In the lab you ran lines like:

Both rest on a single assumption: that the trait performs a random walk in continuous time along the branches of the tree. That random walk is Brownian motion (BM).

BM is the null model of trait evolution: the baseline that the contrast methods of last lecture assume, and the reference that every model next lecture is compared against.

One question to keep in mind:

When a trait "wanders" with no destination, what pattern does it leave at the tips of a tree, and how do we read the rate of wandering back out?

Three steps in this lecture

1. BM as a process. A random walk, its rate $\sigma^2$, and why endpoints are normal.
2. BM on a tree. Shared ancestry becomes the variance–covariance matrix $\mathbf{C}$; tips are multivariate normal.
3. Estimating the rate. Contrasts, maximum likelihood, and REML recover $\sigma^2$ from the tips.

Step 2 is exactly the $\mathbf{C}$ you met in the comparative-methods lecture, now derived from first principles.

• The classic random walk: each instant the trait takes a small step in a random direction.
• Three rules define it:
  • No bias toward any direction (steps average to zero).
  • Each step is independent of every other step.
  • Steps occur in continuous time, not discrete generations.
• Add up many small independent steps and, by the central limit theorem, the displacement is normally distributed.

Independent lineages from a common ancestor at value 0.

• Run the walk for time $t$ and record where each lineage ends up.
• The endpoints are normally distributed: $$X(t) \sim N\!\left(X(0),\; \sigma^2 t\right)$$
• The mean equals the starting value $X(0)$.
• The variance grows linearly with time: $\operatorname{Var}=\sigma^2 t$.
• This single result is the engine behind everything that follows.

• $\sigma^2$ sets how much change accumulates per unit time. It is the one number BM has to estimate.
• After time $t$, the spread of trait values is $N(X(0),\,\sigma^2 t)$: higher $\sigma^2$ means a wider distribution.
• The mean stays at the starting value whatever the rate.
• Units matter: $\sigma^2$ is in (trait units)$^2$ per unit of branch length, so log-transforming a trait changes what $\sigma^2$ means.

The expected value never moves from $X(0)$. The walk has no destination.

$\operatorname{Var}=\sigma^2 t$. Long branches accumulate more change; the uncertainty fans out as $\sqrt{t}$.

Change in one interval is independent of, and normal like, change in any other. Non-overlapping branches evolve independently.

• Add a directional drift parameter $\mu$ and the endpoints become: $$X(t) \sim N\!\left(X(0) + \mu t,\; \sigma^2 t\right)$$
• The mean now slides at rate $\mu$ while the spread still grows as $\sigma^2 t$ (try $\mu \ne 0$ in the simulator three slides back).
• The catch: on an ultrametric tree of living species only, all tips have the same age, so a trend in the mean and a shifted root value are confounded. The trend cannot be estimated from extant tips alone.
• Detecting a trend needs fossils (tips of different ages) or a strong trend relative to $\sigma^2$.

• On a single branch (A), the trait performs BM from the ancestral value $\bar{z}(0)$ to $\bar{z}(t_1)$.
• At a speciation event (B), the two daughter lineages start from the same ancestral value, then evolve independently.
• So two species share all the change that happened before they split, and nothing after.
• Shared history creates covariance; independent history after the split creates variance. This single idea builds the whole tip distribution.

Each lineage does its own BM down the branches. Sister lineages share their path until they split, then diverge.

Time runs left to right; the vertical axis is the trait value. Colour marks the two deep clades.

$\mathbf{C}_{ij}$ = shared branch length from the root to the common ancestor of $i$ and $j$

Diagonal = root-to-tip length (the variance of that tip). Off-diagonal = shared path (the covariance between two tips). A and B share the most; D shares nothing.

• Stack the tip values into a vector $\mathbf{x}=(X_A,\dots,X_n)^\top$. Under BM they follow a multivariate normal: $$\mathbf{x} \sim \mathrm{MVN}\!\left(X(0)\,\mathbf{1},\; \sigma^2\,\mathbf{C}\right)$$
• Two parameters only: the root state $X(0)$ and the rate $\sigma^2$. The tree fixes $\mathbf{C}$.
• This is the likelihood. Everything else (contrasts, ML, REML, PGLS) is a different route to the same multivariate normal.
• For several traits at once, replace $\sigma^2$ with a rate matrix $\mathbf{R}$ (the multivariate BM of last lecture).

• Because the tips are multivariate normal, we can also estimate what the internal nodes most likely were.
• A node's estimate is an inverse-variance weighted average of its descendants: a short-branch descendant has drifted little, so it reflects the ancestor closely and counts more; a long-branch descendant counts less.
• Uncertainty grows toward the root: deeper nodes have wider intervals, and the root state is hardest to estimate.
• The same logic extends to multivariate traits, and to other models (OU, early burst) by swapping $\mathbf{C}$ for the model's own covariance.

• Write down the multivariate-normal likelihood and find the $(\sigma^2, X(0))$ that maximise it.
• The likelihood surface (right) has a single peak: the maximum-likelihood estimate.
• This is what geiger::fitContinuous(model="BM") reports: sigsq ($\hat{\sigma}^2$) and z0 ($\hat{X}(0)$), plus the log-likelihood and AIC.
• The log-likelihood and parameter count are exactly what feed the AIC model comparison you will do next lecture.

Data simulated under a true rate, then we plot the log-likelihood as a function of the candidate rate $\sigma^2$.

• For most models there is no formula for the peak, so software takes iterative steps uphill.
• On a smooth, single-peaked surface (right), it converges in a handful of steps.
• Risk: getting trapped on a local optimum when the surface is rugged (common for OU and multi-rate models next lecture).
• Mitigations:
  • multiple random starting points,
  • simulated annealing,
  • Bayesian MCMC, which explores the whole landscape rather than climbing one hill.

Fitting BM to a single continuous trait (here, log body size on a lizard phylogeny) returns four numbers that summarise the whole fit:

Two free parameters here ($\sigma^2$ and $X(0)$), so $\mathrm{AIC}=2\cdot 2 - 2\ln L$.