IsoplotR: ludwig – R documentation

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ludwig

Linear regression of U-Pb data with correlated errors, taking into account decay constant uncertainties.

Description

Implements the maximum likelihood algorithm for Total-Pb/U isochron regression of Ludwig (1998) and extends the underlying methodology to accommodate U-Th-Pb data and initial U-series disequilibrium.

Usage

ludwig(x, ...)

## Default S3 method:
ludwig(x, exterr = FALSE, alpha = 0.05, model = 1, anchor = 0, ...)

Arguments

`x`	an object of class `UPb`
`...`	optional arguments
`exterr`	propagate external sources of uncertainty (i.e. decay constants)?
`alpha`	cutoff value for confidence intervals
`model`	one of three regression models: `1`: fit a discordia line through the data using the maximum likelihood algorithm of Ludwig (1998), which assumes that the scatter of the data is solely due to the analytical uncertainties. In this case, `IsoplotR` will either calculate an upper and lower intercept age (for Wetherill concordia), or a lower intercept age and common (^{207}Pb/^{206}Pb)_\circ-ratio intercept (for Tera-Wasserburg). If the p-value for the chi-square test is less than `alpha`, then the analytical uncertainties are augmented by a factor √{MSWD}. `2`: fit a discordia line ignoring the analytical uncertainties `3`: fit a discordia line using a modified maximum likelihood algorithm that includes accounts for any overdispersion by adding a geological (co)variance term.
`anchor`	control parameters to fix the intercept age or common Pb composition of the isochron fit. This can be a scalar or a vector. If `anchor[1]=0`: do not anchor the isochron. If `anchor[1]=1`: fix the common Pb composition at the values stored in `settings('iratio',...)`. If `anchor[1]=2`: force the isochron line to intersect the concordia line at an age equal to `anchor[2]`.

Details

The 3-dimensional regression algorithm of Ludwig and Titterington (1994) was modified by Ludwig (1998) to fit so-called 'Total Pb-U isochrons'. These are constrained to a radiogenic endmember composition that falls on the concordia line. In its most sophisticated form, this algorithm does not only allow for correlated errors between variables, but also between aliquots. IsoplotR currently uses this algorithm to propagate decay constant uncertainties in the total Pb-U isochron ages.

Value

LL: the log likelihood of the discordia fit
par: a vector with the lower concordia intercept, the common Pb ratios and (if model=3) the dispersion parameter
cov: the covariance matrix of par
logpar: the logarithms of par
logcov: the logarithms of cov
df: the degrees of freedom of the model fit (n-2 if x$format<4 or 2n-3 if x$format>3, where n is the number of aliquots).
mswd: the mean square of weighted deviates (a.k.a. reduced Chi-square statistic) for the fit.
p.value: p-value of a Chi-square test for the linear fit

References

Ludwig, K.R., 1998. On the treatment of concordant uranium-lead ages. Geochimica et Cosmochimica Acta, 62(4), pp.665-676.

Ludwig, K.R. and Titterington, D.M., 1994. Calculation of ^{230}Th/U isochrons, ages, and errors. Geochimica et Cosmochimica Acta, 58(22), pp.5031-5042.

Examples

f <- system.file("UPb4.csv",package="IsoplotR")
d <- read.data(f,method="U-Pb",format=4)
fit <- ludwig(d)

IsoplotR

Statistical Toolbox for Radiometric Geochronology

v3.8

GPL-3

Authors

Pieter Vermeesch [aut, cre]

Initial release

2021-04-05