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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29 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 ...
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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 $nxn$...
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1answer
120 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 ...
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1answer
47 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 ...
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22 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 ...
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83 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 ...
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1answer
167 views

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

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 ...
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1answer
130 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} ...
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How to overcome the issue of singular within-cluster scatter matrices in clustering using entropy-based feature ranking?

I am trying to implement the entropy-based feature selection method for clustering by Dash and Liu. In this method, features are ranked in importance based on an entropy minimisation procedure and ...
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84 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 ...
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1answer
169 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., ...
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28 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 $\...
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1answer
397 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 ...
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1answer
220 views

Bypassing inverse matrix calculation and the comparison of Gradient Descent and Newton Descent

I am currently reading Pattern Classification (2nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John. In the 5th chapter the Gradient Descent algorithm is introduced with the following notations : $...
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120 views

Distribution of the Sum of an AR(1) Model Time Series

I have the following model for my model $\Delta X_{t} = \mu \Delta t + \rho \Delta X_{t-1} + \sigma \sqrt{\Delta t} Z_t$ with the following initial conditions - $\Delta X_{1} = \mu \Delta t + \...
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1answer
173 views

numerically stable sparse Gaussian process regression (matrix inversion)

In sparse approximations of GP for large data set $(X,\mathbf{y})$ with $n$ samples, usually $m$ inducing points are chosen such that the true covariance matrix is approximated by $K_{nn}\to K_{nm}K_{...
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1answer
283 views

Incremental solution for matrix inverse using Shermann-Morrison in $O(n^2)$

I have been reading a presentation on Value Function Approximation by David Silver (Introduction to Reinforcement Learning Course). On page 43 he finds a solution for linear least squares for an ...
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1answer
898 views

Which is more numerically stable for OLS: pinv vs QR

If I am doing standard OLS and want to calculate beta values (OLS estimators), which of the following is the more numerically stable method? And why? Assuming that the columns of $X$ are already mean-...
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1answer
112 views

Use $X^{-1}Y$ instead of $(𝑋^T𝑋)^{βˆ’1}𝑋^T Y$ to calculate $\beta$ when $X$ is already a square matrix in the least square problem

In the least squares problem $X\beta = Y$, the solution is $\hat{\beta} = (𝑋^T𝑋)^{βˆ’1}𝑋^TY$. I learned that two facts: $𝑋^T𝑋$ is square matrix so that the definition of matrix inversion is ...
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1answer
104 views

Proof of Pearson-Aitken selection formula

I am trying to understand the proof of the Pearson-Aitken selection formula, widely used in statistical genetics. A proof that the formula is general is given by Aitken (1936). However, I failed to ...
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1answer
75 views

Why the representation in the form of $Z'X(X'X)^{-1}X'Z$ can not be simplified into $Z'Z$

Representation similar to $Z'X(X'X)^{-1}X'Z$ frequently appear to e.g. 2SLS. I think that $Z'X(X'X)^{-1}X'Z = Z'XX^{-1}X'^{-1}X'Z = Z'(XX^{-1})(X'^{-1}X')Z = Z'Z$. So why it seems that in the context ...
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1answer
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matrix inequality related to finance

I'm trying to show that, for certain investment strategies, it pays to have more precise estimates of the covariance matrix of your returns. I have always took this for granted, but I've been having ...
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1answer
365 views

How are biases updated when 'batch size' > 1?

This is my network represented in matrices: (a dot represents an arbitrary number) Feed-forwarding: (I omitted nesting it all in an activation function for the sake of brevity) Backpropagation The ...
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1answer
2k views

Matrices: system that is “computationally singular” versus “exactly singular” [closed]

I would like to know the mathematical concepts behind singular matrices. Matrices that do not have inverses in R throw one of two errors. I have provided some examples of both errors below: Error in ...
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1answer
225 views

Explanation of generalization of Newton's Method for multiple dimensions

I've been following the CS 229 lecture videos for machine learning, and in lecture 4 (~14:00), Ng explains Newton's Method for optimization to maximize an objective function ($f$), but doesn't clearly ...
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23 views

Recursively expressing matrix inverse

Let $X$ be an $D \times N$ matrix. Let $I$ be a $D \times D$ identity matrix. Also let $y$ be a $N \times 1$ column vector. Suppose we are trying to solve $(X X ^T + k I) w = Xy$ for a $D$ dimensional ...
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1answer
2k views

How do we know $X'X$ is nonsingular in OLS?

I am currently working through understanding the mechanics of OLS estimates and the hat matrix. One thing I have been searching for without luck is how we know that the term $X'X$ is invertible where $...
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3answers
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Intuition behind $(X^TX)^{-1}$ in closed form of w in Linear Regression

The closed form of w in Linear regression can be written as $\hat{w}=(X^TX)^{-1}X^Ty$ How can we intuitively explain the role of $(X^TX)^{-1}$ in this equation?
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Why does higher variance of independent variables decrease standard errors of the estimator?

