The combination of complex, multiple minerogenic stages and mineral superposition during geological processes has resulted in dynamic spatial distributions and nonstationarity of geological variables. For example, geochemical elements exhibit clear spatial variability and trends with coverage type changes. Thus, bias is likely to occur under these conditions when general regression models are applied to mineral prospectivity mapping (MPM). In this study, we used a spatially weighted technique to improve general logistic regression and developed an improved model, i.e., the improved logistic regression model, based on a spatially weighted technique (ILRBSWT, version 1.0). The capabilities and advantages of ILRBSWT are as follows: (1) it is a geographically weighted regression (GWR) model, and thus it has all advantages of GWR when managing spatial trends and nonstationarity; (2) while the current software employed for GWR mainly applies linear regression, ILRBSWT is based on logistic regression, which is more suitable for MPM because mineralization is a binary event; (3) a missing data processing method borrowed from weights of evidence is included in ILRBSWT to extend its adaptability when managing multisource data; and (4) in addition to geographical distance, the differences in data quality or exploration level can be weighted in the new model.

The main distinguishing characteristic of spatial statistics compared to
classical statistics is that the former has a location attribute. Before
geographical information systems were developed, spatial statistical
problems were often transformed into general statistical problems in which the
spatial coordinates were similar to a sample ID because they only had an
indexing feature. However, even in nonspatial statistics, the reversal or
amalgamation paradox (Pearson et al., 1899; Yule, 1903; Simpson, 1951),
which is commonly called Simpson's paradox (Blyth, 1972), has attracted
significant attention from statisticians and other researchers. In spatial
statistics, some spatial variables exhibit certain trends and spatial
nonstationarity. Thus, it is possible for Simpson's paradox to occur when a
classical regression model is applied, and the existence of unknown
important variables may worsen this condition. The influence of Simpson's
paradox can be fatal. For example, in geology, due to the presence of cover
and other factors that occur post-mineralization, ore-forming elements in
Area I are much lower than those in Area II, while the actual
probability of a mineral in Area I is higher than that in Area II
simply because more deposits were discovered in Area I (Agterberg,
1971). In this case, negative correlations would be obtained between
ore-forming elements and mineralization according to the classical
regression model, whereas high positive correlations can be obtained in both
areas if they are separated. Simpson's paradox is an extreme case of bias
generated from classical models, and it is usually not so severe in
practice. However, this type of bias needs to be considered and care needs
to be taken when applying a classical regression model to a spatial problem.
Several solutions to this issue have been proposed, which can be divided
into three types.

Locations are introduced as direct or indirect independent variables.
This type of model is still a global model, but space coordinates or
distance weights are employed to adjust the regression estimation between
the dependent variable and independent variables (Agterberg, 1964, 1970, 1971; Agterberg
and Cabilio, 1969; Agterberg and Kelly, 1971; Casetti, 1972; Lesage and Pace, 2009, 2011). For example, Reddy et
al. (1991) performed logistic regression by including trend variables to map
the base-metal potential in the Snow Lake area, Manitoba, Canada; Helbich
and Griffith (2016) compared the spatial expansion method (SEM) to other
methods in modeling the house price variation locally in which the regression
parameters are themselves functions of the

Local models are used to replace global models, i.e., geographically weighted models (Fotheringham et al., 2002). Geographically weighted regression (GWR) is the most popular model among the geographically weighted models. GWR models were first developed at the end of the 20th century by Brunsdon et al. (1996) and Fotheringham et al. (1996, 1997, 2002) for modeling spatially heterogeneous processes and have been used widely in geosciences (e.g., Buyantuyev and Wu, 2010; Barbet-Massin et al., 2012; Ma et al., 2014; Brauer et al., 2015).

Trends in spatial variables are reduced. For example, Cheng developed a local singularity analysis technique and a spectrum–area (S-A) model based on fractal–multi-fractal theory (Cheng, 1997, 1999). These methods can remove spatial trends and mitigate the strong effects on predictions of the variables starting at high and low values, and thus they are used widely to weaken the effect of spatial nonstationarity (e.g., Zhang et al., 2016; Zuo et al., 2016; Xiao et al., 2018).

GWR models can be readily visualized and are intuitive, which have made them applied in geography and other disciplines that require spatial data analysis. In general, GWR is a moving-window-based model in which instead of establishing a unique and global model for prediction, it predicts each current location using the surrounding samples, and a higher weight is given when the sample is located closer. The theoretical foundation of GWR is Tobler's observation that “everything is related to everything else, but near things are more related than distant things” (Tobler, 1970).

