Computers & Geosciences

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1 Computers & Geosciences ] (]]]]) ]]] ]]] Contents lists available at ScienceDirect Computers & Geosciences journal homepage: A spatially weighted principal component analysis for multi-element geochemical data for mapping locations of felsic intrusions in the Gejiu mineral district of Yunnan, China Qiuming Cheng a,c,n, Greame Bonham-Carter b, Wenlei Wang a, Shengyuan Zhang a,d, Wenchang Li e, Xia Qinglin c a Department of Earth and Space Science and Engineering, Department of Geography, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3 b Geological Survey of Canada, 601 Booth St., Ottawa, Ontario, Canada K1A 0E8 c State Key Lab of Geological Processes and Mineral Resources, China University of Geosciences, China d Institute of Resources and Environment, Shijiangzhuang University of Economics, China e Yunnan Academy of Geological Survey, Kunming , Yunnan, China article info Article history: Received 12 December 2009 Received in revised form 7 October 2010 Accepted 2 November 2010 Keywords: Fuzzy mask Geochemical data Spatially weighted principal component analysis Stream sediment samples Tin mineralization abstract Principal component analysis (PCA) is frequently used in geosciences for information extraction. In many applications, masking PCA has been used to create subsets of samples or sub-areas to enhance the effect of the main objects of interest. In this paper we suggest how the representativeness of samples or pixels can be quantified using a fuzzy membership function based on fuzzy set theory. In this new method, the relative importance of pixels or samples can be taken into account using a multivariate statistical method such as PCA. A Fuzzy Masking PCA is proposed and implemented in GeoDAS GIS on the basis of a spatially weighted PCA (SWPCA). This paper introduces the mathematical treatment of the fuzzy masking PCA and follows a case study of identifying the locations of intrusive bodies from geochemical data in the Gejiu mineral district in Yunnan, China. Power-law functions based on the inverse distance from mapped felsic intrusions are applied as weighting functions in FMPCA. The results indicate that fuzzy mask PCA increases the signal-noise ratio of the component representing igneous intrusions and decreases the influence of sedimentary rocks. The areas delineated as potential areas for new intrusions (including buried intrusions) are valuable guides for Sn mineral prospecting. & 2010 Elsevier Ltd. All rights reserved. 1. Introduction Principal component analysis (PCA) has become a standard statistical approach for image processing and geochemical data analysis for the following two reasons: (1) to reduce the number of correlated image bands or variables, forming a small number of uncorrelated principal components that represent most of the variability carried by the multiple image bands or variables, and (2) to enhance the interpretability of the components as combinations of multiple bands or variables (Cheng et al., 2006). PCA has been frequently applied in processing geochemical and other types of geoscience data (e.g. Grunsky, 1997; Harris et al., 1997; Chandrjith et al., 2001; Garrett and Grunsky, 2001; Cheng et al., 2006). Several varieties of PCA can be found in the literature, including PCA applied to raw data, to pre-processed data (Xu and Cheng, 2001) and to subsets of data with masks (Ma et al., 1990). n Corresponding author at: Department of Earth and Space Science and Engineering, Department of Geography, York University, 4700 Keele Street, Toronto, Ontario, Canada M3J 1P3. address: qiuming@yorku.ca (Q. Cheng). The foundation of PCA is the correlation (covariance) matrix, which measures the interrelationships among multiple image bands (variables). The concepts of PCA and its relevant terminologies can be found in many references, including the book by Davis (2002). When applying PCA to spatial data in geographic information systems (GIS) and in image processing, a number of improvements and modifications can be applied to the definition of the correlation (covariance) matrix (Cheng, 1999, 2002, 2007; Cheng et al., 2006). For example, Cheng (2002) shows how a high-order correlation coefficient can be defined on the basis of multifractal modeling and Cheng et al. (2006) proposes that this correlation coefficient can be used to construct a correlation matrix with an optimum order property. Other examples of attempting to enhance the principal components include the use of band or variable ratios to enhance the information from particular bands (Frazer and Green, 1987), and the application of masking techniques to restrict the analysis to subgroups of pixels (e.g., Ma et al., 1990; Cheng, 1997, 2000). Specific multispectral bands have also been chosen to contain the feature information of the targets (e.g., Chavez and Kwarteng, 1989; Crosta and Moore, 1989). In order to optimize feature detection and to avoid using irrelevant pixels in the /$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi: /j.cageo

