gilberto a. paula - ime-uspgiapaula/slides_exemplos_semip.pdf · semiparametric models with...

Semiparametric models with applications using R

Gilberto A. Paula

Instituto de Matemática e EstatísticaUniversidade de São Paulo, Brasil

[email protected]

2o Semestre 2016

G. A. Paula (IME-USP) Semiparametric models in R 2o Semestre 2016 1 / 61

Examples

Outline

1 Examples

2 Defining f (x)

3 Additive normal model

4 Semiparametric normal model

5 Packages in R

6 Voltage drop data

7 Boston housing data

8 Bibliography


Examples

Voltage drop data

Description

As a 1st example we will consider the voltage drop data (Montgomeryand Peck, 2001) in which a battery voltage drop in a guided missilemotor is observed over the time of missile flight. It was intended avoltage drop model for using a digital-analog simulation model of themissile. Altogether there are 41 observations.


Examples

Scatter plot of voltage drop data

0 5 10 15 20

810

1214

Time

Volta

ge


Examples


0 5 10 15 20

810

1214

Time

Volta

ge


Examples

Possible model

Description

The data suggest a nonparametric model such as:

Voltagei = α+ f (Timei) + ǫi ,


Examples

Possible model

Description

The data suggest a nonparametric model such as:


where ǫi∼ N(0, σ2) for i = 1, . . . , 41, with f (·) being a continuous,smooth and nonparametric function.


Examples

Boston housing data

Description

As a 2nd example we will consider the Boston housing data that havebeen analyzed by various authors (see, for instance, Belsley et al.1980). The aim of the study is to assess the association of houseprices with the air quality of the neighborhood by using regressionmodels. The outcome variable


Examples

Boston housing data

Description


LMEDV (logarithm of the median house price in USD 1000)


Examples

Boston housing data

Description


LMEDV (logarithm of the median house price in USD 1000)

is related with 13 explanatory variables. Altogether there are 506observations.


Examples

Boston housing data

Illustration

We will work, for the purpose of motivating the semi-parametricmodels, with three explanatory variables:


Examples

Boston housing data

Illustration


NOX (annual average nitric oxide concentration, p.p. 10 million);


Examples

Boston housing data

Illustration



LSTAT (% lower status of the population);


Examples

Boston housing data

Illustration



LSTAT (% lower status of the population);

DIS (weighted distances to five Boston employment centers).


Examples

Plot of LMEDV versus NOX

0.4 0.5 0.6 0.7 0.8

2.0

2.5

3.0

3.5

4.0

NOX

LME

DV


Examples


0.4 0.5 0.6 0.7 0.8

2.0

2.5

3.0

3.5

4.0

NOX

LME

DV


Examples

Plot of LMEDV versus LSTAT

10 20 30

2.0

2.5

3.0

3.5

4.0

LSTAT

LME

DV


Examples


10 20 30

2.0

2.5

3.0

3.5

4.0

LSTAT

LME

DV


Examples

Plot of LMEDV versus DIS

2 4 6 8 10 12

2.0

2.5

3.0

3.5

4.0

DIS

LME

DV


Examples

Possible model

Description

We may try to fit initially the following semi-parametric model:

LMEDVi = α+ βNOXi + f1(LSTATi) + f2(DISi) + ǫi ,


Examples

Possible model

Description


LMEDVi = α+ βNOXi + f1(LSTATi) + f2(DISi) + ǫi ,

where ǫiiid∼ N(0, σ2) for i = 1, . . . , 506, with f1(·) and f2(·) being

continuous, smooth and nonparametric functions.


Examples

Comparison of snacks

Description

As a 3rd example, we will consider a data set from an experimentdeveloped in School of Public Health - Universidade de São Paulo, inwhich 4 different forms of light snacks (B, C, D and E) were comparedacross 20 weeks with a traditional snack (A).


