Expectile and Quantile Regression

Description

Expectile and quantile regression of models with nonlinear effects e.g. spatial, random, ridge using least asymmetric weighed squares / absolutes as well as boosting; also supplies expectiles for common distributions.

Details

• This package requires the packages BayesX-package, mboost-package, splines-package and quadprog.

Author(s)

Fabian Otto-Sobotka
Carl von Ossietzky University Oldenburg
https://uol.de

Elmar Spiegel
Helmholtz Centre Munich
https://www.helmholtz-munich.de

Sabine Schnabel
Wageningen University and Research Centre
https://www.wur.nl

Linda Schulze Waltrup
Ludwig Maximilian University Munich
https://www.lmu.de

with contributions from

Paul Eilers
Erasmus Medical Center Rotterdam
https://www.erasmusmc.nl

Thomas Kneib
Georg August University Goettingen
https://www.uni-goettingen.de

Goeran Kauermann
Ludwig Maximilian University Munich
https://www.lmu.de

Maintainer: Fabian Otto-Sobotka <fabian.otto-sobotka@uni-oldenburg.de>

References

Fenske N and Kneib T and Hothorn T (2009) Identifying Risk Factors for Severe Childhood Malnutrition by Boosting Additive Quantile Regression Technical Report 052, University of Munich

He X (1997) Quantile Curves without Crossing The American Statistician, 51(2):186-192

Koenker R (2005) Quantile Regression Cambridge University Press, New York

Schnabel S and Eilers P (2009) Optimal expectile smoothing Computational Statistics and Data Analysis, 53:4168-4177

Schnabel S and Eilers P (2011) Expectile sheets for joint estimation of expectile curves (under review at Statistical Modelling)

Sobotka F and Kneib T (2010) Geoadditive Expectile Regression Computational Statistics and Data Analysis, doi: 10.1016/j.csda.2010.11.015.

See Also

mboost-package, BayesX-package

Examples

data(dutchboys)

## Expectile Regression using the restricted approach
ex = expectreg.ls(dist ~ rb(speed),data=cars,smooth="f",lambda=5,estimate="restricted")
names(ex)

## The calculation of expectiles for given distributions
enorm(0.1)
enorm(0.5)

## Introducing the expectiles-meet-quantiles distribution
x = seq(-5,5,length=100)
plot(x,demq(x),type="l")

## giving an expectile analogon to the 'quantile' function
y = rnorm(1000)

expectile(y)

eenorm(y)

Gasoline Consumption

Description

A panel of 18 observations from 1960 to 1978 in OECD countries.

Usage

data("Gasoline")

Format

A data frame with 342 observations on the following 6 variables.

country

a factor with 18 levels AUSTRIA BELGIUM CANADA DENMARK FRANCE GERMANY GREECE IRELAND ITALY JAPAN NETHERLA NORWAY SPAIN SWEDEN SWITZERL TURKEY U.K. U.S.A.

year

the year

lgaspcar

logarithm of motor gasoline consumption per car

lincomep

logarithm of real per-capita income

lrpmg

logarithm of real motor gasoline price

lcarpcap

logarithm of the stock of cars per capita

Source

Online complements to Baltagi (2001).

https://www.wiley.com/legacy/wileychi/baltagi/

References

Baltagi, Badi H. (2001) "Econometric Analysis of Panel Data", 2nd ed., John Wiley and Sons.

Gibraltar, B.H. and J.M. Griffin (1983) ???Gasoline demand in the OECD: An application of pooling and testing procedures???, European Economic Review, 22(2), 117???137.

Examples

data(Gasoline)

expreg<-expectreg.ls(lrpmg~rb(lcarpcap),smooth="fixed",data=Gasoline,
lambda=20,estimate="restricted",expectiles=c(0.01,0.05,0.2,0.8,0.95,0.99))

plot(expreg)

Semiparametric M-Quantile Regression

Description

Robust M-quantiles are estimated using an iterative penalised reweighted least squares approach. Effects using quadratic penalties can be included, such as P-splines, Markov random fields or Kriging.