2 2 Q. Cheng et al. / Computers & Geosciences ] (]]]]) ]]] ]]] calculation, masks are often included in PCA. For example, in order to identify alteration related to mineralization, masks can be used to eliminate pixels covered by water, snow and ice, or heavy vegetation (Ma et al., 1990; Cheng, 1997, 2000). Other types of masks include desert masks (Olson et al., 1983), cloud masks (Saunders and Kriebel, 1998), and sunlight masks (Arino et al., 1993). Most of these masks are binary, and PCA is calculated either with or without certain pixels. However, there are some practical situations where the inclusion of a pixel cannot be decided on the basis of a yes or no scheme. For example, when vegetation cover in an image is used to identify alteration related to mineralization, the proportion of vegetation cover in a pixel can be used to define whether the pixel is fully or only partially occupied by vegetation. A binary mask for this situation is usually defined with a threshold value of vegetation coverage. The decision on the threshold often affects the definition of the mask. If the threshold is set too high, the mask will be too small, and if the threshold is set too low, the mask will be too large. Alternatively, a fuzzy mask can be utilized to represent the relative importance of pixels on the basis of vegetation coverage. To implement the fuzzy mask in PCA, the mathematical model needs to be altered. Construction of fuzzy masks is also a necessary task in the implementation of fuzzy mask PCA (FMPCA). This paper will introduce a new way to implement FMPCA. The method will be validated using a case study of mapping locations of felsic igneous intrusions, an essential factor for Sn Cu mineralization, using stream sediment geochemical data in the Gejiu mineral district in southern Yunnan, China. 2. Fuzzy mask PCA Using multivariate statistical analysis methods such as PCA to characterize spatial patterns and extract components often requires selection of samples that are representative of certain populations. For example, if the study is mainly for characterizing certain rock types, pixels should be chosen from the areas where these types of rocks are likely to occur. The pixels in the areas where the rock types are known not to occur should be eliminated from the PCA model in order to reduce the influence of noise from irrelevant pixels and a binary mask is often used to deal with this situation. Most commercial GIS and image processing packages have two standard methods for selecting pixels. The first method is to set a rectangular window so that processing only applies to pixels within the window. The other method sets a polygonal mask, again restricting processing to pixels within the mask. Both of these methods use a binary mask with two values: 1 and 0. In the current paper, this idea will be extended to a more general form by weighting the contributions of the samples (or pixels) with a spatial weighting factor. A brief introduction to the principles of the method is given next. Let A, B, and C be the images (variables) to be processed, and their values at location (i,j) are A ij, B ij, and C ij, respectively. A weighting factor is defined as an image W (variable) with values from 0 to 1, with 0rW ij r1(cheng, 2000). The weighted correlation coefficient between images A and B is defined as (Cheng, 2000) P Wij ða ij AÞðB ij BÞ RðA, BÞ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi P qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Wij ða ij AÞ 2 P ð1þ Wij ðb ij BÞ 2 where A and B are the weighted mean values of A ij and B ij (A ¼ P W ij A ij = P W ij, B ¼ P W ij B ij = P W ij ). The effect of applying the weight W to the correlation coefficient canbeseenfromthe following properties: 1. R(A,B) is a symmetrical correlation index, 2. 1rR(A,B)r1, and R(A,B)¼1 or 1 if and only if A and B have a positive or negative linear relationship, 3. if W ij ¼constant, R(A,B) reduces to the ordinary correlation coefficient, and 4. if W ij is a binary image with two values 1 and 0, then W is equivalent to the ordinary mask. The first three properties ensure that definition (Eq. (1)) meets the basic properties required for a correlation coefficient. The last two properties demonstrate that definition Eq. (1) is a generalized form compared with the ordinary correlation coefficient and the correlation coefficient with a binary mask applied. The ordinary mask treatment becomes a special case of the spatial weighting approach. Similarly, the eigenvalues, eigenvectors, and loadings