Questions tagged [determinant]

determinant of a matrix. For purely mathematical questions about determinant, better ask at mathSE

Filter by
Sorted by
Tagged with
1 vote
0 answers
32 views

How to derive conditional destribution of MVN variable

I am working with following model specifications (Regression_ Modelle, Methoden und Anwendungen-Springer-Verlag Berlin Heidelberg (2009), p. 147): $$Y \sim MVN(X\beta, \sigma^2I)$$ $$\beta|\sigma^2 \...
BlankerHans's user avatar
2 votes
1 answer
51 views

Different notions of width for multivariate Gaussian distribution [closed]

Given a multivariate Gaussian distribution $\mathcal{N}(\mu, \Sigma)$ where $\mu$ is a real vector and $\Sigma$ an associated covariance matrix, I am looking for a good notion of width. There seem to ...
0 votes
0 answers
58 views

Determinant of High Dimensional Correlation Matrix

According to [1] : The determinant of the correlation matrix will equal 1.0 only if all correlations equal 0, otherwise the determinant will be less than 1 [...] When the measures are uncorrelated, ...
Liam F-A's user avatar
2 votes
0 answers
24 views

Wishart conditionned by the determinant

I am interested in the density of the Wishart distribution under the constraint that the determinant of the outcome is 1. It suffice do divide the density of the Wishart by the marginal density of the ...
Chevallier's user avatar
1 vote
0 answers
61 views

is it fair to use a subset of eigenvalues to evaluate the multidimensional variance

I want to find a single metric to assess how spread (or how much variance) a multidimensional dataset (a large number of features) is. I learned that the determinant (or pseudo-determinant) of the ...
qliang's user avatar
  • 21
1 vote
1 answer
17 views

Is there additional value of using repeated measurements in this specific case?

I am working on a paper on smoking and depression in a sample of approximately n=2.000 subjects, and for that, I use longitudinal data (5 waves, 2 years between each wave). For a part of the paper, I ...
Wendela's user avatar
  • 11
2 votes
1 answer
128 views

How to derive the determinant of the variance of a negative multinomial distribution?

The probability mass function of the negative multinomial distribution is: \begin{align*} \mathbb{P}(\boldsymbol{\rm{X}}=\boldsymbol{\rm{x}}|\mathbf{p})=\frac{\Gamma\left(x_0+\sum_{i=1}^{m}x_{i}\right)...
d d's user avatar
  • 31
0 votes
1 answer
518 views

Multicollinearity iff determinant correlation matrix = 0

I'm studying Linear Models again, after finishing my degree some years ago. I found in my old notes that, according to my professor, one can check multicollinearity calculating the determinant of the ...
Francisco Jácome Maura's user avatar
0 votes
0 answers
153 views

Generalized variance of a multivariate normal without calculating determinants

In order to calculate ‘generalized variance’ of a multivariate normal distribution, it is often recommended (e.g., here: https://online.stat.psu.edu/stat505/lesson/1/1.5) to calculate the determinant ...
macleginn's user avatar
  • 405
1 vote
1 answer
323 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 ...
stevew's user avatar
  • 831
1 vote
0 answers
263 views

Impute Missing Correlations from Incomplete Correlation Matrix in R

I am in need of a function that will take an incomplete correlation matrix and return a complete matrix with correlations imputed for missing values. I am working on models that utilize only summary ...
Billy's user avatar
  • 826
0 votes
0 answers
37 views

Is negative Mahalanobis distance proportional to log probability?

Is the following statement true? I was sure it was, but I was told by someone else that it is not. $$ p(\mathbf{x}_i | y_i) = \frac{1}{2\pi^{\frac{D}{2}} |\mathbf{\Sigma}_c|^{\frac{1}{2}} } e^{ -\frac{...
Joff's user avatar
  • 872
1 vote
1 answer
223 views

If all elements of A are greater than x and all elements of B are smaller than a, det(A) is greater than det(B)

Let $A$ and $B$ are two $n \times n$ matrices and $x > 0$ is a scalar. If $\forall \; a \in A \;\;\; a > x$ and $\forall \; b \in B \;\;\; 0 \leq b \leq x$, and assume $A$ and $B$ are both ...
Maryam Bahrami's user avatar
6 votes
1 answer
505 views

Is there a formula for the determinant of the covariance matrix $\mathbf{X_n}^T \mathbf{X_n}$ in the case of multiple regression?

