Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Perspectives of Predictive Modeling Arthur Charpentier
[email protected] http ://freakonometrics.hypotheses.org/
(SOA Webcast, November 2013)
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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
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Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
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Prediction ? Best estimate ? E.g. predicting someone’s weight (Y )
ε is some unpredictable noise
Yb = y is our ‘best guess’...
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with E(ε) = 0 and Var(ε) = σ 2
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Model : Yi = β0 + εi
−→ Density
Consider a sample {y1 , · · · , yn }
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Weight (lbs.)
Predicting means estimating E(Y ).
ε
z }| { Recall that E(Y ) = argmin{kY − ykL2 } = argmin{E [Y − y]2 } y∈R y∈R | {z } least squares
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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
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we can estimate the distribution of Y dF (x) F (y) = P(Y ≤ y) or f (y) = dx x=y
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Confidence intervals can be derived if
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Density
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P(Y ∈ [y− , y+ ]) = 1 − α
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Weight (lbs.)
(related to the idea of “quantifying uncertainty” in our prediction...) 4
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Parametric inference 0.015
E.g. predicting someone’s weight (Y )
−→ maximum likelihood techniques
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90% 0.000
yb− = F −1 (α/2) b θ yb+ = F −1 (1 − α/2) b θ Standard estimation technique :
Density
2. Compute bound estimates
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Assume that F ∈ F = {Fθ , θ ∈ Θ} b 1. Provide an estimate θ
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Weight (lbs.)
n X b θ = argmax log fθ (yi ) θ∈Θ
i=1
|
{z
log likelihood
explicit (analytical) expression for θ b numerical optimization (Newton Raphson)
} 5
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Non-parametric inference
2. Compute bound estimates yb− = Fb−1 (α/2) yb+ = Fb−1 (1 − α/2)
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natural estimator for P(Y ≤ y)
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90% 0.000
#{i such that yi ≤y}
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Density
1. Empirical distribution function n X 1 1(yi ≤ y) Fb(y) = n i=1 | {z }
0.015
E.g. predicting someone’s weight (Y )
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Weight (lbs.)
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Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Prediction using some covariates
Male
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βF −βM
with E(ε) = 0 and Var(ε) = σ 2
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βM
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or Yi = β0 + β1 1(X1,i = F) + εi |{z} |{z}
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 )
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Weight (lbs.)
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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
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assume that Y |X1 = x1 ∼ Fθ(x1 )
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conditional on the covariate
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Weight (lbs.)
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Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Prediction using some (categorical) covariates
0.010 0.005
Density 0.005
Density
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0.015
Prediction of Y when X1 = M
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Prediction of Y when X1 = F
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90%
0.000
90%
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Weight (lbs.)
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Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Linear Models, and Ordinary Least Squares E.g. predicting someone’s weight (Y )
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based on his/her height (X2 ) ● ●
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Linear Model : Yi = β0 + β2 X2,i + εi 200
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Conditional parametric model
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σ2
b = argmin −→ ordinary least squares, β
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E.g. Gaussian Linear Model
β0 +β2 x2
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Y |X2 = x2 ∼ N ( µ(x2 ) , σ 2 (x2 )) | {z } | {z }
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assume that Y |X2 = x2 ∼ Fθ(x2 )
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Weight (lbs.)
with E(ε) = 0 and Var(ε) = σ
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( n X
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Height (in.)
2 [Yi − X T β] i
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b is also the M.L. estimator of β β 10
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Prediction using no covariates
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Height (in.)