Usage

Mqreg(formula, data = NULL, smooth = c("schall", "acv", "fixed"), 
      estimate = c("iprls", "restricted"),lambda = 1, tau = NA, robust = 1.345,
      adaptive = FALSE, ci = FALSE, LSMaxCores = 1)

Arguments

Details

In the least squares approach the following loss function is minimised:

S = \sum_{i=1}^{n}{ w_p(y_i - m_i(p))^2}

with weights

w_p(u) = (-(1-p)*c*(u_i< -c)+(1-p)*u_i*(u_i<0 \& u_i>=-c)+p*u_i*(u_i>=0 \& u_i<c)+p*c*(u_i>=c)) / u_i

for quantiles and

w_p(u) = -(1-p)*c*(u_i< -c)+(1-p)*u_i*(u_i<0 \& u_i>=-c)+p*u_i*(u_i>=0 \& u_i<c)+p*c*(u_i>=c)

for expectiles, with standardised residuals u_i = 0.6745*(y_i - m_i(p)) / median(y-m(p)) and robustness constant c.

Value

An object of class 'expectreg', which is basically a list consisting of:

plot, predict, resid, fitted, effects and further convenient methods are available for class 'expectreg'.

Author(s)

Monica Pratesi
University Pisa
https://www.unipi.it

M. Giovanna Ranalli
University Perugia
https://www.unipg.it

Nicola Salvati
University Perugia
https://www.unipg.it

Fabian Otto-Sobotka
University Oldenburg
https://uol.de

References

Pratesi M, Ranalli G and Salvati N (2009) Nonparametric M-quantile regression using penalised splines Journal of Nonparametric Statistics, 21:3, 287-304.

Otto-Sobotka F, Ranalli G, Salvati N, Kneib T (2019) Adaptive Semiparametric M-quantile Regression Econometrics and Statistics 11, 116-129.

See Also

expectreg.ls, rqss

Examples

m <- Mqreg(dist~rb(speed,"pspline"),data=cars,smooth="f",
                        tau=c(0.05,0.5,0.95),lambda=10)
plot(m,rug=FALSE)

Calculation of the conditional CDF based on expectile curves

Description

Estimating the CDF of the response for a given value of covariate. Additionally quantiles are computed from the distribution function which allows for the calculation of regression quantiles.

Usage

cdf.qp(expectreg, x = NA, qout = NA, extrap = FALSE, e0 = NA, eR = NA,
       lambda = 0, var.dat = NA)

cdf.bundle(bundle, qout = NA, extrap = FALSE, quietly = FALSE)

Arguments

Details

Expectile curves can describe very well the spread and location of a scatterplot. With a set of curves they give good impression about the nature of the data. This information can be used to estimate the conditional density from the expectile curves. The results of the bundle model are especially suited in this case as only one density will be estimated which can then be modulated to over the independent variable x. The density estimation can be formulated as penalized least squares problem that results in a smooth non-negative density. The theoretical values of a quantile regression at this covariate value are also returned for adjustable probabilities qout.

Value

A list consisting of

Author(s)

Goeran Kauermann, Linda Schulze Waltrup
Ludwig Maximilian University Munich
https://www.lmu.de

Fabian Sobotka
Georg August University Goettingen
https://www.uni-goettingen.de

Sabine Schnabel
Wageningen University and Research Centre
https://www.wur.nl

Paul Eilers
Erasmus Medical Center Rotterdam
https://www.erasmusmc.nl

References

Schnabel SK and Eilers PHC (2010) A location scale model for non-crossing expectile curves (working paper)

Schulze Waltrup L, Sobotka F, Kneib T and Kauermann G (2014) Expectile and Quantile Regression - David and Goliath? Statistical Modelling.

See Also

expectreg.ls, expectreg.qp

Examples

d = expectreg.ls(dist ~ rb(speed),data=cars,smooth="f",lambda=5,estimate="restricted",
                 expectiles=c(0.0001,0.001,seq(0.01,0.99,0.01),0.999,0.9999))
e = cdf.qp(d,15,extrap=TRUE)
e 

Data set about the growth of dutch children

Description

Data from the fourth dutch growth study in 1997.

Usage

data(dutchboys)

Format

A data frame with 6848 observations on the following 10 variables.

defnr

identification number

age

age in decimal years

hgt

length/height in cm

wgt

weight in kg

hc

head circumference in cm

hgt.z

z-score length/height

wgt.z

z-score weight

hc.z

z-score head circumference

bmi.z

z-score body mass index

hfw.z

z-score height for weight

z-scores were calculated relative to the Dutch references.

