So a webinar-2013-2

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So a webinar-2013-2

  1. 1. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Perspectives of Predictive Modeling Arthur Charpentier charpentier.arthur@uqam.ca http ://freakonometrics.hypotheses.org/ (SOA Webcast, November 2013) 1
  2. 2. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Agenda • ◦ ◦ • ◦ ◦ • • ◦ ◦ ◦ • Introduction to Predictive Modeling Prediction, best estimate, expected value and confidence interval Parametric versus nonparametric models Linear Models and (Ordinary) Least Squares From least squares to the Gaussian model Smoothing continuous covariates From Linear Models to G.L.M. Modeling a TRUE-FALSE variable The logistic regression R.O.C. curve Classification tree (and random forests) From individual to functional data 2
  3. 3. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 0.015 Prediction ? Best estimate ? E.g. predicting someone’s weight (Y ) −→ q qqq qqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqq qqqq q qqqqqq qq qqqq q qq qqq q 0.010 Consider a sample {y1 , · · · , yn } Density Model : Yi = β0 + εi with E(ε) = 0 and Var(ε) = σ 2 0.005 ε is some unpredictable noise 0.000 Y = y is our ‘best guess’... 100 150 200 250 Weight (lbs.) Predicting means estimating E(Y ). Recall that E(Y ) = argmin{ Y − y y∈R ε L2 } = argmin{E [Y − y]2 } y∈R least squares 3
  4. 4. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Best estimate with some confidence 0.015 E.g. predicting someone’s weight (Y ) Give an interval [y− , y+ ] such that q qq qqq q we can estimate the distribution of Y dF (x) F (y) = P(Y ≤ y) or f (y) = dx x=y 0.000 Confidence intervals can be derived if 0.005 Density 0.010 P(Y ∈ [y− , y+ ]) = 1 − α qqq qqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqq qqqq q qqqqqq qq qqqq q 100 150 200 250 Weight (lbs.) (related to the idea of “quantifying uncertainty” in our prediction...) 4
  5. 5. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Parametric inference 0.015 E.g. predicting someone’s weight (Y ) Assume that F ∈ F = {Fθ , θ ∈ Θ} 0.010 1. Provide an estimate θ Density 2. Compute bound estimates y+ = θ F −1 (1 θ − α/2) 0.005 y− = F −1 (α/2) Standard estimation technique : 0.000 −→ maximum likelihood techniques 90% 100 150 200 250 Weight (lbs.) n θ = argmax θ∈Θ log fθ (yi ) i=1   explicit (analytical) expression for θ  numerical optimization (Newton Raphson) log likelihood 5
  6. 6. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Non-parametric inference natural estimator for P(Y ≤ y) qq qqq q 0.010 qqq qqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqq qqqq q qqqqqq qq qqqq q 0.005 #{i such that yi ≤y} q Density 1. Empirical distribution function n 1 1(yi ≤ y) F (y) = n i=1 0.015 E.g. predicting someone’s weight (Y ) 2. Compute bound estimates 90% (α/2) y+ = F −1 (1 − α/2) 0.000 y− = F −1 100 150 200 250 Weight (lbs.) 6
  7. 7. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Prediction using some covariates βM 1(X1,i = F) + εi 0.010 0.005 β1 Male βF −βM with E(ε) = 0 and Var(ε) = σ 2 0.000 or Yi = β0 + Female Density based on his/her sex (X1 )   β + ε if X = F F i 1,i Model : Yi =  βH + εi if X1,i = M 0.015 E.g. predicting someone’s weight (Y ) 100 150 200 250 Weight (lbs.) 7
  8. 8. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Prediction using some (categorical) covariates E.g. predicting someone’s weight (Y ) 0.015 based on his/her sex (X1 ) Conditional parametric model 0.005 Male 0.000 −→ our prediction will be Female Density   F if X = F θF 1,i i.e. Yi ∼  Fθ if X1,i = M M 0.010 assume that Y |X1 = x1 ∼ Fθ(x1 ) 100 conditional on the covariate 150 200 250 Weight (lbs.) 8
