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Questions tagged [matrix-inverse]

The inverse of a given square matrix, $A$, is the matrix $A^{-1}$ such that $AA^{-1}$ is the identity matrix.

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Pseudoinversion giving poor predictions, converges with smaller matrix

I'm having an issue where running pseudoinversion as a means of predicting the values of certain phenotypes using methylation data gives extremely large numbers, very pooly predicting said values, ...
Epic Cabbage's user avatar
1 vote
2 answers
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Understanding the multicollinearity issue in relation to linear regression

There are 2 issues that multicollinearity in linear regression leads to Interpretability goes for a toss Parameter confidence intervals are wide and it is difficult to find a parameter significant I ...
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$E(X)*E(X'X)^{-1}$ simplifies when X vector contains a constant?

For self study, I've calculated $E(X)*E(X'X)^{-1}=[1,0]$ when $X=[1,X_2]$ for any scalar random variable $X_2$ (with finite mean and variance). Proof: $\begin{pmatrix}1 & E(X_2)\end{pmatrix}\begin{...
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Efficient way to compute covariance matrix of Vector Autoregressive Process of order 1 (VAR)

For a VAR process $$ X_t = A_1 X_{t-1} + \epsilon_t $$ The covariance of $X_t$ can be computed in the following way: $$ \text{vec}(\Sigma) = (I -(A \otimes A))^{-1} \text{vec}(\Sigma_{\epsilon}) $$ ...
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Random correlation matrices

Suppose that we simulate random $n\times n$ correlation matrices by assigning iid $U(-1,1)$ random variables to all off-diagonal entries and accept matrices $\boldsymbol\Sigma$ that are positive ...
Jarle Tufto's user avatar
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Series expansion for a single coefficient of OLS multiple linear regression?

The Matrix/vector formulation of OLS is given by $$\vec \beta = \left( X^T X \right)^{-1} X^T \vec y$$ which is very nice as a matrix equation. But what is the equivalent scalar equation for a single ...
Galen's user avatar
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Collapse of sampled Mahalanobis distance

Let $\{ x^{(1)}, \ldots, x^{(M)}\}$ be $M$ samples from a $n$-dimensional multivariate Gaussian distribution $\pi_{X} = \mathcal{N}(\mu, \Sigma)$. We recall the definition of the squared Mahalanobis ...
Mathieu le provost's user avatar
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Why does a valid Kernel only have to be positive semi-definite instead of positive definite?

I'm currently concerned with the topic of Gaussian Processes. To compute the covariance matrix of the conditional distribution, we have to invert $(K_{XX})^{-1}$, where $K_{XX}$ is a matrix of a ...
rodeo's user avatar
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3 votes
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354 views

Inverse of Matrix with normally distributed Elements?

Let's say we have some Matrix $X \in \mathbb{R}^{n_x \times n_x}$ whose elemets $x_{i,j} \sim \mathcal{N}(\mu, \sigma^2)$ are normally distributed. In other words: $vec(X) \sim \mathcal{N}(m, S)$, ...
Looper's user avatar
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The mean of Gaussian distribution subject to another Gaussian distribution, how to derive it?

I got a fomula $$N(y;cx,R)N(x;\bar{x},\Sigma)=N(y;c\bar{x},S)N(x;g,F)$$ where: \begin{align}S&=c\Sigma c^T+R\\g &=\bar{x}+\Sigma c^Ts^{-1}(y-c\bar{x})\\F &= \Sigma - \Sigma c^T s^{-1} c \...
night3759's user avatar
3 votes
2 answers
532 views

Parameter estimate in linear regression model with singular matrix $X^TX$

I started to study linear regression this autumn and got stuck at the estimation of regression parameters (in matrix form). In parameter estimation of the general linear model, the OLS method is used ...
Planet's user avatar
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6 votes
2 answers
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Why do large LMs use the transpose of the word embeddings matrix in the classification head?