of each image on all components can be calculated from the weighted correlation (covariance) matrix. Unlike traditional mask PCA, which excludes samples from not only the calculation of loadings and components but also from calculation of scores, the new method only excludes the samples from the calculation of loadings and components but not from calculation of scores. Spatial weights are only applied to the determination of the eigenvectors. After eigenvectors determined then the scores of sample on the eigenvectors are calculated as the projection of the original variable values on the eigenvectors. In this step, the scores of the images are calculated without further use of spatial weights. For example, in a special case if ordinary mask (with 1 or 0 values) applied to PCA, the principle components (loadings and components) are calculated according to the correlation (covariance) matrix constructed only using a subset of samples with weighting factor value 1. Therefore the components are contributed only by samples with weighting factor value 1. Based on the component factors one can calculate the projected values (scores) of all samples including those with weighting factor value 0 and not being included in the calculation of components. This way an entire scores map can be constructed. New method has been implemented in GeoDAS GIS (Cheng, 2000). Using GeoDAS, one can define a weighting image with values representing the relative importance of pixel locations. The values on the weighting image can be, for example, the distance from ore deposits, density of ore deposits, distance from contacts or faults, concentration values of trace elements, etc. The scores of images on a principal component could also be taken as a weighting factor. The spatially weighted PCA or FMPCA enhances the influence of pixels with large weights (weights close to 1) and reduces the effect of pixels with small weights (weights close to 0). This method has been used to analyze geochemical anomalies for mineral exploration (Bonham-Carter and Cheng, 2001). In the current paper, the method will be used in a case study of mapping locations of igneous rocks related to Sn Cu mineralization using stream sediment geochemical data in the Gejiu mineral district in southern Yunnan, China. 3. Defining the fuzzy mask Technically, defining a fuzzy mask on the basis of pixel location can be done using some basic GIS functions. For example, a fuzzy mask can be defined according to the distance to objects of special interest such as environmental impact sites or mineral deposits. If the study is related to such distances, then a fuzzy mask can be constructed according to the distance from these point objects. The closer the pixels to these point sites, the closer the weights are to 1, and the farther the pixels are from these sites, the closer the weights are to 0. Certain decay functions can be applied to assign fuzzy weight values to distance. A fuzzy mask can also be defined according to a measure of density, such as deposit density and alteration intensity. Pixels located in an area with a high density of mineral deposits or high intensity of alteration can be given high

3 Q. Cheng et al. / Computers & Geosciences ] (]]]]) ]]] ]]] 3 weights, and those located in areas with a low density of mineral deposits or alteration intensity can be given low weights. Besides defining fuzzy masks with respect to images, fuzzy mask PCA can be applied to point sample data, with sample weights being defined for point features instead of pixels. GeoDAS provides two modes for implementing FMPCA: image and point modes, with either images or tables of attribute data, respectively, being used to run PCA. One advantage of using images as inputs is that the image resolution does not have to be uniform; another is that missing values for some variables, so often a problem in PCA of table data, can first be dealt with by interpolation of each variable to a grid. 4. Mapping the locations of igneous rocks related to Sn Cu mineralization in Gejiu mineral district, Yunnan, China The area chosen for this study was the Gejiu mineral district located in southern Yunnan, approximately 200 km south of the city of Kunming, the capital of Yunnan Province (Fig. 1). The area is known for its world-class Sn mineral deposits and Sn production. A location of study area and a simplified geology map is shown in Fig. 1. The geological units in the main study area consist primarily of a sequence of Paleozoic Mesozoic sedimentary (Gejiu Formation and other formations) and igneous rocks. The Proterozoic lowgrade metamorphic sand-shale rocks are mainly distributed in the southern part of the study area. The Paleozoic strata of carbonate sedimentary rocks are well developed with an extensive coverage in the study area. Mesozoic