2017 Predictive Analytics Symposium

Transcription

1 2017 Predictive Analytics Symposium Session 31, Beyond Crisp Logic--Welcome to the Fuzzy Real World! Presenter: Geoffrey R. Hileman, FSA, MAAA SOA Antitrust Compliance Guidelines SOA Presentation Disclaimer

2 2017 SOA Predictive Analytics Symposium Geof Hileman Fuzzy Logic September 15, 2017

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4 Our world is full of fuzzy questions. Are you old? Am I bald? Is this presentation interesting? Are we on the lake? Is it early in the day? Aren t we just playing games with precision in language? Or does this matter?

5 Which holds more value? Ask Heraclitus 4

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7 The Paradox of the Heap 6

8 Sorites Paradox Sorites Paradox, or Paradox of the Heap: - Premise 1: A billion grains of sand constitutes a heap. - Premise 2: A heap of sand minus one grain is still a heap. - Conclusion: A single grain of sand must still be a heap. Fuzzy logic provides a way out of the paradox, by establishing a range of heapiness from definitely not a heap to definitely a heap. Our last foray into ancient Greek philosophers! We owe this one to Eubulides, also the holder of the honor as the inventor of the terrible joke today is opposite day.

9 This sounds a lot like probability theory. From Kosko: The chief, but superficial similarity, is that both systems describe uncertainty with numbers in the unit interval [0,1]. The key distinction concerns how the system deal simultaneously with a thing A and its opposite A. Fuzziness describes event ambiguity. It measures the degree to which an event occurs, not whether it occurs. Randomness describes the uncertainty of event occurrence. An event occurs or not. Kosko (1989). Fuzziness vs. Probability. International Journal of General Systems.

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11 This is fun, but what does this have to do with my models?

12 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)

13 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

14 Fuzzy Logic: Better fit to Reality

15 Fuzzy Logic: Process (The Fuzzy Controlled Machine) 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). We do this every day! Consider the decision on whether to pull out into traffic

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:

17 Fuzzy Underwriting From the SOA Archives 16

18 Source: The Application of Fuzzy Sets to Group Health Underwriting. Virginia R. Young. Transactions of the Society of Actuaries: 1993 (45). 17

19 Consider desirable characteristics of a prospective group of insureds (health): Rules best represented by non-fuzzy sets: A) Obtaining insurance is incidental to the purpose of the group. B) Employment status determines eligibility for health coverage (actively-at-work). C) A minimum number of people are in the group. D) Benefits are determined automatically. Now consider four categories of risk: 1) Preferred risk: ) Normal risk: 0.50 to ) Substandard risk: 0.25 to ) Unacceptable risk: 0.00 to 0.25 Threshold Criteria 18 Source: The Application of Fuzzy Sets to Group Health Underwriting. Virginia R. Young. Transactions of the Society of Actuaries: 1993 (45).

20 Fuzzy characteristics of a desirable group: E) Young lives flow constantly into the group and old lives out. a/x = Annual % change in age/sex factor over past two years g/s = Annual % change in the group size during the past two years 19 Source: The Application of Fuzzy Sets to Group Health Underwriting. Virginia R. Young. Transactions of the Society of Actuaries: 1993 (45).

21 Fuzzy characteristics of a desirable group: F) Minimum participation in the plan. 20 Source: The Application of Fuzzy Sets to Group Health Underwriting. Virginia R. Young. Transactions of the Society of Actuaries: 1993 (45).

22 Fuzzy characteristics of a desirable group: G) The employer pays some or all of the cost of insurance. Employee Dependent 21 Source: The Application of Fuzzy Sets to Group Health Underwriting. Virginia R. Young. Transactions of the Society of Actuaries: 1993 (45).

23 Fuzzy characteristics of a desirable group: H) A strong, central department of the employer helps the insurer administer billing, enrollment, and certification of eligibility. (Assigned 0 to 1 by the underwriter). I) The industry of the group is acceptable. U/W assigned. J) The policyholder has a good credit rating. U/W assigned. K) Ongoing claims are not large as a proportion of total expected claims. c is the percentage of ongoing claims as a proportion of total expected claims. As in <0.5% is optimal. How quaint! (Ah, 1993). 22 Source: The Application of Fuzzy Sets to Group Health Underwriting. Virginia R. Young. Transactions of the Society of Actuaries: 1993 (45).

24 Fuzzy characteristics of a desirable group: L) Loss ratio Excludes retention 23 Source: The Application of Fuzzy Sets to Group Health Underwriting. Virginia R. Young. Transactions of the Society of Actuaries: 1993 (45).