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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 ) ●
b |X1 = M) = 1 E(Y nM
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Yi = Yb (M)
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b Y ∈ Y (M) ± u1−α/2 · σ b | {z }
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Remark In the linear model, Var(ε) = σ 2 does not depend on X1 . 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 = F ∼ N (µF , σ 2 ) ●
b |X1 = F) = 1 E(Y nF
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Yi = Yb (F)
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Remark In the linear model, Var(ε) = σ 2 does not depend on X1 . 13
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 ) ●
b |X2 = x2 ) = βb0 + βb1 x2 = Yb (x2 ) E(Y b b Y ∈ Y (x2 ) ± u1−α/2 · σ | {z }
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Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Improving our prediction ?
sex as
b (Empirical) residuals, εbi = Yi − X T iβ | {z } bi Y R2 or log-likelihood
covariate
height as covariate no covariates
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parsimony principle ? −→ penalizing the likelihood with the number of covariates Akaike (AIC) or Schwarz (BIC) criteria
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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)
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16
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 ) b |X2 = x2 ) = µ E(Y b(x2 ) = Yb (x2 ) b Y ∈ Y (x2 ) ± u1−α/2 · σ b | {z }
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Gaussian model : E(Y |X = x) = µ(x) (e.g. xT β) and Var(Y |X = x) = σ 2 . 17
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Nonlinearities and missing covariates 250
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E.g. predicting someone’s weight (Y ) ● ●
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model mispecification
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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
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−→ nonlinearities can be related to
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Height (in.)
b = argmin −→ local linear regression, β x
( n X
) 2 ωi (x) · [Yi − X T i β]
i=1 Tb
and set Yb (x) = x β x 18
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Local regression and smoothing techniques ● ●
250
250
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6.0 Height (in.)
6.5
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6.5
Height (in.)
19
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
k-nearest neighbours and smoothing techniques ● ●
250
250
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6.0 Height (in.)
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Height (in.)
20
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Kernel regression and smoothing techniques ● ●
250
250
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6.0 Height (in.)
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5.0
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6.0
6.5
Height (in.)
21
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 | {z } | {z }
−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
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 | {z } | {z }
−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
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, ●
E.g. Binomial Y ∼ B(p(x)) −→ Generalized Linear Model
●
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E.g. Y |X2 = x2 ∼ P e
●
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(see next section) β0 +β2 x2
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Remark With a Poisson model, E(Y |X2 = x2 ) = Var(Y |X2 = x2 ). 24
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Logistic regression
●●●
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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
●● ● ● ● ● ● ● ●● ●●● ●●●●● ●● ●●●
0.8
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6.5
Height (in.)
1.0
Bernoulli variable, 0.8
●●
0.6 0.4
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}
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150
200
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250
Weight (lbs.)
25
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) T Odds ratio = exp x 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 ? b (Newton - Raphson) −→ maximum likelihood β
5.5
g Hei
100
in.) ht (
5.0
26
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Smoothed logistic regression
●●●
● ●
●
0.4 0.0
GLMs are linear since P(Y = 1|X = x) T = exp x β P(Y = 0|X = x)
●● ● ● ● ● ● ● ●● ●●● ●●●●● ●● ●●●
0.8
● ● ●
●● ● ● ● ● ●● ● ●● ● ● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●●
5.0
5.5
6.0
6.5
●
●●
0.8
●● ●●●●●●●●●●●●●●●●● ●●●● ●●● ●●● ● ● ●●
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.)
●
●● ● ●●●●●●●●●● ●●●●●●●●●●● ● ●●●●●●●●●● ●● ● ●
100
150
200
●
●
●
250
Weight (lbs.)
