# Is advanced linear algebra necessary for understanding multivariate statistics and stochastic processes?

I heard that linear algebra especially matrix algebra including singular value decomposition, symmetric, Hermitian, conjugate transpose, unitary geometry, transposes, and spectral theory show up in multivariate statistics and stochastic processes. Is multivariate statistics really dependent on understanding advanced linear algebra? Should I avoid multivariate statistics and stochastic processes if the advanced linear algebra and diagonalization methods of matrices don't make sense to me?

Whenever I read a stochastic processes book or book on multivariate statistics, the matrix algebra used when doing change of basis scares me and confuses me.

I've also looked at a book on probability theory which is a prerequisite for stochastic processes which is applied calculus and all the computations or proofs on for example leverage and Cook's distance seem messy and not elegant. I don't think I'd like mathematical stats.

• What is elementary and what is advanced? Sorry, but the answer does depend on this! Commented Feb 24, 2018 at 18:09
• elementary is like the kind of linear algebra you'd need for a basic undergrad ordinary differential equations class to prove things regarding systems of solutions and dfqs and such and determinants and wronskians and solutions
– user196575
Commented Feb 24, 2018 at 18:50
• do you need multivariate stats for epidemiology?
– user196575
Commented Feb 24, 2018 at 18:51
• I also mean matrix algebra for epi
– user196575
Commented Feb 24, 2018 at 18:58

I will tell you what I got out of linear algebra. My knowledge is not comprehensive.

1. Understand what a column space is. This enables you to truly understand why adding extra variables does not really add to a model that already contains most of the relevant information. It will tell you why a 36 question survey can be reduced to about 3 questions and help you have groups of questions (factor analysis).

2. Understand what an orthogonal projection is. All linear explanatory modeling is the projection of an n dimensional space onto a p dimensional space. The column space of the response is divided into the model and the error. Why does the plot of the error by the response always show a pattern?

3. Avoid using numerically unstable formulas when you try to implement methods you see in text books or papers. The formulas you see in text books for linear regression are not the ones used in practice. Many people who try to roll their own code try to use unstable formulas.

4. See what is similar in different methods. For example GLM, and ANOVA, MANCOVA and other things. Look at all the ANOVA variants. Understanding linear algebra will cut through all that mess.

5. Understand what is happening with methods like ridge regression and leave-one-out cross validation.

6. Understand covariance structure. Why is a random intercept the same as a nested model. What does auto-correlation do to your model?

• do you need multivariate statistics for epidemiology?
– user196575
Commented Feb 24, 2018 at 18:50
• I mean matrix algebra for epidemiology?
– user196575
Commented Feb 24, 2018 at 18:58
• (+1) It'd be worth adding that recommendation algorithms are very dependent on matrix factorizations and their interpretation. Commented Feb 24, 2018 at 19:10