So a webinar-2013-2

Arthur CHARPENTIER - Predictive Modeling - SoA Webinar, 2013

Perspectives of Predictive Modeling

Arthur Charpentier

[email protected]

http ://freakonometrics.hypotheses.org/

(SOA Webcast, November 2013)

1


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


Prediction ? Best estimate ?

E.g. predicting someone’s weight (Y )Consider a sample {y1, · · · , yn} −→Model : Yi = β0 + εi

with E(ε) = 0 and Var(ε) = σ2

ε is some unpredictable noise

Y = y is our ‘best guess’...

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Predicting means estimating E(Y ).

Recall that E(Y ) = argminy∈R

{‖Y − y‖L2} = argminy∈R

{E([

ε︷︸︸︷Y − y]2

)}︸︷︷︸

least squares

3


Best estimate with some confidence

E.g. predicting someone’s weight (Y )Give an interval [y−, y+] such that

P(Y ∈ [y−, y+]) = 1− α

Confidence intervals can be derived ifwe can estimate the distribution of Y

F (y) = P(Y ≤ y) or f(y) = dF (x)dx

∣∣∣∣x=y

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(related to the idea of “quantifying uncertainty” in our prediction...)

4


Parametric inference

E.g. predicting someone’s weight (Y )Assume that F ∈ F = {Fθ,θ ∈ Θ}1. Provide an estimate θ2. Compute bound estimates

y− = F−1θ

(α/2)

y+ = F−1θ

(1− α/2)

Standard estimation technique :−→ maximum likelihood techniques

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θ = argmaxθ∈Θ

{ n∑i=1

log fθ(yi)︸︷︷︸log likelihood

} explicit (analytical) expression for θnumerical optimization (Newton Raphson)

5


Non-parametric inference

E.g. predicting someone’s weight (Y )1. Empirical distribution function

F (y) = 1n

n∑i=1

1(yi ≤ y)︸︷︷︸#{i such that yi≤y}

natural estimator for P(Y ≤ y)2. Compute bound estimates

y− = F−1(α/2)y+ = F−1(1− α/2)

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Prediction using some covariates

E.g. predicting someone’s weight (Y )based on his/her sex (X1)

Model : Yi =

βF + εi if X1,i = FβH + εi if X1,i = M

or Yi = β0︸︷︷︸βM

+ β1︸︷︷︸βF−βM

1(X1,i = F) + εi


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Prediction using some (categorical) covariatesE.g. predicting someone’s weight (Y )

based on his/her sex (X1)Conditional parametric modelassume that Y |X1 = x1 ∼ Fθ(x1)

i.e. Yi ∼

FθFif X1,i = F

FθMif X1,i = M

−→ our prediction will beconditional on the covariate Weight (lbs.)

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Prediction using some (categorical) covariatesPrediction of Y when X1 = F Prediction of Y when X1 = M

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Linear Models, and Ordinary Least Squares

E.g. predicting someone’s weight (Y )based on his/her height (X2)

Linear Model : Yi = β0 + β2 X2,i + εi


Conditional parametric modelassume that Y |X2 = x2 ∼ Fθ(x2)

E.g. Gaussian Linear ModelY |X2 = x2 ∼ N ( µ(x2)︸︷︷︸

β0+β2x2

, σ2(x2)︸︷︷︸σ2

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−→ ordinary least squares, β = argmin{

n∑i=1

[Yi −XTi β]2

}β is also the M.L. estimator of β

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Prediction using no covariates

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11


Prediction using a categorical covariates


E.g. Gaussian linear modelY |X1 = M ∼ N (µM , σ2)

E(Y |X1 = M) = 1nM

∑i:X1,i=M

Yi = Y (M)

Y ∈[Y (M)± u1−α/2︸︷︷︸

1.96

· σ]

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Remark In the linear model, Var(ε) = σ2 does not depend on X1.

12


Prediction using a categorical covariates


E.g. Gaussian linear modelY |X1 = F ∼ N (µF , σ2)

E(Y |X1 = F) = 1nF

∑i:X1,i=F

Yi = Y (F)

Y ∈[Y (F)± u1−α/2︸︷︷︸

1.96

· σ]

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Remark In the linear model, Var(ε) = σ2 does not depend on X1.

13


Prediction using a continuous covariates


E.g. Gaussian linear modelY |X2 = x2 ∼ N (β0 + β1x2, σ

2)

E(Y |X2 = x2) = β0 + β1x2 = Y (x2)

Y ∈[Y (x2)± u1−α/2︸︷︷︸

1.96

· σ]

14


Improving our prediction ?

(Empirical) residuals, εi = Yi−XTi β︸︷︷︸Yi

R2 or log-likelihoodparsimony principle ?−→ penalizing the likelihood with

the number of covariatesAkaike (AIC) orSchwarz (BIC) criteria

no covariates

height as

covariate

sex as

covariate

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−50 0 50 100

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Relaxing the linear assumption in predictionsUse of b-spline function basis to estimate µ(·) where µ(x) = E(Y |X = x)

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5.0 5.5 6.0 6.5

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16


Relaxing the linear assumption in predictions


E.g. Gaussian linear modelY |X2 = x2 ∼ N (µ(x2), σ2)E(Y |X2 = x2) = µ(x2) = Y (x2)

Y ∈[Y (x2)± u1−α/2︸︷︷︸

1.96

· σ]

Gaussian model : E(Y |X = x) = µ(x) (e.g. xTβ) and Var(Y |X = x) = σ2.