27
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
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
●
0.6
●
Y=0
^ Y=1
● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ●● ● ●● ● ● ● ● ● ●● ● ●●●●● ●● ● ● ●●●●
0.4
^ Y=0
●
0.8
Predicted
0.0
if P(Y |X = x) ≤ s, then Yb = 0 if P(Y |X = x) > s, then Yb = 1
● ●●●● ●● ● ●●● ●● ● ● ●● ● ● ●● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ●●● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●●●● ●● ● ● ●●
0.0
0.2
0.4
●
0.6
●
0.8
1.0
0.8
1.0
1.0
Predicted
error
Y =1
error
fine
0.6
fine
0.4
Y =0
●
0.2
Yb = 1 ●
0.0
Yb = 0
True Positive Rate
0.8
●
0.0
0.2
0.4
0.6
False Positive Rate
29
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Observed
Observed
Y=0
●
0.0
0.2
Y=1
●
● ●●●● ●● ● ●● ● ● ●● ● ● ● ● ● ● ● ● ●● ● ●● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●● ● ● ● ●● ● ● ●●●● ●● ● ● ●●
0.0
0.2
0.4
●
0.6
●
0.8
1.0
1.0
Predicted
●
●
0.6 0.4 0.0
0.2
True Positive Rate
0.8
●
R.O.C. curve is {F P (s), T P (s)), s ∈ (0, 1)}
^ Y=1
● ● ● ● ● ● ●● ● ● ●● ● ● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ● ● ● ● ●● ● ●●●●● ● ● ●●●●
0.6
^ Y=0
●
0.8
Predicted
0.4
True positive rate T P (s) = P(Yb (s) = 1|Y = 1) nYb =1,Y =1 = nY =1 False positive rate F P (s) = P(Yb (s) = 1|Y = 0) nYb =1,Y =1 = nY =1
1.0
R.O.C. curve
0.0
0.2
0.4
0.6
0.8
1.0
False Positive Rate
(see also model gain curve) 30
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Classification tree (CART)
250
●
● ●
●
If Y is a TRUE-FALSE variable 200
E(Y |X = x) = pj if x ∈ Aj
●
●
●
●
● ● ● ●
●
● ● ● ●
150
● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
regions of the X-space.
●
100
● ●
●
●
●
●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ●
●
● ● ●
● ●
●
●
● ●
● ● ● ● ●
●
● ● ●
●
● ● ● ● ● ● ● ●
● ●
●
●
● ●
●
● ● ● ● ● ●
● ● ●
● ●
where A1 , · · · , Ak are disjoint
●
●
Weight (lbs.)
prediction is a classification problem.
●
●
● ●
● ●
●
●
● ● ●
●
● ●
● ●
● ●
●
●
5.0
5.5
6.0
6.5
Height (in.)
31
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Classification tree (CART) −→ iterative process
●
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}
●
●
● ● ●
200
● ●
● ● ●
●
150
● ●
●
100
●
& maximize heterogeneity between subsets
●
● ●
●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
maximize homogeneity within subsets
●
●
● ● ● ●
●
●
and A2 = {i, X2,i > s}
●
●
Weight (lbs.)
and A2 = {i, X1,i > s}
● ●
●
●
●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ●
●
● ● ●
● ●
●
●
●
● ●
●
● ● ●
●
● ● ● ● ● ● ● ●
● ●
● ● ● ●
●
●
● ●
●
●
●
● ● ●
● ●
● ●
●
●
● ● ●
●
● ●
● ●
● ●
●
P(Y=1) = 25% P(Y=0) = 75%
●
5.0
5.5
6.0
6.5
Height (in.)
32
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Classification tree (CART)
250
●
● ●
●
●
●
0.317
●
● ● ● ●
●
● ● ● ●
●
150
● ● ●
y∈{0,1}
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ●
●
●
●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ●
●
● ● ●
● ●
●
●
● ● ●
● ● ● ● ●
●
● ● ●
●
● ● ● ● ● ● ● ●
● ●
●
●
● ●
●
● ● ● ● ● ●
● ● ●
●
100
x∈{A1 ,A2 }
●
●
0.523
Weight (lbs.)
E.g. Gini index X nx X nx,y nx,y − · 1− n nx nx
200
Need an impurity criteria
●
●
● ●
● ●
●
●
● ● ●
●
● ●
● ●
● ●
●
0.273
●
5.0
5.5
6.0
6.5
Height (in.)
