A matrix (plural matrices) is a rectangular array of numbers, symbols, or expressions arranged in rows and columns. The individual items in a matrix are called its elements or entries.

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Posterior Conditional on Beta in Bayesian Linear Regression with Factor Analysis

This should be an easy question if you're familiar with the terms involved. I am performing some research using a hierarchical Bayesian regression model that incorporates factor analysis into the Beta ...
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14 views

SVD on normalized correlation matrix [on hold]

Assume that we have a correlation matrix that is normalized to have a unitary vectors as its columns. I know that by applying SVD on such a matrix, besides finding the principal directions, we can ...
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33 views

How to figure out whether PCA can be performed on a data set or not?

I do have idea on the way PCA works but I do not know how to figure out whether a high dimensional data set is suited for PCA compression. I googled for some algorithms but could find any. Are there ...
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152 views

Textbooks on Matrix Calculus?

See this question on Math SE. Short story: I read The Elements of Statistical Learning and got frustrated when I was trying to verify some of the results, e.g., given $$\text{RSS}(\beta) = ...
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1answer
32 views

Is a covariance matrix composed of matrixes derived from separate samples guaranteed to be positive definitive?

I have two samples that partially overlap on the variables they describe. The samples are taken from more or less the same population, and show similar values on the overlapping variables. Based on ...
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1answer
35 views

Markov Process-Variance of time until jump

A Markov process on E = {1, 2} is constructed according to holding time parameters λ1 = 2 and λ2 = 4; the defining Markov chain has transition probabilities p11 = p12 = 0.5 and p21 = 1. How do I ...
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1answer
23 views

I need a matrix in order to calculate g-inverse of it [closed]

I want to calculate g-inverse of a matrix, which has 4 rows and not a square matrix and has no inverse. Please help me find such a sound (good) matrix. I only need a matrix. You can suggest a book or ...
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23 views

Correlation between two quadratic forms of Gaussian random vectors

I want to approximately calculate the correlation between two quadratic forms of two Gaussian random vecotrs (of course these are in fact non-Gaussian densities). Does anyone know the derivation of ...
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29 views

Prove that minimum of the matrix norm is achieved on certain parameters

Given matrix $A\in R^{n\times m}$ prove that minimum of the $||A-xy^T||$, $||B||=tr(B^TB)$, is achieved when $x$ is an eigenvector of $AA^T$, corresponding to its greatest eigenvalue, and $y$ is an ...
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21 views

Linear form arising in expected value of empirical variance of non-independent variables

Consider a normal vector $Y \sim \mathcal N(\mu, V)$ with $\mu \in\mathbb R^n$ and $V\in\mathbb R^{n\times n}$. I am interested in the expected value of $$ {1\over n-1} \left( Y'Y - {1\over n} ...
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Matrices F and c for F test?

In $\mathcal{F}$ test, the hypotheses involve multiple regression coefficients with the $\mathcal{F}_{0}$ test statistic: \begin{equation} z_{j} = \beta_{0} + \beta_{1} x_{1} + \beta_{2} x_{2} + ... + ...
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16 views

List of statistical test where correlation matrix is an important part

I wondered where all the correlation matrix forms an integral part of statistical test. Standalone also it is important (though there is no correlation-matrix tag for posts on this forum). It is ...
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46 views

Why the principal components correspond to the eigenvalues? [duplicate]

Suppose ${\bf{X}} = ({X_1},{X_2},\ldots,{X_n})$ are the original components (also random variables) and ${{\bf{w}}_j} = ({\omega _1},{\omega _2},\ldots,{\omega _n})$ are loadings for the $j$th ...
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31 views

Validating Neural Network

I have a relatively large set of labelled data which I am using to verify how well-trained a neural-network I wrote is. The network uses a competitive learning technique. It has N output neurons, each ...
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1answer
53 views

Correlation matrix is not positive definite… But why?