  9. 9. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Prediction using some (categorical) covariates 0.010 0.005 Density 0.005 Density 0.010 0.015 Prediction of Y when X1 = M 0.015 Prediction of Y when X1 = F 0.000 90% 0.000 90% 100 150 200 Weight (lbs.) 250 100 150 200 250 Weight (lbs.) 9
  10. 10. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Linear Models, and Ordinary Least Squares E.g. predicting someone’s weight (Y ) 250 q based on his/her height (X2 ) q q q Linear Model : Yi = β0 + β2 X2,i + εi 200 2 Conditional parametric model q q q 150 q q 100 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q E.g. Gaussian Linear Model β0 +β2 x2 q q q Y |X2 = x2 ∼ N ( µ(x2 ) , σ 2 (x2 )) q q q q q q q q q q q q assume that Y |X2 = x2 ∼ Fθ(x2 ) q q q q Weight (lbs.) with E(ε) = 0 and Var(ε) = σ q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q σ2 5.0 5.5 6.0 6.5 Height (in.) n [Yi − X T β]2 i −→ ordinary least squares, β = argmin i=1 β is also the M.L. estimator of β 10
  11. 11. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Prediction using no covariates 250 q q q q q q q q q q q q q q q q q q q q q Weight (lbs.) q q q q q q q q q q q q q q q q q q q q 150 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q qq q q q q qq q qq q q q q qq qqqq q q q q q q qq q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q qq q q q q q q qq q q qq q q q qqqqqqq q q q q q q q q qq q q q qq q q q q q q 90% 200 q q q q q q q q 100 q q q q q q q q q q q q q q q q q 5.0 5.5 6.0 6.5 Height (in.) 11
  12. 12. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Prediction using a categorical covariates E.g. predicting someone’s weight (Y ) based on his/her sex (X1 ) E.g. Gaussian linear model Y |X1 = M ∼ N (µM , σ 2 ) q 1 E(Y |X1 = M) = nM q q q q q q q q qq q qq q q q q qq q q q qq q q q q qqqq q q q q q q qq q q qq q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q qq q q qq q q q qqqqqqq q q q q q q q q q q q q q qq q q q q q q q q Yi = Y (M) i:X1,i =M q q q q Y ∈ Y (M) ± u1−α/2 · σ q q q q q q q 1.96 Remark In the linear model, Var(ε) = σ 2 does not depend on X1 . 12
  13. 13. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Prediction using a categorical covariates E.g. predicting someone’s weight (Y ) based on his/her sex (X1 ) E.g. Gaussian linear model Y |X1 = F ∼ N (µF , σ 2 ) q 1 E(Y |X1 = F) = nF q q q q q q q q qq q qq q q q q qq q q q qq q q q q qqqq q q q q q q qq q q qq q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q qq qq q q q q q q q q q q q q q q q q q q q qq q q qq q q q qqqqqqq q q q q q q q q q q q q q qq q q q q q q q q Yi = Y (F) i:X1,i =F q q q q Y ∈ Y (F) ± u1−α/2 · σ q q q q q q q 1.96 Remark In the linear model, Var(ε) = σ 2 does not depend on X1 . 13
  14. 14. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Prediction using a continuous covariates E.g. predicting someone’s weight (Y ) based on his/her height (X2 ) E.g. Gaussian linear model Y |X2 = x2 ∼ N (β0 + β1 x2 , σ 2 ) q E(Y |X2 = x2 ) = β0 + β1 x2 = Y (x2 ) Y ∈ Y (x2 ) ± u1−α/2 · σ q q q q q q q q q qq q qq q q q q qq q qq q q q q qq qqqq q q q q q q qq q q qq q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q qq q q q q q q q q q q qq q q q q q q q qq q q q q q q qq q q qq q q q qqqqqqq q q q q q q q q qq q q q qq q q q q q q q q q q q q q q q q q q 1.96 14
  15. 15. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Improving our prediction ? sex as covariate (Empirical) residuals, εi = Yi − X T β i height as covariate no covariates Yi R2 or log-likelihood −50 0 50 100 −50 0 50 100 parsimony principle ? −→ penalizing the likelihood with the number of covariates Akaike (AIC) or Schwarz (BIC) criteria q q q q qqq q qq q 15