strata are poorly developed with a limited coverage. Tertiary strata are also scattered throughout the study area (Zhuang et al., 1996). Two main types of igneous rocks are mapped: Paleozoic volcanic rocks and Mesozoic intrusive rocks. The former are mainly basalts, including the Ailaoshan basalts extensively spread over the study area (Zhuang et al., 1996). Mesozoic igneous rocks are mainly intrusive rocks. The intrusions are dominated by granite and granitoid rocks including biotitic granite, biotitic monzogranite and biotitic plagiogranite. Mafic and ultramafic intrusive rocks are scattered throughout the study area. There are several batholith outcrops in the area. The Gejiu Batholith is a controlling factor for Fig. 1. Geology of study area is simplified from four map sheets of 1:200,000 scales (308 Geological Exploration Team, 1984). Legends: 1 pink color represents Proterozoic metamorphic rocks, 2 yellow color for Gejiu Formation of Paleozoic carbonate sedimentary rocks, 3 grey color for other sedimentary rocks, 4 brown for mafic igneous rocks including basalts, 5 red color for felsic intrusive rocks, 6 black triangles and green circles, represent Sn and Cu mineral deposits and mineral occurrences, and 7 solid lines for faults systems, respectively. Star in the inset China map indicates location of study area. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article). Sn mineralization, and several large Sn mineral deposits are located near the Gejiu Batholith. The Gejiu Batholith is a granitoid complex located in the center of the study area with an outcrop area of about 450 km 2. The Gejiu Formation is dominated by limestone with minor dolomites and serves as the main country rock hosting most of the discovered Sn deposits. The area has a long history of tectonic and complex structural activity resulting in fault and fold systems at various scales. The main faults and folds in the central parts of the study area have N S and E W orientations. The main orientations of the faults and folds in the western and eastern parts of the study area strike NE SW or NW SE. These fault systems control the general configuration of the mineralization and distribution of ore bodies in the area. The main trend of the mineralization is in NNE SSW orientation in the central area, but the ore fields are concentrated along the intersections of NNE SSW and E W faults. The mineralization is associated with sedimentary country rocks (Gejiu Formation), structures and igneous activities. The Sn polymetallic system of mineralization in the area is believed to be related to Mesozoic intrusions including the granitoid Gejiu Batholith and Bainiuchang Batholith, which intruded into the folded limestone deposited during the Middle Triassic (e.g., the Gejiu Formation). Enrichment of Sn, Cu and other metals occurred in and near the contact zone between the granite and wall rock through metamorphism, contact-metasomatic and filling processes. The types of ores found in the contact zone or in the wall rock surrounding the batholith include skarn ores, interlayer ores in the wall rocks, vein-type ores in the fractures and placer ores on the paleosurface. Although granite intrusion-related mineralization is not the only possible model for the concentration of Sn, the influence of intrusions (e.g., the Gejiu Batholith) on mineralization is indisputable. Delineation and characterization of the contacts of batholiths are therefore important for Sn mineral exploration. The general geology and mineral deposits of the study area have been described previously by Southwest Metallurgy Exploration Company (1984), Yu et al. (1988) and Zhuang et al. (1996). The Gejiu area has been explored for Sn and Cu over the entire last century, and significant exploration has been performed by local geologists since the 1950s. This exploration led to the discovery of several large mineral deposits, including the Laochang and Kafang deposits in the central part of the study area. The area has become well known throughout the world as a Sn mineral province. The area chosen for this study is covered by about 7349 evenly distributed stream sediment samples, each covering a 2 km 2km (4 km 2 ) area. The samples were collected and analyzed by the Chinese National Geochemical Mapping Project as part of the Regional Geochemistry National Reconnaissance (RGNR) Project, which was initiated in 1979 (Xie et al., 1997). For each sample, the concentrations of 39 geochemical elements and 7 oxides were measured. The data used in this paper are geochemical concentration values of 7 major oxides. Further details about the sampling and analysis of the stream sediment data can be found in Xie et al. (1997). The trace elements and their associations with Sn mineralization in the area were previously studied (Cheng, 2007; Cheng and Agterberg, 2009; Cheng et al., 2009a, b, 2010). The main objective of the current study is to delineate felsic intrusions, including unknown intrusions, using the stream sediment geochemical data. The Chinese literature includes substantial research and exploration efforts on the mineralogy, petrology, geochemistry and isotope geochemistry of the Gejiu area. An excellent study using a geophysical model for mineral exploration was carried out by Xiong and Shi (1994), who summarized the main geophysical properties of various rock types in the Gejiu area and proposed a geophysical-geological model for applying geophysical survey techniques to map igneous intrusions and for delineating areas

