Spatially Weighted Technology For Logistic Regression And Its Application In Mineral Prospectivity Mapping

Integrating different sources of data including geological, geochemical, geophysical and remote sensing data is one of the most important way to improve the accuracy of mineral potential mapping. Since the International Union of Geological Sciences recommended six methods for mineral resource assessment and potential mapping in 1978, information integrating methods have become popular. Since the purpose of mineral prediction is about the prediction of a binary event, that is whether a deposit is occurred at certain location, log-linear models which result in posterior probabilities are more suitable in mineral resource potential mapping. At present, weights-of-evidence and logistic regression are two of the most popular methods used for information integration in mineral potential mapping. Unlike traditional and classical statistics and predictions, metalorganic prediction belongs to the field of geostatistics since both of the independent variables and the mineral events have location attributes. GIS has been applied in mineral resource assessments and prediction for a long time and some predictive factors are built up through spatial analysis such as overlay analysis, buffer analysis. However, some more complex forms of spatial relationships were not reflected in prediction model, such as spatial clustering and trends of spatial variables.The complexity of geological processes and the multi-stage performance of mineralization lead to the structural, non-stationary and singularity spatial distribution of geological factor layers, thus the predictive model based on classical statistical methods are often biased. As early as 1960s to 1970s, Professor Frits Agterberg, a geoscientist in Geological Survey had noticed that there are certain spatial trends in some predictive variables which may cause systematic errors in mineral potential mapping, thus he suggested to add coordinate variables in the predictive model in order to express different forms of spatial trends. Professor Qiuming Cheng, a scholar of our country found that the distribution of ore deposits shows great spatial concentration which can be characterized by fractal/multi-fractal and added fractal theory into weights-of-evidence model.In recent years, geographically weighted regression (GWR) has become popular from the field of economic geography. In GWR, the prediction for current location is based on the model established on the nearby samples which are weighted according to the distance between them and the current location. In fact the Tobler’s law is well expressed in this spatially variable coefficient model. Because of the application of local window technology, GWR can well deal with the spatial non-stationary so that it can overcome the disadvantage of the global model. While the GWR model is becoming more and more popular in geosciences, geographically weighted logistic regression has not yet been used in mineral potential mapping. As is mentioned above, the object of mineral potential mapping is binary events, and log-linear models are more preferred compared to linear regression models. In this research, spatially weighted technique will be applied in logistic regression model.Broadly speaking, the geographical weigh is a form of spatial weights, which also include the weight based on the object of interest, as well as the weight determined by exploration degree. Because there are differences in the extent of exploration, it will not make sense if all samples are treated the same, thus the exploration weight will be considered in this research. We use the concept of "spatially weighed" instead of "geographically weighed" because the former include the latter.Missing data or incomplete data is a common difficulty which we have to face in statistics as well as geostatistics including mineral potential mapping. There are some common treatment methods for handling missing data:removing the samples with missing data, removing the variables with missing data and using average value to replace missing data. Nevertheless, an alternative solution is provided in weights-of-evidence, in which the value of 1 and 0 in a binary variable is replaced by positive and negative weights so that the missing data can be replaced by 0. The practices of removing samples and variables are crude and will cause great loss of useful information, and the solution to use average value seems better; the solution for missing data in weighs-of-evidence is even better according to former research. Besides, a set of solutions are provided in weights-of-evidence for the binarization (discretization) of independent variables, and they are widely used in mineral potential mapping. In this research, both the solutions for missing data handling and for independent variable binarization in weights-of-evidence are accepted in spatially weighed logistic regression model.The main contents of this paper are listed as followings.(1) Some log-linear models used in mineral potential mapping, their advantages and disadvantages and relationship. It is wrong for one to equate the assumptions of Naive Bayes and weights-of-evidence, because it ignores that the assumption has been relaxed in weights-of-evidence, i.e. from conditional independence to weak conditional independence. There are both difference and relation between weights-of-evidence and logistic regression, and each has its own advantages and disadvantages, thus we combined these two models in this research and built up a spatially weighed model.(2) Spatially weighted technique and information integration in mineral potential mapping.We illustrated in details the theoretical basis and practical significance to apply spatially weighed technique in mineral potential mapping. We described the influence of spatial non-stationary on mineral potential mapping, reviewed the former research on the trend of spatial variable, spatial structure and singularity, and discussed the forms of spatial weights.(3) Software module development for spatially weighed logistic regression.A software module for improved logistic regression based on spatially weighed technique is developed in Visual Studio by C# language. This module not only include the solution used in weights-of-evidence for missing data handling and variable binarization, and bypass the assumption of conditional independence, but also combine geographical weights and other forms of spatial weights. With respect to the determination for the range and function of geographical weights, one can use either Kriging variation model or spatial t-value method in weights-of-evidence, or combine both of them.(4) Case study Taking the prediction of gold mineralization in southwest Nova Scotia of Canada as an example, the technological process for spatially weighted logistic regression is well illustrated, and a comparative study is done among weights-of-evidence, logistic regression and spatially weighted logistic regression.The main innovations in this paper are as follows:(1) Based on weighted logistic regression, a spatially weighted logistic regression technology was established in GIS, and it can well deal with spatial non-stationary and anisotropy in potential mapping and mineral predicting. The spatial weights include both geographic weight and other forms of weight, for example the exploration condition. (2) It was explored how to obtain a local window (or band width), especially the anisotropy window, and geostatistical spatial variation model was applied to determine local window range and the kernel function for weight attenuation.(3) The solution for missing data processing and variable binaryzation in weights-of-evidence was added in the new model.