Questions tagged [determinant]

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How to write or show statements for a matrix of order 2 or more [migrated]

Let $A$ be an $n \times n$ orthogonal matrix where $n$ is even, $|A|=-1$. Show that $|I-A|=0$. I can prove this statement for a matrix of $n=2$, I can slog and also provide a proof for $n=4$, but how ...
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1answer
30 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 ...
6
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1answer
130 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?
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36 views

GDA producing negative covariance determinant

I'm running Gaussian Discriminant Analysis across a large set of examples (~80k) in $\mathbb{R}^{8}$. I know that the covariance matrix $\Sigma$ is, by definition, positive semi-definite, which means ...
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1answer
36 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 ...
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1answer
290 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 ...
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0answers
13 views

How to find optimum matrix set based on determinant values using python

I am new at programming, so I want to find the optimum set of row values based on maximum determinant logic. 1) Set the 1st Column 'Serial_no' as index. 2) Take first 'N' row values as user input ...
2
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1answer
264 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 $...
3
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1answer
262 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 ...
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53 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}\...
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1answer
207 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, ...
4
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1answer
123 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 ...
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1answer
4k 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 ...
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0answers
1k 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 ...
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164 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 ...
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0answers
39 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{...
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206 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 $\...
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1answer
139 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 &...
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1answer
209 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 ...
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0answers
115 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 &...
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1answer
826 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'...
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1answer
47 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 ...
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1answer
44 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 ...
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1answer
117 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 ...
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1answer
104 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 = \...
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2answers
1k 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 ...
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1answer
2k 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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2answers
1k 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} \...
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1answer
339 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 \...
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1answer
285 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}...
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1answer
2k 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, ...
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1answer
918 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 ...
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3answers
11k 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 ...
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3answers
43k views

Why does correlation matrix need to be positive semi-definite and what does it mean to be or not to be positive semi-definite?

I have been researching the meaning of positive semi-definite property of correlation or covariance matrices. I am looking for any information on Definition of positive semi-definiteness; Its ...
4
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1answer
9k views

Factor analysis: What to do when determinant is almost zero and when KMO for a variable is low?

I'm conducting a factor analysis on 40 interval-level variables, and have two immediate concerns: The determinant is 6.608E-006, which is much lower than the cut-...
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1answer
418 views

MLE estimation of spatial effects radius

I am trying to identify the maximum likelihood estimates in an SDM model (a hedonic home price model, with observations being 5,000 individual homes), $$y=\rho W y + X\beta + WX\lambda + \epsilon$$ ...