Examples


Experiment description

The hydrogenated vegetable fat (hvt) was replaced by canola oil underdifferent proportions:


Examples




A: 22% hvf, 0% canola oil


Examples





B: 0% hvf, 22% canola oil


Examples






C: 17% hvf, 5% canola oil


Examples







D: 11% hvf, 11% canola oil


Examples








E: 5% hvf, 17% canola oil.


Examples








E: 5% hvf, 17% canola oil.

In this analysis we will only consider the variable TEXTURE that will becompared across time among the 5 snack types.


Examples

Mean profiles

5 10 15 20

4050

6070

80

Weeks

Text

ure

ABCDE


Examples

Variation coefficient profiles

5 10 15 20

0.05

0.10

0.15

0.20

0.25

0.30

0.35

Weeks

VC

of T

extu

reABCDE


Examples

Double gamma model

Description

Similarly to Paula (2013) we may consider a semi-parametric doublegamma model:


Examples

Double gamma model

Description


yijkind∼ G(µij , φij);


Examples

Double gamma model

Description



log(µij) = β0 + βi + f (Weeksj);


Examples

Double gamma model

Description




log(φ−1ij ) = γ0 + γi ,


Examples

Double gamma model

Description




log(φ−1ij ) = γ0 + γi ,

for i = 1(A), 2(B), 3(C), 4(D), 5(E), j = 2, 4, . . . , 20 and k = 1, . . . , 15,where φ−1

ij is the dispersion parameter, β0 + βi and γ0 + γi denote thesnack effects whereas f (·) is continuous, smooth and nonparametricfunction.


Defining f (x)

Outline

1 Examples

2 Defining f (x)



5 Packages in R

6 Voltage drop data


8 Bibliography


Defining f (x)

Defining f (x)

How to define f (x)?


Defining f (x)

Defining f (x)


Piecewise-cubic splines


Defining f (x)

Defining f (x)


Piecewise-cubic splinesB-splines


Defining f (x)

Defining f (x)



Natural cubic splines


Defining f (x)

Defining f (x)



Natural cubic splinesP-splines


Defining f (x)

Defining f (x)



Natural cubic splinesP-splinesThin-plate splines


Defining f (x)

Defining f (x)



Natural cubic splinesP-splinesThin-plate splines· · ·


Defining f (x)

Defining f (x)




Kernel


Defining f (x)

Defining f (x)




Kernel

Loess


Defining f (x)

Defining f (x)




Kernel

Loess

Wavelets


Defining f (x)

Defining f (x)




Kernel

Loess

Wavelets

· · ·


Defining f (x)


Definition

Suppose the explanatory variable values are in the interval [a, b], fori = 1, . . . , n, with m internal knots, namely a < t1 < · · · < tm < b,where m ≤ n − 2.


Defining f (x)


Definition

Suppose the explanatory variable values are in the interval [a, b], fori = 1, . . . , n, with m internal knots, namely a < t1 < · · · < tm < b,where m ≤ n − 2.

A simple choice for the nonparametric function f (x) could be thepiecewise-cubic spline, described as

f (x) = β0 + β1x + β2x2 +

m∑

j=1

γj(x − tj)3+,

where

(x − tj)+ =

{

0 se x ≤ tj(x − tj) se x > tj ,


Defining f (x)

Voltage drop data

Suppose m = 2 internal knots at t1 = 6.5 and t2 = 13.


Defining f (x)

Voltage drop data

Suppose m = 2 internal knots at t1 = 6.5 and t2 = 13.

0 5 10 15 20

810

1214

Time

Volta

ge


Defining f (x)

Voltage drop data

Fitting on the interval [0;6.5]


Defining f (x)

Voltage drop data


yi = β0 + β1xi + β2x2i + β3x3

i + ǫi .


Defining f (x)

Voltage drop data


yi = β0 + β1xi + β2x2i + β3x3

i + ǫi .

Fitting on the interval (6.5;13]


Defining f (x)

Voltage drop data


yi = β0 + β1xi + β2x2i + β3x3

i + ǫi .


yi = β0 + β1xi + β2x2i + β3x3

i + γ1(xi − 6.5)3 + ǫi .


