On Bayesian Semiparametric Quantile Regressionaard004/Arashkiama.pdf · Quantile Regression Here...
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On Bayesian Semiparametric Quantile Regression © A. Ardalan University of Auckland 7 Dec 2011 @ Kiama Joint with: M. P. Wand & T. W. Yee [email protected] © A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 1 / 40
Transcript of On Bayesian Semiparametric Quantile Regressionaard004/Arashkiama.pdf · Quantile Regression Here...
On Bayesian Semiparametric Quantile Regression
© A. Ardalan
University of Auckland
7 Dec 2011 @ Kiama
Joint with:M. P. Wand & T. W. Yee
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 1 / 40
Contents
Outline of this document
1 Quantile RegressionQuantile Regression and Asymmetric Laplace distribution
2 Semiparametric Regression and Graphical ModelPenalised SplineGraphical Models
3 Bayesian Quantile Regression
4 Extension
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 2 / 40
Quantile Regression
IntroductionSome motivation
For a regression model, typically, we use the quadratic loss orabsolute loss function and it means, we look at the E [Y |X ] orMedian[Y |X ], respectively.
But sometimes we lose some information with these models andthey are not appropriate for all data. In addition, in some casesthe tails of the distributions are more interest than the center ofthem.
To give a more complete picture of the relationship between theresponse Y and explanatory variables X , we can consider thequantiles of distribution 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.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 3 / 40
Quantile Regression
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Density
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Figure: (X , Y ) have the bivariate normal distributionQ(Y |x)(p) = µ(Y |x) + σ(Y |x)Φ
−1(p).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 4 / 40
Quantile Regression
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75%
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Density
Y
0.0 0.2 0.4
Figure: (X , Y ) have the bivariate normal distributionQ(Y |x)(p) = µ(Y |x) + σ(Y |x)Φ
−1(p)
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 5 / 40
Quantile Regression
Growth chart example
Girls’ height and weight.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 6 / 40
Quantile Regression
Here are three classes
1 Classical Quantile Regression models. UsingAsymmetric L1 loss function, Koenker andBassett (1978).
2 Expectile regression methods. Using Asymmetric L2
loss function, Newey and Powell (1987) andEfron (1991).
3 LMS-type methods. These transform the response tosome parametric distribution, Cole and Green (1992)(e.g., Box-Cox to N(0, 1)).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 7 / 40
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,
−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
−1.5
−1.0
−0.5
0.0
0.5
1.0
1.5
2.0
(b)
x
Influ
ence
Fun
ctio
n
Figure: (a) are symmetric and asymmetric absolute loss function with p = 0.5 (L1regression) and p = 0.75 (asymmetric L1 regression); (b) are the derivatives ofloss functions or Influence functions of (a).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 8 / 40
Quantile Regression
Quantile Regression (QR) and AL distribution
Suppose the pth conditional quantile is Q(Y |X )(p) = Xβ(p).
Estimation
β̂(p)
= arg minβ∈Rk
n∑i=1
ρp(yi − xtiβ).
Quantiles traditionally are estimated by linear programming.Koenker and Machado (1999) considered the followingrepresentation of asymmetric Laplace distribution (AL) in QR.
AL distribution
f (y ;µ, σ, p) =p(1− p)
σexp
{−ρp
(y − µσ
)}(1)
µ ∈ R, σ > 0, 0 < p < 1 and P(Y ≤ µ) = p.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 9 / 40
Quantile Regression
Quantile Regression (QR) and AL distribution
Suppose the pth conditional quantile is Q(Y |X )(p) = Xβ(p).
Estimation
β̂(p)
= arg minβ∈Rk
n∑i=1
ρp(yi − xtiβ).
Quantiles traditionally are estimated by linear programming.Koenker and Machado (1999) considered the followingrepresentation of asymmetric Laplace distribution (AL) in QR.
AL distribution
f (y ;µ, σ, p) =p(1− p)
σexp
{−ρp
(y − µσ
)}(1)
µ ∈ R, σ > 0, 0 < p < 1 and P(Y ≤ µ) = p.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 9 / 40
Quantile Regression
Quantile Regression (QR) and AL distribution
Suppose the pth conditional quantile is Q(Y |X )(p) = Xβ(p).
Estimation
β̂(p)
= arg minβ∈Rk
n∑i=1
ρp(yi − xtiβ).
Quantiles traditionally are estimated by linear programming.
Koenker and Machado (1999) considered the followingrepresentation of asymmetric Laplace distribution (AL) in QR.