Consider the standard simple linear regression model: $$ Y_i = \beta_0 + \beta_1 X_i + \epsilon_i, $$ for $i=1,\dots,n$. In matrix-vector form this is $$ \mathbf{Y} = \mathbf{X_n}\beta + \epsilon, $$ ...
sonicboom's user avatar
  • 920
2 votes
0 answers
240 views

What does the determinant of a homography matrix represent?

I am trying to find one image (needle) within another (haystack). I am using the following OpenCV method, which first matches keypoints with SIFT and then applies homography: https://opencv-python-...
badmax's user avatar
  • 2,191
1 vote
1 answer
54 views

Question regarding Optimal Designs of experiments

I'm a bit unclear on the concept of optimal design of a data matrix $X$. I propose a small example to work through: Suppose $\epsilon_i \sim N(0, \sigma^2)$ are i.i.d., and I have some experiment ...
EzioBosso's user avatar
  • 384
0 votes
0 answers
24 views

Given a function $\phi:X \to Y$ and a target distribution $\pi_Y$ on $Y$, does there exist a distribution $\pi_X$ on $X$ s.t. $\phi(x) \sim \pi_Y$?

More formally, given $X\subseteq\mathbb{R}^n$, $Y\subseteq\mathbb{R}^m$, $\phi: X \to Y$ and distribution $\pi_Y$ on $Y$, does there exist a distribution $\pi_X$ on $X$ such that $\phi(x) \sim \pi_Y$ ...
deasmhumnha's user avatar
  • 1,049
3 votes
1 answer
110 views

Multivariate gaussian bivariate gaussian proof

I'm having trouble seeing how the multivariate gaussian formula evaluates to the bivariate gaussian. See multivariate PDF, source: http://cs229.stanford.edu/section/gaussians.pdf [![multivariate][1]][...
jbuddy_13's user avatar
  • 2,798
1 vote
0 answers
599 views

What's the importance of parallel eigenvectors?

I'm studying eigenvectors. I read that if a matrix is symmetric and if the eigenvalues are real numbers, the eigenvectors will be perpendicular. However, I have no idea what it means (if anything) ...
jbuddy_13's user avatar
  • 2,798
0 votes
1 answer
1k views

Is the determinant of the correlation matrix << 0.00001 necessarily a problem for PCA?

I am running PCA on my data and my KMO value > 0.80 and p-value of Bartlett's test of sphericity < 0.05. However, the determinant of the correlation matrix ( around 10^-30) is very close to zero....
user05's user avatar
  • 1
1 vote
1 answer
420 views

How to prove the determinant of covariance matrix is zero, when n≤p?

If there are n observations on p dimensions, then the covariance matrix will be: But when n≤p, its determinant will be zero. I know it is because it becomes as a singular matrix, but I do not know ...
abba's user avatar
  • 45
6 votes
1 answer
219 views

What is the meaning of $\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$?

What is the meaning of the quantity: $$\varepsilon=\sqrt{\mathrm{var}(X)\mathrm{var}(P)-[\mathrm{cov}(X,P)]^2}$$ Is there, for example, a geometric explanation? Is there a term for it in statistics?
apadana's user avatar
  • 161
2 votes
1 answer
255 views

Linear dependency among columns and rows

Singular matrix is defined as square matrix with the determinant of zero. The determinant of zero occurs when matrix columns are linearly dependent (i.e. one of the columns can be defined as a linear ...
PsychometStats's user avatar
4 votes
1 answer
5k views

Singular Matrix and Linear Dependency

Singular matrix is defined as a square matrix with determinant of zero. I am aware that linear dependency among columns or rows leads to determinant being equal to zero (e.g. one column is a linear ...
PsychometStats's user avatar
3 votes
1 answer
621 views

Understanding how the determinant of the multidimensional normal likelihood can overrule the prior probability

I am doing Bayesian inference. I have a normal prior probability distribution of some theoretical parameter $\theta$ and I am trying to update my knowledge of $\theta$ using some data $D$ and a model $...
rhombidodecahedron's user avatar
10 votes
1 answer
2k views

Reason for absolute value of Jacobian determinant in change-of-variable formula?

