Women in Data Science 2018 Slides--Small Samples, Subgroups, and Topology
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This document discusses using topological data analysis techniques to address challenges with small sample sizes and distinct subgroups in data. It provides three case studies applying topological tools: 1) exploring profoundly gifted students with small educational samples, 2) identifying risk clusters in auto insurance claims data with subgroups, and 3) validating a psychometric survey using topology over traditional factor analysis. Topological data analysis is presented as a robust solution to problems traditional statistical and machine learning algorithms struggle with for small and complex data.
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Women in Data Science 2018 Slides--Small Samples, Subgroups, and Topology
- 1. Human Behavior, Small Samples, and the Problem of Subgroups The Power of Topology 3/5/2018
- 2. Introduction Big data hype Less publicized but very important types of data: 1. Small data 2. Data with distinct subgroups Industries where these are common: Education Insurance Biotechnology/ pharmaceuticals Industrial psychology
- 3. Problems Unique to Small Data Types of small data problems: Rare diseases (100 cases worldwide with unknown genetic causes) Pilot studies (10’s or 100’s of participants) Small educational programs (10’s of students enrolled in the previous year) Main issues: Statistical models require minimum sample sizes to estimate effects with computational issues or wide confidence intervals (singularities, p>>n problems). Machine learning algorithms need to converge for stable estimates and models. Small samples can induce sparsity in the data space, which is problematic for clustering and general data mining techniques. 3
- 4. Problems Unique to Subgroups in Data Types of data in which subgroups are common: Medicine (diverse causes of a given disease, subtypes of disease) Education (different types of students and risk types for failure) Industrial psychology (different personality trait patterns) Main issues: Washing out of effects in a full model Examples: Small subgroup defined by extremely high extraversion and openness related to public speaking outcome Rare genetic variant combination predicting high likelihood of response to a drug within a disease population Defining robust partitions within a piecewise regression model to deal with this phenomenon Mixed results from many methods employing this strategy Difficult with small sample sizes for most piecewise regression models 4
- 5. Unique Solutions: Topology Branch of mainly pure mathematics Study of changes in function behavior on different shapes Identify invariant properties of shapes Classify similarities/ differences between shapes 5 Deep connections to physics and differential equations
- 6. Topological Data Analysis Data as discrete point clouds Topological spaces, called simplicial complexes, built from these: Connect points within a certain distance of each other Topologically similar to a graph Tools using simplicial complexes for data analysis called topological data analysis (TDA) 6 2-d neighborhoods are defined by Euclidean distance. Points within a given circle are mutually connected, forming a simplex.
- 7. Tool 1: Persistent Homology Filtration Series of simplicial complexes based on varying distance thresholds Features appear and disappear as lens changes Nested sequence of features with deep algebraic properties Persistence as length of feature existence in the sequences (plotted as persistence diagrams) Termed persistent homology A bit like an MRI-type examination of data Persistence as organ size and type Gives a comprehensive view of data Persistent homology related to hierarchical clustering Statistical methods to compare datasets0 2 4 6 8 10 0246810 Birth Death 0 2 4 6 8 10 time
- 8. Tool 2: Morse-Smale Clustering Multivariate technique from TDA similar to mode clustering Find peaks and valleys in data by filtering on a defined function: A watershed on mountains Dribbling a soccer ball across a field of hills Separate data based on shared peaks and valleys Many nice developments on convergence and theoretical properties 8
- 9. Tool 3: Homotopy and Path Equivalence Homotopy arrow example Red to blue by wiggling start to finish path Yellow arrow and hole problem Homotopy method in LASSO Wiggles easy regression path to optimal regression path Recent success solving ordinary differential equations Avoids local optima that can trap other regression estimators 9
- 10. Case Study 1: Small Educational Samples Problem set-up 1. Understand subgroups of profoundly gifted students (IQ>160) 2. Explore impact of educational interventions on early career awards Sample 1. 17 profoundly gifted students: Gross’s 2003 sample Intelligence testing data available Early achievement testing (verbal, math) available 2. 16 of these same students with follow-up data related to: Educational intervention data Early career recognition/awards 10
- 11. Data Mining 11 9 3 13 5 1 7 8 14 10 11 12 6 16 15 17 2 4 0204060 Intelligence and Achievement Dendrogram hclust (*, "complete") dist(mydata[, 2:4]) Height Distinct population that separates out very early in the filtration (box) Students with an IQ>200 and achievement scores 5+ grades ahead for math and verbal (multivariate outliers) Corroborates previous evidence of a “high flat” profile distinct from other types of profound giftedness
- 12. Logistic Regression Coefficient Comparison Comparison of 2 machine learning models and 2 topologically-based models Too few observations for traditional logistic regression Multivariate adaptive regression splines (MARS) inadequate fit (R^2=0.27) Bayesian model averaging (BMA) extremely large confidence interval DGLARS and HLASSO (topologically-based) good fit, small confidence intervals, consistent results across replication 12 IQ Score Early English Early Math Early Entry Grade Skip Subject Acceleration Radical Acceleration MARS 0.44 BMA -6.25 5.79 0.97 1.38 2.41 33.10 DGLARS 2.20 4.66 HLASSO 0.02 -0.26 1.44 3.27
- 13. Case Study 2: Actuarial Modeling with Subgroups Problem set-up Understand risk factors associated with auto insurance claims Understand subgroups with different types of risk Sample Open-source Swedish automobile claims dataset from 1977 2182 claims, 6 predictors 13
- 14. Risk Clusters Group 1: relatively high dependence on make and number of claims Group 2: relatively high dependence on bonus and number of years insured Group 3: almost solely dependent on number of claims and geographic zone 14 Three distinct subgroups with varying risk type
- 15. Case Study 3: Psychometric Test Design Set-up: Explore/validate survey measuring identity importance/expression across social contexts Create subscales within the survey Sample: 406 participants in a pilot study 91 test items Random samples of 130 participants taken with replacement as validation samples 15
- 16. Advantages of Topology Over Factor Analysis 16 Loss of information with each projection to a lower-dimensional space (errors) Topological methods work by partitioning existing space into homogenous components (no maps, no error) 2D example
- 18. Insights Gained Some aspects of identity fluid, others are fixed Political and racial/ethnic identity fixed Other types, such as athletic or gender, fairly fluid No statistically significant differences between samples Subscales consistent Validates measure 18
- 19. Conclusions Unique challenges in data science Subgroups Small samples Failure of statistical and machine learning algorithms Topological data analysis as robust solutions 19
- 20. Reference Papers Carlsson, G. (2009). Topology and data. Bulletin of the American Mathematical Society, 46(2), 255-308. Edelsbrunner, H., & Harer, J. (2008). Persistent homology-a survey. Contemporary mathematics, 453, 257-282. Farrelly, C. M. (2017). Extensions of Morse-Smale Regression with Application to Actuarial Science. arXiv preprint arXiv:1708.05712. Accepted as new model by Casualty Actuarial Society, December 2017. Farrelly, C. M. (2018). Topology and Geometry for Small Sample Sizes: An Application to Research on the Profoundly Gifted. Farrelly, C. M., Schwartz, S. J., Amodeo, A. L., Feaster, D. J., Steinley, D. L., Meca, A., & Picariello, S. (2017). The analysis of bridging constructs with hierarchical clustering methods: An application to identity. Journal of Research in Personality, 70, 93-106. Gerber, S., Rübel, O., Bremer, P. T., Pascucci, V., & Whitaker, R. T. (2013). Morse–smale regression. Journal of Computational and Graphical Statistics, 22(1), 193-214. 20