AL distribution
f (y ;µ, σ, p) =p(1− p)
σexp
{−ρp
(y − µσ
)}(1)
µ ∈ R, σ > 0, 0 < p < 1 and P(Y ≤ µ) = p.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 9 / 40
Quantile Regression
Quantile Regression (QR) and AL distribution
Suppose the pth conditional quantile is Q(Y |X )(p) = Xβ(p).
Estimation
β̂(p)
= arg minβ∈Rk
n∑i=1
ρp(yi − xtiβ).
Quantiles traditionally are estimated by linear programming.Koenker and Machado (1999) considered the followingrepresentation of asymmetric Laplace distribution (AL) in QR.
AL distribution
f (y ;µ, σ, p) =p(1− p)
σexp
{−ρp
(y − µσ
)}(1)
µ ∈ R, σ > 0, 0 < p < 1 and P(Y ≤ µ) = p.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 9 / 40
Quantile Regression
Quantile Regression (QR) and AL distribution
Suppose the pth conditional quantile is Q(Y |X )(p) = Xβ(p).
Estimation
β̂(p)
= arg minβ∈Rk
n∑i=1
ρp(yi − xtiβ).
Quantiles traditionally are estimated by linear programming.Koenker and Machado (1999) considered the followingrepresentation of asymmetric Laplace distribution (AL) in QR.
AL distribution
f (y ;µ, σ, p) =p(1− p)
σexp
{−ρp
(y − µσ
)}(1)
µ ∈ R, σ > 0, 0 < p < 1 and P(Y ≤ µ) = p.© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 9 / 40
Quantile Regression
−15 −10 −5 0 5
0.00
0.05
0.10
0.15
0.20
x
AL
Den
sity
P(X ≤ 0) = 0.75P(X > 0) = 0.25
−5 0 5 10 15
0.00
0.05
0.10
0.15
0.20
x
AL
Den
sity
P(X ≤ 0) = 0.25P(X > 0) = 0.75
−5 0 5 10
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
ftpl(l
ocat
ion=
0 ,
scal
e= 1
.2 ,
skew
par=
0.2
5 )
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 10 / 40
Quantile Regression Quantile Regression and Asymmetric Laplace distribution
Quantile Regression and AL distribution
Consider the linear model
yi = xtiβ + εi , for i = 1, . . . , n,
and letεi ∼ AL(0, σ, p) for i = 1, . . . , n.
L(β) =n∏
i=1
f (yi ;β, σ) = exp
{n∑
i=1
ρ
(yi − xtiβ
σ
)}, (2)
ρ(x) = − log(f (x)), (3)
β̂(p)n = arg min
{n∑
i=1
ρ
(yi − xtniβ
σ
): β ∈ Rq
}.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 11 / 40
Quantile Regression Quantile Regression and Asymmetric Laplace distribution
Q(Y |x)(0.75) = a(0.75) + b(0.75)x
−3 −2 −1 0 1 2
−4
−2
0
2
0.75% quantile regression via AL distribution
X
Y
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75%
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 12 / 40
Quantile Regression Quantile Regression and Asymmetric Laplace distribution
Next
First, Penalised Spline.
Then,Hierarchical Bayesian Modellingand Graphical Models.
After that, an example.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 13 / 40
Semiparametric Regression and Graphical Model Penalised Spline
Penalised spline
Consider
yi = f (xi ) + εi where f (xi ) = E (Yi |xi ).
Penalised spline approach is:
f (xi) = β0 + β1xi +K∑
k=1
uk zk(xi)
whereI the uk are penalised coeffs and
I zk(xi ) are spline basis functions.
One way to penalise is
uk ∼ N(0, σ2u).
Our default here is “O’Sullivan spline” for the zk which provides aclose approximation to smoothing splines. Wand & Ormerod(2008)
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 14 / 40
Semiparametric Regression and Graphical Model Penalised Spline
Penalised spline
Consider
yi = f (xi ) + εi where f (xi ) = E (Yi |xi ).
Penalised spline approach is:
f (xi) = β0 + β1xi +K∑
k=1
uk zk(xi)
whereI the uk are penalised coeffs and
I zk(xi ) are spline basis functions.
One way to penalise is
uk ∼ N(0, σ2u).
Our default here is “O’Sullivan spline” for the zk which provides aclose approximation to smoothing splines. Wand & Ormerod(2008)
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 14 / 40
Semiparametric Regression and Graphical Model Penalised Spline
Penalised spline
Consider
yi = f (xi ) + εi where f (xi ) = E (Yi |xi ).
