ps: P-Splines Fits in a GAMLSS Formula

Description Usage Arguments Details Value Warning Author(s) References See Also Examples

View source: R/PS.R

Description

There are several function which use P-spline methodology:

a) pb(), the current version of P-splines which uses SVD in the fitting and therefore is the most reliable

b) pbo() and pbp(), older versions of P-splines. The first uses a simple matrix algebra in the fits. The second is the last version of pb() with SVD but uses different method for prediction.

c) pbc() the new version of cycle P-splines (using SVD)

d) cy() the older version of cycle P-splines.

e) pbm() for fitting monotonic P-splines (using SVD)

f) pbz() for fitting P-splines which allow the fitted curve to shrink to zero degrees of freedom

g) ps() the original P-splines with no facility of estimating the smoothing parameters and

j) pvc() penalised varying coefficient models.

k) pvp() older version of pb() where the prediction was different (it is here in case someone would like to compare the results).

Theoretical explanation of the above P-splines can be found in Eilers et al. (2016)

The functions take a vector and return it with several attributes. The vector is used in the construction of the design matrix X used in the fitting. The functions do not do the smoothing, but assign the attributes to the vector to aid gamlss in the smoothing. The functions doing the smoothing are gamlss.pb(), gamlss.pbo(), gamlss.pbc() gamlss.cy() gamlss.pvc(), gamlss.pbm(), gamlss.pbz and gamlss.ps() which are used in the backfitting function additive.fit.

The function pb() is more efficient and faster than the original penalised smoothing function ps(). After December 2014 the pb() has changed radically to improved performance. The older version of the pb() function is called now pbo(). pb() allows the estimation of the smoothing parameters using different local (performance iterations) methods. The method are "ML", "ML-1", "EM", "GAIC" and "GCV".

The function pbm() fits monotonic smooth functions, that is functions which increase or decrease monotonically depending on the value of the argument mono which takes the values "up" or "down".

The function pbz() is similar to pb() with the extra property that when lambda becomes very large the resulting smooth function goes to a constant rather than to a linear function. This is very useful for model selection. The function is based on Maria Durban idea of using a double penalty, one of order 2 and one of order 1. The second penalty only applies if the effective df are close to 2 (that is if a linear is already selected).

The function pbc() fits a cycle penalised beta regression spline such as the last fitted value of the smoother is equal to the first fitted value. cy() is the older version.

The function pvc() fits varying coefficient models see Hastie and Tibshirani(1993) and it is more general and flexible than the old vc() function which was based on cubic splines.

The function getZmatrix() creates a (random effect) design matrix Z which can be used to fit a P-splines smoother using the re()) function. (The re() is an interface with the random effect function lme of the package nlme.

Usage

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pb(x, df = NULL, lambda = NULL, max.df=NULL, 
    control = pb.control(...), ...)
pbo(x, df = NULL, lambda = NULL, control = pbo.control(...), ...)
pbp(x, df = NULL, lambda = NULL, control = pbp.control(...), ...)
pbo.control(inter = 20, degree = 3, order = 2, start = 10, quantiles = FALSE, 
               method = c("ML", "GAIC", "GCV", "EM", "ML-1"), k = 2, ...)
pb.control(inter = 20, degree = 3, order = 2, start = 10, quantiles = FALSE, 
          method = c("ML", "GAIC", "GCV"), k = 2, ...)
pbp.control(inter = 20, degree = 3, order = 2, start = 10, quantiles = FALSE, 
          method = c("ML", "GAIC", "GCV"), k = 2, ...)
pbc(x,  df = NULL, lambda = NULL, max.df=NULL, 
    control = pbc.control(...), ...)
pbc.control(inter = 20, degree = 3, order = 2, start = 10, 
          method = c("ML", "GAIC", "GCV"), k = 2, sin = TRUE, ...)
cy(x, df = NULL, lambda = NULL, control = cy.control(...), ...)
cy.control(inter = 20, degree = 3, order = 2, start = 10, 
          method = c("ML", "GAIC", "GCV", "EM", "ML-1"), k = 2, ts=FALSE, ...)
pvc(x, df = NULL, lambda = NULL, by = NULL, control = pvc.control(...), ...)          
pvc.control(inter = 20, degree = 3, order = 2, start = 10, quantiles = FALSE, 
             method = c("ML", "GAIC", "GCV"), k = 2, ...) 
pbm(x, df = NULL, lambda = NULL, mono=c("up", "down"), 
            control = pbm.control(...), ...)
pbm.control(inter = 20, degree = 3, order = 2, start = 10, quantiles = FALSE, 
            method=c("ML","GAIC", "GCV"), k=2, kappa = 1e10, ...)
pbz(x, df = NULL, lambda = NULL, control = pbz.control(...), ...)
pbz.control(inter = 20, degree = 3, order = 2, start = c(1e-04, 1e-04), 
     quantiles = FALSE, method = c("ML", "GAIC", "GCV"), k = 2, lim = 3, ...)