When we have a random variable $x$ with a probability density $p(x)$, and a function $y = f(x)$ that is differentiable and can be solved for $x = g(y)$, the change of variable formula leads us to a ...
Durden's user avatar
  • 1,255
0 votes
0 answers
68 views

Name of a second order statistical quantity similar to generalized variance

I am looking for the name of a statistical quantity similar to generalized variance in a 2 dimensional space. Specifically, I define generalized variance as $\epsilon_{a,b}^2 = \left|\begin{matrix}\...
Billyziege's user avatar
5 votes
1 answer
336 views

Why does the determinant of the Hessian grow with n?

Context: I'm trying to understand BIC on a deeper level. I'm using BIC for model/structure selection for Bayesian networks. I'm confused because BIC is an approximation to the likelihood of a model, ...
Lizzie Silver's user avatar
4 votes
1 answer
138 views

How to model for independent determinants in several groups based on follow-up time

I want to answer the research question which determinants are associated with long-term survival after a myocardial infarction (MI) in a prospective patient cohort study. More precisely: I want to ...
Tami's user avatar
  • 320
12 votes
1 answer
10k views

Do the Determinants of Covariance and Correlation Matrices and/or Their Inverses Have Useful Interpretations?

While learning to calculate covariance and correlation matrices and their inverses in VB and T-SQL a few years ago, I learned that the various entries have interesting properties that can make them ...
SQLServerSteve's user avatar
2 votes
0 answers
2k views

Multicollinearity (or not) in exploratory factor analysis

I’m performing an exploratory factor analysis with 28 items, n = 300. I’m confused whether I have a multicollinearity problem or not, and if so whether/how I go about choosing items to remove from the ...
Anneli's user avatar
  • 21
2 votes
0 answers
192 views

Algebraic derivation of canonical correlations

In a paper from 1936, Harold Hotelling (access on JSTOR) defined the concepts of canonical correlations and canonical variates for two sets of variates. In pages 327 and 328, he precisely derives ...
M. Toya's user avatar
  • 477
2 votes
0 answers
41 views

Maximum determinant with $l_2$ norm penalties

Let the random sample $Y_1,\ldots,Y_n\sim N_p(0,\Sigma)$, then the likelihood is given by, \begin{align*} L(\Sigma)=\frac{1}{(2\pi)^{np/2}|\Sigma|^{n/2}} e^{-\frac{1}{2}\sum_i Y'_i\Sigma^{-1}Y_i} \end{...
TPArrow's user avatar
  • 2,295
1 vote
0 answers
326 views

Log Expectation of Inflated Determinant of Wishart Distribution

Let $\Lambda \sim \mathcal W(\nu, \Psi)$, i.e., following a $n \times n$ dimensional Wishart distribution with mean $\nu \Psi$ and degrees of freedom $\nu$. The expectation of the log determinant of $\...
Taemin Kim's user avatar
4 votes
1 answer
320 views

Determinant of a block matrix with sparse elements

I have a positive definite symmetric matrix that looks like $$\pmatrix{A & 0 & 0 & E \\ 0 & B & 0 & F \\ 0 & 0 & C & G \\ E^\prime & F^\prime & G^\prime &...
Wis's user avatar
  • 2,164
0 votes
1 answer
319 views

How to get the determinant of a covariance matrix from its diagonal elements

I am trying to implement a speaker recognition system in MATLAB. I am using Gaussian Mixture Models (GMM) for speaker modelling and maximizing the posterior probabilities for classification. The ...
Sounak's user avatar
  • 103
2 votes
0 answers
221 views

Stability of VAR$(p)$: Prove $\det(I_{Kp}-\tilde{A}z)=\det(I_k-A_1z-\ldots-A_pz^p)$

Given we have a VAR$(p)$ process written in the companion form $$\tilde{y}=\tilde{v}+\tilde{A}\tilde{y}_{t-1}+\tilde{u}_t$$ where $$\tilde{A}=\left(\begin{array}{ccccc} A_1& A_2 & \ldots &...
Stefan Voigt's user avatar
  • 1,330
7 votes
1 answer
1k views

How is the determinant of $(X'X)$ related to variance?

