Function to find endmembers in Pb Isotope groups.

[karan3242]

I want to propose the addition of a function to find endmembers using PCA analysis and geochron slope alignment, as published in Shnyr et al (2026).

Shnyr, E., Kuflik, T., Desai, K., & Eshel, T. (2026). Determining the origins of Phoenician silver: Exploring the potential of machine learning for lead isotope analysis. Journal of Archaeological Science, 188, 106499. https://doi.org/10.1016/j.jas.2026.106499

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Description

We need to implement a new method for endmember expansion and classification within our lead isotope analysis suite. While standard PCA (via statprcomp) identifies the extreme points ($p_a$ and $p_b$) along the first principal component ($c_1$), these two vectors often fail to capture the full geological heterogeneity of ore sources.

Following Albarède et al. (2024b) propose a method to expand these endmembers by identifying points that align parallel to the Geochron.

Proposed Workflow

1. Initial PCA: Perform PCA on 3D isotope data ($^{206}Pb/^{204}Pb, ^{207}Pb/^{204}Pb, ^{208}Pb/^{204}Pb$).
2. Primary Identification: Define $p_a$ at $\min(c_1)$ and $p_b$ at $\max(c_1)$, provided $c_1$ accounts for $>95%$ of total variance.
3. Geochron Expansion: Identify additional endmembers $e_i$ that satisfy the proximity to the Geochron slope ($S \approx 0.626208$).
4. Condition: A vector $\vec{e}_i$ is an endmember if it falls within tolerance $\epsilon$ of the line $p_i$ parallel to the geochron in the $(^{206}Pb/^{204}Pb, ^{207}Pb/^{204}Pb)$ plane.
5. Equation:$$|e_2 - (p_1 + (S \times (e_1 - p_1)))| < \epsilon$$Confidence Threshold: Define sets $E_a$ and $E_b$. A set is considered "confirmed" only if $|E_i| &gt; 2$. If $|E_i| \le 2$, the point is flagged as a "possible endmember" for manual review.

Requirements

Function Name: expand_geochron_endmembers()

Parameters:

• data: 3D lead isotope data frame.

• epsilon: Tolerance value (default 0.01 based on empirical results).

• slope: Geochron slope (default 0.626208).

Outputs: A classified list of confirmed vs. possible endmembers.

Calculated probabilities for provenance based on isotopic similarity to existing ore databases.