scam: Shape constrained additive models (SCAM) and integrated...

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

View source: R/scam.r

Description

This function fits a SCAM to data. Univariate smooths subject to monotonicity, convexity, or monotonicity plus convexity are available as model terms, as well as bivariate smooths with double or single monotonicity. Smoothness selection is estimated as part of the fitting. Confidence/credible intervals are available for each smooth term.

All the shaped constrained smooths have been added to the gam() in package mgcv setup using the smooth.construct function. The routine calls a gam() function for the model set up, but there are separate functions for the model fitting, scam.fit, and smoothing parameter selection, bfgs_gcv.ubre. Any unconstrained smooth available in gam can be taken as model terms.

Usage

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scam(formula, family = gaussian(), data = list(), gamma = 1, 
      sp = NULL, weights = NULL, offset = NULL,optimizer="bfgs", 
      optim.method=c("Nelder-Mead","fd"),scale = 0, knots=NULL,
      not.exp=FALSE, start= NULL, etastart=NULL,mustart= NULL,
      control=list(),AR1.rho=0, AR.start=NULL,drop.unused.levels=TRUE)

Arguments

formula

A SCAM formula. This is exactly like the formula for a GAM (see formula.gam of the mgcv library) except that monotone smooth terms, can be added in the expression of the form
s(x1,k=12,bs="mpi",by=z),
where bs indicates the basis to use for the constrained smooth: the built in options for the monotonic smooths are described in shape.constrained.smooth.terms,

family

A family object specifying the distribution and link to use in fitting etc. See glm and family for more details.

data

A data frame or list containing the model response variable and covariates required by the formula. By default the variables are taken from environment(formula): typically the environment from which gam is called.

gamma

A constant multiplier to inflate the model degrees of freedom in the GCV or UBRE/AIC score.

sp

A vector of smoothing parameters can be provided here. Smoothing parameters must be supplied in the order that the smooth terms appear in the model formula. The default sp=NULL indicates that smoothing parameters should be estimated. If length(sp) does not correspond to the number of underlying smoothing parameters or negative values supplied then the vector is ignored and all the smoothing parameters will be estimated.

weights

Prior weights on the data.

offset

Used to supply a model offset for use in fitting. Note that this offset will always be completely ignored when predicting, unlike an offset included in formula. This conforms to the behaviour of lm, glm and gam.

optimizer

The numerical optimization method to use to optimize the smoothing parameter estimation criterion. "bfgs" for the built in to scam package routine bfgs_gcv.ubre, "optim", "nlm", "nlm.fd" (based on finite-difference approximation of the derivatives). "efs" for the extended Fellner Schall method of Wood and Fasiolo (2017) (rather than minimizing REML as in gam(mgcv) this minimizes the GCV criterion).

optim.method

In case of optimizer="optim" this specifies the numerical method to be used in optim in the first element, the second element of optim.method indicates whether the finite difference approximation should be used ("fd") or analytical gradient ("grad"). The default is optim.method=c("Nelder-Mead","fd").

scale

If this is positive then it is taken as the known scale parameter of the exponential family distribution. Negative value indicates that the scale paraemter is unknown. 0 indicates that the scale parameter is 1 for Poisson and binomial and unknown otherwise. This conforms to the behaviour of gam.

knots

An optional list containing user specified knot values to be used for basis construction. Different terms can use different numbers of knots.

not.exp

if TRUE then notExp() function will be used in place of exp (default value) in positivity ensuring beta parameters re-parameterization.

start

Initial values for the model coefficients.

etastart

Initial values for the linear predictor.

mustart

Initial values for the expected values.

control

A list of fit control parameters to replace defaults returned by scam.control. Values not set assume default values.

AR1.rho

The AR1 correlation parameter. An AR1 error model can be used for the residuals of Gaussian-identity link models. Standardized residuals (approximately uncorrelated under correct model) returned in std.rsd if non zero.