As part of an analysis I am conducting (Structural Equation Modeling) the estimated correlation matrix among some variables ended up looking like this: ...
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1answer
88 views

F-test and Rank test for underidentification

I've been performing a manual 2sls regression and I've come with the following results and that I find a bit suspicious. I've done the F-test of the first-stage regression and I've obtained a score ...
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1answer
63 views

How to generate uniformly random orthogonal matrices of positive determinant?

I've got probably a silly question about which, I must confess, I'm confused. Imagine repeated generating of uniformly distributed random orthogonal (orthonormal) matrix of some size $p$. Sometimes ...
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1answer
49 views

Understanding the marginal distribution of multivariate normal distribution

I am trying to better understand the multivariate normal distribution. Here I try to refer to the conditional distribution part of wiki also the fifth page of this tutorial. I do not quite ...
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22 views

Find repetitive patterns in matrices below

How can I identify the repetitive patterns from the matrices below? My problem is that the patterns in the matrix are different from matrix to matrix (dependent on the input data). I need some machine ...
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24 views

Difficulties in interpreting the Equations in Parameter estimation for linear dynamical system

I have implemented the Kalman Smoothing with Expectation Maximization based on the Paper Parameter Estimation for Linear dynamical system. All notations are based on this paper. The model is an IIR ...
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1answer
53 views

Clustering Matrices

Suppose I have a set of 100 $n \times 2$ matrices that all have the following format: Bid Profit [5.00 7.10] [3.14 6.04] [2.9 10.08] Where the numbers ...
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1answer
52 views

How to create a QQ plot of azimuths to test rotational symmetry of a spherical point dataset?

I am trying to do some Q-Q plots and the Kuiper test for rotational symmetry about the mean direction. These are unit vector, spherical data. What I am struggling to understand is the rotation from ...
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1answer
58 views

Can $x'x$ be written as correlation matrix?

$x'x=$ $$ \begin{bmatrix} \sum_{i=1}^{n}(X_{1i}-\bar X_1)^2&\sum_{i=1}^{n}(X_{1i}-\bar X_1)(X_{2i}-\bar X_1)\cdots & \sum_{i=1}^{n}(X_{1i}-\bar X_1)(X_{ki}-\bar X_k) \\ ...
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1answer
17 views

log multivariate normal differentiation (MLE)

I've come across a lot of explanations of how to differentiate the multivariate normal, but they all appear to skip the step that I'm stuck on. Here's what I've got so far. By logging and removing ...
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34 views

Matrix manipulation (weighted error term)

I have a question about matrix manipulation. I start off with a weighted error term. $$E_{D}(w) = \frac{1}{2}\sum_{n = 1}r_n (t_n - w^T \phi(x_n))^2$$ I differentiate, and set to zero to minimize ...
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17 views

How to run ANOVA on multiple groups of samples, each composed by different variables

I have a $m$ x $n$ matrix, where the $n$ columns are split into multiple classes. If I had only a $1$ x $n$ vector, I would have used ANOVA to evaluate if all subset of columns had the same ...
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27 views

Relation between raw and central moments

This question arose when reading Johansen's likelihood-based inference in cointegrated VAR models, the 2009 reprint, page 146. I will do my best to make my post self-contained. Let $Z_{0t}=\Delta ...
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1answer
24 views

Is this a valid method for unipartite projection of a bipartite graph?

I would like to know if a given method of projecting a bipartite graph exists, and if yes, if there is a formula for transforming the weight matrix. Given a bipartite graph with edges' weights ...
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24 views

Markov Process w/ a non-stochastic matrix?

I've come across a problem which at first appeared to be a markov process however the transition matrix of the graph is non-stochastic. That is, the probabilities among edges leaving a node do not sum ...
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8 views

How to evaluate enrichment of features in groups of samples?

