Lead Isotope Ratio Source Tool

Screen all caller lead-source multivariate baselines using reused Mahalanobis distance and combined covariance; retains ambiguous/no matches, without simple ratio aliases or source identification.

Description

Several lead isotope ratios can be screened jointly when their covariance is known.

Several lead isotope ratios can be screened jointly when their covariance is known. Mahalanobis distance accounts for correlated variation instead of treating every ratio as an independent matching test.^[1]

Screen all caller lead-source multivariate baselines using reused Mahalanobis distance and combined covariance; retains ambiguous/no matches, without simple ratio aliases or source identification.

When to use Lead Isotope Ratio Source Tool

Use this tool to screen all caller lead-source multivariate baselines using reused Mahalanobis distance and combined covariance; retains ambiguous/no matches, without simple ratio aliases or source identification.

Prepare the artifact, site or survey measurements and any independently established calibration or model choices. Keep paired observations aligned and confirm units and coordinate conventions.

How to use Lead Isotope Ratio Source Tool and understand its results

Prepare the artifact, site or survey measurements and any independently established calibration or model choices. Keep paired observations aligned and confirm units and coordinate conventions.
Enter sampleLeadIsotopeRatios, sourceMeanRatioVectors, combinedCovariance, callerMaximumSquaredDistance using the field table and calculation rules, then run the tool.
Read sourceSquaredMahalanobisDistances, compatibleSourceFlags. The explanation below defines the units, signs, ordering and any flags or placeholder values.
If the tool reports an input issue, correct the values and run it again. Apply the stated assumptions when interpreting a successfully returned result.

Input field	Meaning	Value type; see rules below	Requirement
sampleLeadIsotopeRatios	Sample lead isotope ratios	list of number	Required
sourceMeanRatioVectors	Source mean ratio vectors	list of list of number	Required
combinedCovariance	Combined covariance	list of list of number	Required
callerMaximumSquaredDistance	Caller maximum squared distance	number	Required

Supply positive sample ratios and positive source vectors with matching dimensions, up to sixteen ratios per source, plus a compatible symmetric positive-definite combined covariance matrix and nonnegative squared-distance cutoff. The reused distance calculation uses Cholesky factorization. Return squared distances and inclusive compatibility flags for every supplied source, retaining zero or multiple matches.

Paired lists must be nonempty and have equal lengths.

D²_i = (sample−sourceMean_i)ᵀ covariance⁻¹(sample−sourceMean_i)

Units, indexing, boundary conditions and exceptional cases are defined in the calculation rules above.

Output field	Meaning	Value type
sourceSquaredMahalanobisDistances	Source squared mahalanobis distances	list of number
compatibleSourceFlags	Compatible source flags	list of number

json

{
  "input": {
    "sampleLeadIsotopeRatios": [
      1,
      2
    ],
    "sourceMeanRatioVectors": [
      [
        1,
        2
      ],
      [
        3,
        2
      ]
    ],
    "combinedCovariance": [
      [
        1,
        0
      ],
      [
        0,
        1
      ]
    ],
    "callerMaximumSquaredDistance": 1
  },
  "output": {
    "sourceSquaredMahalanobisDistances": [
      0,
      4
    ],
    "compatibleSourceFlags": [
      1,
      0
    ]
  }
}

Limitations and assumptions

The supplied covariance and baselines control the screen. The tool does not estimate covariance, identify a unique provenance or assign posterior source probabilities.

Numeric inputs must be finite. Follow the calculation-specific rules for overflow, underflow and exceptional cases. Matrices are limited to 128 rows and 128 entries per row; other numeric lists are limited to 100,000 entries. Missing required fields, extra fields, wrong value types and violations of the calculation rules are input errors.

Alternative or Complementary methods

Use Strontium Isotope Provenance Ratio Tool: Compare measured strontium ratio to all caller source baselines with independent sigma, retaining overlap and no matches; no ratio aliases, geographic identification or forced provenance. For a complementary task, use Mahalanobis Distance Calculator: Calculate Mahalanobis distance using a symmetric positive-definite covariance matrix.

References

Lead Isotope Ratio Source Tool — tool.io

Similar or alternative tools

Strontium Isotope Provenance Ratio Tool
Compare measured strontium ratio to all caller source baselines with independent sigma, retaining overlap and no matches; no ratio aliases, geographic identification or forced provenance.
Mahalanobis Distance Calculator
Calculate Mahalanobis distance using a symmetric positive-definite covariance matrix.