17


Nonlinearities and missing covariates

E.g. predicting someone’s weight (Y )based on his/her height and sex

−→ nonlinearities can be related tomodel mispecification

E.g. Gaussian linear model

Yi =

β0,F + β2,FX2,i + εi if X1,i = Fβ0,M + β2,MX2,i + εi if X1,i = M

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−→ local linear regression, βx = argmin{

n∑i=1

ωi(x) · [Yi −XTi β]2

}and set Y (x) = xTβx

18


Local regression and smoothing techniques

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k-nearest neighbours and smoothing techniques

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Kernel regression and smoothing techniques

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21


Multiple linear regression

E.g. predicting someone’s misperceptionof his/her weight (Y )based on his/her height︸︷︷︸

X2

and weight︸︷︷︸X3

−→ linear modelE(Y |X2, X3) = β0 + β2X2 + β3X3

Var(Y |X2, X3) = σ2

22


Multiple non-linear regression

E.g. predicting someone’s misperceptionof his/her weight (Y )based on his/her height︸︷︷︸

X2

and weight︸︷︷︸X3

−→ non-linear modelE(Y |X2, X3) = h(X2, X3)Var(Y |X2, X3) = σ2

23


Away from the Gaussian model

Y is not necessarily GaussianY 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))

(see next section)−→ Generalized Linear ModelE.g. Y |X2 = x2 ∼ P

(eβ0+β2x2

)

Remark With a Poisson model, E(Y |X2 = x2) = Var(Y |X2 = x2).

24


Logistic regression

E.g. predicting someone’s misperceptionof his/her weight (Y )

Yi =

1 if prediction > observed weight0 if prediction ≤ observed weight

Bernoulli variable,P(Y = y) = py(1− p)1−y, where y ∈ {0, 1}−→ logistic regressionP(Y = y|X = x) = p(x)y(1− p(x))1−y,where y ∈ {0, 1}

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25


Logistic regression

−→ logistic regressionP(Y = y|X = x) = p(x)y(1− p(x))1−y,where y ∈ {0, 1}

Odds ratio P(Y = 1|X = x)P(Y = 0|X = x) = exp

(xTβ

)

E(Y |X = x) =exp

(xTβ

)1 + exp (xTβ)

Estimation of β ?−→ maximum likelihood β (Newton - Raphson)

26


Smoothed logistic regression

GLMs are linear sinceP(Y = 1|X = x)P(Y = 0|X = x) = exp

(xTβ

)−→ smooth nonlinear function insteadP(Y = 1|X = x)P(Y = 0|X = x) = exp (h(x))

E(Y |X = x) = exp (h(x))1 + exp (h(x))

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27


Smoothed logistic regression

−→ non linear logistic regressionP(Y = 1|X = x)P(Y = 0|X = x) = exp (h(x))

E(Y|X = x) = exp (h(x))1 + exp (h(x))

Remark we do not predict Y here,but E(Y |X = x).

28


Predictive modeling for a {0, 1} variable

What is a good {0, 1}-model ?

−→ decision theory

if P(Y |X = x) ≤ s, then Y = 0if P(Y |X = x) > s, then Y = 1

Y = 0 Y = 1

Y = 0 fine errorY = 1 error fine

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29


R.O.C. curve

True positive rateTP (s) = P(Y (s) = 1|Y = 1)

=nY=1,Y=1

nY=1False positive rateFP (s) = P(Y (s) = 1|Y = 0)

=nY=1,Y=1

nY=1

R.O.C. curve is{FP (s), TP (s)), s ∈ (0, 1)}

(see also model gain curve)

30


Classification tree (CART)

If Y is a TRUE-FALSE variableprediction is a classification problem.

E(Y |X = x) = pj if x ∈ Ajwhere A1, · · · , Ak are disjointregions of the X-space.

31



−→ iterative processStep 1. Find two subset of indiceseither A1 = {i,X1,i < s}

and A2 = {i,X1,i > s}or A1 = {i,X2,i < s}

and A2 = {i,X2,i > s}

maximize homogeneity within subsets& maximize heterogeneity between subsets

32



Need an impurity criteriaE.g. Gini index

−∑

x∈{A1,A2}

nxn

∑y∈{0,1}

nx,ynx·(

1− nx,ynx

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5.0 5.5 6.0 6.510

015

020

025

0

Height (in.)

Wei

ght (

lbs.

)

0.273

0.523 0.317

33



Step k. Given partition A1, · · · , Akfind which subset Aj will be divided,either according to X1

or according to X2

maximize homogeneity within subsets& maximize heterogeneity between subsets ●

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5.0 5.5 6.0 6.5

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Height (in.)

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ght (

lbs.

)

0.273

0.444 0.621

0.111

0.472

34


Visualizing classification trees (CART)

|Y < 124.561

X < 5.692260.2727

0.5231 0.3175

|Y < 124.561

X < 5.39698 X < 5.69226

X < 5.52822 Y < 162.04

0.3429 0.1500

0.4444 0.6207

0.1111 0.4722

35


From trees to forests

Problem CART tree are not robust−→ boosting and bagginguse bootstrap : resample in the dataand generate a classification treerepeat this resampling strategyThen aggregate all the trees

36


A short word on functional data

Individual data, {Yi, (X1,i, · · · , Xk,i, · · · )}Functional data, {Y i = (Yi,1, · · · , Yi,t, · · · )}E.g. Winter temperature, in Montréal, QC

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5 10 15

−20

−15

−10

−5

05

10

Week of winter (from Dec. 1st till March 31st)

Tem

pera

ture

(in

°C

elsi

us)

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0 20 40 60 80 100 120

−30

−20

−10

010

Day of winter (from Dec. 1st till March 31st)

Tem

pera

ture

(in

°C

elsi

us)

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So a webinar-2013-2

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