4 4 Q. Cheng et al. / Computers & Geosciences ] (]]]]) ]]] ]]] for mineral exploration. Cheng et al. (2009a, b) used trace element data and geophysical data (gravity and aeromagnetic data) to interpret the subsurface extensions of the Gejiu Batholith. This study demonstrates how to use fuzzy mask PCA to process the stream sediment geochemical data to map felsic intrusions. To reduce the compositional closure of geochemical data, the following log-ratio transformations (e.g. Aitchison, 1986) were applied prior to PCA. 0 1 Xij ¼ lnx ij 1=7 X7 A ð2þ j ¼ 1 X ij where X ij is the concentration of element oxide j for sample i, and X * ij is the log-ratio transformed data. The correlation coefficients of the transformed elements are shown in Table 1. Ordinary PCA was applied to the log-ratio transformed data. Seven principal components (PCs), their component variances (eigenvalues), and the loadings of elements on each of the first three components were calculated and are shown in Fig. 2(A), Table 2 and Fig. 2(B) (D), respectively. The first three components account for about 80% of the total variance of the data. The general element loadings on the first three components show that they mainly represent different rock types. For example, both loadings and the spatial distribution of the component scores suggest that the first two components (Fig. 2B and C) mainly represent the sedimentary rocks, and metamorphic and volcanic rocks. Since these components are not the main interests of the study the results are not shown here. The third component (Fig. 2D), that accounts for about 10% of the total variance, is dominated by Na 2 O, K 2 O and Al 2 O 3, which have negative loadings, and CaO, which has a positive loading. This component mainly represents felsic intrusions. The distributions of negative scores on the third component are shown in Fig. 3. The areas with large negative score values represent not only the mapped intrusions on the ground but also areas with possible unknown intrusions, including buried intrusions. From these results one can see that the third component mainly representing the igneous intrusions only accounts about 10% of the total variance, whereas the first and the second components account for 43% and 28%, respectively. This confirms that the main rock types in terms of area proportion of the total study area are sedimentary rock and metamorphic and volcanic rocks. Since the main goal of this study is to characterize and to identify spatial distribution of igneous intrusions, it is necessary to enhance the relative importance of the third component and reduce the influences of the other rock types. Therefore, weighting functions can be applied to the samples according to their relationship to igneous intrusions. Several masks were defined and accordingly weighting functions were used as fuzzy masks in spatially weighted PCA. First a binary mask gives all samples fall in the outcropped igneous intrusions weight 1 and rest of the samples weight 0. The effect of the binary mask is that the samples with zero weight are not included in the calculation of the covariance matrix and the loadings. Considering the influence of igneous intrusions, especially shallowly buried intrusions, may be beyond the mapped outcrops, the areas closer to intrusions are more likely to be Table 1 Correlation coefficients of log-ratio transformed element oxides. Si 2 O Na 2 O MgO K 2 O Fe 2 O 3 CaO Al 2 O 3 Si 2 O Na 2 O MgO K 2 O Fe 2 O CaO Al 2 O Table 2 Loadings on principal components. Component Loading PCA1 PCA2 PCA3 Si 2 O Na 2 O MgO K 2 O Fe 2 O CaO Al 2 O Loadings on PCA1 Si 2 O K 2 O Na 2 O MgO 5 6 CaO 7 FeO Al 2 O 3 Loadings on PCA2 Loadings on PCA3 Na 2 O MgO K 2 O CaO Na 2 O K 2 O FeO Al 2 O 3 Si 2 O FeO Al 2 O 3 1 Si 2 O MgO 6 CaO Fig. 2. Results obtained by ordinary PCA to seven log-ratio transformed element oxides. (A) Eigenvalues corresponding to seven principal components. Vertical scale is logarithmic; and (B) (D) Loadings on first, second and third components, respectively. Loadings plotted in (D) are reversed for comparison.