Details

The Fourth Dutch Growth Study is a cross-sectional study that measures growth and development of the Dutch population between ages 0 and 21 years. The study is a follow-up to earlier studies performed in 1955, 1965 and 1980, and its primary goal is to update the 1980 references.

Source

van Buuren S and Fredriks A (2001) Worm plot: A simple diagnostic device for modeling growth reference curves Statistics in Medicine, 20:1259-1277

References

Schnabel S and Eilers P (2009) Optimal expectile smoothing Computatational Statistics and Data Analysis, 53: 4168-4177

Examples

data(dutchboys)

expreg <- expectreg.ls(dutchboys[,3] ~ rb(dutchboys[,2],"pspline"),smooth="f",
                       estimate="restricted",expectiles=c(.05,.5,.95))
plot(expreg)

Expectiles of distributions

Description

Much like the 0.5 quantile of a distribution is the median, the 0.5 expectile is the mean / expected value. These functions add the possibility of calculating expectiles of known distributions. The functions starting with 'e' calculate an expectile value for given asymmetry values, the functions starting with 'pe' calculate vice versa.

Usage

enorm(asy, m = 0, sd = 1)
penorm(e, m = 0, sd = 1)

ebeta(asy, a = 1, b = 1)
pebeta(e, a = 1, b = 1)

eunif(asy, min = 0, max = 1)
peunif(e, min = 0, max = 1)

et(asy, df)
pet(e, df)

elnorm(asy, meanlog = 0, sdlog = 1)
pelnorm(e, meanlog = 0, sdlog = 1)

egamma(asy, shape, rate = 1, scale = 1/rate)
pegamma(e, shape, rate = 1, scale = 1/rate)

eexp(asy, rate = 1)
peexp(e, rate = 1)

echisq(asy, df)
pechisq(e, df)

Arguments

Details

An expectile of a distribution cannot be determined explicitely, but instead is given by an equation. The expectile z for an asymmetry p is: p = \frac{G(z) - z F(z)}{2(G(z) - z F(z)) + z - m} where m is the mean, F the cdf and G the partial moment function G(z) = \int\limits_{-\infty}^{z} uf(u) \mbox{d}u .

Value

Vector of the expectiles or asymmetry values for the desired distribution.

Author(s)

Fabian Otto- Sobotka
Carl von Ossietzky University Oldenburg
https://uol.de

Thomas Kneib
Georg August University Goettingen
https://www.uni-goettingen.de

References

Newey W and Powell J (1987) Asymmetric least squares estimation and testing Econometrica, 55:819-847

See Also

eemq

Examples

x <- seq(0.02,0.98,0.2)

e = enorm(x)
e
penorm(e)

Sample Expectiles

Description

Expectiles are fitted to univariate samples with least asymmetrically weighted squares for asymmetries between 0 and 1. For graphical representation an expectile - expectile plot is available. The corresponding functions quantile, qqplot and qqnorm are mapped here for expectiles.

Usage

expectile(x, probs = seq(0, 1, 0.25), dec = 4)

eenorm(y, main = "Normal E-E Plot",
       xlab = "Theoretical Expectiles", ylab = "Sample Expectiles",
       plot.it = TRUE, datax = FALSE, ...)
       
eeplot(x, y, plot.it = TRUE, xlab = deparse(substitute(x)),
       ylab = deparse(substitute(y)), main = "E-E Plot", ...)

Arguments

Details

In least asymmetrically weighted squares (LAWS) each expectile is fitted independently from the others. LAWS minimizes:

S = \sum_{i=1}^{n}{ w_i(p)(x_i - \mu(p))^2}

with

w_i(p) = p 1_{(x_i > \mu(p))} + (1-p) 1_{(x_i < \mu(p))} .

\mu(p) is determined by iteration process with recomputed weights w_i(p).

Value

Numeric vector with the fitted expectiles.

Author(s)

Fabian Otto-Sobotka
Carl von Ossietzky University Oldenburg
https://uol.de

References

Sobotka F and Kneib T (2010) Geoadditive Expectile Regression Computational Statistics and Data Analysis, doi: 10.1016/j.csda.2010.11.015.