I'm working on a problem (and actually have the answer) but I don't know why this is the answer, can someone explain this equality?. It has to do with the the determinant of the partitioned matrix $(X'...
Joel Sinofsky's user avatar
2 votes
1 answer
61 views

What's special about (x'x-x'Proj(w)x)?

I'm working on my homework and I keep seeing something like $$(x'x-x'W(W'W)^{-1}W'x)$$ I know that $W(W'W)^{-1}W'$ is the projection matrix, but what is so special about subtracting those two inner ...
Joel Sinofsky's user avatar
2 votes
1 answer
51 views

Column1 = Temp_F and Column2 = Temp_C -- are these linearly dependent?

If $X$ is a matrix of size $m$ x $2$, where the first column is a range of Celsius values and the second column is their corresponding Fahrenheit values, would the columns be considered linearly ...
compguy24's user avatar
  • 557
3 votes
1 answer
199 views

Meaning of determinant of matrix of impulse-response functions in a VAR

Suppose there is a structural VAR model, such as: $A y_{t} = B y_{t-1} + \varepsilon_{t}$, where $\varepsilon_{t} \sim N(0, I_{n})$; then the matrix representing the contemporaneous impulse-response ...
Douglas K's user avatar
1 vote
1 answer
118 views

Sampling from an (almost) multivariate normal over matrices

Consider $n$ points in the euclidean plane, $p_i = (x_i,y_i)_{1\leq i \leq n}$. Now consider a $2 \times 2$ matrix $M = \left(\begin{array}{cc}a & b\\c& d\end{array}\right)$ a vector $r = \...
Arthur B.'s user avatar
  • 2,780
0 votes
2 answers
2k views

Applying inferential statistics for census data

Let's assume I have a census data of a population which I would like to study and it has variables such as age, gender, sex, occupation etc and the dependent variable which is community participation ...
Dr. S.Rama Gokula Krishnan's user avatar
12 votes
1 answer
3k 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 ...
ttnphns's user avatar
  • 57.2k
11 votes
2 answers
2k views

Fisher information matrix determinant for an overparameterized model

Consider a Bernoulli random variable $X\in\{0,1\}$ with parameter $\theta$ (probability of success). The likelihood function and Fisher information (a $1 \times 1$ matrix) are: $$ \begin{align} \...
Tyler Streeter's user avatar
1 vote
1 answer
440 views

Determinant of the covariance matrix in a normal distribution

Suppose a $p \times 1$ vector $x \sim N_p(\boldsymbol 0, \boldsymbol \Sigma_1)$. Now, There is another covariance matrix $\boldsymbol \Sigma_2$. We know that $|\boldsymbol \Sigma_2| < |\boldsymbol \...
user154969's user avatar
9 votes
1 answer
621 views

Why people often optimize the determinant of $(X'\Sigma X)^{-1}$

Say I have a random vector $Y\sim N(X\beta,\Sigma)$ and $\Sigma\neq\sigma^2 I$. That is, the elements of $Y$ (given $X\beta$) are correlated. The natural estimator of $\beta$ is $(X'\Sigma^{-1}X)^{-1}...
qoheleth's user avatar
  • 1,462
5 votes
1 answer
5k views

What does Determinant of Covariance Matrix give?

I am representing my 3d data in covariance matrix. I just want to know what the determinant of a covariance matrix gives. If the determinant is positive, zero, negative, high positive, high negative, ...
david's user avatar
  • 53
1 vote
1 answer
2k views

How to calculate Eigenvalue without using eign() function with R? [closed]

I understand how to calculate EIGN by hand but when I try to write code without function EIGN(), I did not have a clue. To calculate Eigenvalue is to count all the possible c in ...
user40596's user avatar
9 votes
3 answers
15k views

Why do we use the determinant of the covariance matrix when using the multivariate normal?

I am not well versed in statistics. I wanted to know why we use the determinant of the covariance matrix instead of having the covariance matrix itself when writing down the multivariate normal ...
Sophie's user avatar
  • 93