  16. 16. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Relaxing the linear assumption in predictions Use of b-spline function basis to estimate µ(·) where µ(x) = E(Y |X = x) 250 q 250 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5.0 q q 100 150 q q q q q q q q q q 150 Weight (lbs.) q q q q q q q 200 q q q q q q Weight (lbs.) 200 q q q 100 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5.5 6.0 Height (in.) 6.5 5.0 5.5 6.0 6.5 Height (in.) 16
  17. 17. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Relaxing the linear assumption in predictions E.g. predicting someone’s weight (Y ) based on his/her height (X2 ) E.g. Gaussian linear model Y |X2 = x2 ∼ N (µ(x2 ), σ 2 ) q q q q q q q q q q qq q qq q q q q qq q qq q q q q q q qqqq q q q q q q qq q q qq q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q qq q q q q q q q q q q qq qq q q q q q q q q q q q qq q q q q q q qq q q qq q q q qqqqqqq q q q q q q q q q q q q q qq q q q q q E(Y |X2 = x2 ) = µ(x2 ) = Y (x2 ) Y ∈ Y (x2 ) ± u1−α/2 · σ q q q q q q q q q q q q q 1.96 Gaussian model : E(Y |X = x) = µ(x) (e.g. xT β) and Var(Y |X = x) = σ 2 . 17
  18. 18. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Nonlinearities and missing covariates 250 q E.g. predicting someone’s weight (Y ) q q q based on his/her height and sex q q 200 model mispecification q q q q q q q q q q q q q q q q 150 q E.g. Gaussian linear model   β 0,F + β2,F X2,i + εi if X1,i = F Yi =  β0,M + β2,M X2,i + εi if X1,i = M q q Weight (lbs.) −→ nonlinearities can be related to q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 100 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5.0 5.5 6.0 6. Height (in.) n ωi (x) · [Yi − X T β]2 i −→ local linear regression, β x = argmin i=1 T and set Y (x) = x β x 18
  19. 19. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Local regression and smoothing techniques q q 250 250 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 100 q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 100 150 q q q q q q q q q q q q q q q q q q q q q q q 150 q q q q q q q q q q q q q q q q q Weight (lbs.) Weight (lbs.) q q q q q q q q 200 200 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5.5 6.0 Height (in.) 6.5 5.0 5.5 6.0 6.5 Height (in.) 19
  20. 20. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 k-nearest neighbours and smoothing techniques q q 250 250 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 150 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 100 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 100 q q q q q q q q q q q q q 150 q q q q q q q q q q q Weight (lbs.) Weight (lbs.) q q q q q q q q 200 200 q q q q q q q q q 5.5 6.0 Height (in.) 6.5 5.0 5.5 6.0 6.5 Height (in.) 20
  21. 21. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Kernel regression and smoothing techniques q q 250 250 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q 150 q q q q q q qq q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5.0 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 100 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 100 q q q q 150 q q q q q q qqq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Weight (lbs.) Weight (lbs.) q q q q q q q q q q q 200 200 q q q q q 5.5 6.0 Height (in.) 6.5 5.0 5.5 6.0 6.5 Height (in.) 21
  22. 22. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Multiple linear regression 10 E.g. predicting someone’s misperception 5 of his/her weight (Y ) 0 based on his/her height and weight −5 X2 −10 250 −→ linear model 200 ht ig We 6.0 150 s.) (lb E(Y |X2 , X3 ) = β0 + β2 X2 + β3 X3 Var(Y |X2 , X3 ) = σ X3 5.5 2 g Hei 100 in.) ht ( 5.0 22
