Principal component analysis transforms a set of correlated variables into uncorrelated orthogonal components. It can be shown that these components extract successively a maximal part of the total variance of the variables. A graphical display, called bi-plot, can be produced which shows the position of the variables in the plane spanned by two components. Let Z(x)=(Z_1(x),...,Z_p(x))^T, x\in D\subset\IR^d, be a multivariate spatial process satisfying the joint second order stationarity hypothesis. The principal components are calculated by means of the matrix Q containing the orthonormal eigenvectors of the variance-covariance matrix, Sigma\in\IR^{p*p}, of the process Z. Due to spatial correlation, the natural estimator, \hat{U}=\frac{1}{n-1}\sum_{k=1}^{n}(Z(x_k)-\bar{Z})(Z(x_k)-\bar{Z})^T, of \Sigma is biased. Thus, the eigenvectors of \Sigma and \hat{U} are not necessary the same. While this bias has in many cases no significant effect on the eigenvectors, it is nevertheless of interest to correct for it. A new unbiased estimator of \Sigma is presented in this poster. Its components are estimated by means of the cross covariogram C_{ij}(x_k-x_l)=Cov(Z_i(x_k),Z_j(x_l)), whose estimation clearly involves intensive calculations. Therefore, the need for approximations. Moreover, asymptotic calculations show that the unbiased estimator can be approximated using only the sill and the range of the corresponding cross covariograms. We illustrate the new method by applying it to a data set containing n=293 measurements of seven heavy metals rates (Zn, Cr, Cd, Co, Hg, Pb and Ni) taken from Lake Geneva sediments.
Keywords: Principal component analysis, spatial correlation, estimation, asymptotic approximation.