Indirect Dating With Mixture Density Networks

K. Blake Vernon

Scientific Computing and Imaging Institute (UU)

Brian Codding

Department of Anthropology (UU)

Scott Ortman

Center for Collaborative Synthesis in Archaeology (CU)

Simon Brewer

School of Environment, Society, and Sustainability (UU)

2025-04-26

kbvernon/saa_2025-mdn

The Goal

Estimate a regional chronology \(p(t)\) that effectively summarizes the individual chronologies \(p_i(t)\) of all sites \(s_i\) in a region of interest.

But Why a Chronology?

Because climate and population are the levers of human history!

And Why Indirect Dating?

Because direct dates are expensive and often destructive .

OK, So Why Not Aoristic Sums?

Because (1) date ranges are never tested and (2) site chronologies are assumed to be independent.

An AI Alternative

We can estimate a regional chronology conditioned on some data \(x\) using a Mixture Density Network:

\[ p(t \mid x) = \sum_{k=1}^{K} \pi_k(x)\; \mathcal{N} (t \mid \mu_k(x),\; \sigma_{k}(x)) \]

with

• \(K\) component Gaussians
  
• \(\pi_k(x)\) mixing weights
  
• \(u_k(x)\) means
  
• \(\sigma_{k}(x)\) standard deviations

Why MDN?

Because it’s…

• Comprehensive: can estimate site and regional chronologies simultaneously.
  
• Flexible: can (i) train on any inputs, even images! and (ii) model any outputs or direct dates, notably radiocarbon and tree ring dates.
• Probabilistic: can quantify our uncertainty.
  
• Smart: can represent complex multi-modal distributions.

Still A Gaggle of Challenges

1. Aligning data sets across sites
   
2. Accommodating variable site sizes
   
3. Addressing inconsistent site definitions
   

Coping Strategy

Intuition: We want to loosen the requirement of direct association and look instead at the wider context.

Spatial Aggregates

For \(q\) direct dates, \(r\) sites with recorded artifacts, and \(m\) artifact types, define aggregate \(q \times m\) artifact matrix \(\boldsymbol{A}\) as:

\[ \boldsymbol{A} = \boldsymbol{WF} \]

where

• \(\boldsymbol{W}\) is a \(q \times r\) spatial weight matrix
  
• \(\boldsymbol{F}\) is a disaggregate \(r \times m\) artifact matrix

Intuition: \(\boldsymbol{A}\) tells us how much stuff is around each direct date.

Spatial Basis Functions

For a regular grid of \(p\) knots, define a \(q \times p\) spatial process matrix \(\boldsymbol{B}\) as:

\[ \boldsymbol{B} = (b(i,j)) \]

where

• \(b(i, j)\) is a spatial basis function that accounts for the distance between direct date \(i \in 1,...,q\) and spatial knot \(j \in 1, ..., p\)

Intuition: \(\boldsymbol{B}\) tells us how direct dates are related in space.

Model Graph

Test Case: Central Mesa Verde

Diagnostics

Regional Chronology

Overspecified with \(K = 128\) mixture components

Site Chronologies

Tree Ring Dates

More Site Chronologies

Tree Ring Dates

These sites are right next to each other.

What Next?

1. Reduce over-fitting .
   
2. Account for durations , not just construction dates.
   
3. Correlate spatial and temporal proximity in a more realistic way.
   
4. Get more data !

Acknowledgments