quantile regression via tpnd - aucklandaard004/arar.pdfefron (1991) introduced the percentile and...

Quantile Regression via TPND

© A. Ardalan

University of Auckland

30 Aug 2011 @ Auckland

Joint with:T. W. Yee

[email protected]

http://www.stat.auckland.ac.nz/~aard004

© A. Ardalan (University of Auckland ) Quantile Regression via TPND 30 Aug 2011 @ Auckland 1 / 31

http://www.stat.auckland.ac.nz/~aard004

Contents

Outline of this document

1 Quantile Regression

2 Expectile regression

3 Two-Piece Normal Distribution

4 Quantile Regression via Two-piece normal distributionQuantile Regression by TPNPenalized Iterratively Reweighted Least Square AlgorithmAsymptotic Normality

5 Summary


Quantile Regression

Introduction to Quantile Regression ISome motivation

Q: Why quantiles?To begin with, people prefer the summary of every thing.Usually, we summarise the data in central tendencies such as mean,median or even mode.Sometimes these summaries are deceptive. In this regard the quantilesenable us to visualise the shape of the distribution rather than the centraltendencies.In brief, quantiles give us a good picture of the distribution of data.


Quantile Regression

Introduction to Quantile Regression IISome motivation

Q: Why quantile regression?

Regression analysis includes any techniques for modeling relationshipbetween dependent variable Y and one or more explanatoryvariables X .

Typically, we use the quadratic loss or absolute loss function and itmeans we look at the E [Y |X ] or Median[Y |X ], respectively.

Sometimes we lose some information and these models are notappropriate for all data. In addition, in some cases the tails of thedistributions are of more interest than the center of them.

To give a more complete picture of the relationship between theresponse and explanatory variables, we can consider the quantiles ofdistribution of [Y |X ] i.e. Q[Y |X ]

The resulting curves are called the quantile regression curves.Clearly, they can be smoothed in some ways.


Quantile Regression

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Quantile Regression

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75%

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Y

Figure: (X , Y ) have the bivariate normal distributionQ(Y |x)(p) = µ(Y |x) + σ(Y |x)Φ

−1(p)


Quantile Regression

age

Den

sity

20 40 60 80

0.0000.0050.0100.0150.0200.0250.030

20 30 40 50 60 70 80

20

30

40

50

60

age

BM

I

0.00 0.06

20

30

40

50

60

Density

BM

I

Figure: The body mass indexes and ages from a random sample of New Zealandadults, n = 700. BMI = their body mass indexes, which is their weight divided bythe square of their height (kg/m2), age = their age (years).


Quantile Regression

age

Den

sity

20 40 60 80

0.0000.0050.0100.0150.0200.0250.030

20 30 40 50 60 70 80 90

20

30

40

50

60

age

BM

I

25%50%75%

0.00 0.06

20

30

40

50

60

Density

BM

I

Figure: The body mass indexes and ages from a random sample of New Zealandadults, n = 700. BMI = their body mass indexes, which is their weight divided bythe square of their height (kg/m2), age = their age (years).


Quantile Regression

Applications of quantile regression come from many fields. Here aresome:

Medical examples include investigating height, weight, body massindex (BMI) as a function of age of the person.

Economics, e.g., it has been used to study determinants of wages,discrimination effects, and trends in income inequality. SeeKoenker (2005) for more references.

Education, e.g., the performance of students in public schools onstandardized exams as a function of socio-economic variables such asparents’ income and educational attainment.

Ecology, e.g., the Melbourne temperature data exhibits bimodalbehaviour.


Quantile Regression

Growth chart example

Girls’ height and weight.


Quantile Regression

Three classes

1 Classical Quantile Regression models. Koenker and Bassett (1978)introduced the classical quantile regression estimator. (UsingAsymmetric L1 loss function)

2 Expectile regression methods. Newey and Powell (1987) andEfron (1991) introduced the percentile and expectile quantileregression estimator. (Using Asymmetric L2 loss function).

3 LMS-type methods. These transform the response to someparametric distribution (e.g., Box-Cox to N(0, 1)). Estimatedquantiles on the transformed scale are back-transformed on to theoriginal scale Cole and Green (1992). A problem with the LMSmethod is to find justification for the underlying method.


Quantile Regression

Classical Quantile Regression

Koenker and Bassett (1978) considered asymmetric L1 loss function

ρp(u) =

{(1− p)(−u) u < 0,p(u) u ≥ 0,

A specific quantile can be found by minimizing the expected loss of Y − uwith respect to u

minu

E (ρp(Y−u)) = minu

{(p − 1)

∫ u

−∞(y − u)dFY (y) + p

∫ ∞u

(y − u)dFY (y)

}.

