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This session will include an overview of the R statistical computing environment as well as a sequence of brief case studies of actuarial analyses in R. Case studies will include examples from loss distribution analysis, ratemaking, loss reserving, and predictive modeling.

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- 1. Actuarial Science as Data ScienceActuarial Modeling in RRevolution Analytics Webinar Jim Guszcza, FCAS, MAAA Deloitte Consulting LLP University of Wisconsin-MadisonMarch 28, 2012
- 2. About Your Presenter• James Guszcza, PhD, FCAS, MAAA• National Predictive Analytics Lead – Deloitte Consulting Actuarial, Risk, Analytics practice• Assistant professor of actuarial science & risk management – U. Wisconsin-Madison• PhD in Philosophy – The University of Chicago• Fellow of the Casualty Actuarial Society• Lots experience building predictive models / analyzing data in and outside of insurance jguszcza@deloitte.com jguszcza@bus.wisc.edu 2 Deloitte Analytics Institute © 2011 Deloitte LLP
- 3. Agenda Introduction Actuarial Science and Data Science R Background Case Studies • Fitting a complex size of loss model • Loss Reserving • Bayesian Hierarchical Modeling • Revolution: Tweedie Regression on big data
- 4. Actuarial Science andData Science
- 5. Not Just Hype“Perhaps the most important cultural trend today: The explosion of data about every aspect of our world and the rise of applied math gurus who know how to use it.” -- Chris Anderson, editor-in-chief of Wired• So behavioral economics is important in insurance for two classes of reasons: • Decision-makers at insurance companies are human • People making insurance purchasing decisions are human5 Deloitte Analytics Institute © 2010 Deloitte LLP
- 6. Brave New World With Such Algorithms In IT• The analysis of data affects:• What we buy• What we read• What we watch• How we network• How we socialize• The opinions we form• Whom we date and marry!6 Deloitte Analytics Institute © 2010 Deloitte LLP
- 7. Clinical vs Actuarial Judgment – the Motion Picture7 Deloitte Analytics Institute © 2010 Deloitte LLP
- 8. Analytics Everywhere • Neural net models are used to predict movie box-office returns based on features of their scripts • Decision tree models are used to help ER doctors better triage patients complaining of chest pain. • Predictive models are used to predict the price of different wine vintages based on variables about the growing season. • Predictive models to help commercial insurance underwriters better select and price risks. • Predict which non-custodial parents are at highest risk of falling into arrears on their child support. • Predicting which job candidates will successfully make it through the interviewing / recruiting process… and which candidates will subsequently retain and perform well on the job. • Predicting which doctors are at highest risk of being sued for malpractice. • Predicting the ultimate severity of injury claims.8 Deloitte Analytics Institute (Deloitte applications in green) © 2010 Deloitte LLP
- 9. At the Center of It All: Data Science Or: “The Collision between Statistics and Computation”• Today the analytics world is different largely due to exponential growth in computing power.• The skill set underlying business analytics is increasingly called data science.• Data science goes beyond: • Traditional statistics • Business intelligence [BI] Image borrowed from Drew Conway’s blog • Information technology http://www.dataists.com/2010/09/the-data-science-venn-diagram9 Deloitte Analytics Institute © 2010 Deloitte LLP