ps(x, df = 3, lambda = NULL, ps.intervals = 20, degree = 3, order = 3)

getZmatrix(x, xmin = NULL, xmax = NULL, inter = 20, degree = 3, order = 2)

Arguments

x

the univariate predictor

df

the desired equivalent number of degrees of freedom (trace of the smoother matrix minus two for the constant and linear fit)

lambda

the smoothing parameter

max.df

the limit of how large the effective degrees of freedom should be allowed to be

control

setting the control parameters

by

a factor, for fitting different smoothing curves to each level of the factor or a continuous explanatory variable in which case the coefficients of the by variable change smoothly according to x i.e. beta(x)*z where z is the by variable.

...

for extra arguments

inter

the no of break points (knots) in the x-axis

degree

the degree of the piecewise polynomial

order

the required difference in the vector of coefficients

start

the lambda starting value if the local methods are used, see below

quantiles

if TRUE the quantile values of x are use to determine the knots

ts

if TRUE assumes that it is a seasonal factor

method

The method used in the (local) performance iterations. Available methods are "ML", "ML-1", "EM", "GAIC" and "GCV"

k

the penalty used in "GAIC" and "GCV"

mono

for monotonic P-splines whether going "up" or "down"

kappa

the smoothing hyper-parameter for the monotonic part of smoothing

ps.intervals

the no of break points in the x-axis

xmin

minimum value for creating the B-spline

xmax

maximum value for creating the B-spline

sin

whether to use the sin penalty or not

lim

at which level the second penalty of order 1 should start

Details

The ps() function is based on Brian Marx function which can be found in his website. The pb(), cy(), pvc() and pbm() functions are based on Paul Eilers's original R functions. Note that ps() and pb() functions behave differently at their default values if df and lambda are not specified. ps(x) by default uses 3 extra degrees of freedom for smoothing x. pb(x) by default estimates lambda (and therefore the degrees of freedom) automatically using a "local" method. The local (or performance iterations) methods available are: (i) local Maximum Likelihood, "ML", (ii) local Generalized Akaike information criterion, "GAIC", (iii) local Generalized Cross validation "GCV" (iv) local EM-algorithm, "EM" (which is very slow) and (v) a modified version of the ML, "ML-1" which produce identical results with "EM" but faster.

The function pb() fits a P-spline smoother.

The function pbm() fits a monotonic (going up or down) P-spline smoother.

The function pbc() fits a P-spline smoother where the beginning and end are the same.

The pvc() fits a varying coefficient model.

Note that the local (or performance iterations) methods can occasionally make the convergence of gamlss less stable compared to models where the degrees of freedom are fixed.

Value

the vector x is returned, endowed with a number of attributes. The vector itself is used in the construction of the model matrix, while the attributes are needed for the backfitting algorithms additive.fit().

Warning

There are occasions where the automatic local methods do not work. One accusation which came to our attention is when the range of the response variable values is very large. Scaling the response variable will solve the problem.

