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7 views

How to obtain all but an indexed row in a matrix - MATLAB? [on hold]

I am estimating a Vector Autoregression with ratios and first differences. Namely, I have T/Y, G/Y and $\Delta$Y. Where T is total taxes, G is government spending and Y is GDP. To compute my Impulse ...
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21 views

Matlab: Help in mitigating the problem: Non-positive definiteness of a matrix

EKUKF toolbox contains a file ut_sigmas.m I am facing a frustrating problem with the covariance matrix $P$ of the measurement when using Cholesky decomposition. ...
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0answers
13 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
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1answer
41 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 ...
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1answer
57 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 ...
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1answer
43 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 ...
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0answers
35 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 ...
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1answer
36 views

Does a polynomial kernel with degree less than 1 satsify mercers condition

Consider the polynomial kernel: $$K(\boldsymbol{x}, \boldsymbol{x}') = (\boldsymbol{x}^{T} \boldsymbol{x}'+c)^{d}$$ This kernel satisfies the mercers theorem/condition. Since I never saw any ...
4
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1answer
171 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 ...
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2answers
73 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 ...
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1answer
24 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 ...
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1answer
93 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 ...
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1answer
31 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 ...
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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 ...
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2answers
245 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 $\ge 0$) ...
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1answer
77 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
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1answer
80 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 ...
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0answers
20 views

Help with Euclidean update rule

I'm trying to figure out the derivation of the Euclidean update rule for NMF on page 2 of http://sig.umd.edu/publications/Tjoa_ICASSP2_201003.pdf but can't seem to eke it out. The end result that they ...
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0answers
54 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 ...
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0answers
146 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 ...
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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 ...
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0answers
55 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 ...
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1answer
109 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 ...
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1answer
333 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 ...
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0answers
99 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
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1answer
482 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 ...
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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
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1answer
285 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
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0answers
126 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 ...
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1answer
90 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 ...
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2answers
1k 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 $\bf X$ ...
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2answers
104 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 ...
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1answer
451 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 ...
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1answer
167 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 $X_1, X_2 , X_3$ ...
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1answer
184 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. ...
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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
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1answer
178 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
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1answer
133 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 ...
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0answers
63 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
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2answers
359 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
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1answer
150 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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0answers
136 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
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1answer
975 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 ...
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2answers
208 views

Adding content information to matrix factorization-based recommender

I'm currently using a matrix factorization method to generate recommendations (for info on this, check: Matrix Factorization Techniques for Recommender Systems). At the moment, my rating estimate is ...
0
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1answer
257 views

Why sparse QR decomposition yield different results from dense QR decomposition in R?

A simple example is given below: mat <- diag(1:6) qr.R(qr(as(mat,"Matrix"))) qr.R(qr(dd)) the diagnal values are of opposite signs. There is a warning message ...
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0answers
335 views

Matrix factorization and gradient descent for recommender systems; user bias?

I've been reading about using Matrix Factorization techniques to do collaborative filtering. A popular thing to do seems to be to add user and item biases into the ratings prediction. What I don't ...
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0answers
105 views

recommender system implicit rating to ordinal scale

There are 4 ways a user can show preference for an article within my news app: number of times an article was viewed; for how long was the article viewed; whether it was favorited or not; and number ...
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0answers
110 views

How to sample from a multivariate normal given the $P^T LDL^T P$ decomposition of $\Sigma$?

How to sample from a multivariate normal given the $P^TLDL^TP$ decomposition of $\Sigma$, and $\mu$? Here, $P$ is a permutation matrix, $L$ is lower-triangular, and $D$ is diagonal Given the ...
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0answers
172 views

Slope Derivation for the variance of a least square problem via Matrix notation

I have a question to solve the following matrix problem: $$E[( (X'X)^{-1}X \epsilon )^T ((X'X)^{-1}X \epsilon )]$$ into the solution $$= \Sigma^{2} (X'X)^{-1}.$$ Where $\Sigma$ is the covariance ...
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0answers
179 views

Matrix factorization vs random walk with restart for recommender systems

Suppose I want to handle "friend recommendation" problem on a large social network graph. I came across random-walk-with-restart as one technique used. I was thinking of using matrix factorization as ...