- 10. Where Do We Want to Be? •Here? Image borrowed from Drew Conway’s blog http://www.dataists.com/2010/09/the-data-science-venn-diagram10 Deloitte Analytics Institute © 2010 Deloitte LLP
- 11. Where Do We Want to Be? •Or Here? Image borrowed from Drew Conway’s blog http://www.dataists.com/2010/09/the-data-science-venn-diagram11 Deloitte Analytics Institute © 2010 Deloitte LLP
- 12. On then, on to R12 Deloitte Analytics Institute © 2010 Deloitte LLP
- 13. R Background
- 14. R Overview R is an open-source, object-oriented statistical programming language. In the past decade, it has become the global lingua franca of statistics.• History: • R is based on the S statistical programming language developed by John Chambers at Bell labs in the 1980’s • R is an open-source implementation of the S language • Developed by Robert Gentlemen and Ross Ihaka at U Auckland • Revolution R is a commercially supported, scalable implementation of R, with parallel processing and big data capabilities• Features: • R is an interactive, object-oriented programming environment • R has advanced graphical capabilities • Statisticians around the world contribute add-on packages14 Deloitte Analytics Institute © 2010 Deloitte LLP
- 15. On the Shoulders of Giants• … therefore prominent people tend say things like this:http://www.nytimes.com/2009/01/07/technology/business-computing/07program.html?pagewanted=all15 Deloitte Analytics Institute © 2010 Deloitte LLP
- 16. Facets of R• In a recent article John Chambers discussed 6 “Facets of R” 1. An interface to computational procedures of many kinds 2. Interactive, hands-on in real time 3. Functional in its model of programming 4. Object-oriented, “everything is an object” 5. Modular, built from standardized pieces 6. Collaborative, a world-wide, open-source effort• Interactive interface: Chambers was influenced by APL • In the days before spreadsheets, APL was very popular in the actuarial community • One of the rare interactive scientific computing environments • Gives user ability to express novel computations • Heavy emphasis on matrices and arrays • But: unlike R, APL had no interface to procedures16 Deloitte Analytics Institute © 2010 Deloitte LLP
- 17. A Network ExteRnality• Hal Varian’s “giant” has grown at an exponential rate.• The open-source nature of R has encouraged top researchers from around the world to contribute new, often highly advanced, packages.• Result: a powerful “network effect”. • The value of a product increases as more people use it.• R has become something like the Wikipedia of the statistics world.17 Deloitte Analytics Institute © 2010 Deloitte LLP
- 18. Adoption in the Actuarial World18 Deloitte Analytics Institute © 2010 Deloitte LLP
- 19. Free from Frees• Jed Frees at the University of Wisconsin-Madison has made R integral to his new book on regression and time series. He maintains a nice website containing R instructions, data, and code. http://instruction.bus.wisc.edu/jfrees/jfreesbooks/Regression%20Modeling/BookWebDec2010/learnR.html19 Deloitte Analytics Institute © 2010 Deloitte LLP
- 20. Case Studies
- 21. Some Everyday Uses of R• Free-form Exploratory Data Analysis • ad hoc data munging, data visualizations, fitting simple models on the fly • Loss models (“exam 4/C”)• Unsupervised Learning • Correlation analysis, principal component / factor analysis, variable clustering, k-means and hierarchical clustering, self-organizing maps, association rules (aka “market basket analysis”), Latent Dirichlet Analysis• Supervised Learning • “statistics paradigm”: GLM, Multilevel/Hierarchical models, quantile regression • “machine learning paradigm: CART, MARS, Random Forests, Neural Networks, Support Vector Machines • Bayesian data analysis (MCMC simulation), causal analysis• Optimization21 Deloitte Analytics Institute © 2010 Deloitte LLP