  23. 23. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Multiple non-linear regression 10 E.g. predicting someone’s misperception 5 of his/her weight (Y ) 0 based on his/her height and weight −5 X2 −→ non-linear model 250 200 ht 6.0 150 s.) (lb Var(Y |X2 , X3 ) = σ −10 ig We E(Y |X2 , X3 ) = h(X2 , X3 ) X3 5.5 2 ght 100 Hei ) (in. 5.0 23
  24. 24. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Away from the Gaussian model Y is not necessarily Gaussian Y can be a counting variable, E.g. Poisson Y ∼ P(λ(x)) Y can be a FALSE-TRUE variable, q E.g. Binomial Y ∼ B(p(x)) −→ Generalized Linear Model q q q q q q E.g. Y |X2 = x2 ∼ P e q q (see next section) β0 +β2 x2 q q q q q q q q qq q qq q q q q qq q q q qq q q q q qqqq q q q q q q qq q q qq q qq q qq q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q qq qq q q q q q qq qq q q q q q q q q q q q q q q q qq q q q q q q q q q q q qqqqqqq q q q q q q q q qq q q q qq q q q q q q q q q q q Remark With a Poisson model, E(Y |X2 = x2 ) = Var(Y |X2 = x2 ). 24
  25. 25. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Logistic regression qqq q q q 0.4 of his/her weight (Y )   1 if prediction > observed weight Yi =  0 if prediction ≤ observed weight 0.0 E.g. predicting someone’s misperception qq q q q q q q q q q q q q q q q q q q q qq 0.8 q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q qq qq q qq qq qq q q 5.0 5.5 6.0 6.5 Height (in.) 1.0 Bernoulli variable, qq 0.4 0.6 0.8 q P(Y = y|X = x) = p(x)y (1 − p(x))1−y , where y ∈ {0, 1} 0.2 −→ logistic regression 0.0 P(Y = y) = py (1 − p)1−y , where y ∈ {0, 1} q qqqqqqqqqqq qqqq qq q q qqqqqqqq qq q q q q q q qqqqq qqqqqqqqqqqq q qq q qqqqq qqqqq qqqqq q 100 150 200 q q q 250 Weight (lbs.) 25
  26. 26. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Logistic regression 1.0 −→ logistic regression 0.8 y P(Y = y|X = x) = p(x) (1 − p(x)) 1−y , 0.6 where y ∈ {0, 1} P(Y = 1|X = x) Odds ratio = exp xT β P(Y = 0|X = x) 0.4 0.2 0.0 250 200 ht ig We 6.0 150 s.) (lb exp xT β E(Y |X = x) = 1 + exp (xT β) Estimation of β ? 5.5 g Hei 100 in.) ht ( 5.0 −→ maximum likelihood β (Newton - Raphson) 26
  27. 27. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Smoothed logistic regression qqq q q q 0.4 0.0 GLMs are linear since P(Y = 1|X = x) = exp xT β P(Y = 0|X = x) qq q q q q q q q q q q q q q q q q q q q qq 0.8 q q q qq q q q q q q q q q q q q q q q q q q q q q qq q q q qq qq q qq qq qq q q 5.0 5.5 6.0 6.5 q qq 0.8 q qqqqqqqqqqq qqqq qq q q qqqqqqqq qq q q q 0.2 0.0 exp (h(x)) E(Y |X = x) = 1 + exp (h(x)) 0.4 0.6 −→ smooth nonlinear function instead P(Y = 1|X = x) = exp (h(x)) P(Y = 0|X = x) 1.0 Height (in.) q q q qqqqq qqqqqqqqqqqq q qq q qqqqq qqqqq qqqqq q 100 150 200 q q q 250 Weight (lbs.) 27
  28. 28. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Smoothed logistic regression 1.0 −→ non linear logistic regression P(Y = 1|X = x) = exp (h(x)) P(Y = 0|X = x) 0.8 0.6 0.4 0.2 E(Y|X = x) = exp (h(x)) 1 + exp (h(x)) 0.0 250 200 ht ig We s.) (lb Remark we do not predict Y here, 6.0 150 5.5 ght Hei 100 but E(Y |X = x). ) (in. 5.0 28
  29. 29. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Predictive modeling for a {0, 1} variable 1.0 What is a good {0, 1}-model ? Observed Observed −→ decision theory 0.2 Y=1 q 0.6 q Y=0 ^ Y=1 qq qqq qqqqq qq qq q qq qqq qqqq qqq qq qqq qqqq qq q q qqq qq q qq qq q 0.4 ^ Y=0 q 0.8 Predicted 0.0   if P(Y |X = x) ≤ s, then Y = 0  if P(Y |X = x) > s, then Y = 1 q qqqqqqqqqqqqqqqqq q q qqqqqqqqqqqqq qq q qqq qqqq qqq q qq qqqqqq q qqq q q q qq qq qqq q q 0.0 0.2 0.4 q 0.6 q 0.8 1.0 0.8 1.0 1.0 Predicted error Y =1 error fine 0.6 fine 0.4 Y =0 q 0.2 Y =1 q 0.0 Y =0 True Positive Rate 0.8 q 0.0 0.2 0.4 0.6 False Positive Rate 29