All literature, guides and tutorials describing the construction of language models have used two separate matrices for the input and output projections: To project one-hot token IDs into hidden ...
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Updating the Posterior Psi parameter for an Inverse Wishart Distribution

I am fitting a Mixed Multivariate Normal Distribution where the mixing occurs over the mean $\mu_j$ and the covariance matrix $H$ with mixing parameter $B_j$. The number of mixing elements is denotes ...
Thomas_R's user avatar
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Why the columns of design matrix for linear models with additive Gaussian noise linear independent?

This question comes from page 142 of the book "Pattern Recognition and Machine Learning" by Christopher M. Bishop. Since it takes pages to arrive at the result, I excerpt the major settings ...
zzzhhh's user avatar
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compute Dirichlet distribution parameter from known mean distribution

For a particular Bayesian study I am going to apply Dirichlet distribution as my proposal random number generator. I am going to update the distribution parameter every trial based on a given ...
Rezgar Arabzadeh's user avatar
3 votes
1 answer
319 views

Natural gradients with Moore–Penrose inverse of the Fisher information matrix

I'd like to show you my rough sketch for scaling up natural gradients to deep neural networks that appears to be easy to automate just like automatic differentiation. I think there must be a flaw ...
all feedback welcome's user avatar
5 votes
2 answers
128 views

What do you do in regression if $(X^\text{T} X)^{-1}$ does not exist?

Assume we want to solve $X z=b$ for $z$ where $X$ is a non-square matrix and $z$ and $b$ are column vectors. In case the system is overdetermined, we have no solution and can look for a solution ...
Botond's user avatar
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2 votes
1 answer
136 views

Invertibility of the Gram matrix of a convex combination

Let's assume two real valued matrices $A,B\in R^{w\times d}$ for which $d>w$ and they both have full (column) rank. I am interested in the invertibility of the Gram matrix $$H(t):=(A+t(B-A))(A+t(B-...
Mehr's user avatar
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2 answers
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How to obtain least squares when $X^TX$ cannot be inverted

This work is all theoretical and for school, so we were only provided this information to work with, no actual y values. I have a simple linear model I have been asked to translate into a matrix, ...
barnsm2's user avatar
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how can an inverse correlation matrix have <1 in diagonals. the correlations x1, x2 are about 0.5 [duplicate]

The inverse correlation matrix was calculated with the minverse function. The results show that in the diagonal the values are less than 1. how is this possible? the x1,x2 independent variables are ...
stephen's user avatar
3 votes
1 answer
250 views

Inverse of the outer product of some vectors with their transpose

Assume i have $n$ 3D unit vectors $v_s$, with different values. Then i define a matrix $T$ as: $$ T = \frac{1}{n} \sum_{s=1}^{n} v_s \times v_s' $$ where $v_s$ are $3\times 1$ vectors and therefore $...
Denis's user avatar
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1 vote
1 answer
367 views

Condition for covariance matrix to be non-invertible

Context: I'm working on a machine learning problem where I'm using multivariate normal likelihood which requires calculating determinant and inverting the covariance matrix. I'm trying to generate ...
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How to derive an inverse of Gaussian Kernel

As an example, say I have a function (Gaussian process kernel): $$K(x_i, x_j) \equiv \exp(-\alpha |x_i-x_j|^2)+\beta \delta_{ij}$$ Is there a way to analytically express $K^{-1}(x_i,x_j)$, s.t. ...
CWC's user avatar
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3 votes
1 answer
131 views

How to prove $(P^{-1} + B^T R^{-1} B)^{-1} B^T R^{-1} = PB^T(BPB^T + R)^{-1}$

It is Equation C.5 from https://www.seas.upenn.edu/~cis520/papers/bishop_appendix_C.pdf I tried right multiply both sides with $(BPB^T + R)$, but not sure how to continue from there.
zyxue's user avatar
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1 vote
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336 views

How can nuisance parameters in Fisher matrix can deteriorate the useful constraints?

