In this paper, we introduce a notion of a self-similar action of a group $G$ on a $k$-graph $\Lambda $ and associate it a universal C$^\ast $-algebra ${{\mathcal{O}}}_{G,\Lambda }$. We prove that ${{\mathcal{O}}}_{G,\Lambda }$ can be realized as the Cuntz–Pimsner algebra of a product system. If $G$ is amenable and the action is pseudo free, then ${{\mathcal{O}}}_{G,\Lambda }$ is shown to be isomorphic to a “path-like” groupoid C$^\ast $-algebra. This facilitates studying the properties of ${{\mathcal{O}}}_{G,\Lambda }$. We show that ${{\mathcal{O}}}_{G,\Lambda }$ is always nuclear and satisfies the universal coefficient theorem; we characterize the simplicity of ${{\mathcal{O}}}_{G,\Lambda }$ in terms of the underlying action, and we prove that, whenever ${{\mathcal{O}}}_{G,\Lambda }$ is simple, there is a dichotomy: it is either stably finite or purely infinite, depending on whether $\Lambda $ has nonzero graph traces or not. Our main results generalize the recent work of Exel and Pardo on self-similar graphs.

No abstract

Predictive modelling of mineral prospectivity, a critical, but challenging procedure for delineation of undiscovered prospective targets in mineral exploration, has been spurred by recent advancements of spatial modelling techniques and machine learning algorithms. In this study, a set of machine learning methods, including random forest (RF), support vector machine (SVM), artificial neural network (ANN), and a deep learning convolutional neural network (CNN), were employed to conduct a data-driven W prospectivity modelling of the southern Jiangxi Province, China. A total of 118 known W occurrences derived from long-term exploration of this brownfield area and eight evidential layers of multi-source geoscience information related to W mineralization constituted the input datasets. This provided a data-rich foundation for training machine learning models. The optimal configuration of model parameters was trained by a grid search procedure and validated by 10-fold cross-validation. The resulting predictive models were comprehensively assessed by a confusion matrix, receiver operating characteristic curve, and success-rate curve. The modelling results indicate that the CNN model achieves the best classification performance with an accuracy of 92.38%, followed by the RF model (87.62%). In contrast, the RF model outperforms the rest of ML models in overall predictive performance and predictive efficiency. This is characterized by the highest value of area under the curve and the steepest slope of success-rate curve. The RF model was chosen as the optimal model for mineral prospectivity in this region as it is the best predictor. The prospective zones delineated by the prospectivity map occupy 9% of the study area and capture 66.95% of the known mineral occurrences. The geological interpretation of the model reveals that previously neglected Mn anomalies are significant indicators. This implies that enrichment of ore-forming material in the host rocks may play an important role in the formation process of wolframite and can represent an innovative exploration criterion for further exploration in this area.

IntroductionMammographic density is similar among women at risk of either sporadic or BRCA1/2-related breast cancer. It has been suggested that digitized mammographic images contain computer-extractable information within the parenchymal pattern, which may contribute to distinguishing between BRCA1/2 mutation carriers and non-carriers.MethodsWe compared mammographic texture pattern features in digitized mammograms from women with deleterious BRCA1/2 mutations (n = 137) versus non-carriers (n = 100). Subjects were stratified into training (107 carriers, 70 non-carriers) and testing (30 carriers, 30 non-carriers) datasets. Masked to mutation status, texture features were extracted from a retro-areolar region-of-interest in each subject’s digitized mammogram. Stepwise linear regression analysis of the training dataset identified variables to be included in a radiographic texture analysis (RTA) classifier model aimed at distinguishing BRCA1/2 carriers from non-carriers. The selected features were combined using a Bayesian Artificial Neural Network (BANN) algorithm, which produced a probability score rating the likelihood of each subject’s belonging to the mutation-positive group. These probability scores were evaluated in the independent testing dataset to determine whether their distribution differed between BRCA1/2 mutation carriers and non-carriers. A receiver operating characteristic analysis was performed to estimate the model’s discriminatory capacity.ResultsIn the testing dataset, a one standard deviation (SD) increase in the probability score from the BANN-trained classifier was associated with a two-fold increase in the odds of predicting BRCA1/2 mutation status: unadjusted odds ratio (OR) = 2.00, 95% confidence interval (CI): 1.59, 2.51, P = 0.02; age-adjusted OR = 1.93, 95% CI: 1.53, 2.42, P = 0.03. Additional adjustment for percent mammographic density did little to change the OR. The area under the curve for the BANN-trained classifier to distinguish between BRCA1/2 mutation carriers and non-carriers was 0.68 for features alone and 0.72 for the features plus percent mammographic density.ConclusionsOur findings suggest that, unlike percent mammographic density, computer-extracted mammographic texture pattern features are associated with carrying BRCA1/2 mutations. Although still at an early stage, our novel RTA classifier has potential for improving mammographic image interpretation by permitting real-time risk stratification among women undergoing screening mammography.Electronic supplementary materialThe online version of this article (doi:10.1186/s13058-014-0424-8) contains supplementary material, which is available to authorized users.

No abstract

No abstract

No abstract

Purpose Radiomic texture analysis is typically performed on images acquired under specific, homogeneous imaging conditions. These controlled conditions may not be representative of the range of imaging conditions implemented clinically. We aim to develop a two‐stage method of radiomic texture analysis that incorporates the reproducibility of individual texture features across imaging conditions to guide the development of texture signatures which are robust across mammography unit vendors. Methods Full‐field digital mammograms were retrospectively collected for women who underwent screening mammography on both a Hologic Lorad Selenia and GE Senographe 2000D system. Radiomic features were calculated on manually placed regions of interest in each image. In stage one (robustness assessment), we identified a set of nonredundant features that were reproducible across the two different vendors. This was achieved through hierarchical clustering and application of robustness metrics. In stage two (classification evaluation), we performed stepwise feature selection and leave‐one‐out quadratic discriminant analysis (QDA) to construct radiomic signatures. We refer to this two‐state method as robustness assessment, classification evaluation (RACE). These radiomic signatures were used to classify the risk of breast cancer through receiver operator characteristic (ROC) analysis, using the area under the ROC curve as a figure of merit in the task of distinguishing between women with and without high‐risk factors present. Generalizability was investigated by comparing the classification performance of a feature set on the images from which they were selected (intravendor) to the classification performance on images from the vendor on which it was not selected (intervendor). Intervendor and intravendor performances were also compared to the performance obtained by implementing ComBat, a feature‐level harmonization method and to the performance by implementing ComBat followed by RACE. Results Generalizability, defined as the difference between intervendor and intravendor classification performance, was shown to monotonically decrease as the number of clusters used in stage one increased (Mann–Kendall P < 0.001). Intravendor performance was not shown to be statistically different from ComBat harmonization while intervendor performance was significantly higher than ComBat. No significant difference was observed between either of the single methods and the use of ComBat followed by RACE. Conclusions A two‐stage method for robust radiomic signature construction is proposed and demonstrated in the task of breast cancer risk assessment. The proposed method was used to assess generalizability of radiomic texture signatures at varying levels of feature robustness criteria. The results suggest that generalizability of feature sets monotonically decreases as reproducibility of features decreases. This trend suggests that considerations of feature robustness in feature selection methodology could improve classifier generalizability in multif...