Defining f (x)

Voltage drop data


yi = β0 + β1xi + β2x2i + β3x3

i + ǫi .


yi = β0 + β1xi + β2x2i + β3x3

i + γ1(xi − 6.5)3 + ǫi .

Fitting on the interval (13;20]


Defining f (x)

Voltage drop data


yi = β0 + β1xi + β2x2i + β3x3

i + ǫi .


yi = β0 + β1xi + β2x2i + β3x3

i + γ1(xi − 6.5)3 + ǫi .


yi = β0 + β1xi + β2x2i + β3x3

i + γ1(xi − 6.5)3 + γ2(xi − 13)3 + ǫi .


Defining f (x)

Voltage drop data


yi = β0 + β1xi + β2x2i + β3x3

i + ǫi .


yi = β0 + β1xi + β2x2i + β3x3

i + γ1(xi − 6.5)3 + ǫi .


yi = β0 + β1xi + β2x2i + β3x3

i + γ1(xi − 6.5)3 + γ2(xi − 13)3 + ǫi .

The parameter vector β = (β0, β1, β2, β3, γ1, γ2)⊤ may be estimated by

least-squares.


Defining f (x)

B-splines

Definition

A more flexible class that contains candidates for f (x) is the B-splinesclass, defined as


Defining f (x)

B-splines

Definition


f (x) =q

∑

j=1

Nj(x)τj , x ∈ [a, b],


Defining f (x)

B-splines

Definition


f (x) =q

∑

j=1

Nj(x)τj , x ∈ [a, b],

where Nj(x) are the B-spline basis functions and τj are coefficients.


Defining f (x)


Definition

NCS (see, for instance, Green and Silverman, 1994) may beexpressed as B-splines and have the following properties:


Defining f (x)


Definition


the explanatory variable values have distinct values, namelya ≤ t1 < · · · < tq ≤ b,


Defining f (x)


Definition



f (x) is a cubic spline in the intervals [t1, t2], . . . , [tq−1, tq],


Defining f (x)


Definition




f (x) is linear in the intervals [a, t1] and [tq, b],


Defining f (x)


Definition





f (x), f ′(x) and f ′′(x) are continuous.


Defining f (x)


Definition






Therefore, for NCS one has m = q − 2 internal knots.


Defining f (x)


Definition






Therefore, for NCS one has m = q − 2 internal knots.

NCS may also be defined for arbitrary m internal knots.


Defining f (x)

P-splines

Definition

P-splines (Eilers and Marx, 1996) form a flexible class of B-splinesdefined as


Defining f (x)

P-splines

Definition


f (x) =q

∑

j=1

Nj,k (x)τj , x ∈ [a, b],


Defining f (x)

P-splines

Definition


f (x) =q

∑

j=1

Nj,k (x)τj , x ∈ [a, b],

where Nj,k (x) are the B-spline basis functions of degree k (de Boor,1978), for k = 0, 1, 2, . . ., τj are coefficients, m is the number of internalknots, namely a < t1 < · · · < tm < b, and m = q + k + 1.


Defining f (x)

P-splines

Basis function

De Boor’s B-splines basis functions are expressed as


Defining f (x)

P-splines

Basis function


Nj,0(x) ={

1 tj ≤ x ≤ tj+1

0 otherwise


Defining f (x)

P-splines

Basis function


Nj,0(x) ={

1 tj ≤ x ≤ tj+1

0 otherwise

and

Nj,k (x) =(x − tj)(tj+k − tj)

Nj,k−1(x) +(tj+k+1 − x)(tj+k+1 − tj+1)

Nj+1,k−1(x),

for j = 1, . . . , q and k = 1, 2, 3, . . . .


Defining f (x)

Penalization

Why to penalize?

The aim of penalization is to reduce the parametric space solution inorder to avoid overfitting.


