We describe Fibres of Failure (FiFa), a method to classify failure modes of predictive processes using the Mapper algorithm from Topological Data Analysis. Our method uses Mapper to build a graph model of input data stratified by prediction error. Groupings found in high-error regions of the Mapper model then provide distinct failure modes of the predictive process. We demonstrate FiFa on misclassifications of MNIST images with added noise, and demonstrate two ways to use the failure mode classification: either to produce a correction layer that adjusts predictions by similarity to the failure modes; or to inspect members of the failure modes to illustrate and investigate what characterizes each failure mode.
The method builds on , an algorithm from Topological Data Analysis that constructs a graph (or simplicial complex) model of arbitrary data. has had success in a wide range of application areas, from medical research studying cancer, diabetes, asthma and many more topics @cite_57 @cite_2 @cite_11 @cite_23 , genetics and phenotype studies @cite_5 @cite_7 @cite_50 @cite_47 @cite_59 , to hyperspectral imaging, material science, sports and politics @cite_14 @cite_15 @cite_54 @cite_18 . Of note for our approach are in particular the contributions on cancer, diabetes and fragile X syndrom @cite_57 @cite_5 @cite_2 where was used to extract new subgroups from a segmentation of the input space.
@cite_18 @cite_14 @cite_7 @cite_15 @cite_54 @cite_57 @cite_23 @cite_2 @cite_50 @cite_5 @cite_47 @cite_59 @cite_11
1989757660 1486210143 1808652302 1965973321
This paper applies topological methods to study complex high dimensional data sets by extracting shapes (patterns) and obtaining insights about them. Our method combines the best features of existing standard methodologies such as principal component and cluster analyses to provide a geometric representation of complex data sets. Through this hybrid method, we often find subgroups in data sets that traditional methodologies fail to find. Our method also permits the analysis of individual data sets as well as the analysis of relationships between related data sets. We illustrate the use of our method by applying it to three very different kinds of data, namely gene expression from breast tumors, voting data from the United States House of Representatives and player performance data from the NBA, in each case finding stratifications of the data which are more refined than those produced by standard methods. This paper tackles the problem of computing topological invariants of geometric objects in a robust manner, using only point cloud data sampled from the object. It is now widely recognised that this kind of topological analysis can give qualitative information about data sets which is not readily available by other means. In particular, it can be an aid to visualisation of high dimensional data. Standard simplicial complexes for approximating the topological type of the underlying space (such as Cech, Rips, or a-shape) produce simplicial complexes whose vertex set has the same size as the underlying set of point cloud data. Such constructions are sometimes still tractable, but are wasteful (of computing resources) since the homotopy types of the underlying objects are generally realisable on much smaller vertex sets. We obtain smaller complexes by choosing a set of 'landmark' points from our data set, and then constructing a "witness complex" on this set using ideas motivated by the usual Delaunay complex in Euclidean space. The key idea is that the remaining (non-landmark) data points are used as witnesses to the existence of edges or simplices spanned by combinations of landmark points. Our construction generalises the topology-preserving graphs of Martinetz and Schulten [MS94] in two directions. First, it produces a simplicial complex rather than a graph. Secondly it actually produces a nested family of simplicial complexes, which represent the data at different feature scales, suitable for calculating persistent homology [ELZ00, ZC04]. We find that in addition to the complexes being smaller, they also provide (in a precise sense) a better picture of the homology, with less noise, than the full scale constructions using all the data points. We illustrate the use of these complexes in qualitatively analyzing a data set of 3 × 3 pixel patches studied by David [LPM03]. Objective To summarize literature describing approaches aimed at automatically identifying patients with a common phenotype. @PARASPLIT Materials and methods We performed a review of studies describing systems or reporting techniques developed for identifying cohorts of patients with specific phenotypes. Every full text article published in (1) Journal of American Medical Informatics Association , (2) Journal of Biomedical Informatics , (3) Proceedings of the Annual American Medical Informatics Association Symposium , and (4) Proceedings of Clinical Research Informatics Conference within the past 3 years was assessed for inclusion in the review. Only articles using automated techniques were included. @PARASPLIT Results Ninety-seven articles met our inclusion criteria. Forty-six used natural language processing (NLP)-based techniques, 24 described rule-based systems, 41 used statistical analyses, data mining, or machine learning techniques, while 22 described hybrid systems. Nine articles described the architecture of large-scale systems developed for determining cohort eligibility of patients. @PARASPLIT Discussion We observe that there is a rise in the number of studies associated with cohort identification using electronic medical records. Statistical analyses or machine learning, followed by NLP techniques, are gaining popularity over the years in comparison with rule-based systems. @PARASPLIT Conclusions There are a variety of approaches for classifying patients into a particular phenotype. Different techniques and data sources are used, and good performance is reported on datasets at respective institutions. However, no system makes comprehensive use of electronic medical records addressing all of their known weaknesses. In a time of information glut, observations about complex systems and phenomena of interest are available in several applications areas, such as biology and text. As a consequence, scientists have started searching for patterns that involve interactions among the objects of analysis, to the effect that research on models and algorithms for network analysis has become a central theme for knowledge discovery and data mining (KDD). The intuitions behind the plethora of approaches rely upon few basic types of networks, identified by specific local and global topological properties, which we term "pure" topology types.In this paper, (1) we survey pure topology types along with existing sampling algorithms that generate them, (2) we introduce novel algorithms that enhance the diversity of samples, and address the case of cellular topologies, (3) we perform statistical studies of the stability of the properties of pure types to alternative generative algorithms, and a joint study of the separability of pure types, in terms of their embedding in a space of metrics for network analysis, widely adopted in the social and physical sciences.We conclude with a word of caution to the practitioners, who sample pure topology types to assess the "statistical significance" of their findings, e.g., the p-value of the clustering coefficient is sensitive to the sampling algorithm used. We find that different pure types share similar topological properties. Further, real world networks hardly present the variability profile of a single pure type. We suggest the assumption of "mixtures of types" as an alternative starting point for developing models and algorithms for network analysis.