See Also

expectreg.ls, quantile

Examples

data(dutchboys)

expectile(dutchboys[,3])

x = rnorm(1000)

expectile(x,probs=c(0.01,0.02,0.05,0.1,0.2,0.5,0.8,0.9,0.95,0.98,0.99))

eenorm(x)

Quantile and expectile regression using boosting

Description

Generalized additive models are fitted with gradient boosting for optimizing arbitrary loss functions to obtain the graphs of 11 different expectiles for continuous, spatial or random effects.

Usage

expectreg.boost(formula, data, mstop = NA, expectiles = NA, cv = TRUE, 
BoostmaxCores = 1, quietly = FALSE)

quant.boost(formula, data, mstop = NA, quantiles = NA, cv = TRUE, 
BoostmaxCores = 1, quietly = FALSE)

Arguments

Details

A (generalized) additive model is fitted using a boosting algorithm based on component-wise univariate base learners. The base learner can be specified via the formula object. After fitting the model a cross-validation is done using cvrisk to determine the optimal stopping point for the boosting which results in the best fit.

Value

An object of class 'expectreg', which is basically a list consisting of:

plot, predict, resid, fitted and effects methods are available for class 'expectreg'.

Author(s)

Fabian Otto- Sobotka
Carl von Ossietzky University Oldenburg
https://uol.de

Thomas Kneib, Elmar Spiegel
Georg August University Goettingen
https://www.uni-goettingen.de

References

Fenske N and Kneib T and Hothorn T (2009) Identifying Risk Factors for Severe Childhood Malnutrition by Boosting Additive Quantile Regression Technical Report 052, University of Munich

Sobotka F and Kneib T (2010) Geoadditive Expectile Regression Computational Statistics and Data Analysis, doi: 10.1016/j.csda.2010.11.015.

See Also

expectreg.ls, gamboost, bbs, cvrisk

Examples

ex <- expectreg.boost(dist ~ bbs(speed),cars, mstop=200, 
                      expectiles=c(0.1,0.5,0.95),quietly=TRUE)
fitted(ex)

Expectile regression of additive models

Description

Additive models are fitted with least asymmetrically weighted squares or quadratic programming to obtain expectiles for parametric, continuous, spatial and random effects.

Usage

expectreg.ls(formula, data = NULL, estimate = c("laws", "restricted", "bundle", "sheets"),
smooth = c("schall", "ocv", "gcv", "cvgrid", "aic", "bic", "lcurve", "fixed"), 
lambda = 1, expectiles = NA, ci = FALSE, LAWSmaxCores = 1, ...)

expectreg.qp(formula, data  = NULL, id = NA, smooth = c("schall", "acv", "fixed"), 
             lambda = 1, expectiles = NA) 

Arguments

Details

In least asymmetrically weighted squares (LAWS) each expectile is fitted independently from the others. LAWS minimizes:

S = \sum_{i=1}^{n}{ w_i(p)(y_i - \mu_i(p))^2}

with

w_i(p) = p 1_{(y_i > \mu_i(p))} + (1-p) 1_{(y_i < \mu_i(p))} .

The restricted version fits the 0.5 expectile at first and then the residuals. Afterwards the other expectiles are fitted as deviation by a factor of the residuals from the mean expectile. This algorithm is based on He(1997). The advantage is that expectile crossing cannot occur, the disadvantage is a suboptimal fit in certain heteroscedastic settings. Also, since the number of fits is significantly decreased, the restricted version is much faster.

The expectile bundle has a resemblence to the restricted regression. At first, a trend curve is fitted and then an iteration is performed between fitting the residuals and calculating the deviation factors for all the expectiles until the results are stable. Therefore this function shares the (dis)advantages of the restricted.

The expectile sheets construct a p-spline basis for the expectiles and perform a continuous fit over all expectiles by fitting the tensor product of the expectile spline basis and the basis of the covariates. In consequence there will be most likely no crossing of expectiles but also a good fit in heteroscedastic scenarios.

The function expectreg.qp also fits a sheet over all expectiles, but it uses quadratic programming with constraints, so crossing of expectiles will definitely not happen. So far the function is implemented for one nonlinear or spatial covariate and further parametric covariates. It works with all smoothing methods.

Value

An object of class 'expectreg', which is basically a list consisting of:

plot, predict, resid, fitted, effects and further convenient methods are available for class 'expectreg'.