- 22. Case Study #1Loss Distribution Modeling
- 23. Modeling a Non-Trivial Loss Distribution• A typical actuarial problem: modeling a highly skew and ambiguous loss 8 e-06 distribution 6 e-06• Traditional medium of analysis: spreadsheets. 4 e-06• Why limit ourselves? 2 e-06 0 e+00 0 e+00 1 e+06 2 e+06 3 e+06 4 e+06 5 e+06 loss23 Deloitte Analytics Institute © 2010 Deloitte LLP
- 24. Case Study #2Loss Reserving
- 25. Three Approaches to Loss Reserving • A garden-variety loss triangle: Cumulative Losses in 1000s AY premium 12 24 36 48 60 72 84 96 108 120 CL Ult CL LR CL res 1988 2,609 404 986 1,342 1,582 1,736 1,833 1,907 1,967 2,006 2,036 2,036 0.78 0 1989 2,694 387 964 1,336 1,580 1,726 1,823 1,903 1,949 1,987 2,017 0.75 29 1990 2,594 421 1,037 1,401 1,604 1,729 1,821 1,878 1,919 1,986 0.77 67 1991 2,609 338 753 1,029 1,195 1,326 1,395 1,446 1,535 0.59 89 1992 2,077 257 569 754 892 958 1,007 1,110 0.53 103 1993 1,703 193 423 589 661 713 828 0.49 115 1994 1,438 142 361 463 533 675 0.47 142 1995 1,093 160 312 408 601 0.55 193 1996 1,012 131 352 702 0.69 350 1997 976 122 576 0.59 454chain link 2.365 1.354 1.164 1.090 1.054 1.038 1.026 1.020 1.015 1.000 12,067 1,543chain ldf 4.720 1.996 1.473 1.266 1.162 1.102 1.062 1.035 1.015 1.000growth curve 21.2% 50.1% 67.9% 79.0% 86.1% 90.7% 94.2% 96.6% 98.5% 100.0% • Let’s use R to forecast outstanding losses using three methods: • Replicate the above chain-ladder spreadsheet calculation – easy! • Use the Over-dispersed Poisson GLM model • Longitudinal data analysis using growth curves 25 Deloitte Analytics Institute © 2010 Deloitte LLP
- 26. What Do You See? • Let’s look at the loss triangle with fresh eyes. • We would like to do stochastic reserving the “right” way. • What considerations come to mind? Cumulative Losses in 1000s AY premium 12 24 36 48 60 72 84 96 108 120 CL Ult CL LR CL res 1988 2,609 404 986 1,342 1,582 1,736 1,833 1,907 1,967 2,006 2,036 2,036 0.78 0 1989 2,694 387 964 1,336 1,580 1,726 1,823 1,903 1,949 1,987 2,017 0.75 29 1990 2,594 421 1,037 1,401 1,604 1,729 1,821 1,878 1,919 1,986 0.77 67 1991 2,609 338 753 1,029 1,195 1,326 1,395 1,446 1,535 0.59 89 1992 2,077 257 569 754 892 958 1,007 1,110 0.53 103 1993 1,703 193 423 589 661 713 828 0.49 115 1994 1,438 142 361 463 533 675 0.47 142 1995 1,093 160 312 408 601 0.55 193 1996 1,012 131 352 702 0.69 350 1997 976 122 576 0.59 454chain link 2.365 1.354 1.164 1.090 1.054 1.038 1.026 1.020 1.015 1.000 12,067 1,543chain ldf 4.720 1.996 1.473 1.266 1.162 1.102 1.062 1.035 1.015 1.000growth curve 21.2% 50.1% 67.9% 79.0% 86.1% 90.7% 94.2% 96.6% 98.5% 100.0% 26 Deloitte Analytics Institute © 2010 Deloitte LLP
- 27. Some Essential Features of Loss Reserving Cumulative Losses in 1000s AY premium 12 24 36 48 60 72 84 96 108 120 CL Ult CL LR CL res 1988 2,609 404 986 1,342 1,582 1,736 1,833 1,907 1,967 2,006 2,036 2,036 0.78 0 1989 2,694 387 964 1,336 1,580 1,726 1,823 1,903 1,949 1,987 2,017 0.75 29 1990 2,594 421 1,037 1,401 1,604 1,729 1,821 1,878 1,919 1,986 0.77 67 1991 2,609 338 753 1,029 1,195 1,326 1,395 1,446 1,535 0.59 89• Repeated measures 1992 2,077 257 569 754 892 958 1,007 1,110 0.53 103 1993 1,703 193 423 589 661 713 828 0.49 115 1994 1,438 142 361 463 533 675 0.47 142 1995 1,093 160 312 408 601 0.55 193 1996 1,012 131 352 702 0.69 350 1997 976 122 576 0.59 454 • The dataset is inherently longitudinal in nature. chain link chain ldf growth curve 2.365 1.354 1.164 1.090 1.054 1.038 1.026 1.020 1.015 4.720 1.996 1.473 1.266 1.162 1.102 1.062 1.035 1.015 1.000 1.000 21.2% 50.1% 67.9% 79.0% 86.1% 90.7% 94.2% 96.6% 98.5% 100.0% 12,067 1,543• A “Bundle” of time series • Loss triangle: a collection of time series that are “related” to one another… • … no guarantee that the same development pattern is appropriate to each one• Non-linear • Each year’s loss development pattern in inherently non-linear • Ultimate loss (ratio) is an asymptote• Incomplete information • Few loss triangles contain all of the information needed to make forecasts • Most reserving exercises must incorporate judgment and/or background information Loss reserving is inherently Bayesian27 Deloitte Analytics Institute © 2010 Deloitte LLP