AR.start

logical variable of same length as data, TRUE at first observation of an independent section of AR1 correlation. Very first observation in data frame does not need this. If NULL (default) then there are no breaks in AR1 correlaion.

drop.unused.levels

as with gam by default unused levels are dropped from factors before fitting.

Details

A shape constrained additive model (SCAM) is a generalized linear model (GLM) in which the linear predictor is given by strictly parametric components plus a sum of smooth functions of the covariates where some of the functions are assumed to be shape constrained. For example,

log(E(Y_i))=X_i*b+f_1(x_1i)+m_2(x_2i)+f_3(x_3i)

where the independent response variables Y_i follow Poisson distribution with log link function, f_1, m_2, and f_3 are smooth functions of the corresponding covariates, and m_2 is subject to monotone increasing constraint.

All available shape constrained smooths are decsribed in shape.constrained.smooth.terms.

Residual auto-correlation with a simple AR1 correlation structure can be dealt with, for Gaussian models with identity link. Currently, the AR1 correlation parameter should be supplied (rather than estimated) in AR1.rho. AR.start input argument (logical) allows to set independent sections of AR1 correlation. Standardized residuals (approximately uncorrelated under correct model) are returned in std.rsd if AR1.rho is non zero.

Value

The function returns an object of class "scam" with the following elements (this agrees with gamObject):

aic

AIC of the fitted model: the degrees of freedom used to calculate this are the effective degrees of freedom of the model, and the likelihood is evaluated at the maximum of the penalized likelihood, not at the MLE.

assign

Array whose elements indicate which model term (listed in pterms) each parameter relates to: applies only to non-smooth terms.

bfgs.info

If optimizer="bfgs", a list of convergence diagnostics relating to the BFGS method of smoothing parameter selection. The items are: conv, indicates why the BFGS algorithm of the smoothness selection terminated; iter, number of iterations of the BFGS taken to get convergence; grad, the gradient of the GCV/UBRE score at convergence; score.hist, the succesive values of the score up until convergence.

call

the matched call.

coefficients

the coefficients of the fitted model. Parametric coefficients are first, followed by coefficients for each spline term in turn.

coefficients.t

the parametrized coefficients of the fitted model (exponentiated for the monotonic smooths).

conv

indicates whether or not the iterative fitting method converged.

CPU.time

indicates the real and CPU time (in seconds) taken by the fitting process in case of unknown smoothing parameters

data

the original supplied data argument. Only included if the scam argument keepData is set to TRUE (default is FALSE).

deviance

model deviance (not penalized deviance).

df.null

null degrees of freedom.

df.residual

effective residual degrees of freedom of the model.

edf

estimated degrees of freedom for each model parameter. Penalization means that many of these are less than 1.

edf1

alternative estimate of edf.

efs.info

If optimizer="efs", a list of convergence diagnostics relating to the extended Fellner Schall method fot smoothing parameter selection. The items are: conv, indicates why the efs algorithm of the smoothness selection terminated; iter, number of iterations of the efs taken to get convergence; score.hist, the succesive values of the score up until convergence.

family

family object specifying distribution and link used.

fitted.values

fitted model predictions of expected value for each datum.

formula

the model formula.

gcv.ubre

the minimized GCV or UBRE score.

dgcv.ubre

the gradient of the GCV or UBRE score.

iter

number of iterations of the Newton-Raphson method taken to get convergence.

linear.predictors

fitted model prediction of link function of expected value for each datum.

method

"GCV" or "UBRE", depending on the fitting criterion used.

min.edf

Minimum possible degrees of freedom for whole model.

model

model frame containing all variables needed in original model fit.

nlm.info

If optimizer="nlm" or optimizer="nlm.fd", a list of convergence diagnostics relating to the BFGS method of smoothing parameter selection. The items are: conv, indicates why the BFGS algorithm of the smoothness selection terminated; iter, number of iterations of BFGS taken to get convergence; grad, the gradient of the GCV/UBRE score at convergence.