5 Q. Cheng et al. / Computers & Geosciences ] (]]]]) ]]] ]]] 5 Fig. 3. Scores of samples on third principal component. Areas with negative high values of scores or yellow to red coloured areas highlight locations of felsic intrusions. Black polygons are mapped intrusions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article). Considering that the influence of heat from intrusions usually decays exponentially as distance from the intrusions, here we suggest the use of power-law functions to define weighting factors. As shown in Fig. 4, the weighting functions are expressed as powerlaw functions of the distance d from the sample (in pixels) to the intrusions: w¼(1 d/66) b, where d is the distance from the intrusions with a maximum range of 66 km and b is the exponent determining the decay rate of the power-law function. For example, the following five decay functions were defined with different exponents: (1) a constant function with w¼1 and b¼0; (2) a linear decay function with value 0rwr1 and b¼1; (3) and (4) a nonlinear function from 1 to 0 following power-law relations and b¼8 and 16, respectively; and (5) a binary function where 1 represents being within an intrusion and 0 represents being outside of an intrusion, which corresponds to b¼n. As the value of b increases, the weighting function for the intensity of the influence of intrusions increases. These weighting functions are used in FMPCA to calculate the correlation coefficient matrix and then the eigenvalues and component loadings. Therefore, the eigenvalues and loadings of elements on principal components are influenced by the mask. However, using these components to recalculate the projected scores of the samples on these components will no longer use the mask. The use of the mask affects only the determination of the combination of components (loadings) and their relative importance (eigenvalues) and does not affect the calculation of scores. This effect is the main difference between the fuzzy mask PCA introduced in the current paper and the ordinary mask used in traditional image processing. In traditional image processing, the mask removes samples from all aspects of data processing. The results obtained using the binary masks are shown in Figs Fig. 4(A) shows the weighting functions, and Fig. 4(B) displays the eigenvalues of FMPCA with various weighting functions applied to the samples. Fig. 5 shows the loadings of elements on the third component (the felsic intrusions component). Fig. 6 displays the distributions of scores of samples on the third component (Fig. 5). The variances (eigenvalues) of the third component with or without the masks are significantly different. For example, the second and third eigenvalues calculated without a mask is 2.01 and 0.78, respectively, but the second and third eigenvalues obtained with a binary mask become 1.56 and 1.35, respectively (Fig. 4b). The eigenvalues of the third component was increased; those of the second component decreased, but first components calculated with the binary mask were slightly reduced. This result indicates that the main influence of the binary mask is an increase in the third component and decrease in the first two main components especially the second component, which confirms that the third component is associated with felsic intrusions. The eigenvalues calculated using linear weighting Fig. 4. (A) Weighting functions defined with distance from intrusions as variable. Functions are, w¼(1 d/66) b, where d is distance from intrusions, and b¼0, 1, 8, 16, N, respectively. (B) Eigenvalues obtained using SWPCA with weighting functions as shown in (A). Na 2 O Loadings on PCA3 K 2 O Al 2 O 3 affected by intrusions than areas farther away. Therefore, the distance from intrusions could be a factor to represent the relative importance of samples in support of identifying both known and shallowly buried igneous intrusions. A weighting scheme was taken to define weighting functions on the basis of distance from intrusions. For comparison, we used the no mask case and a binary mask case as references to evaluate the effectiveness of linear and non-linear distance-based weighting functions. Si 2 O MgO Fe 2 O 3 CaO Fig. 5. Loadings on third principal component calculated using SWPCA with various weighting functions defined in (A).