Author(s)

Fabian Otto-Sobotka
Carl von Ossietzky University Oldenburg
https://uol.de

Thomas Kneib
Georg August University Goettingen
https://www.uni-goettingen.de

Sabine Schnabel
Wageningen University and Research Centre
https://www.wur.nl

Paul Eilers
Erasmus Medical Center Rotterdam
https://www.erasmusmc.nl

Linda Schulze Waltrup, Goeran Kauermann
Ludwig Maximilians University Muenchen
https://www.lmu.de

References

Schnabel S and Eilers P (2009) Optimal expectile smoothing Computational Statistics and Data Analysis, 53:4168-4177

Sobotka F and Kneib T (2010) Geoadditive Expectile Regression Computational Statistics and Data Analysis, doi: 10.1016/j.csda.2010.11.015.

Schnabel S and Eilers P (2011) Expectile sheets for joint estimation of expectile curves (under review at Statistical Modelling)

Frasso G and Eilers P (2013) Smoothing parameter selection using the L-curve (under review)

See Also

rb, expectreg.boost

Examples

library(expectreg)
ex = expectreg.ls(dist ~ rb(speed),data=cars,smooth="b",lambda=5,expectiles=c(0.01,0.2,0.8,0.99))
ex = expectreg.ls(dist ~ rb(speed),data=cars,smooth="f",lambda=5,estimate="restricted")
plot(ex)


explaws <- expectreg.ls(dist~rb(speed,"pspline"),data=cars,smooth="gcv",
                        expectiles=c(0.05,0.5,0.95))
print(explaws)
plot(explaws)

###expectile regression using a fixed penalty
plot(expectreg.ls(dist~rb(speed,"pspline"),data=cars,smooth="fixed",
     lambda=1,expectiles=c(0.05,0.25,0.75,0.95)))
plot(expectreg.ls(dist~rb(speed,"pspline"),data=cars,smooth="fixed",
     lambda=0.0000001,expectiles=c(0.05,0.25,0.75,0.95)))
    #As can be seen in the plot, a too small penalty causes overfitting of the data.
plot(expectreg.ls(dist~rb(speed,"pspline"),data=cars,smooth="fixed",
     lambda=50,expectiles=c(0.05,0.25,0.75,0.95)))
    #If the penalty parameter is chosen too large, 
    #the expectile curves are smooth but don't represent the data anymore.

Malnutrition of Childen in India

Description

Data sample from a 'Demographic and Health Survey' about malnutrition of children in india. Data set only contains 1/10 of the observations and some basic variables to enable first analyses.

Usage

data(india)

Format

A data frame with 4000 observations on the following 6 variables.

stunting

A numeric malnutrition score with range (-600;600).

cbmi

BMI of the child.

cage

Age of the child in months.

mbmi

BMI of the mother.

mage

Age of the mother in years.

distH

The distict in India, where the child lives. Encoded in the region naming of the map india.bnd.

Source

https://dhsprogram.com

References

Fenske N and Kneib T and Hothorn T (2009) Identifying Risk Factors for Severe Childhood Malnutrition by Boosting Additive Quantile Regression Technical Report 052, University of Munich

Examples

data(india)

expreg <- expectreg.ls(stunting ~ rb(cbmi),smooth="fixed",data=india,
lambda=30,estimate="restricted",expectiles=c(0.01,0.05,0.2,0.8,0.95,0.99))
plot(expreg)

Regions of India - boundary format

Description

Map of the country india, represented in the boundary format (bnd) as defined in the package BayesX-package.

Usage

data(india.bnd)

Format

The format is: List of 449 - attr(*, "class")= chr "bnd" - attr(*, "height2width")= num 0.96 - attr(*, "surrounding")=List of 449 - attr(*, "regions")= chr [1:440] "84" "108" "136" "277" ...

Details

For details about the format see read.bnd.

Source

Jan Priebe University of Goettingen https://www.bnitm.de/forschung/forschungsgruppen/implementation/ag-gesundheitsoekonomie/team

Examples

data(india)
data(india.bnd)

drawmap(data=india,map=india.bnd,regionvar=6,plotvar=1)

Methods for expectile regression objects

Description

Methods for objects returned by expectile regression functions.