Additive normal model

Outline

1 Examples

2 Defining f (x)



5 Packages in R

6 Voltage drop data


8 Bibliography




Description

First, we will assume the following nonparametric model:




Description


yi = f (ti) + ǫi ,




Description


yi = f (ti) + ǫi ,

where f (t) is a continuous, smooth and nonparametric function and

ǫiiid∼ N(0, σ2), for i = 1, . . . , n.




Penalization

A suggestion is to use the second derivative penalization.




Penalization

A suggestion is to use the second derivative penalization. So, theobjective function to be minimized is given by




Penalization


SP(f, λ) =n

∑

i=1

{yi − f (ti)}2 + λ

∫ b

a[f ′′(x)]2dx ,




Penalization


SP(f, λ) =n

∑

i=1

{yi − f (ti)}2 + λ

∫ b

a[f ′′(x)]2dx ,

where f = (f (t1), . . . , f (tq))⊤, [a, b] denotes the data interval and λ > 0is the smoothing parameter.




Penalization


SP(f, λ) =n

∑

i=1

{yi − f (ti)}2 + λ

∫ b

a[f ′′(x)]2dx ,

where f = (f (t1), . . . , f (tq))⊤, [a, b] denotes the data interval and λ > 0is the smoothing parameter.

The solution is a natural cubic spline with knots at the distinct valuesa ≤ t1 < · · · < tq ≤ b.




Smoothing parameter

One has the following λ interpretation:




Smoothing parameter


when λ → 0 minimizing SP(f, λ) leads to a data interpolation;




Smoothing parameter



when λ → ∞ one has to impose f ′′(x) = 0 so the solution leads toa linear function for f (x);




Smoothing parameter



when λ → ∞ one has to impose f ′′(x) = 0 so the solution leads toa linear function for f (x);

then 0 < λ < ∞.



Semiparametric normal model

Penalization

One has for B-splines the following solution (see, for instance, Wood,2006):




Penalization


∫ b

a[f ′′(x)]2dx = τ⊤Kτ ,




Penalization


∫ b

a[f ′′(x)]2dx = τ⊤Kτ ,

where K is a (q × q) non-negative definite smoothing matrix that doesnot depend on τ .



Outline

1 Examples

2 Defining f (x)



5 Packages in R

6 Voltage drop data


8 Bibliography




Description

We will assume now the following partially linear model:




Description


yi = x⊤

i β + f (ti) + ǫi ,




Description


yi = x⊤

i β + f (ti) + ǫi ,

where x i = (xi1, . . . , xip)⊤ contains values of explanatory variables,

β = (β1, . . . , βp)⊤, f (ti) = N⊤

i τ is a B-spline and ǫiiid∼ N(0, σ2), for

i = 1, . . . , n.




Description


yi = x⊤

i β + f (ti) + ǫi ,


β = (β1, . . . , βp)⊤, f (ti) = N⊤


i = 1, . . . , n.

Objective function

The penalized least-squares function becomes




Description


yi = x⊤

i β + f (ti) + ǫi ,


β = (β1, . . . , βp)⊤, f (ti) = N⊤


i = 1, . . . , n.

Objective function

The penalized least-squares function becomes

SP(θ, λ) = (y − Xβ − Nτ )⊤(y − Xβ − Nτ ) + λτ⊤Kτ ,

where θ = (β⊤, τ⊤)⊤.




Iterative process

One has the following iterative process:




Iterative process


starting with β(0) as the parametric least-squares solution;




Iterative process



τ (0) = (N⊤N + λK)−1N⊤(y − Xβ(0));




Iterative process



τ (0) = (N⊤N + λK)−1N⊤(y − Xβ(0));back-fitting (Gauss-Seidel) algorithm:




Iterative process




β(m+1) = (X⊤X)−1X⊤{y − Nτ (m)}




Iterative process




β(m+1) = (X⊤X)−1X⊤{y − Nτ (m)}

τ (m+1) = (N⊤N + λK)−1N⊤{y − Xβ(m+1)},




Iterative process




β(m+1) = (X⊤X)−1X⊤{y − Nτ (m)}

τ (m+1) = (N⊤N + λK)−1N⊤{y − Xβ(m+1)},

for m = 0, 1, 2, . . . and λ fixed.




