The Tau Model for Data Redundancy and Information Combination in Earth Sciences: Theory and Application

Krishnan, Sunderrajan

doi:10.1007/s11004-008-9165-5

Cited by 45 publications

(28 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…However, this can be more complicated for Bordley's formula, where there is not the restriction S w ¼1. For this formula, one of the most common approaches is to assume unitary weights, which correspond to the assumption of conditional independence among sources of information (Journel, 2002;Krishnan, 2008). Different weight choices are explored in the following test cases.…”

Section: Weights Selectionmentioning

confidence: 99%

See 1 more Smart Citation

3D multiple-point statistics simulation using 2D training images

Comunian¹,

Renard²,

Straubhaar³

2012

Computers & Geosciences

129

101

View full text Add to dashboard Cite

a b s t r a c tOne of the main issues in the application of multiple-point statistics (MPS) to the simulation of threedimensional (3D) blocks is the lack of a suitable 3D training image. In this work, we compare three methods of overcoming this issue using information coming from bidimensional (2D) training images. One approach is based on the aggregation of probabilities. The other approaches are novel. One relies on merging the lists obtained using the impala algorithm from diverse 2D training images, creating a list of compatible data events that is then used for the MPS simulation. The other (s2Dcd) is based on sequential simulations of 2D slices constrained by the conditioning data computed at the previous simulation steps. These three methods are tested on the reproduction of two 3D images that are used as references, and on a real case study where two training images of sedimentary structures are considered. The tests show that it is possible to obtain 3D MPS simulations with at least two 2D training images. The simulations obtained, in particular those obtained with the s2Dcd method, are close to the references, according to a number of comparison criteria. The CPU time required to simulate with the method s2Dcd is from two to four orders of magnitude smaller than the one required by a MPS simulation performed using a 3D training image, while the results obtained are comparable. This computational efficiency and the possibility of using MPS for 3D simulation without the need for a 3D training image facilitates the inclusion of MPS in Monte Carlo, uncertainty evaluation, and stochastic inverse problems frameworks.

show abstract

Section: Weights Selectionmentioning

confidence: 99%

“…It is known that the weights should not be constant (Krishnan, 2008), because they may depend on the data configuration. However, the complete cpdf (Eq.…”

Section: Weights Selectionmentioning

confidence: 99%

3D multiple-point statistics simulation using 2D training images

Comunian¹,

Renard²,

Straubhaar³

2012

Computers & Geosciences

129

101

View full text Add to dashboard Cite

show abstract

“…The modeling assumption with respect to the factors of Equation (6) of the τ -model [23][24][25] is:…”

Section: Weights-of-evidence Vs the τ -Or ν-Modelmentioning

confidence: 99%

Targeting: Logistic Regression, Special Cases and Extensions

Schaeben

2014

IJGI

View full text Add to dashboard Cite

Logistic regression is a classical linear model for logit-transformed conditional probabilities of a binary target variable. It recovers the true conditional probabilities if the joint distribution of predictors and the target is of log-linear form. Weights-of-evidence is an ordinary logistic regression with parameters equal to the differences of the weights of evidence if all predictor variables are discrete and conditionally independent given the target variable. The hypothesis of conditional independence can be tested in terms of log-linear models. If the assumption of conditional independence is violated, the application of weights-of-evidence does not only corrupt the predicted conditional probabilities, but also their rank transform. Logistic regression models, including the interaction terms, can account for the lack of conditional independence, appropriate interaction terms compensate exactly for violations of conditional independence. Multilayer artificial neural nets may be seen as nested regression-like models, with some sigmoidal activation function. Most often, the logistic function is used as the activation function. If the net topology, i.e., its control, is sufficiently versatile to mimic interaction terms, artificial neural nets are able to account for violations of conditional independence and yield very similar results. Weights-of-evidence cannot reasonably include interaction terms; subsequent modifications of the weights, as often suggested, cannot emulate the effect of interaction terms.

show abstract

“…At each grid cell, two probabilities are available to simulate a geologic unit: (1) the conditional probability obtained using the search tree partitioning method, in the horizontal direction, and (2) the conditional probabilities obtained using the 1D vertical variograms and indicator kriging method in the vertical direction (Deutsch and Journel, 1998). The two probabilities are combined using the tau model (Krishnan and others, 2005;Krishnan, 2008), which provides a solution for integrating partially redundant data. The geologic unit is then assigned by drawing a discrete category from the resulting posterior probability distribution.…”

Section: Geostatistical Modelmentioning

confidence: 99%