Usage

## S3 method for class 'expectreg'
print(x, ...)

## S3 method for class 'expectreg'
summary(object,...)

## S3 method for class 'expectreg'
predict(object, newdata = NULL, with_intercept = T, ...)

## S3 method for class 'expectreg'
x[i]

## S3 method for class 'expectreg'
residuals(object, ...)
## S3 method for class 'expectreg'
resid(object, ...)

## S3 method for class 'expectreg'
fitted(object, ...)
## S3 method for class 'expectreg'
fitted.values(object, ...)

## S3 method for class 'expectreg'
effects(object, ...)

## S3 method for class 'expectreg'
coef(object, ...)
## S3 method for class 'expectreg'
coefficients(object, ...)

## S3 method for class 'expectreg'
confint(object, parm = NULL, level = 0.95, ...)

Arguments

Details

These functions can be used to extract details from fitted models. print shows a dense representation of the model fit.

[ can be used to define a new object with a subset of covariates from the original fit.

The function coef extracts the regression coefficients for each covariate listed separately. For the function expectreg.boost this is not possible.

Value

[ returns a new object of class expectreg with a subset of covariates from the original fit.

resid returns the residuals in order of the response.

fitted returns the overall fitted values \hat{y} while effects returns the values for each covariate in a list.

coef returns a list of all regression coefficients separately for each covariate.

Author(s)

Fabian Otto- Sobotka
Carl von Ossietzky University Oldenburg
https://uol.de

Elmar Spiegel
Georg August University Goettingen https://www.uni-goettingen.de

References

Schnabel S and Eilers P (2009) Optimal expectile smoothing Computational Statistics and Data Analysis, 53:4168-4177

Sobotka F and Kneib T (2010) Geoadditive Expectile Regression Computational Statistics and Data Analysis, doi: 10.1016/j.csda.2010.11.015.

See Also

expectreg.ls, expectreg.boost, expectreg.qp

Examples

data(dutchboys)

expreg <- expectreg.ls(hgt ~ rb(age,"pspline"),data=dutchboys,smooth="f",
                       expectiles=c(0.05,0.2,0.8,0.95))

print(expreg)

coef(expreg)

new.d = dutchboys[1:10,]
new.d[,2] = 1:10

predict(expreg,newdata=new.d)

Regions of northern Germany - boundary format

Description

Map of northern Germany, represented in the boundary format (bnd) as defined in the package BayesX-package.

Usage

data(northger.bnd)

Format

The format is: List of 145 - attr(*, "class")= chr "bnd" - attr(*, "height2width")= num 1.54 - attr(*, "surrounding")=List of 145 - attr(*, "regions")= chr [1:145] "1001" "1002" "1003" "1004" ...

Details

For details about the format see read.bnd.

Source

Thomas Kneib
Georg August University Goettingen
https://www.uni-goettingen.de

Examples

data(northger.bnd)

drawmap(map=northger.bnd,mar.min=NULL)

The "expectiles-meet-quantiles" distribution family.

Description

Density, distribution function, quantile function, random generation, expectile function and expectile distribution function for a family of distributions for which expectiles and quantiles coincide.

Usage

pemq(z,ncp=0,s=1)
demq(z,ncp=0,s=1)
qemq(q,ncp=0,s=1)
remq(n,ncp=0,s=1)
eemq(asy,ncp=0,s=1)
peemq(e,ncp=0,s=1)

Arguments

Details

This distribution has the cumulative distribution function: F(x;ncp,s) = \frac{1}{2}(1 + sgn(\frac{x-ncp}{s}) \sqrt{1 - \frac{2}{2 + (\frac{x-ncp}{s})^2}})

and the density: f(x;ncp,s) = \frac{1}{s}( \frac{1}{2 + (\frac{x-ncp}{s})^2} )^\frac{3}{2}

It has infinite variance, still can be scaled by the parameter s. It has mean ncp. In the canonical parameters it is equal to a students-t distribution with 2 degrees of freedom. For s = \sqrt{2} it is equal to a distribution introduced by Koenker(2005).

Value

demq gives the density, pemq and peemq give the distribution function, qemq gives the quantile function, eemq computes the expectiles numerically and is only provided for completeness, since the quantiles = expectiles can be determined analytically using qemq, and remq generates random deviates.