Some time ago, reading Advanced Data Analysis from an Elementary Point of View by Cosma Rohilla Shalizi, I found this statement in a section about significant coefficients: Moreover, at a fixed ...
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1answer
298 views

Use of Moore-Penrose Inverse in Likelihood Computation

I've been running MLE on a mixed-Gaussian model (let's just focus on the multivariate Gaussian case for now). In order to take the inverse of the covariance matrix, I've been using MATLAB's pinv() ...
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1answer
58 views

Newton-Raphson Error

According to Agresti(2013) pg 364-365, iterative methods such as Newton-Raphson methods, $ \begin{aligned} \beta^\text{new} &= \beta^\text{old} + (X^{T}WX)^{-1}X^{T}(V) \end{aligned} $ help to ...
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70 views

Are the Inverses of two asymptotically equivalent matrices themselves asymptotically equivalent

Suppose $M_n = P_n + op(1)$. Is it the case that $M_n^{-1} = P_n^{-1} + op(1)$, if both $M_n^{-1}$ and $P_n^{-1}$ exist with probability going to 1 as $n$ increases? Can the Continuous Mapping ...
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3answers
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Obtaining the possible least squares solutions when $X^TX$ is not invertible

If $X^TX$ is not invertible, what is the set of solutions for the least squares estimator $\hat{\beta_1}$ in the below? $Y_i = \beta_0 +\beta_1(x_i-\bar{x}) +\epsilon_i$ I got as far as writing out ...
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1answer
232 views

“Undefined real result” with inverse() function in WinBUGS

Using WinBUGS, I'd like to generate the posterior distribution for a function of several other variables (stochastic nodes). One step in building up this function involves taking the inverse of a ...
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120 views

Expectation of a Matrix Raised to a Power

I have been able to find little literature related to the topic online (e.g., A note on the Expected Value of an Inverse Matrix such that E($\frac{1}{X})$ $\geq$ ($\frac{1}{E(X)}$), though, vaguely, I ...
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1answer
82 views

Least Square Solution to Linear Regression Problem

Given two column-vectors, A = [0; 1.6818; 2.8284; 3.8337; 4.7568; 5.6234] and B = [0; 984.7; 1590.7; 2029.1; 2251.9; 2254.45], I ...
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312 views

Section on Moore-Penrose Pseudoinverse in the Deep Learning book

I am reading the following section in the Deep Learning book by Goodfellow, Bengio and Courville. I have some questions. If $A\in\mathbb{R}^{m\times n}$ and $Ax=y$, then we can have left inverse ...
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1answer
528 views

Why can't we cancel these two matrices in the OLS estimator?

When I have a multivariable linear regression model for sample element $i$: $$y_i=\beta_1x_{1,i}+...+\beta_kx_{k,i}+\epsilon_i,$$ then the OLS estimator is determined by the following equation: $$X^Ty=...
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1answer
203 views

Why forward and reverse transforms for PPCA are so different?

In probabilistic PCA (PPCA), they model dimensionality reduction as probabilistic model $t = Wx + \mu + \epsilon$ where $W$ is non-swuare matrix of size $(d \times q)$, $q < d$ $x$ is a vector ...
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1answer
64 views

Under-constrained models and invertibility of covariance matrix

In Goodfellow et al.'s Deep Learning, the authors write on page 232: [$\mathbf{X^\top X}$] can be singular whenever the data-generating distribution truly has no variance in some direction, or when ...
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1answer
1k views

The probability limit of an inverse matrix

all: I am considering a question regarding the calculation of the probability limit for an inverse matrix. Specifically, suppose we have a non-singular and squared matrix $M$ with dimension $2\times ...
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1answer
768 views

Usefulness of convexity of linear regression when there is no closed form solution

The optimisation problem in linear regression, $f(\beta) = ||y-X\beta||^2$ is convex (as it is a quadratic function), and when $(X^TX)$ is invertible, we have a unique solution which we can calculate ...
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1answer
2k views

Non-Singularity due to inclusion of non-zero lambda in ridge regression [duplicate]

There were many similar questions on this site , related to this but none were exactly to the point I wanted to ask So the question is relates to ridge regression and This link where there is a ...
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0answers
343 views

Solving an inverse problem with machine learning

I am running up against a very tough inverse problem that I suspect might be solvable using machine learning. Here is the problem. Overview I am studying an object $X$ which, internally, is ...
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1answer
146 views

Invert singular matrix for design of experiments and regression

I have a matrix, $X'X$, which is singular meaning that I cannot invert it. I need the inverse of this matrix to perform two independent things. I need it for the design of experiments, in R using ...
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1answer
618 views

How to show that demeaning the data in design matrix does not change the hat matrix

When I have a design matrix $$X = \begin{bmatrix} 1 & x_{11} & \ldots & x_{1k}\\ 1 & x_{21} & \ldots & x_{2k}\\ \vdots & \vdots & \ddots & \vdots\\ 1 & ...
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1answer
279 views

Spectral Decomposition of a symmetric matrix times a diagonal matrix

I would like to find the inverse of the sum of a Kronecker product and a diagonal matrix. I found this answered here, but I don't see how the last step is valid. Because of the fact that $C$ ...
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0answers
523 views

How to compare diagonal elements of precision matrix (the inverted correlation matrix)?

Let $$C=\begin{pmatrix}C_{11} & C_{12}\\ C_{21} & C_{22}\end{pmatrix}$$ be a $p\times p$ correlation matrix with positive entries, where $C_{11}$ is a $q\times q$ matrix. Define $D=C^{-1}=(d_{...
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0answers
231 views

when can I substitute an inverse with a pseudo-inverse in an estimator

Short Version: can I substitute the Moore-Penrose generalized inverse of a matrix (R function ginv()) for a matrix inverse (R function ...
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0answers
129 views

Generalised inverses - Solve normal equations

I am still not fully able to handle the concept of generalised inverses when applied to OLS. Is there a way to show that $\beta$ = $\ (X'X)^- X'y + (I_kβˆ’(X'X)^-X'X)z $ solves the normal equations: ...