33
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
Classification tree (CART)
250
●
Step k. Given partition A1 , · · · , Ak
● ●
●
0.472
●
●
● ● ●
●
● ● ● ●
●
● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
100
● ●
●
●
●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ●
●
● ● ●
● ●
●
●
● ●
● ● ● ● ●
●
● ● ●
●
● ● ● ● ● ● ● ●
● ●
●
●
● ●
●
● ● ● ● ● ●
● ● ●
●
maximize homogeneity within subsets & maximize heterogeneity between subsets
●
● ●
150
or according to X2
●
0.621 ●
0.444
Weight (lbs.)
either according to X1
200
find which subset Aj will be divided,
●
●
● ●
● ●
●
● ● 0.111
● ● ● ●
●
● ●
● ●
●
0.273
●
5.0
5.5
6.0
6.5
Height (in.)
34
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
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
From trees to forests ● ●
250
Problem CART tree are not robust −→ boosting and bagging
● ● ●
● ●
● ●
● ● ●
● ● ● ●
● ●
● ●
● ● ● ● ● ●
●
●
● ● ● ●
● ●
0.808
Then aggregate all the trees
● ●
● 0.520 ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ●
100
● ●
●
● ●
● ●
● ●
●
● ●
●
●
● ● ● ● ● ●
150
repeat this resampling strategy
●
● ● ● ●
● ●
Weight (lbs.)
and generate a classification tree
0.467
200
use bootstrap : resample in the data
0.576
●
●
● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ●
● ●
● ● ●
●
● ● ● ● ●
● ●
● ●
● ●
● ●
● ●
● ● ●
● ●
● ● ● ●
● ● ● ● ● ● ● ● ●
●
● ● ● ● ● ●
●
● ● ●
● ●
● ●
● ●
●
●
● ● ●
● ● ● ● ● ● ● ●
●
● ●
● ●
● ● ● ●
● ●
●
● ●
● 0.040
● ● ● ● ● ● ● ●
● ●
● ●
● ●
● ● ● ●
● ●
●
0.217
● ●
5.0
5.5
6.0
6.5
Height (in.)
36
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
10
A short word on functional data ●
● ●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
−15
● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
Individual data, {Yi , (X1,i , · · · , Xk,i , · · · )}
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
●
●
●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ●
●
5 0 −5 −10
●
● ● ● ● ● ● ● ● ●
●
−20
Temperature (in °Celsius)
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
●
●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ● ●
●
●
●
● ●
●
5
10
15
Week of winter (from Dec. 1st till March 31st)
Functional data, {Y i = (Yi,1 , · · · , Yi,t , · · · )} ● ●● ●
0 −10 −20 −30
Temperature (in °Celsius)
10
E.g. Winter temperature, in Montréal, QC
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0
20
40
60
80
100
120
Day of winter (from Dec. 1st till March 31st)
37
Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
A short word on functional data Daily dataset
2001
1982 ●
1974 1999 2007
0
PCA Comp. 1
40
50
●
● ●
●
2006
2009
●
●
1957 ●
2011
1997
20
●
●
−50
●
1986
2005 ●
1961
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JANUARY
DECEMBER
FEBRUARY
MARCH
●
1979 ●
Comp.2
0
●
●
●
●
1994
1996
1966 1964 1973 2010 1955
1978
●
20
40
60
80
100
120
Day of winter (from Dec. 1st till March 31st)
1987 ●
●●
1953 ●
●
1962 1954
1991
●
●
●
1989 ●
1960
0
●
2000 ●
1988 1963 1965 1959 1990
●
●
30
●
●
1992 1971
2004
●
●
●
●
2002
1998 ●
1977
1972 1985
20
●
●
●
1984
●
●
2008 ●
1968 ●
10
1956 1995 ●
●
2003 ●
1969 1975
1983
●
●
1976
●
●
1993 ●
1980 1967
−20
●
●
−20
−10
0
10 Comp.1
20
30
40
50
−30
−20
1981
0
●
●
−10
1958 1970
PCA Comp. 2
●
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Day of winter (from Dec. 1st till March 31st)
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Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013
To go further...
forthcoming book entitled Computational Actuarial Science with R
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