In mineral prospectivity mapping (MPM), the dependent variables are binary,
and logistic regression is used instead of linear regression; therefore, it
is necessary to apply geographically weighted logistic regression (GWLR)
instead. GWLR is a type of geographically weighed generalized linear
regression model (Fotheringham et al., 2002) that is included in the
software module GWR 4.09 (Nakaya, 2016). However, the function module for
GWLR in current software can only manage data in the form of a tabular
dataset containing the fields with dependent and independent variables and

Another problem with applying GWR 4.09 for MPM is that it cannot handle missing data (Nakaya, 2016). Weights of evidence (WofE) is a widely used model for MPM (Bonham-Carter et al., 1988, 1989; Agterberg, 1989; Agterberg et al., 1990) that mitigates the effects of missing data. However, WofE was developed assuming that conditional independence is satisfied among evidential layers with respect to the target layer; otherwise, the posterior probabilities will be biased, and the number of estimated deposits will be unequal to the known deposits. Agterberg (2011) combined WofE with logistic regression and proposed a new model that can obtain an unbiased estimate of the number of deposits while also avoiding the effect of missing data. In this study, we employed the Agterberg (2011) solution to account for missing data.

One more improvement of the ILRBSWT is that a mask layer is included in the new model to address data quality and exploration-level differences between samples.

Conceptually, this research originated from the thesis of Zhang (2015; in Chinese), which showed better efficiency for mapping intermediate and felsic igneous rocks (Zhang et al., 2017). This work elaborates on the principles of ILRBSWT and provides a detailed algorithm for its design and implementation process with the code and software module attached. In addition, processing missing data is not a technique covered in GWR modeling presented in prior research, and a solution borrowed from WofE is provided in this study. Finally, ILRBSWT performance in MPM is tested by predicting Au ore deposits in the western Meguma terrane, Nova Scotia, Canada.

Linear regression is commonly used for exploring the relationship between a response variable and one or more explanatory variables. However, in MPM and other fields, the response variable is binary or dichotomous, so linear regression is not applicable and thus a logistic model is advantageous.

In MPM, the dependent variable (

If there are

The solution can be obtained by taking the first partial derivative of

The Newton iterative method can be used to solve the nonlinear equations:

In practice, vector data are often used, and sample size (area) has to be
considered. In this condition, weighted logistic regression modeling should
be used instead of a general logistic regression. It is also preferable to
use a weighted logistic regression model when a logical regression should be
performed for large sample data because weighted logical regression can
significantly reduce matrix size and improve computational efficiency
(Agterberg, 1992). Assuming that there are four binary explanatory variable
layers and the study area consists of 1000

GWLR is a local-window-based model in which logistic regression is
established at each current location in the GWLR. The current location is
changed using the moving window technique with a loop program. Suppose that

As mentioned in the Introduction, there are primarily two improvements for ILRBSWT compared to GWLR: the capacity to manage different types of weights and the special handling of missing data.

If a diagonal element in

In addition to geographic factors, representativeness of a sample, e.g., differences in the level of exploration, is also considered in this study.

Suppose that there are

Missing data are a problem in all statistics-related research fields. In MPM,
missing data are also prevalent due to ground coverage and limitations of the
exploration technique and measurement accuracy. Agterberg and
Bonham-Carter (1999) compared the following commonly used missing data
processing solutions: (1) removing variables containing missing data, (2)
deleting samples with missing data, (3) using 0 to replace missing data, and
(4) replacing missing data with the mean of the corresponding variable. To
efficiently use existing data, both (1) and (2) are clearly not good
solutions as more data will be lost. Solution (3) is superior to (4) in the
condition that work has not been done and real data are unknown; with respect
to missing data caused by detection limits, solution (4) is clearly a better
choice. Missing data are primarily caused by the latter in MPM, and
Agterberg (2011) pointed out that missing data were better addressed in a
WofE model. In WofE, the evidential variable takes the value of positive
weight (

If “1” and “0” are still used to represent the binary condition of the
independent variable instead of

A raster dataset is used for ILRBSWT modeling. With regular grids, the distance between any two grid points can be calculated easily and distance templates within a certain window scope can be obtained, which is highly efficient for data processing. The circle and ellipse are used for isotropic and anisotropic local window designs, respectively.

Weight template for a circular local window with a half-window size of nine grids in which w1 to w30 represent different weight classes that decrease with distances and 0 indicates that the grid is weighted as 0. Gradient colors ranging from red to green are used to distinguish the weight classes for grid points.

Construction of the distance template based on an elliptic local
window:

Suppose that there are

We still use

The ILRBSWT method primarily focuses on two problems, i.e., spatial
nonstationarity and missing data. We use the moving window technique to
establish local models instead of a global model to overcome spatial
nonstationarity. The spatial

User interface design.