I have a Fisher matrix $F$ which has the matrix blocks form like this : $$ F=\begin{bmatrix} A & B\\ C & D \end{bmatrix} $$ The block $A$ is the most important block, in the sense the ...
user avatar
0 votes
1 answer
38 views

Problem dealing with OLS estimator

I'm an econometrics student and I'm having a little trouble with lineal algebra. I have seen that the OLS estimator, given the following regression in matrix form: $$ y=X \beta+u $$ Is: $$ \hat{\beta}=...
Nacho's user avatar
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How to simplify fallowing martrices' Expected value elements in the equation

I have a matrix equation with four variables inside, $x^1_{00}$, $x^1_{tt}$ and $x^2_{tt}$, $x^2_{tt}$. $x^1_{00}$, $x^1_{tt}$ come from the same distribution , they are only shifted by timelag $t$. ...
Daniel Wiczew's user avatar
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0 answers
129 views

Distribution of solution to linear system

I have a random symmetric matrix $ A \in \mathbb{R}^{M \times M}$, and random vector $b \in \mathbb{R}^M$. I also have access to expressions for the mean and variance of each element of $A$ and $b$ (...
user3204752350982's user avatar
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0 answers
331 views

When is the Optimal weighting matrix in GMM singular?

currently I am trying to estimate a simple linear regression: \begin{equation} y_t = X \beta + \varepsilon_t, \end{equation} where I try to find 4 coefficients and one specific predictor is an ...
user326465's user avatar
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0 answers
23 views

Singularity of a modelmatrix that consists of one predictor raised to multiple powers

I was playing around a little bit with linear regression models in R and wanted to try out whether I can get a perfect fit for a linear model where I have a response $$Y = \left( \begin{array}{c} y_1\...
jonaden's user avatar
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1 vote
1 answer
743 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/...
Akusa's user avatar
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0 answers
67 views

Finding inverse matrix $(X'X)^{-1}$ with $X$ as design matrix [duplicate]

I'm relatively new to all this and I am trying to figure out how I can derive the matrix $(X'X)^{-1}$ when I have given $x_1, x_2, x_3$ and $y$. $X$ is the design matrix in that case but not sure how ...
statsquestions's user avatar
3 votes
1 answer
298 views

linear regression and obfuscation matrix - just not clicking for me?

Here is the basis of my question. We have an exercise to take a data set of made up insurance claims and train a linear regression model to predict future claims; however, because it is sensitive data,...
seuadr's user avatar
  • 33
3 votes
1 answer
192 views

completing a square

If I have a density function of the form $p(x) \propto \exp(−q(x)/2)$ where $q(x)$ has the following quadratic function $$q(x)=x^Tx+y^Ty-[x^TA+y^TB][A^TA+B^TB+\beta\mathbb{I}]^{-1}[A^Tx+B^Ty]$$ where $...
Dalek's user avatar
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2 votes
1 answer
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What is an example of a systems problem that can be solved by either 'matrix inversion' or an 'iterative procedure' to arrive at the same result?

I recently heard over a radio program (in French) that a given problem - can't recall it exactly, but it involved solving large systems - could be solved "through matrix inversion" or "...
StatsScared's user avatar
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20 votes
1 answer
7k views

Relationship between Cholesky decomposition and matrix inversion?

I've been reviewing Gaussian Processes and, from what I can tell, there's some debate whether the "covariance matrix" (returned by the kernel), which needs to be inverted, should be done so ...
jbuddy_13's user avatar
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3 votes
1 answer
319 views

What dimensionality reduction methods allow a lower dimensional reconstruction of the original data besides PCA via invertible transformations?

In eigenfaces, one used the inverse transformation PCA is capable of doing to reconstruct the low dimensional face image. In tsne one may not reconstruct the original dataset to produce something akin ...
user avatar
-2 votes
1 answer
207 views

Why doesn't $(e^{A})^{-1} = e^{-A}$ hold for a symmetric matrix in Python?