Effective degrees of freedom

From the iterative process at the convergence one has that






f = Nτ






f = Nτ

= N(N⊤N + λK)−1N⊤{y − Xβ}






f = Nτ

= N(N⊤N + λK)−1N⊤{y − Xβ}

= H(λ){y − Xβ}.






f = Nτ

= N(N⊤N + λK)−1N⊤{y − Xβ}

= H(λ){y − Xβ}.

So, as suggested by Hastie and Tibshirani (1990) one may take






f = Nτ

= N(N⊤N + λK)−1N⊤{y − Xβ}

= H(λ){y − Xβ}.


df(λ) = tr{H(λ)}






f = Nτ

= N(N⊤N + λK)−1N⊤{y − Xβ}

= H(λ){y − Xβ}.


df(λ) = tr{H(λ)}

= tr{N(N⊤N + λK)−1N⊤}






f = Nτ

= N(N⊤N + λK)−1N⊤{y − Xβ}

= H(λ){y − Xβ}.


df(λ) = tr{H(λ)}

= tr{N(N⊤N + λK)−1N⊤}

= tr{N⊤N(N⊤N + λK)−1}.




Model selection

The Akaike Information Criterion (AIC) and the Bayesian InformationCriterion (BIC) are, respectively, defined as




Model selection


AIC(λ) = −2L(θ, σ2) + 2{p + df(λ) + 1};




Model selection


AIC(λ) = −2L(θ, σ2) + 2{p + df(λ) + 1};

BIC(λ) = −2L(θ, σ2) + log(n){p + df(λ) + 1},




Model selection


AIC(λ) = −2L(θ, σ2) + 2{p + df(λ) + 1};

BIC(λ) = −2L(θ, σ2) + log(n){p + df(λ) + 1},

for given λ.




Estimator of the variance

For σ2 one has (given λ) the following estimator:






σ2 =

∑ni=1(yi − yi)

2

{n − p − df(λ)}.






σ2 =

∑ni=1(yi − yi)

2

{n − p − df(λ)}.

Choosing the smoothing parameter

Minimizing the generalized cross-validation score






σ2 =

∑ni=1(yi − yi)

2

{n − p − df(λ)}.



GCV(λ) =n∑n

i=1(yi − yi)2

{n − df(λ)}2 ,






σ2 =

∑ni=1(yi − yi)

2

{n − p − df(λ)}.



GCV(λ) =n∑n

i=1(yi − yi)2

{n − df(λ)}2 ,

or minimizing (jointly) AIC(λ) and df(λ) for a grid of λ values.



Alternative penalization

P-splines

Eilers and Marx (1996) proposes the alternative penalization




P-splines


SP(θ, λ) = (y − Xβ − Nτ )⊤(y − Xβ − Nτ ) + λ

q∑

j=d+1

[∆dτj ]2,




P-splines



q∑

j=d+1

[∆dτj ]2,

where N is the de Boor’s basis and ∆dτj is the penalty difference termof order d .




P-splines



q∑

j=d+1

[∆dτj ]2,


In matrix notation

SP(θ, λ) = (y − Xβ − Nτ )⊤(y − Xβ − Nτ ) + λτ⊤D⊤

d Ddτ ,




P-splines



q∑

j=d+1

[∆dτj ]2,


In matrix notation

SP(θ, λ) = (y − Xβ − Nτ )⊤(y − Xβ − Nτ ) + λτ⊤D⊤

d Ddτ ,

where Dd is the penalty difference matrix of order d .



P-splines

Penalization examples



P-splines


∆τj = τj − τj−1

D1 =

−1 1 0 00 −1 1 00 0 −1 1

.



P-splines



D1 =

−1 1 0 00 −1 1 00 0 −1 1

.

∆2τj = τj − 2τj−1 + τj−2

D2 =

1 −2 1 0 00 1 −2 1 00 0 1 −2 1

.



