Session 102 L, Getting Warm and Fuzzy Beyond Traditional Set Theory. Moderator: Douglas T. Norris, FSA, MAAA, Ph.D.

Transcription

1 Session 102 L, Getting Warm and Fuzzy Beyond Traditional Set Theory Moderator: Douglas T. Norris, FSA, MAAA, Ph.D. Presenters: Jeff T. Heaton David L. Snell, ASA, MAAA

2 Getting Warm and Fuzzy Beyond Traditional Set Theory SOA Annual Meeting - session Austin - October 13, :00 PM 3:15 PM Dave Snell, ASA, ARA, ACS, CHFC, CLU, FLMI, MAAA, MCP technology evangelist, RGA Reinsurance Company Jeff Heaton, FLMI, ARA, ACS data scientist, RGA Reinsurance Company

3 Actuaries want data that is: Complete Accurate Precise Consistent Structured Dream on! 2

4 Actuaries Get Data that is: Incomplete Approximate Imprecise Inconsistent Unstructured Welcome to the real world! 3

5 Our Mathematical Heritage is a Strength Thousands of years of mathematical training Geometry (Elements) Euclid of Alexandria, c 300 B.C Laws of Motion, (F = ma) Newton C.E. Principia Mathematica (axioms, inference rules, symbolic logic all mathematical truths can be proven) Whitehead and Russell We held the world in our hand 4

6 or it can be a Weakness A 0-dimensional point does not exist or a 1-dimensional line; or a 2-dimensional plane and dimensions do not have to be integers (Hausdorff Besicovitch, 1918) Force = mass * acceleration F = m * dv/dt + v * dm/dt (Einstein, 1905) Gödel's incompleteness theorem (1931) shatters Principia Mathematica 5

7 Fuzzy Logic Just for Kids? or a new paradigm for better models of the real world 6

8 What is Fuzzy Logic? Reality! It is not crisp logic. Crisp Logic is a new name for Boolean Logic (George Boole, 1847) Binary logic Set membership is 0 (false, out) or 1 (true, in) Fuzzy Logic allows interim values (Lotfi Zadeh, 1965) Set membership can be between 0 (completely out) and 1 (all in) 7

9 Fuzzy Logic: Linguistic Variables "measure what is measurable, and make measurable what is not so" - Galileo Galilei, around 1630 CE Linguistic Variables allow the use of descriptive terms such as underweight, or obese to describe normally numeric variables. [Discussion: a safe example] 8

10 Fuzzy Logic: Membership 9 Set Membership (µ) Short 0.4 Average 0.9 Tall 0.8 [Discussion: Tallness]

11 Fuzzy Logic: Membership 10 Set Membership (µ) Short 0.4 Average 0.9 Tall 0.8 [Discussion: Tallness] Sum 1

12 Fuzzy Logic: Hedging Variables "All animals are equal, but some animals are more equal than others - Animal House, by George Orwell,

13 Fuzzy Logic: Better fit to Reality [Discussion: reference ranges] source: author s subset of excellent (award winning) Wikipedia image //upload.wikimedia.org/wikipedia/commons/th umb/c/cb/blood_values_sorted_by_mass_and_ molar_concentration.png contributed by Mikael Häggström, MD and released under the Attribution-Share Alike 3.0 Unportedlicense 12

14 Fuzzy Logic: Overlapping Ranges Slide source: Fuzzy Logic in R, by Jeff Heaton (Forecasting & Futurism Newsletter, July, 2014) 13

15 Fuzzy Logic: Process 1. Fuzzification convert your input and output to linguistic values, utilizing ranges and membership functions. 2. Apply rules (from your experience or knowledge base) using fuzzy logic. 3. Defuzzification convert your results to the form you want (often a numeric result). 14

16 Fuzzy Logic: Rule Sets IF BMI is Obese AND BP is HyperTension AND Diabetes is TRUE THEN Rating is Uninsurable IF InterestRate is Very High AND SurrenderCharge is Low THEN LapseRate is High IF InterestRate is Low AND SurrenderCharge is Moderate THEN LapseRate is Low Bonus tip: prevent combinatorial explosion with Combs method: 15

17 Example using R Install.packages( sets ) Library(sets) sets_options('universe', + seq(from=1,to=9,by=.5)) #note: + indicates a line continuation 16

18 Example using R (continued) vars<-set( + int=fuzzy_partition(varnames + =c(low=2,norm=4,hi=6,vhi=8),sd=1), + unemp=fuzzy_partition(varnames + =c(low=3,norm=4,hi=5,vhi=6),sd=.8), + lapse=fuzzy_partition(varnames + =c(low=3,med=5,hi=9),sd=2)) #sd=standard deviation # also note that in R, there are 4 equivalent assignment operators: x=5, x<-5, 5->x, assign( x,5) 17