not.exp

if TRUE then notExp() function will be used in place of exp (default value) in positivity ensuring beta parameters re-parameterization.

nsdf

number of parametric, non-smooth, model terms including the intercept.

null.deviance

deviance for single parameter model.

offset

model offset.

optim.info

If optimizer="optim", a list of convergence diagnostics relating to the BFGS method of smoothing parameter selection. The items are: conv, indicates why the BFGS algorithm of the smoothness selection terminated; iter, number of iterations of BFGS taken to get convergence; optim.method, the numerical optimization method used.

prior.weights

prior weights on observations.

pterms

terms object for strictly parametric part of model.

R

Factor R from QR decomposition of weighted model matrix, unpivoted to be in same column order as model matrix.

residuals

the working residuals for the fitted model.

scale.estimated

TRUE if the scale parameter was estimated, FALSE otherwise.

sig2

estimated or supplied variance/scale parameter.

smooth

list of smooth objects, containing the basis information for each term in the model formula in the order in which they appear. These smooth objects are returned by the smooth.construct objects.

sp

estimated smoothing parameters for the model. These are the underlying smoothing parameters, subject to optimization.

std.rsd

Standardized residuals (approximately uncorrelated under correct model) if AR1.rho non zero

termcode

an integer indicating why the optimization process of the smoothness selection terminated (see bfgs_gcv.ubre).

terms

terms object of model model frame.

trA

trace of the influence matrix, total number of the estimated degrees of freedom (sum(edf)).

var.summary

A named list of summary information on the predictor variables. See gamObject.

Ve

frequentist estimated covariance matrix for the parameter estimators.

Vp

estimated covariance matrix for the parameters. This is a Bayesian posterior covariance matrix that results from adopting a particular Bayesian model of the smoothing process.

Ve.t

frequentist estimated covariance matrix for the reparametrized parameter estimators obtained using the delta method. Particularly useful for testing whether terms are zero. Not so useful for CI's as smooths are usually biased.

Vp.t

estimated covariance matrix for the reparametrized parameters obtained using the delta method. Paricularly useful for creating credible/confidence intervals.

weights

final weights used in the Newton-Raphson iteration.

cmX

column means of the model matrix (with elements corresponding to smooths set to zero).

contrasts

contrasts associated with a factor.

xlevels

levels of a factor variable used in the model.

y

response data.

Author(s)

Natalya Pya <nat.pya@gmail.com> based partly on mgcv by Simon Wood

References

Pya, N. and Wood, S.N. (2015) Shape constrained additive models. Statistics and Computing, 25(3), 543-559

Pya, N. (2010) Additive models with shape constraints. PhD thesis. University of Bath. Department of Mathematical Sciences

Wood, S.N. (2011) Fast stable restricted maximum likelihood and marginal likelihood estimation of semiparametric generalized linear models. Journal of the Royal Statistical Society (B) 73(1):3-36

Wood S.N. (2006a) Generalized Additive Models: An Introduction with R. Chapman and Hall/CRC Press.

Wood, S.N. (2006b) On confidence intervals for generalized additive models based on penalized regression splines. Australian and New Zealand Journal of Statistics. 48(4): 445-464.

Wood, S.N. and M. Fasiolo (2017) A generalized Fellner-Schall method for smoothing parameter optimization with application to Tweedie location, scale and shape models. Biometrics 73 (4), 1071-1081

See Also

scam-package, shape.constrained.smooth.terms, gam, s, plot.scam, summary.scam, scam.check, predict.scam