This can be shown by setting the derivative of the expected loss functionto 0 and letting qp the pth quantile of the random variable Y .The pth sample quantile can be obtained by solving

qp = arg minq∈R

n∑i=1

ρp(yi−q) = arg minq∈R

(p − 1)∑yi<q

(yi − q) + p∑yi≥q

(yi − q)

.© A. Ardalan (University of Auckland ) Quantile Regression via TPND 30 Aug 2011 @ Auckland 13 / 31

Quantile Regression

Suppose the pth conditional quantile function is Q(Y |X )(p) = Xβ(p).Solving the sample analog gives the estimator of β.

β(p) = arg minβ∈Rk

n∑i=1

(ρp(Yi − xiβ)).

The intuition of this, is the same as solving the population quantile.Quantiles traditionally are estimated by linear programming.Koenker and Machado (1999) considered the following represention ofasymmetric Laplace distribution that can be applied in quantile regression

f (y ;µ, σ, p) =p(1− p)

σexp

{−ρp

(y − µσ

)}(1)

The log-likelihood under the assumption that the εi come from this densityis

`(β) = n log

(p(1− p)

σ

)+ exp

{n∑

i=1

−ρp(

Yi − xiβ

σ

)}(2)


Expectile regression

Percentile and Expectile Regression

Newey and Powell (1987) have considered the asymmetric quadratic lossfunction as an alternative of asymmetric L1 loss function and they defineda new concept and they called it expectile

ρ[2]ω (u) =

{(1− ω)(u)2 u < 0,ω(u)2 u ≥ 0,

An expectile is the minimization of the quantity E[ρ[2]ω (Y − µ)

]wrt µ.

In fact, it is a generalization of mean.Expectiles are similar to quantiles except that they are defined by tailexpectations; see Newey and Powell (1987).The ωth sample expectile can be obtained by solving

µω = arg minµ∈R

n∑i=1

ρ[2]ω (yi − µ)2,



Consider the linear model

yi = xiβ + εi , for i = 1, . . . , n.

Letri (b) = yi − xTi β

be a residual.We would like to compute β

(α)by minimizing

Sω(b) =∑n

i=1 ρ[2]ω {ri (b)}.

We can represent Sω(b) as a matrix weighted model,

Sw (b) = (y − Xb)TW(b)(y − Xb),

where w = ω1−ω and W(b) = diag[w(ri (b))] ia a diagonal matrix and,

w(ri (β)) =

{1 ri (b) ≤ 0,w ri (b) > 0,



Sw (b) is strictly convex and continuously differentiable as a function of b,see Efron (1991). This implies that the minimizer βw exists uniquely, andequals to solution of

Sw (b) ≡ ∇bSw (b) = 0,

and β therefore is the solution of

Sw (b) = 0

and iterative methods are needed to actually solve.The second derivative of Sw (b) is

Sw (b) ≡(∂2Sw (b)

∂bj∂bj ′

)j ,j ′=1,2,...,K

Efron (1991): the usual Newton-Raphson formula suggests bNEW thesolution of Sw (b) = 0, where

bNEW − b = −S−1w (b)Sw (b)



−2 −1 0 1 2

0.0

0.5

1.0

1.5

(a)

x

Loss

Fun

ctio

n

ABSA−ABS

y = x

−2 −1 0 1 2

0.0

0.5

1.0

1.5

2.0

(b)

x

Loss

Fun

ctio

n

SquaredA−Squared

−2 −1 0 1 2

−1.5

−1.0

−0.5

0.0

0.5

1.0

1.5

(c)

x

Influ

ence

Fun

ctio

n

−2 −1 0 1 2

−3

−2

−1

0

1

2

3

(d)

x

Influ

ence

Fun

ctio

n

Figure: Loss functions for (a) quantile regression with p = 0.5 (L1 regression) andp = 0.75 (asymmetric absolute loss function); (b) expectile regression withω = 0.5 (least squares) and ω = 0.75 (asymmetric least squares). (c) and (d) arederivatives of loss functions or Influence functions of (a) anb (b) respectively.


Two-Piece Normal Distribution

Two-piece normal distributionDistribution properties

The two-piece normal distribution (TPN) has a density

f (y ; p, µ, σ) =1

σ√

2π

exp

{− (y−µ)2

8p2σ2

}, y ≤ µ,

exp{− (y−µ)2

8(1−p)2σ2

}, y > µ

(3)

Here, −∞ < µ <∞ is the location parameter, σ > 0 is the scaleparameter and 0 < p < 1 is the shape parameter.Moreover, µ is the pth quantile of distribution, i.e. P(Y ≤ µ) = p.The expected information matrix (EIM) is,

14p(1−p)σ2 0 −2

σ√π

1p(1−p)

0 2σ2 0

−2σ√π

1p(1−p) 0 3

p(1−p)

. (4)