25 Fuzzy characteristics of a desirable group: M) The group does not change insurers often. 24 Source: The Application of Fuzzy Sets to Group Health Underwriting. Virginia R. Young. Transactions of the Society of Actuaries: 1993 (45).

26 Combining into an underwriting rating Created a linear combination of employer administrative ability, industry, and ongoing claims burden. Considered the interactions between various criteria. Combined into a single criterion: 25 Source: The Application of Fuzzy Sets to Group Health Underwriting. Virginia R. Young. Transactions of the Society of Actuaries: 1993 (45).

27 Source: The Application of Fuzzy Sets to Group Health Underwriting. Virginia R. Young. Transactions of the Society of Actuaries: 1993 (45). 26

28 Fuzzy Matching 27

29 Will the real Bob Clemente please stand up? Name Matches Bill Virdon B,l,l,n (4) Bob Skinner B,o,b,n,n,e (6) Roberto Clemente O,b,e,t,o,c,l,e,m,e,n,t,e (13) Roman Meijas R,o,m,n,M,e (6) Joe Christopher O,e,C,t,e (5) Harry Bright B,t (2) Dick Schofield C,c,o,e (4) Harry Simpson M,o,n (3) Rocky Nelson O,c,N,e,o,n (6) Dick Stuart c,t (2) 28

30 R package stringdist can produce a variety of fuzzy text string metrics. > stringdist(c("bob Clemente"),c("Bill Virdon","Bob Skinner","Roberto Clemente", "Ed Clemente")) [1] Default method optimal string alignment > stringdist(c("bob Clemente"),c("Bill Virdon","Bob Skinner","Roberto Clemente", "Ed Clemente"), method='qgram') [1] Q-Gram still confused by Ed > stringdist(c("bob Clemente"),c("Bill Virdon","Bob Skinner","Roberto Clemente", "Ed Clemente"), method='cosine') [1] Cosine distance between Q-gram profiles finally gets it More info: 29

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32 And not just names, of course! - Addresses (123 S. Main St. vs 123 South Main Street) - City / County Name (St. vs Saint) - Spellcheck / Autocomplete

33 Fuzzy Logic in R Example Source: 32

34 library(sets) sets_options("universe", seq(1, 100, 0.5)) variables <- set( temperature = fuzzy_partition(varnames = c(cold = 30, good = 70, hot = 90), sd = 5.0), humidity = fuzzy_partition(varnames = c(dry = 30, good = 60, wet = 80), sd = 5.0), precipitation = fuzzy_partition(varnames = c(no.rain = 30, little.rain = 60, rain = 90), sd = 7.5), weather = fuzzy_partition(varnames = c(bad = 40, ok = 65, perfect = 80), FUN = fuzzy_cone, radius = 10)) 33

35 # Fuzzy rules rules <- set( fuzzy_rule(temperature %is% good && humidity %is% dry && precipitation %is% no.rain, weather %is% perfect), fuzzy_rule(temperature %is% hot && humidity %is% wet && precipitation %is% rain, weather %is% bad), fuzzy_rule(temperature %is% cold, weather %is% bad), fuzzy_rule(temperature %is% good humidity %is% good precipitation %is% little.rain, weather %is% ok), fuzzy_rule(temperature %is% hot && precipitation %is% little.rain, weather %is% ok), fuzzy_rule(temperature %is% hot && humidity %is% dry && precipitation %is% little.rain, weather %is% ok) ) 34

36 model <- fuzzy_system(variables, rules) print(model) plot(model) 35

37 example.1 <- fuzzy_inference(model, list(temperature = 75, humidity = 0, precipitation = 70)) gset_defuzzify(example.1,"centroid") plot(example.1) > gset_defuzzify(example.1,"centroid") [1] 65 36

38 example.2 <- fuzzy_inference(model, list(temperature = 30, humidity = 0, precipitation = 70)) gset_defuzzify(example.2,"centroid") plot(example.2) > gset_defuzzify(example.2,"centroid") [1]

39 Source: An Overview of Insurance Uses of Fuzzy Logic. Arnold F. Shapiro. c1.pdf?sgwid= p And Beyond Fuzzy concepts have extended into a variety of areas: Linguistic variables and fuzzy set theory Fuzzy arithmetic ( fuzzy interest rate calculations ) Fuzzy inference systems Fuzzy clustering Fuzzy programming (i.e. optimization, not computer code) Fuzzy regression 38