8: Fitting Models of Discrete Character Evolution
- Page ID
- 21627
\( \newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} } \) \( \newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}} \)\(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\) \(\newcommand{\id}{\mathrm{id}}\) \( \newcommand{\Span}{\mathrm{span}}\) \( \newcommand{\kernel}{\mathrm{null}\,}\) \( \newcommand{\range}{\mathrm{range}\,}\) \( \newcommand{\RealPart}{\mathrm{Re}}\) \( \newcommand{\ImaginaryPart}{\mathrm{Im}}\) \( \newcommand{\Argument}{\mathrm{Arg}}\) \( \newcommand{\norm}[1]{\| #1 \|}\) \( \newcommand{\inner}[2]{\langle #1, #2 \rangle}\) \( \newcommand{\Span}{\mathrm{span}}\)\(\newcommand{\AA}{\unicode[.8,0]{x212B}}\)
In this chapter, the Felsenstein’s pruning algorithm was presented and show to be used to calculate the likelihoods of Mk and extended-Mk models on phylogenetic trees. I have also described both ML and Bayesian frameworks that can be used to test hypotheses about character evolution. This chapter also includes a description of the “total garbage” test, which will tell you if your data has information about evolutionary rates of a given character.
- 8.1: The Evolution of Limbs and Limblessness
- Squamates had lost their limbs repeatedly over their evolutionary history. This is a pattern that has been known for decades, but analyses have been limited by the lack of a large, well-supported species-level phylogenetic tree of squamates. In the past few years have phylogenetic trees been produced at a scale broad enough to take a comprehensive look at this question. Such efforts to reconstruct this section of the tree of life provide exciting potential to revisit old questions with new data.
- 8.2: Fitting Mk models to Comparative Data
- The equations in Chapter 7 give us enough information to calculate the likelihood for comparative data on a tree. To understand how this is done, we can first consider the simplest case, where we know the beginning state of a character, the branch length, and the end state. We can then apply the method across an entire tree using a pruning algorithm, which will allow calculation of the likelihood of the data given the model and phylogenetic tree.
- 8.5: Exploring Mk - the "total garbage" test
- One problem that arises sometimes in maximum likelihood optimization happens when instead of a peak, the likelihood surface has a long flat “ridge” of equally likely parameter values. In the case of the Mk model, it is common to find that all values of q greater than a certain value have the same likelihood. This is because above a certain rate, evolution has been so rapid that all traces of the history of evolution of that character have been obliterated.
- 8.7: Appendix - Felsenstein's Pruning Algorithm
- Felsenstein’s pruning algorithm (1973) is an example of dynamic programming, a type of algorithm that has many applications in comparative biology. In dynamic programming, we break down a complex problem into a series of simpler steps that have a nested structure. This allows us to reuse computations in an efficient way and speeds up the time required to make calculations.