Projection Pursuit Multivariate Transform
- PDF / 4,275,583 Bytes
- 23 Pages / 439.37 x 666.142 pts Page_size
- 97 Downloads / 194 Views
Projection Pursuit Multivariate Transform Ryan M. Barnett · John G. Manchuk · Clayton V. Deutsch
Received: 15 January 2013 / Accepted: 14 October 2013 / Published online: 13 November 2013 © International Association for Mathematical Geosciences 2013
Abstract Transforming complex multivariate geological data to a Gaussian distribution is an important and challenging problem in geostatistics. A variety of transforms are available for this goal, but struggle with high dimensional data sets. Projection pursuit density estimation (PPDE) is a well-established nonparametric method for estimating the joint density of multivariate data. A central component of the PPDE algorithm transforms the original data toward a multivariate Gaussian distribution. The PPDE approach is modified to map complex data to a multivariate Gaussian distribution within a geostatistical modeling context. Traditional modeling may then take place on the transformed Gaussian data, with a back-transform used to return simulated variables to their original units. This approach is referred to as the projection pursuit multivariate transform (PPMT). The PPMT shows the potential to be an effective means for modeling high dimensional and complex geologic data. The PPMT algorithm is developed before discussing considerations and limitations. A case study compares modeling results against more common techniques to demonstrate the value and place of the PPMT within geostatistics.
1 Introduction Geostatistical modeling of multiple geologic variables commonly rely on the multivariate Gaussian (multi-Gaussian) distribution due to its mathematical tractability
B
R.M. Barnett ( ) · J.G. Manchuk · C.V. Deutsch Centre for Computational Geostatistics, Department of Civil and Environmental Engineering, University of Alberta, 3-133 NREF Building, Edmonton, Alberta, T6G 2W2 Canada e-mail: [email protected] J.G. Manchuk e-mail: [email protected] C.V. Deutsch e-mail: [email protected]
338
Math Geosci (2014) 46:337–359
Fig. 1 Schematic representation of multivariate complexities, including heteroscedastic (left), nonlinear (middle left), and constraint (middle right) features. These features contrast against the elliptical form of multi-Gaussian variables (right)
(Isaaks 1990; Journel and Huijbregts 1978; Verly 1984). As geologic variables are non-Gaussian in nature, a variety of techniques may be used to transform them to a multi-Gaussian form. Traditional modeling methods may then proceed, with associated back-transforms used to reintroduce the original complexity to simulated realizations. The widely applied normal score transformation (Bliss 1934; Deutsch and Journel 1998; Verly 1984) will guarantee that variables are made univariate Gaussian, however, multivariate complexities such as heteroscedasticity, nonlinearity, and constraints may persist (Fig. 1). A multi-Gaussian distribution is also presented in Fig. 1 for comparison to the complex cases. Consider that the elliptical nature of a multi-Gaussian distribution is fully parameterized by its covari
Data Loading...