6 6 Q. Cheng et al. / Computers & Geosciences ] (]]]]) ]]] ]]] Fig. 6. Scores of samples on third principal component calculated using SWPCA with binary mask. Areas with negative high values of scores or yellow to red coloured areas highlight locations of felsic intrusions. Black polygons are mapped intrusions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article). Fig. 7. Scores of samples on third principal component calculated using non-linear weighting function (b¼16). Areas with negative high values of scores or yellow to red coloured areas highlight locations of felsic intrusions. Black polygons are mapped intrusions. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article). functions are similar to those calculated without a mask, implying that the linear weighting function has an insignificant influence on the results. The loadings of the elements on the third component calculated with non-linear weighting functions show systematic changes, and even the signs of the loadings are changed. For example, the loading signs of Fe 2 O 3 and CaO changed between the mask-less PCA and the binary mask PCA. The composites of third components calculated with or without fuzzy masks are Na 2 O, K 2 O and Al 2 O 3 with positive loadings and CaO, MgO, Fe 2 O 3 and Si 2 O with negative loadings. The reason for Si 2 O showing negative loading might be due to multiple sources such as granites and sedimentary rocks (silt and sand stones and river fluvial deposits). The composites of third component clearly indicate the factor of granitic intrusions. In term of influence of fuzzy masks we can further check the relative changes of loadings of elements on the third component in the following sections. The spatial distribution of scores on the third component calculated using a binary mask (Fig. 6) shows that the areas with high scores represent the locations of felsic intrusions. The results obtained using a binary mask highlight most of the known intrusions and some areas in the metamorphic rocks (Figs. 3 and 6). The coverage of the high scores is relatively small compared with that delineated on the scores obtained using the ordinary PCA without a mask. In order to compare the effectiveness of other types of weighting factors defined and shown in Fig. 4(A), FMPCA was applied to the same log-ratio transformed data. The loadings obtained with and without weighting factors are shown in Fig. 6. When the decay intensity of the weighting function is increased from a constant function (b¼0) to more rapid decay rates (b¼8, 16, N), the loadings increase positively for K 2 O, Na 2 O and Al 2 O 3, increase negatively for Si 2 O, MgO and Fe 2 O 3, and decrease for CaO (Fig. 4A). These changes demonstrate that the non-linear weighting functions defined in Fig. 4(A) enhance the patterns indicating felsic intrusions and reduce the influence of sedimentary carbonate rocks. The scores on the third component (Fig. 7) calculated using the non-linear weighting functions also confirm that enhanced scores are mainly in the northern part of the study area where the sedimentary rocks of the Gejiu Formation dominate. The differences between the scores calculated with or without the non-linear mask (Fig. 8) illustrate that in the northern areas scores are enhanced, whereas in the rest of study area, the scores are negative or depleted. Therefore, SWPCA with a non-linear weighting function can provide more information for delineation of intrusions, especially in areas with the sedimentary rocks of the Fig. 8. Differences between scores on third principal component calculated using non-linear weighting function (b¼16) and using ordinary PCS without mask. Black polygons are mapped intrusions. Gejiu Formation. This is significant for mineral exploration in the area since felsic intrusions in the Gejiu Formation are essential for Sn mineralization. Most of the large Sn mineral deposits have been found in the outer contact zones of the intrusions and the Gejiu Formation. To interpret the spatial associations between the score distributions and other geological features, a comprehensive map was created with scores as background and other geological features superimposed (Fig. 9). The top graph in Fig. 9 shows the high score values in yellow to red colors. These areas mainly represent intrusions. Most areas with high scores are surrounded by high fault density, but these faults do not pass through these intrusions. Most Sn minerals occur in the fault systems close to intersections with the intrusions. When dragged over the digital elevation model (DEM) at 90 m resolution, these intrusions generally show positive topographical responses (bottom graph in Fig. 9). The 3D graph was created using ArcGIS 3D Analyst with colors defined according to the values of scores as seen in the top graph and the ground elevation from DEM. 5. Discussion and conclusions It has been shown that spatially weighted PCA provides a new way of implementing PCA with constraints on samples according to