Penalised spline approach is:
f (xi) = β0 + β1xi +K∑
k=1
uk zk(xi)
whereI the uk are penalised coeffs and
I zk(xi ) are spline basis functions.
One way to penalise is
uk ∼ N(0, σ2u).
Our default here is “O’Sullivan spline” for the zk which provides aclose approximation to smoothing splines. Wand & Ormerod(2008)© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 14 / 40
Semiparametric Regression and Graphical Model Penalised Spline
Bracket notation:
Conditional distribution
[y |x ] = density of y given x ,
e.g.
[x |α, β] =1
βαΓ(α)xα−1 exp−x
βmeans
fX (x ;α, β) = [x ;α, β]
or[x ] ∼ Gamma(α, β).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 15 / 40
Semiparametric Regression and Graphical Model Penalised Spline
Hierarchical Bayesian Modelling
For ordinary nonparametric regression
Model
[yi |β0,β1, u1, u2 . . . , uk , σ2u, σ
2ε ] ∼ N(β0 + β1x +
K∑i=1
ukzk(xi ), σ2ε )
[uk |σ2u
]∼ N(0, σ2u),
[β0] ∼ N(0, 108), [β1] ∼ N(0, 108),
[σ2ε], [σ2u] ∼ Inverse Gamma(
1
100,
1
100)
or[σ2u] ∼ Half Cauchy(scale = 25)
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 16 / 40
Semiparametric Regression and Graphical Model Penalised Spline
Hierarchical Bayesian Modelling
For ordinary nonparametric regression
Model
[yi |β0,β1, u1, u2 . . . , uk , σ2u, σ
2ε ] ∼ N(β0 + β1x +
K∑i=1
ukzk(xi ), σ2ε )
[uk |σ2u
]∼ N(0, σ2u),
[β0] ∼ N(0, 108), [β1] ∼ N(0, 108),
[σ2ε], [σ2u] ∼ Inverse Gamma(
1
100,
1
100)
or[σ2u] ∼ Half Cauchy(scale = 25)
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 16 / 40
Semiparametric Regression and Graphical Model Penalised Spline
Hierarchical Bayesian Modelling
For ordinary nonparametric regression
Model
[yi |β0,β1, u1, u2 . . . , uk , σ2u, σ
2ε ] ∼ N(β0 + β1x +
K∑i=1
ukzk(xi ), σ2ε )
[uk |σ2u
]∼ N(0, σ2u),
[β0] ∼ N(0, 108), [β1] ∼ N(0, 108),
[σ2ε], [σ2u] ∼ Inverse Gamma(
1
100,
1
100)
or[σ2u] ∼ Half Cauchy(scale = 25)
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 16 / 40
Semiparametric Regression and Graphical Model Penalised Spline
Hierarchical Bayesian Modelling
Matrix Notation
X =
1 x1...
...1 xn
and Z = [zk(xi )].
[y |β,u, σ2ε , σ2u] ∼ N(Xβ + Zu, σ2ε I ),[u|σ2u
]∼ N(0, σ2uI ),
[β] ∼ N(0, 108I ).
For estimating,
qp(x) = pth quantile of y given x,
we just have
[y |β,u, σ2ε , σ2u] ∼ AL(Xβ + Zu, σ∗ε , p).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 17 / 40
Semiparametric Regression and Graphical Model Penalised Spline
Hierarchical Bayesian Modelling
Matrix Notation
X =
1 x1...
...1 xn
and Z = [zk(xi )].
[y |β,u, σ2ε , σ2u] ∼ N(Xβ + Zu, σ2ε I ),[u|σ2u
]∼ N(0, σ2uI ),
[β] ∼ N(0, 108I ).
For estimating,
qp(x) = pth quantile of y given x,
we just have
[y |β,u, σ2ε , σ2u] ∼ AL(Xβ + Zu, σ∗ε , p).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 17 / 40
Semiparametric Regression and Graphical Model Graphical Models
Graphical Models
A graphical model is
a probabilistic modelfor which a graph denotes theconditional independence structurebetween random variables.
They are commonly used in probabilitytheory, statistics–particularly Bayesianstatistics–and machine learning.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 18 / 40
Semiparametric Regression and Graphical Model Graphical Models
Graphical Models
A graphical model is
a probabilistic modelfor which a graph denotes theconditional independence structurebetween random variables.