Author(s)

Fabian Otto- Sobotka
Carl von Ossietzky University Oldenburg
https://uol.de

Thomas Kneib
Georg August University Goettingen
https://www.uni-goettingen.de

References

Koenker R (2005) Quantile Regression Cambridge University Press, New York

See Also

enorm

Examples

x <- seq(-5,5,length=100)
plot(x,demq(x))
plot(x,pemq(x,ncp=1))

z <- remq(100,s=sqrt(2))
plot(z)

y <- seq(0.02,0.98,0.2)
qemq(y)
eemq(y)

pemq(x) - peemq(x)

Default expectreg plotting

Description

Takes a expectreg object and plots the estimated effects.

Usage

## S3 method for class 'expectreg'
plot(x, rug = TRUE, xlab = NULL, ylab = NULL, ylim = NULL, 
legend = TRUE, ci = FALSE, ask = NULL, cex.main = 2, mar.min = 5, main = NULL, 
cols = "rainbow", hcl.par = list(h = c(260, 0), c = 185, l = c(30, 85)), 
ylim_spat = NULL, ylim_factor = NULL, range_warning = TRUE, add_intercept = TRUE, ...)

Arguments

Details

The plot function gives a visual representation of the fitted expectiles separately for each covariate.

Value

No return value, only graphical output.

Author(s)

Fabian Otto- Sobotka
Carl von Ossietzky University Oldenburg
https://uol.de

Elmar Spiegel
Georg August University Goettingen
https://www.uni-goettingen.de

References

Schnabel S and Eilers P (2009) Optimal expectile smoothing Computational Statistics and Data Analysis, 53:4168-4177

Sobotka F and Kneib T (2010) Geoadditive Expectile Regression Computational Statistics and Data Analysis, doi: 10.1016/j.csda.2010.11.015.

See Also

expectreg.ls, expectreg.boost, expectreg.qp

Examples

data(dutchboys)

expreg <- expectreg.ls(hgt ~ rb(age,"pspline"),data=dutchboys,smooth="f",
                       expectiles=c(0.05,0.2,0.8,0.95))
plot(expreg)

Restricted expectile regression of additive models

Description

A location-scale model to fit generalized additive models with least asymmetrically weighted squares to obtain the graphs of different expectiles or quantiles for continuous, spatial or random effects.

Usage

quant.bundle(formula, data = NULL, smooth = c("schall", "acv", "fixed"), 
             lambda = 1, quantiles = NA, simple = TRUE)

Arguments

Details

In least asymmetrically weighted squares (LAWS) each expectile is fitted by minimizing:

S = \sum_{i=1}^{n}{ w_i(p)(y_i - \mu_i(p))^2}

with

w_i(p) = p 1_{(y_i > \mu_i(p))} + (1-p) 1_{(y_i < \mu_i(p))} .

The restricted version fits the 0.5 expectile at first and then the residuals. Afterwards the other expectiles are fitted as deviation by a factor of the residuals from the mean expectile. This algorithm is based on He(1997). The advantage is that expectile crossing cannot occur, the disadvantage is a suboptimal fit in certain heteroscedastic settings. Also, since the number of fits is significantly decreased, the restricted version is much faster.

The expectile bundle has a resemblence to the restricted regression. At first, a trend curve is fitted and then an iteration is performed between fitting the residuals and calculating the deviation factors for all the expectiles until the results are stable. Therefore this function shares the (dis)advantages of the restricted.

The quantile bundle uses either the restricted expectiles or the bundle to estimate a dense set of expectiles. Next this set is used to estimate a density with the function cdf.bundle. From this density quantiles are determined and inserted to the calculated bundle model. This results in an estimated location-scale model for quantile regression.

Value

An object of class 'expectreg', which is basically a list consisting of:

plot, predict, resid, fitted and effects methods are available for class 'expectreg'.

Author(s)

Fabian Otto- Sobotka
Carl von Ossietzky University Oldenburg
https://uol.de

Thomas Kneib
Georg August University Goettingen
https://www.uni-goettingen.de

Sabine Schnabel
Wageningen University and Research Centre
https://www.wur.nl

Paul Eilers
Erasmus Medical Center Rotterdam
https://www.erasmusmc.nl

References

Schnabel S and Eilers P (2009) Optimal expectile smoothing Computational Statistics and Data Analysis, 53:4168-4177

He X (1997) Quantile Curves without Crossing The American Statistician, 51(2):186-192

Schnabel S and Eilers P (2011) A location scale model for non-crossing expectile curves (working paper)

Sobotka F and Kneib T (2010) Geoadditive Expectile Regression Computational Statistics and Data Analysis, doi: 10.1016/j.csda.2010.11.015.