Author(s)

Mikis Stasinopoulos d.stasinopoulos@londonmet.ac.uk, Bob Rigby and Paul Eilers

References

Eilers, P. H. C. and Marx, B. D. (1996). Flexible smoothing with B-splines and penalties (with comments and rejoinder). Statist. Sci, 11, 89-121.

Eilers, Paul HC, Marx, Brian D and Durban, Maria, (2016) Twenty years of P-splines. SORT-Statistics and Operations Research Transactions, 39, 149–186.

Hastie, T. J. and Tibshirani, R. J. (1993), Varying coefficient models (with discussion),J. R. Statist. Soc. B., 55, 757-796.

Rigby, R. A. and Stasinopoulos D. M. (2005). Generalized additive models for location, scale and shape,(with discussion), Appl. Statist., 54, part 3, pp 507-554.

Rigby, R. A., Stasinopoulos, D. M., Heller, G. Z., and De Bastiani, F. (2019) Distributions for modeling location, scale, and shape: Using GAMLSS in R, Chapman and Hall/CRC. An older version can be found in https://www.gamlss.com/.

Stasinopoulos D. M. Rigby R.A. (2007) Generalized additive models for location scale and shape (GAMLSS) in R. Journal of Statistical Software, Vol. 23, Issue 7, Dec 2007, https://www.jstatsoft.org/v23/i07/.

Stasinopoulos D. M., Rigby R.A., Heller G., Voudouris V., and De Bastiani F., (2017) Flexible Regression and Smoothing: Using GAMLSS in R, Chapman and Hall/CRC.

(see also https://www.gamlss.com/).