They are commonly used in probabilitytheory, statistics–particularly Bayesianstatistics–and machine learning.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 18 / 40
Semiparametric Regression and Graphical Model Graphical Models
Graphical Models
A graphical model is
a probabilistic modelfor which a graph denotes theconditional independence structurebetween random variables.
They are commonly used in probabilitytheory, statistics–particularly Bayesianstatistics–and machine learning.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 18 / 40
Semiparametric Regression and Graphical Model Graphical Models
Graphical Models
Graphical models (probabilistic graphical models)framework is also very useful for semiparametricregression, especially when the problem isnon-standard.
There are two main types:1 Directed acyclic graphs (DAGs), also known as Bayesian networks,2 Undirected graphs, also known as Markov random fields.
Of these, DAGs are more immediately relevant tosemiparametric regression.We use the same conventions as Bishop (2006).
I Random nodes are denoted by open circles. ©I Non-random nodes are shown as small solid circles. •I Observed (“evidence”) nodes are distinguished from parameter
(“hidden”) nodes using shading.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 19 / 40
Semiparametric Regression and Graphical Model Graphical Models
Graphical Models
Graphical models (probabilistic graphical models)framework is also very useful for semiparametricregression, especially when the problem isnon-standard.There are two main types:
1 Directed acyclic graphs (DAGs), also known as Bayesian networks,2 Undirected graphs, also known as Markov random fields.
Of these, DAGs are more immediately relevant tosemiparametric regression.We use the same conventions as Bishop (2006).
I Random nodes are denoted by open circles. ©I Non-random nodes are shown as small solid circles. •I Observed (“evidence”) nodes are distinguished from parameter
(“hidden”) nodes using shading.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 19 / 40
Semiparametric Regression and Graphical Model Graphical Models
Graphical Models
Graphical models (probabilistic graphical models)framework is also very useful for semiparametricregression, especially when the problem isnon-standard.There are two main types:
1 Directed acyclic graphs (DAGs), also known as Bayesian networks,2 Undirected graphs, also known as Markov random fields.
Of these, DAGs are more immediately relevant tosemiparametric regression.
We use the same conventions as Bishop (2006).
I Random nodes are denoted by open circles. ©I Non-random nodes are shown as small solid circles. •I Observed (“evidence”) nodes are distinguished from parameter
(“hidden”) nodes using shading.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 19 / 40
Semiparametric Regression and Graphical Model Graphical Models
Graphical Models
Graphical models (probabilistic graphical models)framework is also very useful for semiparametricregression, especially when the problem isnon-standard.There are two main types:
1 Directed acyclic graphs (DAGs), also known as Bayesian networks,2 Undirected graphs, also known as Markov random fields.
Of these, DAGs are more immediately relevant tosemiparametric regression.We use the same conventions as Bishop (2006).
I Random nodes are denoted by open circles. ©I Non-random nodes are shown as small solid circles. •I Observed (“evidence”) nodes are distinguished from parameter
(“hidden”) nodes using shading.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 19 / 40
Semiparametric Regression and Graphical Model Graphical Models
Graphical Models
Graphical models (probabilistic graphical models)framework is also very useful for semiparametricregression, especially when the problem isnon-standard.There are two main types:
1 Directed acyclic graphs (DAGs), also known as Bayesian networks,2 Undirected graphs, also known as Markov random fields.
Of these, DAGs are more immediately relevant tosemiparametric regression.We use the same conventions as Bishop (2006).
I Random nodes are denoted by open circles. ©
I Non-random nodes are shown as small solid circles. •I Observed (“evidence”) nodes are distinguished from parameter
(“hidden”) nodes using shading.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 19 / 40
Semiparametric Regression and Graphical Model Graphical Models
Graphical Models
Graphical models (probabilistic graphical models)framework is also very useful for semiparametricregression, especially when the problem isnon-standard.There are two main types:
1 Directed acyclic graphs (DAGs), also known as Bayesian networks,2 Undirected graphs, also known as Markov random fields.
Of these, DAGs are more immediately relevant tosemiparametric regression.We use the same conventions as Bishop (2006).
I Random nodes are denoted by open circles. ©I Non-random nodes are shown as small solid circles. •
I Observed (“evidence”) nodes are distinguished from parameter(“hidden”) nodes using shading.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 19 / 40
Semiparametric Regression and Graphical Model Graphical Models
Graphical Models
Graphical models (probabilistic graphical models)framework is also very useful for semiparametricregression, especially when the problem isnon-standard.There are two main types:
1 Directed acyclic graphs (DAGs), also known as Bayesian networks,2 Undirected graphs, also known as Markov random fields.