P-splines



D1 =

−1 1 0 00 −1 1 00 0 −1 1

.

∆2τj = τj − 2τj−1 + τj−2

D2 =

1 −2 1 0 00 1 −2 1 00 0 1 −2 1

.

∆3τj = τj − 3τj−1 + 3τj−2 − τj−3

D3 =

−1 3 −3 1 0 00 −1 3 −3 1 00 0 −1 3 −3 1

.


Packages in R

Outline

1 Examples

2 Defining f (x)



5 Packages in R

6 Voltage drop data


8 Bibliography


Packages in R

Packages in R

Packages in R

Some packages for fitting semi-parametric regression models availablefrom CRAN at http://CRAN.R-project.org:


Packages in R

Packages in R

Packages in R


gamlss (Righy and Stasinopoulos, 2005)


Packages in R

Packages in R

Packages in R



mgcv (Wood, 2015)


Packages in R

Packages in R

Packages in R



mgcv (Wood, 2015)

ssym (Vanegas and Paula, 2015)


Voltage drop data

Outline

1 Examples

2 Defining f (x)



5 Packages in R

6 Voltage drop data


8 Bibliography


Voltage drop data


0 5 10 15 20

810

1214

Time

Volta

ge


Voltage drop data

Fitted model

Description

We will fit by the package ssym the following model:



Voltage drop data

Fitted model

Description



where α is an intercept, f (·) is a continuous, smooth and

nonparametric function and ǫiiid∼ N(0, σ2) for i = 1, . . . , 41.


Voltage drop data

Fitted model

Description



where α is an intercept, f (·) is a continuous, smooth and

nonparametric function and ǫiiid∼ N(0, σ2) for i = 1, . . . , 41.

Suggestion: (n13 + 3) knots.


Voltage drop data

> require(ssym)> fit1.battery = ssym.l(voltage ~ ncs(time), data=battery,family="Normal")> summary(fit1.battery)

Family: NormalSample size: 41Quantile of the Weights0% 25% 50% 75% 100%1 1 1 1 1

************************** Median/Location submodel ********************************** Parametric component

Estimate Std.Err z-value Pr(>|z|)(Intercept) 10.904 0.0542 201.3309 < 2.2e-16 *********** Nonparametric component

Smooth.param Basis.dimen d.f. Statistic p-valuencs(time) 4.243 5.000 4.931 2709 <2e-16 ***

**** Deviance: 41


Voltage drop data

************************* Skewness/Dispersion submodel ******************************* Parametric component

Estimate Std.Err z-value Pr(>|z|)(Intercept) -2.3484 0.2209 -10.6329 < 2.2e-16 ***

**** Deviance: 42.2

*******************************************************************Overall goodness-of-fit statistic: 0.152165

-2*log-likelihood: 20.068AIC: 33.931BIC: 45.808

> np.graph(fit1.battery,which=1,xlab="Time", ylab="Voltage")> np.graph(fit1.battery,which=1,xlab="Time", ylab="Voltage",obs=TRUE)


Voltage drop data

Voltage 95% confidence band

0 5 10 15 20

−4−2

02

4

Voltage

Non

para

met

ric e

stim

ate

0 5 10 15 20

−4−2

02

4

Voltage

Non

para

met

ric e

stim

ate


Voltage drop data

Voltage 95% confidence band

0 5 10 15 20

−4−2

02

4

Voltage

Non

para

met

ric e

stim

ate

0 5 10 15 20

−4−2

02

4

0 5 10 15 20

−4−2

02

4

Voltage

Non

para

met

ric e

stim

ate


Boston housing data

Outline

1 Examples

2 Defining f (x)



5 Packages in R

6 Voltage drop data


8 Bibliography


Boston housing data


0.4 0.5 0.6 0.7 0.8

2.0

2.5

3.0

3.5

4.0

NOX

LME

DV


Boston housing data


10 20 30

2.0

2.5

3.0

3.5

4.0

LSTAT

LME

DV


Boston housing data

Possible model

Description


LMEDVi = α+ βNOXi + f (LSTATi) + ǫi ,


Boston housing data

Possible model

Description


LMEDVi = α+ βNOXi + f (LSTATi) + ǫi ,

where ǫiiid∼ N(0, σ2) for i = 1, . . . , 506, with f (·) being a continuous,

smooth and nonparametric function.