  30. 30. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 q Observed Observed Y=0 T P (s) = P(Y (s) = 1|Y = 1) nY =1,Y =1 = nY =1 False positive rate ^ Y=1 0.0 0.2 Y=1 q qq qqq qqqqq qq qq q qq qqq qqqq qqq qq qqq qqqq q q q qqq qq q qq 0.6 ^ Y=0 q 0.8 Predicted 0.4 True positive rate 1.0 R.O.C. curve q qqqqqqqqqqqqqqqqq q q qqqqqqqqqqqqq qq q qq qqqqqq q qqq qqqq qqq q qq q q q q q 0.0 0.2 0.4 q 0.6 q 0.8 1.0 F P (s) = P(Y (s) = 1|Y = 0) nY =1,Y =1 = nY =1 q q 0.6 0.4 0.0 0.2 True Positive Rate 0.8 q R.O.C. curve is {F P (s), T P (s)), s ∈ (0, 1)} 1.0 Predicted 0.0 0.2 0.4 0.6 0.8 1.0 False Positive Rate (see also model gain curve) 30
  31. 31. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Classification tree (CART) 250 q q q q If Y is a TRUE-FALSE variable 200 E(Y |X = x) = pj if x ∈ Aj q q q q q q q q q q q q q 150 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q regions of the X-space. q 100 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q where A1 , · · · , Ak are disjoint q q Weight (lbs.) prediction is a classification problem. q q q q q q q q q q q q q q q q q q q q 5.0 5.5 6.0 6.5 Height (in.) 31
  32. 32. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Classification tree (CART) −→ iterative process q 250 P(Y=1) = 38.2% P(Y=0) = 61.8% Step 1. Find two subset of indices either A1 = {i, X1,i < s} or A1 = {i, X2,i < s} q q q q q 200 q q q q q q 150 q q q 100 q & maximize heterogeneity between subsets q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q maximize homogeneity within subsets q q q q q q q q and A2 = {i, X2,i > s} q q Weight (lbs.) and A2 = {i, X1,i > s} q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q P(Y=1) = 25% P(Y=0) = 75% q 5.0 5.5 6.0 6.5 Height (in.) 32
  33. 33. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Classification tree (CART) 250 q q q q q q y∈{0,1} 0.317 q q q q q q q q q q q 150 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 100 x∈{A1 ,A2 } nx,y nx,y · 1− nx nx q q 0.523 Weight (lbs.) E.g. Gini index nx − n 200 Need an impurity criteria q q q q q q q q q q q q q q q q q q q 0.273 q 5.0 5.5 6.0 6.5 Height (in.) 33
  34. 34. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Classification tree (CART) 250 q Step k. Given partition A1 , · · · , Ak q q q 0.472 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 100 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q maximize homogeneity within subsets & maximize heterogeneity between subsets q q q 150 or according to X2 q 0.621 q 0.444 Weight (lbs.) either according to X1 200 find which subset Aj will be divided, q q q q q q q q q 0.111 q q q q q q q q q q 0.273 q 5.0 5.5 6.0 6.5 Height (in.) 34
  35. 35. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 Visualizing classification trees (CART) Y < 124.561 | Y < 124.561 | X < 5.39698 0.3429 X < 5.69226 0.1500 X < 5.69226 0.2727 X < 5.52822 0.4444 0.5231 0.3175 Y < 162.04 0.6207 0.1111 0.4722 35
  36. 36. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 From trees to forests q q 250 Problem CART tree are not robust −→ boosting and bagging q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.808 Then aggregate all the trees q q q 0.520 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 100 q q q q q q q q q q q q q q q q q q q q 150 repeat this resampling strategy q q q q q q q Weight (lbs.) and generate a classification tree 0.467 200 use bootstrap : resample in the data 0.576 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.040 q q q q q q q q q q q q q q q q q q q q q 0.217 q q 5.0 5.5 6.0 6.5 Height (in.) 36
