantage of this constraint? Background: I ...
6
votes
1answer
270 views

What norm of the reconstruction error is minimized by the low-rank approximation matrix obtained with PCA?

I am a bit confused between $\parallel \cdot \parallel_2$ and $\parallel \cdot \parallel_F$. Given a PCA approximation of matrix $X$ with a matrix $\hat X$, we know that $\hat X$ is the best ...
1
vote
2answers
103 views

Maximally reducing the rank of a matrix by removing some rows or columns

I have a $N \times M$ matrix, and the rank of matrix, $r$, is near $\min(M,N)$. I want to minimize the rank by removing some of the rows or columns to get $r \ll \min(M,N)$. The goal is to achieve ...
0
votes
1answer
31 views

Independent component analysis with nonnegative mixing matrix

In independent component(s?) analysis, I have the observed signal, $O$, the mixing matrix, $A$, and the source matrix, $S$, with $O ≈ AS$ I've found some literature on ICA with the sources assumed to ...
0
votes
1answer
156 views

Oaxaca Decomposition: Unexplained Constant

I am doing an Oaxaca decomposition of the Log Wage Differential between Whites and non-Whites. I would like to find out if there is any interpretation for the constant term under the unexplained ...
3
votes
1answer
38 views

WH=V matrix decomposition that allows negative values

I'm looking for a matrix factorization method that is able to decompose a matrix: V => W * H V has dimensions ...
0
votes
0answers
22 views

Problems with calculation of numerical identification w.r.t. ANOVA smooth for large scale matrices

Suppose we have two (centered) Spline-matrices $\boldsymbol{B_1}$, $\boldsymbol{B_1}$. Then $\boldsymbol{X_1} = [\boldsymbol{B_1},\boldsymbol{B_2}]$ contrains lower order smooths and ...
4
votes
2answers
388 views

How to choose an optimal number of latent factors in non-negative matrix factorization?

Given a matrix $\mathbf V^{m \times n}$, Non-negative Matrix Factorization (NMF) finds two non-negative matrices $\mathbf W^{m \times k}$ and $\mathbf H^{k \times n}$ (i.e. with all elements $\ge 0$) ...
1
vote
1answer
97 views

Intuition behind matrix factorization formulations?

I'm reading this paper about matrix factorization. In the paper they propose to use this factorization for the adjacency (or similarity) matrix $G$ using the following formulation: $G = U \Lambda U^T$ ...
2
votes
1answer
108 views

Matrix Factorization Recommendation Systems with Only “Like” Ratings

I'm trying to build a recommendation system, but I only have data on what my users have "liked", i.e. all non-missing data has the same numeric value. Is it possible for me to use matrix ...
1
vote
0answers
75 views

Cholesky decomposition and confidence ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. (This question succeeds this one.) What I'm ...
0
votes
0answers
178 views

Trying to use Cholesky decomposition of covariance matrix to sample error ellipsoid

I'm trying to construct an error ellipsoid from a covariance matrix (which exists for a 3D point) and then sample consistent xyz points in this region. In a previous question when I asked about this ...
0
votes
0answers
27 views

clustering versus projection - what are the best example/scenario to explain their differences

I am dealing with non statistician who know are crunching data but don't have a deep understanding of statistics. I am trying to introduce them to non-matrix factorization methods but it has been ...
1
vote
0answers
62 views

Is there a way to perform SVD in a sequential manner?

My neurology experiment has a spike detector outputting 40 sample long spike waveforms. I'm using a dictionary method for sorting the spikes in real time. To ...
1
vote
1answer
146 views

Trying to understand non-negative matrix factorization (NMF)

I'm trying to understand how NMF is derived, and I got the basic idea of NMF, that is, it tries to approximate the original matrix $V$ with $WH$, where $V$ are non-negative, and $W,H$ are constrained ...
3
votes
1answer
409 views

Evaluating matrix factorization algorithms for Netflix

I've been trying to implement Simon Funk's movie recommendation algorithm explained here. I understand how the user and item factors are computed. However the evaluation method is not clearly ...
2
votes
0answers
107 views

Matrix decomposition

I have a symmetric matrix $V$ of order 10. I want to decompose $V$ in such a way that $$V=SS'$$ with $S$ being non triangular. The matrix $S$ has some restrictions that 45 cells have ‘0’ values and ...
2
votes
1answer
559 views

Plotting error ellipsoid from 3x3 covariance matrix in R?