$e^A$ is just the $A$ matrix with all of its elements exponentiated, called a matrix exponential. It follows that the inverse $(e^{A})^{-1} = e^{-A}$ for square matrices, although I could find nothing ...
develarist's user avatar
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0 votes
1 answer
843 views

Proving that the hat matrix is unchanged even when the predictors are multiplied by constants

I know that the hat matrix $H = X(X^T X)^{-1} X^T$, and that $\hat{Y} = HY$. When we have some non-zero constants that we multiply each respective predictor by, which just multiplies every column in ...
Quinky's user avatar
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2 votes
0 answers
35 views

Why is it not possible to simplify $b=(X'X)^{-1}X'y$ [duplicate]

Why is the following not possible: $$ b=(X'X)^{-1}X'y = X^{-1}(X')^{-1}X'y=X^{-1}y $$ While this term $(AB)^{-1}=B^{-1}A^{-1}$ applies to any two matrices as long as both are of full rank and are $nxn$...
cascom's user avatar
  • 41
4 votes
1 answer
1k views

Moore Penrose Pseudo-Inverse Fast Algorithm in R [closed]

I want to apply Moore Penrose Pseudo-Inverse on my matrix, which is a 20,000 * 20,000 symmetric matrix with rank 19,999. I found ginv() function from the ...
Jeffrey's user avatar
  • 107
0 votes
1 answer
1k views

How to do multiple linear regressions with overlapping predictors efficiently

I'm using covariance matrices to do a linear regression. I have the same number of predictors and dependent variables (3000 of each). For each $i$, $X_{i}$ and $Y_{i}$ have a kind of symmetry. I ...
James Pinkerton's user avatar
0 votes
1 answer
174 views

NN type/architecture needed for inverse covariance matrix approximation

The idea is to construct a neural network (NN) that takes N series of financial returns as input and returns the (approximation of the) inverse of the sample covariance matrix (an N times N matrix). I ...
BGa's user avatar
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1 vote
0 answers
956 views

Normalization for solving linear equations

Suppose I want to solve a linear equation system in the form of $$A x = b$$ to get $x$, where $A$ are $n$ by $n$ matrix and $b$ is $n$ by 1 vector. Is there any normalization procedure necessary ...
DiveIntoML's user avatar
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5 votes
1 answer
2k views

Updating the inverse covariance matrix after deleting the i-th column and row of the covariance matrix

Suppose I have a covariance matrix $A_{n\times n}$ and $A^{-1}_{n\times n}$ is its inverse. Then I randomly exclude the $i$-th row and column, where $1\le i\le n \in \mathbb{N}$, obtaining the new ...
Ga13's user avatar
  • 280
7 votes
2 answers
1k views

Least Squares removing first $k$ observations Woodbury formula?

Given the matrix $X_{n,p}$ from the least squares problem $$ \mathbf{X} \cdot \mathbf{\beta} = z $$ Where the normal equation is: $$ \mathbf{\hat{\beta}} = \left(\mathbf{X}^T \mathbf{X}\right)^{-1} ...
imbr's user avatar
  • 205
0 votes
0 answers
527 views

Squared Multiple Correlation (SMC) of my correlation matrix tend towards 1. How to interpret this?

In order to be able to conduct exploratory factor analysis, I want to carry out parallel analysis to determine the number of factors to be extracted. To do so, I want to extract the eignevalues of the ...
DomB's user avatar
  • 541
1 vote
1 answer
508 views

Upper bound trace of inverse of covariance matrix

Let C be the covariance matrix from any normal distribution. If the trace of C is upper-bounded by a constant k (i.e., tr(C)<=k), can I find an upper bound for the trace of the inverse of C (i.e., ...
Tiger07's user avatar
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0 answers
138 views

Fast inverse of positive definite matrix subtracted by its Nystrom approximation

Assume a positive definite symmetric covariance matrix $$C_{n,n}$$ and let its Nystrom approximation be $$\hat{C}_{n,n}=C_{n,q} C^{-1}_{q,q} C_{q,n}$$ for some $q<n$ Inverting $C_{n,n}$ is of $\...
Wis's user avatar
  • 2,204
7 votes
1 answer
3k views

Generalized Least Squares using Moore Penrose pseudo inverse

I'm using GLS to fit a model where some independent variables are strongly correlated. Therefore my covariance matrix is singular. I have found that Moore-Penrose pseudo inverse can be used to find an ...
Patrick's user avatar
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