Abstract
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.
Highlights
The objective of potential modeling or targeting [1] is to identify locations, i.e., pixels or voxels, for which the probability of an event spatially referenced in this way, e.g., a well-defined type of ore mineralization, is relatively maximum, i.e., is larger than in neighbor pixels or voxels
Lacking conditional independence can be exactly compensated for by corresponding interaction terms included in the logistic regression model, and the resulting logistic regression model with interaction terms is optimum for continuous predictor variables if the joint distribution of the target variable and the predictor variables is of a log-linear form
Targeting or potential modeling applies regression or regression-like models to estimate the conditional probability of a target variable given predictor variables
Summary
The objective of potential modeling or targeting [1] is to identify locations, i.e., pixels or voxels, for which the probability of an event spatially referenced in this way, e.g., a well-defined type of ore mineralization, is relatively maximum, i.e., is larger than in neighbor pixels or voxels. Conceptual models of ore deposits have been compiled by [2] They may be read as factor models (in the sense of mathematical statistics), and a proper factor model may be turned into a regression-type model when using the factors as spatially-referenced predictors, which are favorable to or prohibitive of the target event. The pixels or voxels initially provide the physical support of the predictors and the target and will be assigned the predicted conditional probability and the associated estimation errors, respectively. If the spatial resolution provided by the pixels or voxels is poor with respect to the area or volume of the actual physical support of the predictors or target, the numerical results of any kind of mathematical method of targeting are rather an artifact of the inappropriate spatial resolution. Potential modeling applies the assumption of independently identically distributed random variables Their distribution does not depend on the location.
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