Two-Piece Normal Distribution

−6 −4 −2 0 2 4 6

0.00

0.05

0.10

0.15

0.20

0.25

0.30

x

TP

N D

ensi

ty

P(X ≤ 0) = 0.75P(X > 0) = 0.25

−6 −4 −2 0 2 4 6

0.00

0.05

0.10

0.15

0.20

0.25

0.30

x

TP

N D

ensi

ty

P(X ≤ 0) = 0.25P(X > 0) = 0.75

−2 0 2 4

0.0

0.2

0.4

0.6

0.8

1.0

Blue is density, red is cumulative distribution function

Purple lines are the 10,20,...,90 percentilesx

ftpn(

loca

tion=

0 ,

scal

e= 1

.2 ,

skew

par=

0.2

5 )


Quantile Regression via Two-piece normal distribution

Quantile Regression by TPN

Consider the linear model

yi = xiβ + εi , for i = 1, . . . , n.

and letεi ∼ TPN(0, σ, p) for i = 1, . . . , n

L(β) =n∏

i=1

f (yi ;β, σ) =n∑

i=1

ρ

(yi − xtiβ

σ

)(5)

ρ(x) = − log(f (x)) (6)

β(p)n = arg min

{n∑

i=1

ρ

(yi − xtiβ

σ

): β ∈ Rq

}



The matrix representation is

L(β) = (1

σ√

2π)n exp

{− 1

σ2(y − Xβ)TW(y − Xβ)

}(7)

where, W is a diagonal matrix, and the diagonal elements are

w(ii) =

1

8p2(y − Xβ) ≤ 0

18(1−p)2 (y − Xβ) > 0

`(β) = n log(σ)− 1

σ2(y − Xβ)TW(y − Xβ) (8)

U(β) =∂`(β)

∂β= 2

XTW(y − Xβ)

σ2(9)

U(β) =∂2`(β)

(∂β2= −2

XTWX

σ2(10)

U(β) = U(β) + U(β)(β − β) (11)



Iteratively Reweighted Least Square Algorithm

β(new) = β(old) −[U(β)

]−1U(β) (12)

the[U(β)

]−1is not continuous, so we can subtitute it by EIM of TPND.

Then we have (Fisher scoring algorithm)

β(new) = β(old) − [EIM]−1U(β) (13)

and so

β(new) = β(old) + 8σ2p(1− p)(XTX

)(−1)(XTW(old)(y − Xβ(old))

σ2

)=

(XTX

)(−1)XT[Xβ(old) + 8p(1− p)W(old)

(y − Xβ(old))

)]︸︷︷︸

z

=(XTX

)(−1)XTz (14)



−3 −2 −1 0 1 2 3

−3

−2

−1

0

1

2

3

0.25% quantile regression via TPNL distribution

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25%

Q(Y |x)(0.25) = a(0.25) + b(0.25)x



−2 0 2 4

−2

−1

0

1

2

3

0.75% quantile regression via TPNL distribution

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Y

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75%

Q(Y |x)(0.75) = a(0.75) + b(0.75)x



Penalized Iteratively Reweighted Least Square Algorithm

In a similar way we have

β(new) = β(old) +

(NTN

4σ2p(1− p)− λD

)(−1)(NTW(old)(y −Nβ(old))

σ2− λβ(old)D

)=

(NTN

4σ2p(1− p)− λD

)(−1)

NT

[Nβ(old) − 2

σ2W(old)

(y −Nβ(old)

)]︸︷︷︸

z

=

(NTN

4σ2p(1− p)+ λD

)(−1)

NT z (15)

λ is the smoothed parameter.

N is natural cubic spline matrix.

D is the penalty term.



10 20 30 40 50

10

15

20

25

30

35

40

yesterday

toda

y

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Figure: Melbourne temperature data (◦C). These are daily maximumtemperatures during 1981–1990, n = 3650. Y = each day’s maximumtemperature, X = the previous day’s maximum temperature.



10 20 30 40 50

10

15

20

25

30

35

40

yesterday

toda

y

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15 %25 %

35 %

45 %

55 %

65 %

75 %

85 %



Asymptotic NormalityConsider,

B1) max1≤k≤n

x′nk(X′nXn)−1xnk → 0 as n→ 0, (16)

B2) limn→∞

n−1(X′nXn)−1 = V, finite and p.d. (17)

Then we haveTheorem: As n→∞, under the assumptions B1 and B2, the limiting

distribution of√

n(β(p)n − β(p)) is a multivariate normal distribution with

variance-covariance matrix 4σ2p(1− p)V−1.


Summary

Summary

In this talk we have briefly explored QR.

We introduce the two piece normal distribution which it is anasymmetric distribution.

One of the applications of this family of distribution is in quantileregression and it can be a good alternative for expectile regression (L2

asymmetric loss) and since the location of it is the pth quantile canbe a good alternative for classical quantile regression (Asymmetric L1)in some cases.

However this method is not robust.

A package is under construction.


Summary

Thank You


quantile regression via tpnd - aucklandaard004/arar.pdfefron (1991) introduced the percentile and...

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