Boston housing data

> require(ssym)> require(MASS)> fit1.boston= ssym.l(log(medv) ~ nox + psp(lstat), data=Boston,family="Normal")

> summary(fit1.boston)

Family: NormalSample size: 506Quantile of the Weights0% 25% 50% 75% 100%1 1 1 1 1

************************** Median/Location submodel ********************************** Parametric component

Estimate Std.Err z-value Pr(>|z|)(Intercept) 3.1251 0.0650 48.0810 <2e-16 ***nox -0.1543 0.1106 -1.3954 0.1629

******** Nonparametric component

Smooth.param Basis.dimen d.f. Statistic p-valuepsp(lstat) 17.1 11.000 7.282 731.9 <2e-16 ***


Boston housing data

**** Deviance: 506

************************* Skewness/Dispersion submodel ******************************* Parametric component

Estimate Std.Err z-value Pr(>|z|)(Intercept) -2.9854 0.0629 -47.4859 < 2.2e-16 ***

**** Deviance: 762.68

*******************************************************************Overall goodness-of-fit statistic: 0.110987

-2*log-likelihood: -74.654AIC: -54.09BIC: -10.632

> np.graph(fit1.boston, which=1, xlab="Lstat",ylab="Estimate of f(Lstat)")> envelope(fit1.boston)


Boston housing data

f(Lstat) 95% confidence band

10 20 30

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Boston housing data

Normal probability plot

−3 −2 −1 0 1 2 3

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Bibliography

Outline

1 Examples

2 Defining f (x)



5 Packages in R

6 Voltage drop data


8 Bibliography


Bibliography

References

References


Bibliography

References

References

Belsley DA, Kuh E and Welsch RE (1980). RegressionDiagnostics. Identifying Influential Data and Sources ofCollinearity. Wiley, New York.


Bibliography

References

References


De Boor C (1978). A Practical Guide to Splines. AppliedMathematical Sciences. Springer-Verlag, New York.


Bibliography

References

References



Eilers PHC and Marx BD (1996). Flexible smoothing withB-splines and penalties. Statistical Science, 11, 89-121.


Bibliography

References

References




Green PJ and Silverman BW (1994). Nonparametric Regressionand Generalized Linear Models. Chapman and Hall, London.


Bibliography

References

References




Green PJ and Silverman BW (1994). Nonparametric Regressionand Generalized Linear Models. Chapman and Hall, London.

Hastie TJ and Tibshirani RJ (1990). Generalized Additive Models.Chapman and Hall, London.


Bibliography

References

References


Bibliography

References

References

Montgomery DC, Peck EA and Vining GG (2001). Introduction toLinear Regression Analysis, 3rd Edition. Wiley, New York.


Bibliography

References

References


Paula GA (2013). On diagnostics in double generalized linearmodels. Computational Statistics & Data Analysis, 68, 44-51.


Bibliography

References

References



Righy, R. A. e Stasinopoulos, D. M. (2005). Generalized additivemodels for location, scale and shape (with discussion). AppliedStatistics 54, 507-554.


Bibliography

References

References




Vanegas LH and Paula GA (2015). ssym: Fitting Semi-parametricLog-symmetric Regression Models. R package version 1.5.3.http://CRAN.R-project.org/package=ssym.


Bibliography

References

References




Vanegas LH and Paula GA (2015). ssym: Fitting Semi-parametricLog-symmetric Regression Models. R package version 1.5.3.http://CRAN.R-project.org/package=ssym.

Wood SN (2015). mgcv: Mixed GAM Computation Vehicle withGCV/AIC;REML. Smoothness Estimation R package version1.8-7. http://CRAN.R-project.org/package=mgcv.


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