See Also

gamlss, gamlss.ps, cs

Examples

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#==============================
# pb() and ps() functions
data(aids)
# fitting a smoothing cubic spline with 7 degrees of freedom
# plus the a quarterly  effect  
aids1<-gamlss(y~ps(x,df=7)+qrt,data=aids,family=PO) # fix df's 
aids2<-gamlss(y~pb(x,df=7)+qrt,data=aids,family=PO) # fix df's
aids3<-gamlss(y~pb(x)+qrt,data=aids,family=PO) # estimate lambda
with(aids, plot(x,y))
with(aids, lines(x,fitted(aids1),col="red"))
with(aids, lines(x,fitted(aids2),col="green"))
with(aids, lines(x,fitted(aids1),col="yellow"))
rm(aids1, aids2, aids3)
#=============================
## Not run: 
# pbc()
# simulate data
set.seed(555)
x = seq(0, 1, length = 100)
y = sign(cos(1 * x * 2 * pi + pi / 4)) + rnorm(length(x)) * 0.2
plot(y~x)
m1<-gamlss(y~pbc(x)) 
lines(fitted(m1)~x)
rm(y,x,m1)
#=============================
# the pvc() function
# function to generate data
genData <- function(n=200)
 {
f1 <- function(x)-60+15*x-0.10*x^2
f2 <- function(x)-120+10*x+0.08*x^2
set.seed(1441)
x1 <- runif(n/2, min=0, max=55)
x2 <- runif(n/2, min=0, max=55)
y1 <- f1(x1)+rNO(n=n/2,mu=0,sigma=20)
y2 <- f2(x2)+rNO(n=n/2,mu=0,sigma=30)
 y <- c(y1,y2)
 x <- c(x1,x2)
 f <- gl(2,n/2)
da<-data.frame(y,x,f)
da
}
da<-genData(500)
plot(y~x, data=da, pch=21,bg=c("gray","yellow3")[unclass(f)])
# fitting models
# smoothing x
m1 <- gamlss(y~pb(x), data=da)
# parallel smoothing lines
m2 <- gamlss(y~pb(x)+f, data=da)
# linear interaction
m3 <- gamlss(y~pb(x)+f*x, data=da)
# varying coefficient model
m4 <- gamlss(y~pvc(x, by=f), data=da)
GAIC(m1,m2,m3,m4)
# plotting the fit
lines(fitted(m4)[da$f==1][order(da$x[da$f==1])]~da$x[da$f==1]
         [order(da$x[da$f==1])], col="blue", lwd=2)
lines(fitted(m4)[da$f==2][order(da$x[da$f==2])]~da$x[da$f==2]
         [order(da$x[da$f==2])], col="red", lwd=2)
rm(da,m1,m2,m3,m4)
#=================================
# the rent data
# first with a factor
data(rent)
plot(R~Fl, data=rent, pch=21,bg=c("gray","blue")[unclass(rent$B)])
r1 <- gamlss(R~pb(Fl), data=rent)
# identical to model
r11 <- gamlss(R~pvc(Fl), data=rent)
# now with the factor
r2 <- gamlss(R~pvc(Fl, by=B), data=rent)
lines(fitted(r2)[rent$B==1][order(rent$Fl[rent$B==1])]~rent$Fl[rent$B==1]
                [order(rent$Fl[rent$B==1])], col="blue", lwd=2)
lines(fitted(r2)[rent$B==0][order(rent$Fl[rent$B==0])]~rent$Fl[rent$B==0]
                [order(rent$Fl[rent$B==0])], col="red", lwd=2)
# probably not very sensible model
rm(r1,r11,r2)
#-----------
# now with a continuous variable
# additive model
 h1 <-gamlss(R~pb(Fl)+pb(A), data=rent)
# varying-coefficient model
 h2 <-gamlss(R~pb(Fl)+pb(A)+pvc(A,by=Fl), data=rent)
AIC(h1,h2)
rm(h1,h2)
#-----------
# monotone function
set.seed(1334)
x = seq(0, 1, length = 100)
p = 0.4
y = sin(2 * pi * p * x) + rnorm(100) * 0.1
plot(y~x)
m1 <- gamlss(y~pbm(x))
points(fitted(m1)~x, col="red")
yy <- -y
plot(yy~x)
m2 <- gamlss(yy~pbm(x, mono="down"))
points(fitted(m2)~x, col="red")
#==========================================
# the pbz() function
# creating uncorrelated data
set.seed(123)
y<-rNO(100)
x<-1:100
plot(y~x)
#----------------------
# ML estimation
m1<-gamlss(y~pbz(x))
m2 <-gamlss(y~pb(x))
AIC(m1,m2)
op <- par( mfrow=c(1,2))
term.plot(m1, partial=T)
term.plot(m2, partial=T)
par(op)
# GAIC estimation
m11<-gamlss(y~pbz(x, method="GAIC", k=2))
m21 <-gamlss(y~pb(x, method="GAIC", k=2))
AIC(m11,m21)
op <- par( mfrow=c(1,2))
term.plot(m11, partial=T)
term.plot(m21, partial=T)
par(op)
# GCV estimation
m12<-gamlss(y~pbz(x, method="GCV"))
m22 <-gamlss(y~pb(x, method="GCV"))
AIC(m12,m22)
op <- par( mfrow=c(1,2))
term.plot(m12, partial=T)
term.plot(m22, partial=T)
par(op)
# fixing df is more trycky since df are the extra df 
m13<-gamlss(y~pbz(x, df=0))
m23 <-gamlss(y~pb(x, df=0))
AIC(m13,m23)
# here the second penalty is not take effect therefore identical results 
m14<-gamlss(y~pbz(x, df=1))
m24 <-gamlss(y~pb(x, df=1))
AIC(m14,m24)
# fixing lambda
m15<-gamlss(y~pbz(x, lambda=1000))
m25 <-gamlss(y~pb(x, lambda=1000))
AIC(m15,m25)
#--------------------------------------------------
# prediction 
m1<-gamlss(y~pbz(x), data=data.frame(y,x))
m2 <-gamlss(y~pb(x), data=data.frame(y,x))
AIC(m1,m2)
predict(m1, newdata=data.frame(x=c(80, 90, 100, 110)))
predict(m2, newdata=data.frame(x=c(80, 90, 100, 110)))
#---------------------------------------------------

## End(Not run)

gamlss documentation built on March 31, 2021, 5:10 p.m.