Questions tagged [linear-algebra]

A field of mathematics concerned with the study of finite dimensional vector spaces, including matrices and their manipulation, which are important in statistics.

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Why the sudden fascination with tensors?

I've noticed lately that a lot of people are developing tensor equivalents of many methods (tensor factorization, tensor kernels, tensors for topic modeling, etc) I'm wondering, why is the world ...
18k views

What is the intuition behind SVD?

I have read about singular value decomposition (SVD). In almost all textbooks it is mentioned that it factorizes the matrix into three matrices with given specification. But what is the intuition ...
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What is an intuitive explanation for how PCA turns from a geometric problem (with distances) to a linear algebra problem (with eigenvectors)?

I've read a lot about PCA, including various tutorials and questions (such as this one, this one, this one, and this one). The geometric problem that PCA is trying to optimize is clear to me: PCA ...
37k views

Is every covariance matrix positive definite?

I guess the answer should be yes, but I still feel something is not right. There should be some general results in the literature, could anyone help me?
19k views

Reference book for linear algebra applied to statistics?

I have been working in R for a bit and have been faced with things like PCA, SVD, QR decompositions and many such linear algebra results (when inspecting estimating weighted regressions and such) so I ...
22k views

Why does inversion of a covariance matrix yield partial correlations between random variables?

I heard that partial correlations between random variables can be found by inverting the covariance matrix and taking appropriate cells from such resulting precision matrix (this fact is mentioned in ...
24k views

Why does Andrew Ng prefer to use SVD and not EIG of covariance matrix to do PCA?

I am studying PCA from Andrew Ng's Coursera course and other materials. In the Stanford NLP course cs224n's first assignment, and in the lecture video from Andrew Ng, they do singular value ...
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Proof that if covariance is zero then there is no linear relationship

I get that a zero covariance doesn´t imply independence, but everybody says that if there is dependence and the covariance is zero then it is a non linear dependence. People base their interpretation ...
1k views

What is the problem with $p > n$?

I know that this is the solving system of linear equation problem. But my question is why it is a problem the number of observation is lower than the number of predictors how can that thing happen? ...
269 views

Null distribution of subspaces similarity, or what is the distribution of $\mathrm{tr}(AA'BB')$?

What is the distribution of $\mathrm{tr}(AA'BB')$ where $A$ and $B$ are two random matrices of $d \times k$ size with orthonormal columns? Maybe the expected value is easier to compute? A fallback ...
6k views

completing the square for Gaussian multivariate estimation

I have been trying to derive the posterior distribution in the case of weighted Bayesian regression in the case of multivariate normal distribution for a few days and have been stuck. I am not sure if ...
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QR factorization and linear regression

I have been reading "Generalized Additive Models an Introduction with R" by Simon Wood and have come across a section I'm having trouble with. On page 13 it is stated that the model or design matrix ...
162 views

Mixed Model Equations

In this paper on page 1924 it is stated that $$\text{Var}(u \mid y) = \sigma^2[G - GZ^\top H^{-1}ZG]$$ can be written as \text{Var}(u \mid y) = \sigma^...