stats.rmd

# Statistical and mathematical fundamentals

Phylogenetic biology draws on a variety of themes in statistics and mathematics. Given the diverse backgrounds of people coming to phylogenetic biology, one of the challenges of introducing people to this field is ensuring a foundation in these fundamentals. Many people have a solid working grasp of many of these topics, others need a refresher, and some have never seen them.

Throughout the text I have covered some of these fundamentals, but here I provide additional resources.

## Probability theory and General statistics {#stats-general}

I put together this short summary of statistical concepts commonly encountered in biology - https://bitbucket.org/caseywdunn/statistics/raw/master/statistics_in_biology.pdf. It is short and technical, with a focus on the relationship between concepts, such as the exponential family of distributions.

## Bayes theorem {#stats-bayes}

Grant Sanderson of 3Blue1Brown provides an excellent introduction to Bayes Theorem and conditional probability in this video - https://youtu.be/HZGCoVF3YvM. Note that he uses $E$ for Evidence rather than $D$ for Data as is common practice in phylogenetic applications.

This video provides additional perspective, including its application to medical testing - https://youtu.be/R13BD8qKeTg.

The MCMC robot by Paul Lewis is an excellent interactive introduction to MCMC sampling - https://phylogeny.uconn.edu/mcmc-robot/.


## Linear algebra {#linear-algebra}

Linear algebra is a rich field of math that springs from a surprisingly simple 
starting point - linear combinations of 
vectors, often represented as rows and columns of matrices. From basic 
operations such as vector addition and scalar multiplication, one can build 
matrix operations and many sophisticated data analyses. Many familiar data 
analysis methods, such as regression, solving systems of equations, 
Principal Component Analyses (PCA), and common clustering methods, are built from linear 
algebra under the hood. So are many aspects of phylogenetic analysis, including 
the models of evolution themselves and the representation of the phylogenetic 
trees in matrix forms that computers can easily manupulate.

Though linear algebra is fundamental to many topics at the intersection of 
science, mathematics, and computation, most science students have far less 
exposure to this critical field than to statistics, calculus, and other 
quantitative topics. This is too bad, as even a basic working knowledge of 
linear algebra can provide great insight into the tools scientists use daily 
and allow us to better apply, implement, and extend them.

The following resources are a good place to start in your own studies of 
linear algebra:

- The Essence of Linear Algebra video series by Grant Sanderson (3Blue1Brown) - https://www.3blue1brown.com/topics/linear-algebra.

- Linear Algebra: Theory, Intuition, Code, by Mike X Cohen. This book is 
accessible to scientists with little past math experience, and is very 
reasonably priced. https://www.amazon.com/Linear-Algebra-Theory-Intuition-Code/dp/9083136604.

- Data Driven Science and Engineering, by Steve Brunton and Nathan Kutz - http://www.databookuw.com/ .
This book shows how linear algebra is applied to many powerful scientific data analyses.
The authors have excellent companion videos organized into playlists at https://www.youtube.com/c/Eigensteve/playlists, including singular value decomposition for Chapter 1, Fourier analyses for chapter 2.