Multiple variables that are correlated should be jointly simulated and the resulting realizations should reproduce the experimental data statistics (i.e. histogram, variogram, correlation coefficients). Multivariate transforms such as principal component analysis (PCA), minimum/maximum autocorrelation factors (MAF) and projection pursuit multivariate transform (PPMT) are commonly used to independently simulate correlated variables without the requirement of fitting a linear model of coregionalization to the direct and cross variograms of the variables. These transforms, however, operate at different spatial lags \({\mathbf{h }}\); that is, while PCA and PPMT generate factors that are only pairwise \(({\mathbf{h }}=0)\) uncorrelated, MAF generates factors that are both pairwise uncorrelated and have zero cross correlations at one chosen lag \(({\mathbf{h }}\ne 0)\). In addition, PCA and MAF, due to being linear transforms, do not reproduce complex features (i.e. nonlinearity, heteroskedasticity, constraints) that exist in the multivariate distributions of the data; however, PPMT, being a multivariate Gaussian transform, accounts for these features. We show in a case study that these multivariate transforms reproduce the univariate and multivariate statistics of the experimental data. The correlated variables (Cd, Co, Cr, Cu, Ni, Pb and Zn) from the Jura data are transformed into uncorrelated factors using multivariate transforms. The factors are then independently simulated, and the performance of each multivariate transform is quantitatively assessed. The best reproduction of the experimental data statistics is obtained in the case where PPMT is used along with the MAF transform.

应当对关联的多个变量进行联合仿真，并且得到的实现应重现实验数据的统计数据（即直方图，变异函数，相关系数）。通常使用多元变换（例如主成分分析（PCA），最小/最大自相关因子（MAF）和投影追踪多元变换（PPMT））来独立地模拟相关变量，而无需将共区域化的线性模型拟合为直接和交叉变量的方差图。但是，这些变换在不同的空间滞后\（{\ mathbf {h}} \）下运行；也就是说，虽然PCA和PPMT生成的因子仅成对\（（{{\ mathbf {h}} = 0）\）如果不相关，则MAF会生成成对不相关并且在一个选定滞后\（（{{\ mathbf {h}} \ ne 0）\）处具有零互相关的因子。。另外，由于是线性变换，PCA和MAF不会重现数据多元分布中存在的复杂特征（即非线性，异方差，约束）。但是，PPMT是多元高斯变换，说明了这些功能。我们在一个案例研究中显示，这些多变量转换可重现实验数据的单变量和多变量统计数据。使用多变量转换将来自Jura数据的相关变量（Cd，Co，Cr，Cu，Ni，Pb和Zn）转换为不相关的因子。然后独立地模拟这些因素，并定量评估每个多元变换的性能。在将PPMT与MAF变换一起使用的情况下，可以获得实验数据统计数据的最佳再现。