  37. 37. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 10 A short word on functional data q q q q −15 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q Individual data, {Yi , (X1,i , · · · , Xk,i , · · · )} q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 0 −5 −10 q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q −20 Temperature (in °Celsius) q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 5 10 15 Week of winter (from Dec. 1st till March 31st) Functional data, {Y i = (Yi,1 , · · · , Yi,t , · · · )} qq qq 0 −10 −20 −30 Temperature (in °Celsius) 10 E.g. Winter temperature, in Montréal, QC q q qq q qq q q qq q q q qq q q q q qq q qq qqqq q q q q q qqqqqqqqq q qqq qqq qq q q q qq q q qq q qq q q qq qqqqq q q qq q q qqq qqqqqqqqqqqqqq qqq qqqqqq q qqq q q q qq q q qq qq q qq q qq q qq q q q qqq q q q q q q q q qqq q qq qq qqqqqq qqqqqqqqqqqq qqqqqqqq q qq q q qqqq q qq q q q q qqqq qqqqqqqq qqqqqqq q q qq q q q q q q qq q q q qqqq q qq qqq q q qq q qq q qqq q q q q q qqqq q qqqqqqqqqqqq q qqqqqqq qq qqqqq q q q qqqqq qq qqq q q q q qq q q q q qq qq qqqq qqqqqqq q qq q q q q q qq qqqqqqq qqqqq qqqqqqqq qq q q q q qqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqq qq q qqqq q qq qqq qq q q q q q q qqq qqq q q qq qqq qq q qq qq q q q q q qqqqq q qq q qq qq qqqqq qqqqqqqq qq qqqq qqqqqqq qqqqqqqqqqq q qqqq qqqqqqqqq q qq qq qqq qqqqqq qq q qqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqq q qqq q q q qq qqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqq q qqqqq qqqqqqqqqqqqqq qqq q qqq qq q qqq q qqqqqqqqqq qq qqqqqqq qqqqqqqqqqqqq qqq q qqq qqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqq q q q qq q qqqqqqqqqqqqqq q qqqqqqqqqq qqqqqqq q q qqq q qq qq q qqq q qqq q q qqq q q qq qqq qqqqqqqqqqq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqq qqqqqqqq qqqq q qqqqqq q qq qq q q qq qqqq q qq q qq q qq qq qqqq qq qqqq q q q q q qqq qq qq qqq q qqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqq qq q qqqqqq qqqqq qqqqqqqq qqqqq q q qq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q q q qq q qq qqqq qq qqqqqqqqqq qqqqqqqqqqqq qqqqqqqqqqq qqqq qqqqqqq qqqqqq qq qqqqqq q q q q q q q q q q q q q qq qq q q q q qqq qqqqqqqqq qqqqqqqqqqqqqqqqqqq q q qqq q qq qqq qqqqqqqqqqqqq qqqqqqq qqqqqqqqqqqqqqqq q qqqqqqq q qqq qqqq qq qq qqqqq q q qqqq q q q q qqqqqqq qqq qqqqq qq q qqq qqqqqq qqqq q q qq q q qqqqq q q q qq qqqqqqqq qqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qqqq qq q q q qqqqq q q qq q q q q q q q q qq qqqq qqq qqqqqq q qq q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qq q qqqq qq qq qqqq qqqq q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qq qq qqqqqqqqqqqqqqqqqqqqqqq q q qqqqqqqqqqqqqq q qqq q qq qqqqq qqqqq q qqqq q q qqq q q qq qqqqqqqq qqqqq qqqq qq qq q qq q qqq qq qqqq q q q qqq q qq q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqq qq qqqqq q qqqqqqqq q qqqqqq qqqqqqqqqqqqqqqqq qqqqqq qqqq qqq q qqq qq q qq q q qqq q qqq q qqqq qq qqqq q qqqqqqqqq qqqqqq q qq q q qq q q qq qq q q q q q qqqq q qqqq qq q q q qqqqq qq q q q q qq q qqqqqqqqqqqq qqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqq q qqqqqqqqq qqq qqqqqqq qqqqq qqqqqqqq qqqqqqqqqqqqq qq q qq qq q q q qq qqq q qqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q q q qq qqqq q q q q q qqqqq q q qqq qqq q qq qqq q qqqq qqqqqq q q qqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqq q q q q q q qqq qqq q q qq q q q qqqqqq qqq qqqqqqq qqqqqqq q qqq q q q q q q q q q qq q qqqqq qqqq q qqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqq qqqqqqqqqqq qq q qqqq q qq q q q q q qq qqqqqqqqqqqqqqq qq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqq qqqqq qqqqqqqqqqqq q q qqq qq qq q qqqq qqqqq q qq q q q q