Statistics: Mean Square Weighted Deviation

Mean square weighted deviation is a statistical method used extensively in geochronology.

The Mean Square Weighted Deviation (MSWD) is a measure of goodness of fit that takes into account the relative importance of both the integral and external reproducibility. In general when:

MSWD = 1 if the age data fit a univariate normal distribution in t (for the arithmetic mean age) or log(t) (for the geometric mean age) space, or if the compositional data fit a bivariate normal distribution in [log(U/He),log(Th/He)]-space (for the central age).

MSWD < 1 if the observed scatter is less than that predicted by the analytical uncertainties. In this case, the data are said to be "underdispersed", indicating that the analytical uncertainties were overestimated.

MSWD > 1 if the observed scatter exceeds that predicted by the analytical uncertainties. In this case, the data are said to be "overdispersed". This situation is the rule rather than the exception in (U-Th)/He geochronology, indicating an incomplete understanding of the isotope system. Several reasons have been proposed to explain the overdispersion of (U-Th)/He data, including unevenly distributed U-Th distributions and radiation damage.

Often the geochronologist will determine a series of age measurements on a single sample, with the measured value $\ {x_i}$ having a weighting $\ {w_i}$ and an associated error $\sigma_{x_{i}}$ for each age determination. As regards weighting, one can either weight all of the measured ages equally, or weight them by the proportion of the sample that they represent. For example, if two thirds of the sample was used for the first measurement and one third for the second and final measurement then one might weight the first measurement twice that of the second.

The arithmetic mean of the age determinations is:

$\overline{x} = \frac{\sum_{i=1}^N x_i}{N}$

but this value can be misleading unless each determination of the age is of equal significance.

When each measured value can be assumed to have the same weighting, or significance, the biased and unbiased (or "population" and "sample", respectively) estimators of the variance are computed as follows:

$\sigma^2 = \frac{\sum_{i=1}^N (x_i - \overline{x})^2}{N} {\rm \ \ and\ \ } s^2 = \frac{N}{N-1}\cdot\sigma^2 = \frac{N}{N^2-N}\cdot\sum_{i=1}^N (x_i - \overline{x})^2.$

The standard deviation is the square root of the variance.

When individual determinations of an age are not of equal significance it is better to use a weighted mean to obtain an 'average' age, as follows:

$\overline{x}^{\,*} = \frac{\sum_{i=1}^N w_i x_i}{\sum_{i=1}^N w_i}$

The biased weighted estimator of variance can be shown to be:

$\sigma^2 = \frac{\sum_{i=1}^N w_i (x_i - \overline{x}^{\,*})^2}{\sum_{i=1}^N w_i}$

which can be computed on the fly as

$\sigma^2 = \frac{\sum_{i=1}^N w_i x_i^2 \cdot \sum_{i=1}^N w_i - (\sum_{i=1}^N w_i x_i)^2} {(\sum_{i=1}^N w_i)^2}$

The unbiased weighted estimator of the sample variance can be computed as follows:

$s^2 = \frac{\sum_{i=1}^N w_i}{{(\sum_{i=1}^N w_i})^2 - {\sum_{i=1}^N w_i^2} } \ . \ {\sum_{i=1}^N w_i (x_i - \overline{x}^{\,*})^2}$

Again the corresponding standard deviation is the square root of the variance.

The unbiased weighted estimator of the sample variance can also be computed on the fly as follows:

$s^2 = \frac{\sum_{i=1}^N w_i x_i^2 \cdot \sum_{i=1}^N w_i - (\sum_{i=1}^N w_i x_i)^2}{(\sum_{i=1}^N w_i)^2 - \sum_{i=1}^N w_i^2 }$

The unweighted mean square of the weighted deviations (unweighted MSWD) can then be computed, as follows:

MSWD $_u = \frac{1}{N-1} \ . \ \sum_{i=1}^N\frac{ (x_i - \overline{x})^2}{\sigma_{x_i}^2 }$

By analogy the weighted mean square of the weighted deviations (weighted MSWD) can be computed, as follows:

MSWD $_w = \frac{\sum_{i=1}^N w_i}{(\sum_{i=1}^N w_i)^2 - \sum_{i=1}^N w_i^2 } \ . \ \sum_{i=1}^N \frac{w_i . (x_i - \overline{x}^{\,*})^2}{(\sigma_{x_i})^2 }$

Statistics

Tuesday, November 19, 2013

Mean Square Weighted Deviation

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