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Southwest Actuarial Forum

Southwest Actuarial Forum. Interactions and Saddles. Serhat Guven, FCAS, MAAA June 10, 2011. Agenda. Background Interactions Saddles Validation. Simple Model: Age + Gender. Male. Female. Youthful. β 0 + β Y. β 0 + β Y + β F. Adult. β 0. β 0 + β F. Mature. β 0 + β M.

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Southwest Actuarial Forum

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  1. Southwest Actuarial Forum Interactions and Saddles Serhat Guven, FCAS, MAAA June 10, 2011

  2. Agenda • Background • Interactions • Saddles • Validation

  3. Simple Model: Age + Gender Male Female Youthful β0+ βY β0+ βY + βF Adult β0 β0+ βF Mature β0+ βM β0+ βM + βF Seniors β0+ βS β0+ βS + βF 5 Parameters Background SIMPLE MODEL • Relationship between rating levels of one factor is constant for all levels of other rating variables • Assume two rating variables • Age: Youthful, Adult (Base), Mature, Senior • Gender: Male (Base), Female

  4. Simple Model: Age + Gender Male Female Youthful β0+ βY β0+ βY + βF Adult β0 β0+ βF Mature β0+ βM β0+ βM + βF Seniors β0+ βS β0+ βS + βF 5 Parameters Background SIMPLE MODEL Difference between male and female is the same regardless of age

  5. Simple Model: Age + Gender Male Female Youthful β0+ βY β0+ βY + βF Adult β0 β0+ βF Mature β0+ βM β0+ βM + βF Seniors β0+ βS β0+ βS + βF 5 Parameters Interactions INTERACTION MODEL • Relationship between rating levels of one rating factor is different for all levels of another rating variable • Assume two rating variables • Age: Youthful, Adult (Base), Mature, Senior • Gender: Male (Base), Female Interaction: Age + Gender+Age.Gender 8 Parameters

  6. Background INTERACTION MODEL Interaction: Age + Gender+Age.Gender Youthful males are significantly higher than youthful females – resulting structure is still volatile 8 Parameters

  7. Background • Why are interactions present • Because that's how the factors behave • Because the multiplicative model can go wrong at the edges • 1.5 * 1.4 * 1.7 * 1.5 * 1.8 * 1.5 * 1.8 = 26! • Interaction challenges • Identification • Given n factors there are n choose 2 possible interactions to study • Simplification • Once an interaction has been identified the structure needs to be simplified to avoid overfitting

  8. Gender b b - b b b b b b b b - - - - - - - - - - b - - - - - - - - - - b - - - - - - - - - - - - - - - - - - - - - Age b - - - - - - - - - - b - - - - - - - - - - b - - - - - - - - - - b - - - - - - - - - - b - - - - - - - - - - b - - - - - - - - - - Interactions • Age x Gender • Main effects construction

  9. Interactions • Imbalances within the cells exists – suggesting some form of interaction

  10. Gender b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b - - - - - - - - - - - Age b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b b b b - b b b b b b b Interactions • Age x Gender • Full Interaction

  11. Interactions • Rating Area x Vehicle Value • Interaction needs to be simplified Vehicle Value Vehicle Value Vehicle Value b b - b b b b b b b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b Rating Area - - - - - - - - - - - - - - - - - - - - - - Rating Area b - - - - - - - - - - Rating Area b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b b - - - - - - - - - - b b b - b b b b b b b

  12. Interactions • Rating Area x Vehicle Value • Simplification emphasizes interaction strength

  13. Saddles • Quadrant saddle: revisiting an simple main effect model

  14. Saddles • Quadrant saddle: interaction terms twist the paper

  15. Saddles

  16. Saddles

  17. Saddles

  18. Saddles

  19. Saddles • Transforming predictors into single parameter variates Response Factor Levels

  20. b Saddles • Transforming predictors into single parameter variates Response Factor Levels

  21. Simple Model: Age + Gender Variate Model: a1*bAge+ a2*bGender Male Male Female Female Youthful Youthful β0+ a1*βY β0+ βY β0+ βY + βF β0+ a1*βY + a2*βF Adult Adult β0 β0 β0+ a2* βF β0+ βF Mature Mature β0+ a1*βM β0+ βM β0+ βM + βF β0+ a1*βM + a2*βF Seniors Seniors β0+ βS β0+ a1*βS β0+ βS + βF β0+ a1*βS + a2* βF 3 Parameters 5 Parameters Saddles • Parameterization • Fitted values from both structures are the same • Variates • X-Values are the beta parameters from the simple model • Coefficient should be 1.000

  22. Saddles • Parameterization • Nonlinear terms introduced via interaction among variates Full Saddle: a1*bAge + a2*bGender + a3*bAge * bGender

  23. Saddles • Advantages • Non linear variate parameter • Easier to validate • Simpler shape • Useful in identifying interactions in more volatile structures • High dimensional factors • N-way interaction structures • Severity models • Disadvantages • Difficult to detect interactions when a factor is not a main effect

  24. Saddles • Rating Area x Vehicle ValueRevisited

  25. Saddles • Rating Area x Vehicle ValueRevisited

  26. Saddles • Rating Area x Vehicle ValueRevisited

  27. Saddles • Driver Age x Vehicle Age

  28. Validation • Model Comparison • Auto frequency: Out of sample

  29. Validation • Model Comparison • Auto frequency: Out of time

  30. Validation • Model Comparison • Renewals: Out of sample

  31. Conclusion • Interactions are an important part of the model creation process • Interactions have their own challenges • Identification • Simplification • Dimensionality • Effort needed in simplifying and validating identified interaction • Saddles use the framework of variate vectors in the design matrix to quickly simplify and validate new interaction terms

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