7 Q. Cheng et al. / Computers & Geosciences ] (]]]]) ]]] ]]] 7 Fig. 9. Top graph: superimposed with faults (black lines), Sn and Cu mineral deposits (black triangles and light blue dots), and intrusions as polygons. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article). Bottom graph: scores map dragged over digital elevation model DEM. Black lines are faults and polygons are intrusions. DEM data is in 90 m resolution. their spatial properties in relation to the objective of study. The definition and construction of weighting functions are flexible depending upon the properties of samples. Also, fuzzy masks are affected subjectively by user-defined weighting functions based on opinions or experience. Since the choice of weighting functions are so flexible it is impossible to thoroughly evaluate which weighting functions will be the most effective. One recommendation is to define weighting functions based on physical meaning and the results should be compared. In the case study introduced in this paper, we defined a set of distance-based weighting functions and their effectiveness was compared by investigating the changes on relative importance of components (eigenvalues), contribution of variables (loadings) and spatial distribution of scores. These are only some examples of how the results can be evaluated. Application of a distance-based fuzzy mask in conjunction with PCA in this study generated some interesting results showing good fit to the known outcropped igneous intrusions, but more importantly the potential distributions of igneous intrusions especially within the Gejiu formation of sedimentary rocks. Although these results still need field verification by mapping and/or drilling in the case of hidden intrusions, the results have already provided guidelines for further mapping. Some of the results have been used by the local geological survey for planning further mapping and mineral prospecting activities. This study has demonstrated that FMPCA can utilize weighting functions to represent the relative representativeness of samples or pixels. These weighting functions affect the calculation of the correlation coefficient matrix. The variances (eigenvalues) and loadings of components are calculated from the adjusted correlation coefficient matrix. Scores of samples or pixels are calculated based on the principal components. The FMPCA was applied to the case study of mapping intrusions in Gejiu mineral district. Various weighting functions were constructed based on the inverse distance from mapped intrusions and validated by creating components for mapping locations of felsic intrusions, especially unknown intrusions. The defined non-linear decay functions were effective for enhancing the third component for mapping felsic intrusions. This technique enhanced the variance (eigenvalue) of the component, decreased the loadings of CaO, and increased the loadings for compounds associated with felsic intrusions (Si 2 O, Na 2 O, MgO, Fe 2 O 3 and Al 2 O 3 ). This method is expected to become a common tool for analyzing geochemical data and other types of images. Factors such as inverse values of CaO, gravity field values, inverse scores of other components, and the density of faults can be used as weighting functions. Application of statistical methods such as multivariate statistical analysis often involves experiments design and sampling. Some of the experimental designs are control designs so that the samples collected have high quality and can be used equally in statistical analysis. But in other cases when the design involves natural experiments with uncontrolled factors and the sample collections are constrained so that some of the samples may not be representative. In this case, one chooses a subset of samples and removes the rest from the subsequent analysis. In general, if samples are representative of the population of study then these samples should be included in statistical analysis. One of the main advantages of using standard statistical analysis methods is its objective results. However, in many real-world applications, sample collections are limited and constrained by factors such as availability and costs. The samples may have variable quality, for instance, some samples with poor data quality or outliers have to be filtered from those used in the statistical analysis. This is the reason that in practical work exploratory data analysis is often needed to explore the data before they can be used in multivariate analysis. Besides the data quality of samples, representativeness of samples for the primary objective is also a consideration central to statistical analysis. The processes of selecting samples can be objective or subjective depending upon modeler s knowledge about the samples. The results are influenced by the determination of the samples. To utilize properties of samples in statistical analysis is essential for enhancing the results. The fuzzy mask method introduced in this paper provides an option for users to define weights to represent the relative representativeness of samples in statistical analysis. It is true that weights defined and applied to each sample will alter the calculation of the PCA, so that the results are influenced by how weights are defined. This might be one of the disadvantages of using fuzzy concepts in statistics: for example, the results obtained by fuzzy mask PCA are indeed influenced by the weighting function. This type of method should be used with caution and the weighting functions must be defined meaningfully. Acknowledgements The authors sincerely thank the two anonymous reviewers for their constructive comments which have improved the manuscript. The research was financially supported by a Distinguished Young Researcher Grant ( ), a Strategic Research Grant ( ) awarded by the Natural Science Foundation of China, a High-Tech Research and Development Grant (2009AA06Z110, 2008AA121103) by the Ministry of Science and Technology of China and Grants from Ministry of Education of China (Nos. IRT0755 and ). References Aitchison, J., The statistical analysis of compositional data. Chapman and Hall, London, New York, 416 pp.

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