Of these, DAGs are more immediately relevant tosemiparametric regression.We use the same conventions as Bishop (2006).
I Random nodes are denoted by open circles. ©I Non-random nodes are shown as small solid circles. •I Observed (“evidence”) nodes are distinguished from parameter
(“hidden”) nodes using shading.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 19 / 40
Semiparametric Regression and Graphical Model Graphical Models
DAG = Directed Acyclic Graph
x1 x2
x3 x4
x5
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 20 / 40
Semiparametric Regression and Graphical Model Graphical Models
Probability Structures and DAGs
x1 x2
x3 x4
x5
Joint density function of x1, . . . , x5
≡ [x1, x2, x3, x4, x5]
= [x1][x2][x3|x1, x2][x4|x2][x5|x3, x4]
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 21 / 40
Bayesian Quantile Regression
DAGs and Hierarchical Bayesian Models
Bayesian simple quantile regression:
[yi |β0, β1, σ2]
ind .∼ AL(β0 + β1x , σ, p), 1 ≤ i ≤ n,
[β0] ∼ N(µβ0, σ2β0
), [β1] ∼ N(µβ1, σ2β1
),
[σ2] ∼ Inverse Gamma(A,B) = IG (A,B).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 22 / 40
Bayesian Quantile Regression
● ● ● ● ● ●
µβ0 σβ0
2 µβ1 σβ1
2A B
β0 β1 σ2
yi
nxi●
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 23 / 40
Bayesian Quantile Regression
Hierarchical Bayes Model for QR Model:
Bayesian quantile Model:
[yi |xi ,β,u, σε]ind .∼ AL(β0 + β1xi +
K∑k=1
ukzk(xi ), σε, p),
[U|σ2u] ∼ N(0, σ2uI)
[β0] ∼ N(µβ0 , σ2β0), [β1] ∼ N(µβ1 , σ
2β1),
[σ2u] ∼ IG (Au,Bu), [σ2ε ] ∼ IG (Aε,Bε).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 24 / 40
Bayesian Quantile Regression
σε2 β σspl
2
uspl
yi
● ● ● ● ● ●
Aε Bε µβ σβ2 Aspl Bspl
xi●
n
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 25 / 40
Bayesian Quantile Regression
Example : The Canadian age–income data
20 30 40 50 60
1213
1415
Age (years)
Log
(ann
ual i
ncom
e)
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25 %
Scatterplot of log(income) versus age for a sample of n = 205 Canadianworkers with posterior (solid) and 95% credible intervals for the 25%quantile regression.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 26 / 40
Bayesian Quantile Regression
parameter trace lag 1 acf density summary
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Series x[[plotInd]][, j]
0.1 0.15 0.2 0.25
posterior mean: 0.161
95% credible interval:
(0.14,0.185)
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Series x[[plotInd]][, j]
0 0.05 0.1 0.15 0.2 0.25
posterior mean: 0.101
95% credible interval:
(0.0621,0.159)
degrees of
freedom for f●
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Series x[[plotInd]][, j]
10 15 20 25
posterior mean: 17.1
95% credible interval:
(14.1,20.4)
first quartile
of x●
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Series x[[plotInd]][, j]
13.1 13.2 13.3 13.4 13.5 13.6
posterior mean: 13.4
95% credible interval:
(13.2,13.5)
second quart.
of x●
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Series x[[plotInd]][, j]
13.2 13.4 13.6 13.8
posterior mean: 13.5
95% credible interval:
(13.3,13.7)
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Series x[[plotInd]][, j]
13 13.2 13.4 13.6 13.8 14
posterior mean: 13.5
95% credible interval:
(13.3,13.7)
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 27 / 40
Bayesian Quantile Regression
Now,
A Semiparametric Regressionmodel.