qqqqq q qqq qq qqqqq qqqq qq qqq q qq qqq qqqqq q q qqqqqqqq q qqq q qqq q q qq qqq q qq qqqqq q q qqqqqqq qq qq qq qq qq qq q qq q qq qqq q qqqqqqq qq qq q qqqq qq q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqq qqqqqqqqqqqqqqqqqqqq qqqqqq qqqq qqqqqqqqqqqqqq qq qq qq q q q q q q qq qqqqq q qqq qqq q qq qq q qqq qqqqq qq qqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqq qq qqqqqqqqqqqqqqqqq qq q q q q q qq q q qq qq q q qqq qqq q qq qq q q q q q q q qqqq qqqqqq qqqqqq qqqq q qqq q q qq q q q qqqq q qqqqqqq q q q qqqq qqqqqqqq qq q qqqqq q q q qqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqq qq q q q q q q qqq q q q q qqq q qqq qqqqq qq qqqq q q qq qqqqq qqqq q qqq q qqqqq q q q qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqqq q qqqqq q q q qqq q q q qq q qqqq qq qqqqqqqq qq q qqqqqqqqqqq qqqqqqqqqqqqq qq qqq q q qqqq q q qqqqqq q q q qq q q q q q q qqq q q q qqq qqqqqq qqqq q qqqqqqqqqq q qqq q qqqqqqqq qq qq qq qq qqqq qqqqq qq qqqq qq qq q qqqqqqqqqq q q q q q q q qqqq q q q qqq q qqqqqqqqqqqqqqqqqq q qqqqqqqqq qqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqq qqqqqqqqqq q q q qqq q qq q qqq qq qq qq qqqq q q q q qqq q qqq q qq qqqqqqqqqqqqq qqqqqqqqq qq q qqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqqqqqqqqqqq qqqqqqq q qq q qqqq q qqqq qqqqq qqqqq q q q q qq q q q q qq qq qq qq q q qq qqq qqqqq q q qqqqqqqqq q qqqqqq q qq q q qqqq qq q q qq q q q qqqq q q qq q q q qq qq q qq qqqq qqqqqqqqqqq qqqqqqqqqqqqqqqq qqqqqqqqqqqqqqqqqq q q q qq q q q q qqqqq qqqqq qqqqqqq q q q q q q qqqq qqqqq q q q q qqqqq qqq qqqq q q q q q q qqqq q q q q q q qq qqq q q q q q q q qq q q q qqqqqqq qqq qq qqqqq qqqqqqqqqqqqqqq q qq qqq qqqqqqqqqqqqqqq qq q q qq q qq q q q q q q q q q qq q q qq qqqqqq qqqqqqq qqq qq q qqq qq q q qq q qqq q qqqq qqqq qq q qqqq qqqqqqqqqqqqqq qq q q q q q q q q q qqqq qqq qq q q q q qqq qq q q qq qqq q q q q q qqqqqq q qqq q q q q qq qqq q q q qq q q q q qqqq q q qqqq qqqqqq qq q qq q q q qqq q qq qq q q q q q q qq q q q q qqq q q q qqq q q q qqqqqq q q q q q q q q q q qq 0 20 40 60 80 100 120 Day of winter (from Dec. 1st till March 31st) 37
  38. 38. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 A short word on functional data Daily dataset 2001 1982 q 1974 1999 2007 0 PCA Comp. 1 40 50 q q q q 2006 2009 q q 1957 q 2011 1997 20 q q −50 q 1986 2005 q 1961 qq qq q q q qqqq qqqq q qq qq q qq q q q qq qqq qqqq q q q q qq qq q q qqqqq q qqq q qq q q q q qqqqqqqqqqq q q q q q q qq q qqq q qq q qqqqqqq q q qqq q q qq q q qqqq qqq q q q qqqq q q qq q q q q qq q qq qq qq qq q qqqq q qq qqqq qqq qq q qq qqq q qq q qqqq q qq q q q q qq q q q q q qq q qqq qq q qq qq qq qq q qqq qqq q qqq qq qq q q qq q q q qq q qq q q q q qq qq qq qq q q q q qq q q q q q q qq qqqqq qq q qq qq qq qq q q q qq qq qq qq q q q q q qq q q q q q q q qq q q q qq q q qqq q qqqq qqq q q qq qq q q qq q q qq q q q qq q q qq q q qq q qqq q q q q qq q q qqq q qqqq q qqq q q q q qq q q qq q q qq q q q qqqqq qq q q q q q q q qqq q qqq q q q q q q q qq q q q qqq qqq q q q q q q q q q qqqq qq q qqq qqq qqq q q q qq q qqqq qqq q qqq qqq q q qq q qqq q qqqqqq q q q qq q q qq qqq q qqqq qqq qq q qq qq q qqqq q q q qq q q q q q qq qqqq q q qqq q qqq qqqq q q q qqq q q q qq q qqq q qqqqqqq q q q q q qq q q q qqq q q q qqq q q q q q q q qqq q q qq q qq q qq q qqq q qqqq qqqq q q qq qqq q q q q qq qqq q qq qq q q qq qqq q qq q q qq q qq qq qq qqqqq q q qqq q qqq qqq q qqqq q q qq q qqqq qqqq qq q q qqq q qq q qqqqqqq q q qq qq q q q qq q q qq q qq q q q q q qqqq qqqq qq q q q q qqq q qqq q q qq qqqqqqq q qqqq qq qqqq qq q q q qq q q q q qq qqq qq q qq qqq q qq q qq q q q q q q q q qq q q q q qq qq q