19 Example using R (continued) rules<-set( + fuzzy_rule(int %is% low + && unemp %is% low, lapse %is% low), + fuzzy_rule((int %is% hi + int %is% vhi) + && (unemp %is% hi + unemp %is% vhi), lapse %is% hi)) #rules take the form fuzzy_rule(antecedent, consequent) 18

20 Example using R (continued) sys<-fuzzy_system(vars,rules) Plot(sys) 19

21 Example using R (continued) #fuzzify fz_inf<-fuzzy_inference(sys, + list(int=2.5, unemp=3)) plot(fz_inf) 20

22 Example using R (continued) gset_defuzzify(fz_inf,'centroid') [1] gset_defuzzify(fz_inf,'meanofmax') #returns 3 for this example gset_defuzzify(fz_inf,'smallestofmax') #returns 2 for this example gset_defuzzify(fz_inf,'largestofmax') #returns 4 for this example 21

23 Example using R (continued) fz_inf<-fuzzy_inference(sys,list(int=7, unemp=5)) plot(fz_inf) #defuzzify gset_defuzzify(fz_inf,'centroid') #output is gset_defuzzify(fz_inf,'meanofmax') #output is 8 gset_defuzzify(fz_inf,'smallestofmax') #output is 7 gset_defuzzify(fz_inf,'largestofmax') #output is 9 22

24 Example using Python There are lots of Python Add-ins for fuzzy logic: Pyfuzzy, scikitfuzzy, Peach 0.3.1, gfuzzy C:\>pip install scikit-fuzzy This example is from 23

25 Fuzzy Logic: Similar to Crisp 24

26 Fuzzy Logic: Fuzzy Rules NOT x = (1 - truth(x)) x AND y = minimum(truth(x), truth(y)) x OR y = maximum(truth(x), truth(y)) Other definitions exist, but these are common, easy and recommended by Lotfi Zadeh 25

27 Fuzzy Logic Defuzzification Centroid center of gravity Bisector divides the region into two sub-regions of equal area Middle, Smallest, and Largest of Maximum (MOM, SOM, LOM) key off the maximum value assumed by the aggregate membership function. [Discussion with program examples] 26

28 Fuzzy Logic: Applications Although first proposed in the United States, fuzzy logic was most enthusiastically accepted in Asia. 27

29 Fuzzy Logic the new mark of quality Sensor Logic $29.99 Fuzzy Logic $

30 Fuzzy Logic: Summary Closer match to the way humans think Linguistic variables introduce both clarity and flexibility Fuzzification can handle incomplete and inconsistent data Rules sets can be cleaner and fewer in number Defuzzification produces quantifiable result 29

31 Fuzzy Logic: It s the Future As actuaries, we have a natural inclination towards precision. Precision is not truth. Henri Matisse, c We must exploit our tolerance for imprecision. Lotfi Zadeh,

32 Fuzzy Logic: Recommended Reading Shapiro, Arnold and Koissi, Marie-Claire, [2015] Risk Assessment Applications of Fuzzy Logic, 2013] Applying Fuzzy Logic to Risk Assessment and Decision- Making, CAS/CIA/SOA Joint Risk Management Section. Shang, Kailan and Hossen, Zakir [2013] Applying Fuzzy Logic to Risk Assessment and Decision-Making, CAS/CIA/SOA Joint Risk Management Section. L.A. Zadeh, Outline of a new approach to the analysis of complex systems and decision processes, IEEE Trans. Syst., Man, Cybernetics, SMC-3 (1973), pp Snell, David,[2014] Warm and Fuzzy And Real!, Forecasting & Futurism section newsletter Issue 9 (and Part 2 in issue 10) Heaton, Jeff, [2014] Fuzzy Logic in R, F&F newsletter Issue 9 Klir, George and Yuan, Bo [1995], Fuzzy Sets and Fuzzy Logic Theory and Applications, Prentice Hall P T R, Upper Saddle River, New Jersey,1995 in Ross, Timothy [2010] Fuzzy Logic with Engineering Applications, Third Edition, John Wiley and Sons, Ltd., UK. Ostaszewski, Krzysztof M, [1993] An Investigation into Possible Applications of Fuzzy Set Methods in Actuarial Science, Society of Actuaries Search for fuzzy logic on the SOA website for a current list of actuarial papers. 31

33 Getting Warm and Fuzzy Beyond Traditional Set Theory SOA Annual Meeting - session Austin - October 13, :00 PM 3:15 PM Dave Snell, ASA, ARA, ACS, CHFC, CLU, FLMI, MAAA, MCP technology evangelist, RGA Reinsurance Company Jeff Heaton, FLMI, ARA, ACS data scientist, RGA Reinsurance Company