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
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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. Abstract Analytical chemistry is rapidly changing. Indeed we acquire always more data in order to go ever further in the exploration of complex samples. Hyperspectral imaging has not escaped this trend. It quickly became a tool of choice for molecular characterisation of complex samples in many scientific domains. The main reason is that it simultaneously provides spectral and spatial information. As a result, chemometrics has provided many exploration tools (PCA, clustering, MCR-ALS …) well-suited for such data structure at early stage. However we are today facing a new challenge considering the always increasing number of pixels in the data cubes we have to manage. The idea is therefore to introduce a new paradigm of Topological Data Analysis in order explore hyperspectral imaging data sets highlighting its nice properties and specific features. With this paper, we shall also point out the fact that conventional chemometric methods are often based on variance analysis or simply impose a data model which implicitly defines the geometry of the data set. Thus we will show that it is not always appropriate in the framework of hyperspectral imaging data sets exploration. Abstract Topological data analysis is a new method for the analysis and modeling of complex data sets, based on the mathematical notion of shape. It is particularly appropriate for problems arising in biology, due to the flexible representation of data sets as network models. In this paper, we show how the method works, give two examples, and discuss the modeling framework that can be constructed based on it. Author Summary: Hyperspectral remote sensing plays an increasingly important role in many scientific domains and everyday life problems. Indeed, this imaging concept ends up in applications as varied as catching tax-evaders red-handed by locating new construction and building alterations, searching for aircraft and saving lives after fatal crashes, detecting oil spills for marine life and environmental preservation, spying on enemies with reconnaissance satellites, watching algae grow as an indicator of environmental health, forecasting weather to warn about natural disasters and much more. From an instrumental point of view, we can say that the actual spectrometers have rather good characteristics, even if we can always increase spatial resolution and spectral range. In order to extract ever more information from such experiments and develop new applications, we must, therefore, propose multivariate data analysis tools able to capture the shape of data sets and their specific features. Nevertheless, actual methods often impose a data model which implicitly defines the geometry of the data set. The aim of the paper is thus to introduce the concept of topological data analysis in the framework of remote sensing, making no assumptions about the global shape of the data set, but also allowing the capture of its local features. Infected hosts differ in their responses to pathogens; some hosts are resilient and recover their original health, whereas others follow a divergent path and die. To quantitate these differences, we propose mapping the routes infected individuals take through “disease space.” We find that when plotting physiological parameters against each other, many pairs have hysteretic relationships that identify the current location of the host and predict the future route of the infection. These maps can readily be constructed from experimental longitudinal data, and we provide two methods to generate the maps from the cross-sectional data that is commonly gathered in field trials. We hypothesize that resilient hosts tend to take small loops through disease space, whereas nonresilient individuals take large loops. We support this hypothesis with experimental data in mice infected with Plasmodium chabaudi, finding that dying mice trace a large arc in red blood cells (RBCs) by reticulocyte space as compared to surviving mice. We find that human malaria patients who are heterozygous for sickle cell hemoglobin occupy a small area of RBCs by reticulocyte space, suggesting this approach can be used to distinguish resilience in human populations. This technique should be broadly useful in describing the in-host dynamics of infections in both model hosts and patients at both population and individual levels. Type 2 diabetes (T2D) is a heterogeneous complex disease affecting more than 29 million Americans alone with a rising prevalence trending toward steady increases in the coming decades. Thus, there is a pressing clinical need to improve early prevention and clinical management of T2D and its complications. Clinicians have understood that patients who carry the T2D diagnosis have a variety of phenotypes and susceptibilities to diabetes-related complications. We used a precision medicine approach to characterize the complexity of T2D patient populations based on high-dimensional electronic medical records (EMRs) and genotype data from 11,210 individuals. We successfully identified three distinct subgroups of T2D from topology-based patient-patient networks. Subtype 1 was characterized by T2D complications diabetic nephropathy and diabetic retinopathy; subtype 2 was enriched for cancer malignancy and cardiovascular diseases; and subtype 3 was associated most strongly with cardiovascular diseases, neurological diseases, allergies, and HIV infections. We performed a genetic association analysis of the emergent T2D subtypes to identify subtype-specific genetic markers and identified 1279, 1227, and 1338 single-nucleotide polymorphisms (SNPs) that mapped to 425, 322, and 437 unique genes specific to subtypes 1, 2, and 3, respectively. By assessing the human disease–SNP association for each subtype, the enriched phenotypes and biological functions at the gene level for each subtype matched with the disease comorbidities and clinical differences that we identified through EMRs. Our approach demonstrates the utility of applying the precision medicine paradigm in T2D and the promise of extending the approach to the study of other complex, multifactorial diseases. Topological methods are emerging as a new set of tools for the analysis of large genomic datasets. They are mathematically grounded methods that extract information from the geometric structure of data. In the last few years, applications to evolutionary biology, cancer genomics, and the analysis of complex diseases have uncovered significant biological results, highlighting their utility for fulfilling some of the current analytic needs of genomics. In this review, the state of the art in the application of topological methods to genomics is summarized, and some of the present limitations and possible future developments are reviewed. Fragile X syndrome (FXS), due to mutations of the FMR1 gene, is the most common known inherited cause of developmental disability as well as the most common single-gene risk factor for autism. Our goal was to examine variation in brain structure in FXS with topological data analysis (TDA), and to assess how such variation is associated with measures of IQ and autism-related behaviors. To this end, we analyzed imaging and behavioral data from young boys (n = 52; aged 1.57–4.15 years) diagnosed with FXS. Application of topological methods to structural MRI data revealed two large subgroups within the study population. Comparison of these subgroups showed significant between-subgroup neuroanatomical differences similar to those previously reported to distinguish children with FXS from typically developing controls (e.g., enlarged caudate). In addition to neuroanatomy, the groups showed significant differences in IQ and autism severity scores. These results suggest that despite arising from a single gene mutation, FXS may encompass two biologically, and clinically separable phenotypes. In addition, these findings underscore the potential of TDA as a powerful tool in the search for biological phenotypes of neuropsychiatric disorders. Hum Brain Mapp 35:4904–4915, 2014. © 2014 Wiley Periodicals, Inc. Recent developments in applied mathematics are bringing new tools that are capable to synthesize knowledge in various disciplines, and help in finding hidden relationships between variables. One such technique is topological data analysis (TDA), a fusion of classical exploration techniques such as principal component analysis (PCA), and a topological point of view applied to clustering of results. Various phenomena have already received new interpretations thanks to TDA, from the proper choice of sport teams to cancer treatments. For the first time, this technique has been applied in soil science, to show the interaction between physical and chemical soil attributes and main soil-forming factors, such as climate and land use. The topsoil data set of the Land Use Land Cover Area Frame survey (LUCAS) was used as a comprehensive database that consists of approximately 20,000 samples, each described by 12 physical and chemical parameters. After the application of TDA, results obtained were cross-checked against known grouping parameters including five types of land cover, nine types of climate and the organic carbon content of soil. Some of the grouping characteristics observed using standard approaches were confirmed by TDA (e.g., organic carbon content) but novel subtle relationships (e.g., magnitude of anthropogenic effect in soil formation), were discovered as well. The importance of this finding is that TDA is a unique mathematical technique capable of extracting complex relations hidden in soil science data sets, giving the opportunity to see the influence of physicochemical, biotic and abiotic factors on topsoil formation through fresh eyes.