- 28. Origin of the Approach: Dave’s Idea + Random Effects + =28 Deloitte Analytics Institute © 2010 Deloitte LLP
- 29. And Now it’s Bayesian• Fully Bayesian model• Provides posterior credible intervals (“range of reasonable reserves”)• Add further hierarchical structure to simultaneously model loss development for multiple companies. (Wayne’s idea!) 29 Deloitte Analytics Institute © 2010 Deloitte LLP
- 30. Case Study #3Hierarchical Bayes Ratemaking
- 31. Workers Comp Ratemaking• We have 7 years of Workers Comp data • Data from Klugman [1992 Bayes book] • 128 workers comp classes (types of business) • 7 years of summarized data • Given: total payroll, claim count by class • (payroll is a measure of “exposure” in this domain) • Problem: use years 1-6 data to predict year 731 Deloitte Analytics Institute © 2010 Deloitte LLP
- 32. Empirical Bayes “Credibility” Approach• Naïve approach: • Calculate average year 1-6 claim frequency by class • Use these 128 averages as estimates for year 7.• Better approach: build empirical Bayes hierarchical model. • “Bühlmann-Straub credibility model” • “Shrinks” low-credibility classes towards the grand mean • Use Douglas Bates’ lme4 package (UW-Madison again!) clmcnti ~ Poi ( payrolli λ j[ i ] ) ( λ j ~ N µλ , σ λ 2 )32 Deloitte Analytics Institute © 2010 Deloitte LLP
- 33. Shrinkage Effect of Empirical Bayes Model• Top row: estimated claim frequencies from un-pooled Modeled Claim Frequency by C model. Poisson Models: No Pooling and Simple • Separately calculate #claims/payroll by class no pool• Bottom row: estimated claim frequencies from Poisson hierarchical (credibility) model.• Credibility estimates are “shrunk” towards the grand mean. hierach • Dotted line: shrinkage between 5=10%. • Solid line: shrinkage > 10% 0.00 grand mean 0.05 0.10 Claim Frequency33 Deloitte Analytics Institute © 2010 Deloitte LLP
- 34. clmcnti ~ Poi ( payrolli λ j[ i ] )Now Specify a Fully Bayesian Model ( λ j ~ N µλ , σ λ 2 )• Here we specify a fully Bayesian model. • Use the rjags package • JAGS: Just Another Gibbs Sampler • We’re standing on the shoulders of giants named David Spiegelhalter, Martyn Plummer, …34 Deloitte Analytics Institute © 2010 Deloitte LLP
- 35. clmcnti ~ Poi ( payrolli λ j[ i ] )Now Specify a Fully Bayesian Model ( λ j ~ N µλ , σ λ 2 )• Here we specify a fully Bayesian model. • Poisson regression with an offset35 Deloitte Analytics Institute © 2010 Deloitte LLP
- 36. clmcnti ~ Poi ( payrolli λ j[ i ] )Now Specify a Fully Bayesian Model ( λ j ~ N µλ , σ λ 2 )• Here we specify a fully Bayesian model. • Allow for overdispersion36 Deloitte Analytics Institute © 2010 Deloitte LLP
- 37. clmcnti ~ Poi ( payrolli λ j[ i ] )Now Specify a Fully Bayesian Model ( λ j ~ N µλ , σ λ 2 )• Here we specify a fully Bayesian model. • Allow for overdispersion37 Deloitte Analytics Institute © 2010 Deloitte LLP
- 38. clmcnti ~ Poi ( payrolli λ j[ i ] )Now Specify a Fully Bayesian Model ( λ j ~ N µλ , σ λ 2 )• Here we specify a fully Bayesian model. • “Credibility weighting” (aka shrinkage) results from giving class-level intercepts a probability sub-model.38 Deloitte Analytics Institute © 2010 Deloitte LLP