I'm hoping to be able to take a 3x3 covariance matrix and turn this into an error ellipsoid but so far I haven't been able to achieve this. I'm very new to R (in fact turned to it to attempt to solve ...
0
votes
0answers
51 views

How to re-construct a matrix from SVD [duplicate]

I have a Audio time-series, to which I'm trying to detect the most significant parts of the signal, i.e. the voiced parts and forget the unvoiced parts. $$ T = [0, 0, 1, 1, .....n] $$ I then ...
2
votes
1answer
348 views

QR factorization and linear regression

I have been reading "Generalized Additive Models an Introduction with R" by Simon Wood and have come across a section I'm having trouble with. On page 13 it is stated that the model or design matrix ...
2
votes
0answers
152 views

Confused about Cholesky and eigen decomposition

I'm looking to generate correlated random variables. I have a symmetric, positive definite matrix. So I know that you can use the Cholesky decomposition, however I keep being told that this only works ...
1
vote
1answer
113 views

In non-negative matrix tri-factorization, initialization not possible because matrix is singular

I have implemented the non-negative matrix tri-factorization algorithm (link to paper). If is similar to the more widely known NMF (non-negative matrix factorization), but incorporates prior knowledge ...
10
votes
2answers
2k views

Why PCA of data by means of SVD of the data?

This question is about an efficient way to compute principal components. Many texts on linear PCA advocate using singular-value decomposition of the casewise data. That is, if we have data $\bf X$ ...
1
vote
2answers
113 views

IV estimator: efficient implementation?

I would like to implement (in R) an instrumental variable (IV) estimator, that takes the most general form (here not 2SLS or GMM!): $$ \beta_{IV} = (Z'X)^{-1}Z'Y $$ I could code this in the naive ...
1
vote
1answer
567 views

Matrix Factorization Model for recommender systems how to determine number of latent features?

I am trying to design a matrix factorization technique for a simple user-item, rating recommender system. I have 2 questions about this. First in a simple implementation that I saw of matrix ...
4
votes
1answer
184 views

Why do PCA and Factor Analysis return different results in this example?

The following question is about an Exercise 14.15 from "The Elements of Statistical Learning" by Hastie, Friedman and Tibshirani. Generate $200$ observations of three variates $X_1, X_2 , X_3$ ...
1
vote
1answer
190 views

What are the differences between these two kinds of PCA?

The book "Elements of Statistical Learning" describes Principal Components Analysis through SVD as follows: $$X = UDV^T$$ Then $ UD $ are the Principal Components and $ V $ are the directions. ...
0
votes
0answers
36 views

Reducing data size to cross correlate with another data set?

I have three matrices A, B, C. A is a matrix of 200 X 32. There are 2000 such different A matrices which make up the B matrix. B is a matrix of 2000 x A. That is there are 2000 x 200 rows in matrix B ...
0
votes
1answer
193 views

Non-Orthogonality in PCA? [duplicate]

i) What is the main role of "only" trying to find orthogonal components in PCA? I can understand, that we would not want a zero-solution as well as find directions that are orthogonal in order to ...
0
votes
1answer
156 views

Truncated SVD matrix reconstruction: what is the meaning of the real values?

Im my algorithm, I am working with Singular Value Decomposition (SVD). I have an input matrix $A_{in} \in \{0,1\}^{(m * n)} $, made by $n$ rows and $m$ colums. All the entries are 0 or 1. I ...
4
votes
0answers
91 views

How to visualize the low-rank visualization result?

In a numerical low rank decomposition, whether it is non-negative matrix factorization(NMF), or binary matrix factorization(BMF), or non-negative sparse PCA, we have two low-rank matrices to ...
0
votes
2answers
404 views

Cholesky decomposition in error covariance [closed]

I try to implement Unscented Kalman Filter. Everything seems to be done correctly but I do receive an error about Cholesky decomposition ...
2
votes
1answer
157 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 ...
1
vote
0answers
149 views

Harmonic or dummy seasonal model

Within the BFAST package in R, one of the parameters that it gives is the choice of seasonal model parameter (harmonic, dummy, or none). I understand what none does; However, I didn't really ...
0
votes
1answer
1k views

Time series deseasonalization

How do you know when deseasonalization is not necessary? That is, from what I understand, if you want to just look at the trend and irregular components of a time series, then you just need to remove ...