Timepiece X: moments

Inferring demographic history from the frequency spectrum – without solving a single PDE.

The Mechanism at a Glance

moments is a method for demographic inference – learning a population’s history (size changes, splits, migrations) from patterns in its DNA variation. It works by computing how the site frequency spectrum (SFS) evolves through time under different demographic scenarios, then finding the scenario that best matches observed data.

If the other Timepieces in this book work by reconstructing genealogies (the tree structure connecting samples to their ancestors), moments takes a fundamentally different approach. It summarizes the genetic data into a compact statistic – the frequency spectrum – and works entirely in this summary space. Think of it as reading the time from the position of the hands without ever opening the case to see the gears. The mechanism is different, but the information comes from the same source: the coalescent process shaped by population history.

The key innovation: while its predecessor dadi solves a partial differential equation (the Wright-Fisher diffusion) on a frequency grid, moments bypasses this entirely. It derives ordinary differential equations that govern the SFS entries directly. No grid, no PDE, no numerical diffusion artifacts. Just a clean system of ODEs that you can solve with standard numerical methods.

Primary Reference

[Jouganous et al., 2017]

The four gears of moments:

  1. The Frequency Spectrum (the dial) – The summary statistic at the heart of everything: a histogram of allele frequencies across your sample. Different demographic histories produce different spectral signatures, and this is what moments reads.

  2. Moment Equations (the escapement and gear train) – The mathematical engine: a system of ODEs describing how each SFS entry changes under drift, mutation, selection, and migration. This is where coalescent theory and population genetics theory come together.

  3. Demographic Inference (the mainspring) – The optimization machinery: finding the demographic parameters that maximize the likelihood of the observed data. This is where the model meets reality.

  4. Linkage Disequilibrium (a complication) – A second lens: two-locus statistics that capture information invisible to the SFS alone, especially about recent admixture and recombination rate variation.

These gears mesh together into a complete inference framework:

Observed DNA variation
        |
        v
+-----------------------+
|  FREQUENCY SPECTRUM   |
|                       |
|  Summarize variation  |
|  as allele frequency  |
|  histogram (SFS)      |
+-----------------------+
        |
        v
+-----------------------+
|  MOMENT EQUATIONS     |
|                       |
|  Compute expected SFS |
|  under a demographic  |
|  model (forward in    |
|  time via ODEs)       |
+-----------------------+
        |
        v
+-----------------------+
|  DEMOGRAPHIC          |
|  INFERENCE            |
|                       |
|  Find parameters that |
|  maximize P(data |    |
|  model) via numerical |
|  optimization         |
+-----------------------+
        |
        v
Demographic history:
population sizes, split
times, migration rates

Prerequisites for this Timepiece

  • Coalescent Theory – the relationship between population size and genetic diversity

  • Basic familiarity with differential equations is helpful (we’ll explain as we go)

  • No prior knowledge of the site frequency spectrum is needed – we build it from scratch

Chapters