q q q qqq qqqqqqqqqq q qq qqqq qq q qq qq qq q qqqq q q q q q qq q qqq q qq qqq qq q q q qqqqqq q q qqqq qq qqq q q qq q qq q q qqq q q qq q q q q q qq qq qqqqqqqq qq q qq q q q qqq q qq qq qq qq q q q q qq qq q q q q q q q qq q q qqq q qqqqqq q q qqqq qq qq q qq q q q qq q qq qqqq qqqq qq q qq q qq q qq q q qq qqq q q q qq q q q qq qq q q q qqqq qq qqq q q q q qqq q qq q qq q qq q qq qq qq q qqqqq qq qqq q q qqqqqq q q qqq q qq q qq q q q qq q q qqqq q qq 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q q qqq qq q qq q q q q q q qq q q qqq q qq qq q JANUARY DECEMBER FEBRUARY MARCH q 1979 q Comp.2 0 q q q q 1994 1996 1966 1964 1973 2010 1955 1978 q 20 40 60 80 100 120 Day of winter (from Dec. 1st till March 31st) 1987 q qq 1953 q q 1962 1954 1991 q q q 1989 q 1960 0 q 2000 q 1988 1963 1965 1959 1990 q q 30 q q 1992 1971 2004 q q q q 2002 1998 q 1977 1972 1985 20 q q q 1984 q q 2008 q 1968 q 10 1956 1995 q q 2003 q 1969 1975 1983 q q 1976 q q 1993 q 1980 1967 −20 q q −20 −10 0 10 Comp.1 20 30 40 50 −30 −20 1981 0 q q −10 1958 1970 PCA Comp. 2 q q q q q q q q q q q q qq q q q qq qq q q q q q qq q q q q q qq q q q q q qq qq qq q q q q q q q qq q q q qq qq q q q q q q q qq q q q qq q q q q q q q q q q q qq q q q q q q qq qq q q q q q q q q q q qq q q qq q qq q q q q q qq q q qq q qq q q q qq q q q q qq q q q qq q q qq q q q q q qq q qq q qq q qqqq q qq q q q q qqq q q q q qq q q q q q q q q q qqq q q qq q q qq q q q q q q q q q q q q q q qq q q qq q q q q qq q qq q q q q q q q q q q qq q 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q q qq q q q q q q q q qq q q q q q qq qq q qq q q q qq q qq q q q qq q q q qq qq q q qq q q q q q q q q q q q q q q qq q q q q q qq qqq q q q q qq q q q qq q q q q q qqq qq qq q q q qq q qq q q qq q qq q q qq qq q q qq q q q q q q qq qq q q q q q q qqq q q q q q q q q q q qqq qq q qq q q q q q q q q qq q q qq q qq q qq q q q qq q qq q qq q q q q q qq q q q q q q q qq qq q q q q q q q qq qq q q q q q qq q q q q q q q qq qq q q q q qqq qq q q qq q q q q q qq q q qq q q q qqqq qq q q q q q q qq q q q qq q qq qq q qq q qqqq qq q q qq qq q q q q qq q qq q qq q q q q q qq q qq q q q q q q qq q q q qq q q q q q qq qq q q q q q q q qq qq q q q q q qq q q q q qq q q qq q qqq q q qq q q q qqq q q q q q qq q qq q q qq q q qq q q q q q q qq q q q qq q qq q qq q q qq q q q q qqq q q q q qqq q q q q q qq q qq q q qq q q q qq q q q q q q qq q qq q q qq q q q q q q q qq q qq qq q qq q q q qq qq q q q q q qq q q q q qq qq q qq qq q q q q q q qq q qq q q qq q qq qq qq q qq q q qq q q q qq q q qq q q q q q qq q qq q q q q q q q q q qq q q q q q qq q q qq q qq q q q q q q q q qq q q q q q q q qqq q q q q q q q q q qq q q q qq q q qq q q q q q qqq q q q q q q q q q q qq q qqq q q q qq qqq q q q qq q q q q qq q qqq q q qq qq qq q qq q q q q q q q qqq qq q qq qq q q q q qq qq qq q qqq q q qq qq q q qqq q q q q qq q qq q q qq q q qq q q qq qq qq q q qq qq q q q qq q qq q q qq q q q q q qq q q q q q q qqq q q q qq qq q qq q qqq q q q q qq q qqq q q q q q qqq q q q q q qq q qq q qq q q q q q q q q q q qqq q q q q q q q q q qq q qq q q q qq qq q q qq q q q q qqq qq qq q q qq qq qqqqq q q q q q q qq q q q q q q q qq q qq q q qq q q q q q q q q q q qq q qq q q q q qq q q qq qq qq q q qq qqq qq qq q q qq qq qq q q qq q q q q qq q q q qqq qq q qq q qq qq q q q q q q qq q qqq q qq qq qqq qq q q qq qq q q q q q q q q q q q q q q q q q q q q 0 20 40 60 80 100 120 Day of winter (from Dec. 1st till March 31st) 38
  39. 39. Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013 To go further... forthcoming book entitled Computational Actuarial Science with R 39

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