Evidential layers used to map Au deposits in this study: buffer of
anticline axes

Before performing spatially weighted logistical regression with ILRBSWT 1.0, data preprocessing is performed using the ArcGIS 10.2 platform and GeoDAS 4.0 software. All data are originally stored in grid format, which should be transformed into ASCII files with the ArcToolbox in ArcGIS 10.2; after modeling with ILRBSWT 1.0, the resulting data will be transformed back into grid format

As shown in Fig. 3, the main interface for ILRBSWT 1.0 is composed of four parts.

The upper left part is for the layer input settings where independent
variable layers, dependent variable layers, and global weight layers should
be assigned. Layer information is shown at the upper right corner, including
row numbers, column numbers, grid size, ordinate origin, and the expression
for missing data. The local window parameters and weight attenuation
function can be defined as follows. Using the drop-down list, we prepared a
circle or ellipse to represent various isotropic and anisotropic spatial
conditions, respectively. The corresponding window parameters should be set
for each window type. For the ellipse, it is necessary to set parameters
composed of the initial length of the equivalent radius (initial major
radius), final length of the equivalent radius (largest major radius),
increase in the length of the equivalent radius (growth rate), threshold of
the spatial

The test data used in this study were obtained from the case study reported
in Cheng (2008). The study area (

The four independent variables described previously were also used for
ILRBSWT modeling in this study (see Fig. 4a to d), and they were
uniformed in the ArcGIS grid format with a cell size of 1 km

Study area (A and B) where there are missing geochemical data in area B.

Exploration-level weights can be determined based on prior knowledge about data quality, e.g., different scales may exist throughout the whole study area; however, these weights can also be calculated quantitatively. The density of known deposits is a good index for the exploration level; i.e., the research is more comprehensive when more deposits are discovered. The exploration-level weight layer for the study area was obtained using the kernel density tool provided by the ArcToolbox in ArcGIS 10.2, as shown in Fig. 6.

Exploration-level weights.

Both empirical and quantitative methods can be used to determine the local window parameters and attenuation function for geographical weights. The variation function in geostatistics, which is an effective method for describing the structures and trends in spatial variables, was applied in this study. To calculate the variation function for the dependent variable, it is necessary to first map the posterior probability using the global logistic regression method before determining the local window type and parameters from the variation function. Variation functions were established in four directions to detect anisotropic changes in space. If there are no significant differences among the various directions, a circular local window can be used for ILRBSWT, as shown in Fig. 1; otherwise, an elliptic local window should be used, as shown in Fig. 2. The specific parameters for the local window in the study area were obtained as shown in Fig. 7, and the final local window and geographical weight attenuation were determined as indicated in Fig. 8a and b, respectively.

Experimental variogram fitting in different directions; the
green lines denote the variable ranges determined for azimuths of

Nested spherical model for different directions. The green lines in

Posterior probability maps obtained for Au deposits by

Prediction–area (P-A) plots for discretized posterior probability maps obtained by logistic regression and ILRBSWT.

Using the algorithm described in Sect. 3.2, ILRBSWT was applied to the study area according to the parameter settings in Fig. 3. The estimated probability map obtained for Au deposits by ILRBSWT is shown in Fig. 9b, while Fig. 9a presents the results obtained by logistic regression. As shown in Fig. 8, ILRBSWT better manages missing data than logistic regression, as the Au deposits in the north part of the study area (with missing data) fit better within the region with higher posterior probability in Fig. 9b than in Fig. 9a.

To evaluate the predictive capacity of the newly developed and traditional methods, the posterior probability maps obtained through logistic regression and ILRBSWT shown in Fig. 9a and b were divided into 20 classes using the quantile method. Prediction–area (P-A) plots (Mihalasky and Bonham-Carter, 2001; Yousefi et al., 2012; Yousefi and Carranza, 2015a) were then made according to the spatial overlay relationships between Au deposits and the two classified posterior probability maps in Fig. 10a and b, respectively. In a P-A plot, the horizontal ordinate indicates the discretized classes of a map representing the occurrence of deposits. The vertical scales on the left and right sides indicate the percentage of correctly predicted deposits from the total known mineral occurrences and the corresponding percentage of the delineated target area from the total study area (Yousefi and Carranza, 2015a). As shown in Fig. 10a and b, with the decline of the posterior probability threshold for the mineral occurrence from left to right on the horizontal axis, more known deposits are correctly predicted, and in the meantime more areas are delimited as the target area; however, the growth in the prediction rates for deposits and corresponding occupied area is similar before the intersection point in Fig. 10a, while the former shows a higher growth rate than the latter in Fig. 10b. This difference suggests that ILRBSWT can predict more known Au deposits than logistic regression for delineating targets with the same area and indicates that the former has a higher prediction efficiency than the latter.