- 39. clmcnti ~ Poi ( payrolli λ j[ i ] )Now Specify a Fully Bayesian Model ( λ j ~ N µλ , σ λ 2 )• Here we specify a fully Bayesian model. • Put a diffuse prior on all of the hyperparameters • Fully Bayesian model • Bayes or Bust!39 Deloitte Analytics Institute © 2010 Deloitte LLP
- 40. clmcnti ~ Poi ( payrolli λ j[ i ] )Now Specify a Fully Bayesian Model ( λ j ~ N µλ , σ λ 2 )• Here we specify a fully Bayesian model. • Replace year-7 actual values with missing values • We model the year-7 results … produce 128 posterior density estimates • Can compare actual claims with Bayesian posterior probabilities40 Deloitte Analytics Institute © 2010 Deloitte LLP
- 41. A Credible Result• Let’s rank the top 30 WC classes by the median of the posterior predictive density of year-7 claim count.• 87% of the top 30 classes have actual year-7 claim count falling within the 90% posterior credible interval.41 Deloitte Analytics Institute © 2010 Deloitte LLP
- 42. Case Study #4Big Data in Revolution R
- 43. Big Data Headed Our Way• Credibility concerns and a Bayesian outlook are part and parcel of actuarial science.• But for many actuaries, working with “big data” is a much more pressing concern. • Many millions of personal lines policy terms • Premium, loss, credit, billing transactions • Telematics data • … much more to come• Base R handles data in memory • This is beautiful for “small data” problems like doing loss reserving on summarized data • But breaks down for many industrial datasets• So on to Revolution-R43 Deloitte Analytics Institute © 2011 Deloitte LLP
- 44. The kaggle Allstate Claim Prediction Challenge Data44 Deloitte Analytics Institute © 2011 Deloitte LLP
- 45. Loading the Data• Data volume: • 13M rows • ~ 40 cols• Took about 6-7 minutes to load • Perform some variable transformations on the fly to minimize passes though the data.• Data saved on disk in “xdf” file format for easy access and interactive modeling.45 Deloitte Analytics Institute © 2011 Deloitte LLP
- 46. Viewing the Data• Data characteristics: • 13,184,290 rows • A few dozen predictive variables (mostly blinded) • Target variable: claim amount• kaggle competition goal: build a model that segments well out-of-sample• Let’s use the 2005-6 data to predict the 2007 data• (Just a quick model to get a sense of Revolution R’s scalability)• Tweedie regression models fit in seconds46 Deloitte Analytics Institute © 2011 Deloitte LLP
- 47. Helpful Resources• Edward (Jed) Frees – Regression modeling with actuarial and financial applications http://www.amazon.com/Regression-Actuarial-Financial-Applications- International/dp/0521135966• Andrew Gelman / Jennifer Hill - Data Analysis using Regression and Multilevel/Hierarchical Models http://www.amazon.com/Analysis-Regression-Multilevel- Hierarchical- Models/dp/052168689X/ref=sr_1_1?s=books&ie=UTF8&qid=1332961819&sr=1-1• Venables and Ripley – Modern Applied Statistics in S http://www.amazon.com/Modern- Applied-Statistics- Computing/dp/1441930086/ref=sr_1_1?s=books&ie=UTF8&qid=1332961867&sr=1-1• Hastie, Tibshirani, Friedman – the Elements of Statistical Learning http://www.amazon.com/The-Elements-Statistical-Learning- Prediction/dp/0387848576/ref=sr_1_1?s=books&ie=UTF8&qid=1332961913&sr=1-1• Gelman, Carlin, Stern, Ruin – Bayesian Data Analysis http://www.amazon.com/Bayesian- Analysis-Edition-Chapman-Statistical/dp/158488388X/ref=tag_dpp_lp_edpp_ttl_in• John Kruschke – Doing Bayesian Data Analysis http://www.amazon.com/Doing-Bayesian- Data-Analysis- Tutorial/dp/0123814855/ref=sr_1_3?s=books&ie=UTF8&qid=1332961975&sr=1-347 Deloitte Analytics Institute © 2011 Deloitte LLP

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