I have a matrix composed by features on rows and samples on columns. Each value in the matrix correspond to a value of activation of the feature $i$ in the sample $j$. I want to evaluate enrichment ...
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24 views

Correlation of Distance Matrix

I have a matrix with 15 samples and ~10,000 data points (all z-scores). I calculated a distance matrix with euclidean distances using R. Is it valid to calculate and present a correlation on this ...
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1answer
36 views

R t-test, 1 response, multiple independent variables

I'd like to perform a t-test between groups 'A' and 'B'. The difficulty is that although there is only one response variable, there are many observations, and the grouping (A or B) differs with each ...
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12 views

Creating a readable matrix with a netCDF file

I'm still learning R and I'm having trouble getting R to read and organize data in a way for me to perform analysis on. I have taken data from a free online weather database here: ...
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1answer
40 views

Similarity between two samples (2D matrices)

I have two matrices: A= 4000x78 where 78 is the dimension of features and 4000 is the number of samples. B= 1000x78 where 78 is the dimension of features and 1000 is the number of samples again. ...
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33 views

Closed form solution for the diagonals of the symmetric matrix in RAM specification

Question: Is there a closed-form solution to finding the residual variances of endogenous variables using RAM specification (for SEM and Path Analysis)? Problem: Suppose someone wants to represent ...
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1answer
69 views

Whitening transformation for skewness?

Let $X$ be an $(m,n)$-matrix interpreted as a two dimensional array with each column representing $m$ samples from a random variable, with known covariance matrx $M$ and mean equal to $0$, it is ...
2
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1answer
68 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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17 views

Local covariance matrix in Locally Linear Embedding

In Locally Linear Embedding (LLE), when contructing the weight matrix, denote $\mathbf{x}_i$'s neighboring data points centered w.r.t. $\mathbf{x}_i$ by ...
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1answer
77 views

Justification for default contr.poly() polynomial contrasts in R

In R, one can use contr.poly() to create a set of orthogonal contrast codes for testing linear and higher order effects of categorical factors: ...
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49 views

Similarity metric for 2 sets of vectors

I'm trying to determine the similarity between two sentences. I have vectors for each word in a corpus, and using cosine distance of the two vectors, I can get quite a good "similarity" score ...
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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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36 views

Represent decision tree as matrices

Is it possible to use matrix operations to generate node membership for decision trees? In a binary decision tree, each node represents a condition for a single variable. Ignoring the more complicated ...
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1answer
91 views

Linear Algebra (trace): How to pull out $X^T[…]X$?

In fallowing equation - (1) $tr(X^TA^TAX+X^TLX)$ equals to $tr(X^T[A^TA+L]X)$. My question is how about in the following equation (2) $tr(X^TA^TAX+X^TXL)$, can we pull out $tr(X^T[...]X)$ like first ...
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1answer
75 views

R - Constrained optimization for a function taking a matrix

I would like to do constrained optimization for a function which takes a matrix as input. The matrix is sparse, representing a weighted adjacency matrix , and only the weights shall be subject to ...
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20 views

How to get covariance matrix of the coefficient estimates of compound regression model in SPSS

AFter running regression model in SPSS, I get the compound model: ln(diamter)=1.126*height +3.102. I would like to have the covariance matrix of the coefficient estimates of this model and get the ...
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1answer
60 views

machine learning with linear regression algorithm

I'm noob in machine learning, but I'm trying to know more about it. I have a question about a prediction model (predict for every page when the number of click). I try to use kNNimpute to handle with ...
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212 views

interpreting PERMANOVA (adonis function) output?

I am trying to look at expression of some genes and relate the dist matrix (Sm) to a number of different factors that I collected on the individuals (e.g., litter size, licking behavior, group housing ...
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1answer
29 views

Difference between recentred and scaled eigenvalues and the Tracy Widom distribution

I have been generating correlation matrices from independent normal data simulated using the MASS package. I do this k times and extract the eigenvalues of the matrices. I was interested in comparing ...
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2answers
79 views

Apply function generating random numbers to a matrix (R) [closed]

I have a function which basically creates a random number. Now I want to apply the function to a matrix for given conditions to replace the value in the matrix with the random number created by the ...
2
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1answer
59 views

Variance expressed in Quadratic Form

Given a vector $\pmb x$ of length $n$, $\pmb x=\{x_1,...,x_n\}$ the variance is proportional to $\pmb x^\top \pmb x - \frac{1}{n}{\left(\sum_{i=1}^n x_i\right)}^2$ I'm trying to determine $(i,j)$ ...