ISSN:
1573-8868
Keywords:
covariance matrix
;
correlation matrix
;
observations below detection limits
;
maximum likelihiood estimation
;
log-normal distribution function
;
mean
;
variances
;
marginal maximum likelihood estimation
Source:
Springer Online Journal Archives 1860-2000
Topics:
Geosciences
,
Mathematics
Notes:
Abstract Multivariate statistical analyses have been extensively applied to geochemical measurements to analyze and aid interpretation of the data. Estimation of the covariance matrix of multivariate observations is the first task in multivariate analysis. However, geochemical data for the rare elements, especially Ag, Au, and platinum-group elements, usually contain observations the below detection limits. In particular, Instrumental Neutron Activation Analysis (INAA) for the rare elements produces multilevel and possibly extremely high detection limits depending on the sample weight. Traditionally, in applying multivariate analysis to such incomplete data, the observations below detection limits are first substituted, for example, each observation below the detection limit is replaced by a certain percentage of that limit, and then the standard statistical computer packages or techniques are used to obtain the analysis of the data. If a number of samples with observations below detection limits is small, or the detection limits are relatively near zero, the results may be reasonable and most geological interpretations or conclusions are probably valid. In this paper, a new method is proposed to estimate the covariance matrix from a dataset containing observations below multilevel detection limits by using the marginal maximum likelihood estimation (MMLE) method. For each pair of variables, sayY andZ whose observations containing below detection limits, the proposed method consists of three steps: (i) for each variable separately obtaining the marginal MLE for the means and the variances, $$\widetilde{\widetilde\mu }_Y $$ , $$\widetilde{\widetilde\mu }_Z $$ , $$\widetilde{\widetilde\sigma }_{YY} $$ , and $$\widetilde{\widetilde\sigma }_{ZZ} $$ forY andZ: (ii) defining new variables by $$C = (Y - \widetilde{\widetilde\mu }_Y )/\sqrt {\widetilde{\widetilde\sigma }_{YY} } $$ and $$D = (Z - \widetilde{\widetilde\mu }_Z )/\sqrt {\widetilde{\widetilde\sigma }_{ZZ} } $$ and lettingA=C+D andB=C−D, and obtaining MLE for variances, $$\widetilde\sigma _ + $$ and $$\widetilde\sigma _ - $$ forA andB; (iii) estimating the correlation coefficient ϱYZ by $$\widetilde\rho _{YZ} = (\widetilde\sigma _ + - \widetilde\sigma _ - )/(\widetilde\sigma _ + + \widetilde\sigma _ - )$$ and the covariance σ YZ by $$\widetilde\sigma _{YZ} = \bar \rho _{YZ} \sqrt {\widetilde{\widetilde\sigma }_{YY} \widetilde{\widetilde\sigma }_{YY} .} $$ . The procedure is illustrated by using a precious metal geochemical data set from the Fox River Sill, Manitoba, Canada.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF00891047
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