It would be a little inconvenient to consider the ratios of both predicted known deposits and occupied area. Mihalasky and Bonham-Carter (2001) proposed a normalized density, i.e., the ratio of the predicted rate of known deposits to its corresponding occupied area. The intersection point in a P-A plot is the crossing of two curves. The first is fitted from scatterplots of the class number of the posterior probability map and the rate of predicted deposit occurrences (the “prediction rate” curves in Fig. 10). The second is fitted according to the class number of the posterior probability map and corresponding accumulated area rate (the “area” curves in Fig. 10). At the interaction point, the sum of the prediction rate and corresponding occupied area rate is 1; the normalized density at this point is more commonly used to evaluate the performance of a certain spatial variable in indicating the occurrence of ore deposits (Yousefi and Carranza, 2015a). The intersection point parameters for both models are given in Table 1. As shown in the table, 71 % of the known deposits are correctly predicted with 29 % of the total study area delineated as the target area when the logistic regression is applied; if ILRBSWT if applied, 74 % of the known deposits can be correctly predicted with only 26 % of the total area delineated as the target area. The normalized densities for the posterior probability maps obtained from the logistic regression and ILRBSWT are 2.45 and 2.85, respectively; the latter performed significantly better than the former.

Parameters extracted from the intersection points in Fig. 10a and b.

Because of potential spatial heterogeneity, the model parameter estimates obtained based on the total equal-weight samples in the classical regression model may be biased, and they may not be applicable for predicting each local region. Therefore, it is necessary to adopt a local window model to overcome this issue. The presented case study shows that ILRBSWT can obtain better prediction results than classical logistic regression because of the former's sliding local window model, and their corresponding intersection point values are 2.85 and 2.45, respectively. However, ILRBSWT has advantages. For example, characterizing 26 or 29 % of the total study area as a promising prospecting target is too high in terms of economic considerations. If instead 10 % of the total area is mapped as the target area, the proportions of correctly predicted known deposits obtained by ILRBSWT and logistic regression are 44 and 24 %, respectively. The prediction efficiency of the former is 1.8 times larger than the latter.

In this study, we did not separately consider the influences of spatial heterogeneity, missing data, and degree of exploration weight, so we cannot evaluate the impact of each factor. Instead, the main goal of this work was to provide the ILRBSWT tool, thereby demonstrating its practicality and overall effect. Zhang et al. (2017) applied this model to mapping intermediate and felsic igneous rocks and proved the effectiveness of the ILRBSWT tool in overcoming the influence of spatial heterogeneity specifically. In addition, Agterberg and Bonham-Carter (1999) showed that WofE has the advantage of managing missing data, and we have taken a similar strategy in ILRBSWT. We did not fully demonstrate the necessity of using exploration weight in this work, which will be a direction for future research. However, it will have little influence on the description and application of the ILRBSWT tool as it is not an obligatory factor, and users can individually decide if the exploration weight should be used.

Similar to WofE and logistic regression, ILRBSWT is a data-driven method, and thus it inevitably suffers from the same problems as data-driven methods, e.g., the information loss caused by data discretization and exploration bias caused by the training sample location. However, it should be noted that evidential layers are discretized in each local window instead of the total study area, which may cause less information loss. This can also be regarded as an advantage of the ILRBSWT tool. With respect to logistic regression and WofE, some researchers have proposed solutions to avoid the information loss resulting from spatial data discretization by performing continuous weighting (Pu et al., 2008; Yousefi and Carranza, 2015b, c), and these concepts can be incorporated into further improvements of the ILRBSWT tool in the future.

Given the problems in existing MPM models, this research provides an ILRBSWT tool. We have proven its operability and effectiveness through a case study. This research is also expected to provide software tool support for geological exploration researchers and workers in overcoming the nonstationarity of spatial variables, missing data, and differences in exploration degree, which should improve the efficiency of MPM work.

The software tool ILRBSWT v1.0 in this research was developed using C#, and the source codes and executable programs (software tool) are prepared in the folders “source code for ILRBSWT in C#” and “executable programs for ILRBSWT”, respectively. They can be found in the Supplement.

The data used in this research are sourced from the demo
data for GeoDAS software
(

The supplement related to this article is available online at:

The authors declare that they have no conflict of interest.

This study benefited from joint financial support from the National Natural Science Foundation of China (nos. 41602336 and 71503200), the China Postdoctoral Science Foundation (nos. 2017T100773 and 2016M592840), the Shaanxi Provincial Natural Science Foundation (no. 2017JQ7010), and Fundamental Research from Northwest A&F University in 2017 (no. 2017RWYB08). The first author thanks former supervisors Qiuming Cheng and Frits Agterberg for fruitful discussions of spatial weights and for providing constructive suggestions. Thanks also to the anonymous referees for their helpful suggestions and corrections. Edited by: Lutz Gross Reviewed by: two anonymous referees