See Also

rb, expectreg.boost

Examples

qb = quant.bundle(dist ~ rb(speed),data=cars,smooth="f",lambda=5)
plot(qb)

qbund <- quant.bundle(dist ~ rb(speed),data=cars,smooth="f",lambda=50000,simple=FALSE)

Creates base for a regression based on covariates

Description

Based on given observations a matrix is created that creates a basis e.g. of splines or a markov random field that is evaluated for each observation. Additionally a penalty matrix is generated. Shape constraint p-spline bases can also be specified.

Usage

rb(x, type = c("pspline", "2dspline", "markov", "krig", "random", 
"ridge", "special", "parametric", "penalizedpart_pspline"), B_size = 20, 
B = NA, P = NA, bnd = NA, center = TRUE, by = NA, ...)

mono(x, constraint = c("increase", "decrease", "convex", "concave", "flatend"), 
by = NA)

Arguments

Details

Possible types of bases:

pspline

Penalized splines made upon B_size equidistant knots with degree 3. The penalization matrix consists of differences of the second order, see diff.

2dspline

Tensor product of 2 p-spline bases with the same properties as above.

markov

Gaussian markov random field with a neighbourhood structure given by P or bnd.

krig

'kriging' produces a 2-dimensional base, which is calculated as exp(-r/phi)*(1+r/phi) where phi is the maximum euclidean distance between two knots divided by a constant.

random

A 'random' effect is like the 'markov' random field based on a categorial variable, and since there is no neighbourhood structure, P = I.

ridge

In a 'ridge' regression, the base is made from the independent variables while the goal is to determine significant variables from the coefficients. Therefore no penalization is used (P = I).

special

In the 'special' case, B and P are user defined.

parametric

A parametric effect.

penalizedpart_pspline

Penalized splines made upon B_size equidistant knots with degree 3. The penalization matrix consists of differences of the second order, see diff. Generally a P-spline of degree 3 with 2 order penalty can be splited in a linear trend and the deviation of the linear trend. Here only the wiggly deviation of the linear trend is kept. It is possible to combine it with the same covariate of type parametric

Value

List consisting of:

Warning

The pspline is now centered around its mean. Thus different results compared to old versions of expectreg occure.

Author(s)

Fabian Otto- Sobotka
Carl von Ossietzky University Oldenburg
https://uol.de

Thomas Kneib, Elmar Spiegel
Georg August University Goettingen
https://www.uni-goettingen.de

Sabine Schnabel
Wageningen University and Research Centre
https://www.wur.nl

Paul Eilers
Erasmus Medical Center Rotterdam
https://www.erasmusmc.nl

References

Fahrmeir L and Kneib T and Lang S (2009) Regression Springer, New York

See Also

quant.bundle, expectreg.ls

Examples

x <- rnorm(100)

bx <- rb(x,"pspline")

y <- sample(10,100,replace=TRUE)

by <- rb(y,"random")

Update given expectreg model

Description

Updates a given expectreg model with the specified changes

Usage

## S3 method for class 'expectreg'
update(object, add_formula, data = NULL, estimate = NULL, 
smooth = NULL, lambda = NULL, expectiles = NULL, delta_garrote = NULL, ci = NULL, 
...)

Arguments

Details

Re-estimates the given model, with the specified changes. If nothing is specified the characteristics of the original model are used. Except lambda here the default 1 is used as initial value.

Value

object of class expectreg

Author(s)

Elmar Spiegel
Helmholtz Zentrum Muenchen
https://www.helmholtz-munich.de

See Also

update, update.formula

Examples

data(india)

model1<-expectreg.ls(stunting~rb(cbmi),smooth="fixed",data=india,lambda=30,
                     estimate="restricted",expectiles=c(0.01,0.05,0.2,0.8,0.95,0.99))
plot(model1)

# Change formula and update model
add_formula<-.~.+rb(cage)
update_model1<-update(model1,add_formula)
plot(update_model1)

# Use different asymmetries and update model
update_model2<-update(model1,expectiles=c(0.1,0.5,0.9))
plot(update_model2)