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 28 / 40
Bayesian Quantile Regression
Spinal Bone Mineral Data
age (years)
spin
al b
one
min
eral
den
sity
0.5
1.0
1.5
2.0
10 15 20 25
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© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 29 / 40
Bayesian Quantile Regression
Model for Spinal Bone Mineral Data and its DAG
Bayesian semiparametric quantile model:
[yij |β,usbj ,uspl , σ2sbj , σ2spl , σε]ind .∼ AL(βT xi + ui ,sbj + f (agei ,j ;σ
2spl), σε, p),
[usbj |σ2sbj ] ∼ N(0, σ2
sbjI), [uspl |σ2spl ] ∼ N(0, σ2
spl I)
[β] ∼ N(0, σ2βI), [σ2
sbj ] ∼ IG (Asbj ,Bsbj),
[σ2spl ] ∼ IG (Aspl ,Bspl), [σ2
ε ] ∼ IG (Aε,Bε).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 30 / 40
Bayesian Quantile Regression
σε2 σsbj
2 σspl2
usbj uspl
y
β
● ● ● ● ● ●
Aε Bε Asbj Bsbj Aspl Bspl
x●
age●
●
σβ2
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 31 / 40
Bayesian Quantile Regression
Nonparametric Quantile Regression with Missingness inPredictor
yi = f (xi) + εi , εi i .i .d . AL(0, σ2ε , p), 1 ≤ i ≤ n
xiind .∼ N(µx , σ
2x), but some are missing
(completely at random).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 32 / 40
Bayesian Quantile Regression
Hierarchical Bayes Model for Missingness Example
Bayesian quantile Model:
[yi |xi ,β,u, σε]ind .∼ AL(β0 + β1xi +
K∑k=1
ukzk(xi ), σε, p),
[xi |µx , σ2x ] ∼ N(µx , σ2x), [U|σ2u] ∼ N(0, σ2uI)
[β] ∼ N(0, σ2βI), [µx ] ∼ N(0, σ2µx ),
[σ2x ] ∼ IG (Ax ,Bx), [σ2u] ∼ IG (Au,Bu), [σ2ε ] ∼ IG (Aε,Bε).
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 33 / 40
Bayesian Quantile Regression
● ● ● ● ● ●
µβ0 σβ0
2 µβ1 σβ1
2A B
β0 β1 σ2
y
xobs
µx σx2
xmis
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 34 / 40
Bayesian Quantile Regression
Example on Missing Data
0.2 0.4 0.6 0.8 1.0
−1.
5−
1.0
−0.
50.
00.
51.
01.
5
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fully observedx value unobserved
true f25 %
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 35 / 40
Bayesian Quantile Regression
parameter trace lag 1 acf density summary
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Series x[[plotInd]][, j]
0.42 0.44 0.46 0.48 0.5 0.52
posterior mean: 0.476
95% credible interval: (0.455,0.497)
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Series x[[plotInd]][, j]
0.12 0.14 0.16 0.18 0.2
posterior mean: 0.161
95% credible interval: (0.147,0.177)
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Series x[[plotInd]][, j]
0.08 0.1 0.12 0.14
posterior mean: 0.109
95% credible interval: (0.0961,0.124)
degrees of
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Series x[[plotInd]][, j]
14 16 18 20 22 24
posterior mean: 19.1
95% credible interval: (17.1,21.4)
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Series x[[plotInd]][, j]
−1.3 −1.2 −1.1 −1
posterior mean: −1.1
95% credible interval: (−1.16,−1.03)
second quart.
of x●
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Series x[[plotInd]][, j]
−0.85 −0.8 −0.75 −0.7 −0.65 −0.6
posterior mean: −0.734
95% credible interval: (−0.804,−0.666)
third quartile
of x ●●
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Series x[[plotInd]][, j]
0.3 0.4 0.5 0.6 0.7 0.8
posterior mean: 0.556
95% credible interval: (0.459,0.678)
Girls’ height and weight.© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 36 / 40
Bayesian Quantile Regression
parameter trace lag 1 acf density summary
x10mis
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Series x[[plotInd]][, j]
0 0.2 0.4 0.6 0.8
posterior mean: 0.543
95% credible interval:
(0.133,0.696)
x18mis
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Series x[[plotInd]][, j]
0.2 0.4 0.6 0.8 1
posterior mean: 0.427
95% credible interval:
(0.297,0.895)
x27mis
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Series x[[plotInd]][, j]
0 0.5 1 1.5
posterior mean: 0.507
95% credible interval:
(0.178,0.756)
x44mis
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Series x[[plotInd]][, j]
0 0.2 0.4 0.6 0.8 1
posterior mean: 0.489
95% credible interval:
(0.223,0.773)
x59mis
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Series x[[plotInd]][, j]
0 0.5 1 1.5
posterior mean: 0.467
95% credible interval:
(0.252,0.783)
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 37 / 40
Extension
Extension
1 Longitudinal data see Wand (2003)
2 Additive models
3 Additive mixed models
4 Measurement error
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 38 / 40
Extension
Thank You
© A. Ardalan (University of Auckland ) On Bayesian Semiparametric Quantile Regression 7 Dec 2011 @ Kiama 39 / 40