Examples

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##=============================
## Gaussian model, two smooth terms: unconstrained and increasing...
 ## simulating data...
require(scam)
set.seed(4)
n <- 200
x1 <- runif(n)*6-3
f1 <- 3*exp(-x1^2) # unconstrained term
x2 <- runif(n)*4-1;
f2 <- exp(4*x2)/(1+exp(4*x2)) # monotone increasing smooth
y <- f1+f2 +rnorm(n)*.5
dat <- data.frame(x1=x1,x2=x2,y=y)
## fit model, get results, and plot...
b <- scam(y~s(x1,bs="cr")+s(x2,bs="mpi"),data=dat)
print(b)
summary(b)
plot(b,pages=1,shade=TRUE)

##===============================
## Gaussian model, two smooth terms: increasing and mixed (decreasing and convex)...
 ## simulating data...
set.seed(5)
n <- 200
x1 <- runif(n)*4-1;
f1 <- exp(4*x1)/(1+exp(4*x1)) # increasing smooth
x2 <- runif(n)*3-1;
f2 <- exp(-3*x2)/15  # decreasing and convex smooth
y <- f1+f2 + rnorm(n)*.4
dat <- data.frame(x1=x1,x2=x2,y=y)
  ## fit model, results, and plot...
b <- scam(y~ s(x1,bs="mpi")+s(x2, bs="mdcx"),data=dat)
b
summary(b)
plot(b,pages=1,scale=0,shade=TRUE)

##=================================
## Not run: 
## using the extended Fellner-Schall method for smoothing parameter selection...
b0 <- scam(y~ s(x1,bs="mpi")+s(x2,bs="mdcx"),data=dat,optimizer="efs")
summary(b0)

## using optim() for smoothing parameter selection...
b1 <- scam(y~ s(x1,bs="mpi")+s(x2,bs="mdcx"),data=dat,optimizer="optim")
summary(b1)

b2 <- scam(y~ s(x1,bs="mpi")+s(x2,bs="mdcx"),data=dat,optimizer="optim",
           optim.method=c("BFGS","fd"))
summary(b2)

## using nlm()...
b3 <- scam(y~ s(x1,bs="mpi")+s(x2,bs="mdcx"),data=dat,optimizer="nlm")
summary(b3)

## End(Not run)

##===================================
## Poisson model ....
 ## simulating data...
set.seed(2)
n <- 200
x1 <- runif(n)*6-3
f1 <- 3*exp(-x1^2) # unconstrained term
x2 <- runif(n)*4-1;
f2 <- exp(4*x2)/(1+exp(4*x2)) # monotone increasing smooth
f <- f1+f2
y <- rpois(n,exp(f))
dat <- data.frame(x1=x1,x2=x2,y=y)
  ## fit model, get results, and plot...
b <- scam(y~s(x1,bs="cr")+s(x2,bs="mpi"),
      family=poisson(link="log"),data=dat,optimizer="efs")
summary(b)
plot(b,pages=1,shade=TRUE)
scam.check(b)

## Gamma model...
   ## simulating data...
set.seed(6)
n <- 300
x1 <- runif(n)*6-3
f1 <- 1.5*sin(1.5*x1) # unconstrained term
x2 <- runif(n)*4-1;
f2 <- 1.5/(1+exp(-10*(x2+.75)))+1.5/(1+exp(-5*(x2-.75))) # increasing smooth
x3 <- runif(n)*6-3;
f3 <- 3*exp(-x3^2)  # unconstrained term
f <- f1+f2+f3
y <- rgamma(n,shape=1,scale=exp(f))
dat <- data.frame(x1=x1,x2=x2,x3=x3,y=y)
   ## fit model, get results, and plot...
b <- scam(y~s(x1,bs="ps")+s(x2,k=15,bs="mpi")+s(x3,bs="ps"),
          family=Gamma(link="log"),data=dat,optimizer="efs")
b
summary(b)
par(mfrow=c(2,2))
plot(b,shade=TRUE)

## Not run: 
## bivariate example...
 ## simulating data...
   set.seed(2)
   n <- 30
   x1 <- sort(runif(n)*4-1)
   x2 <- sort(runif(n))
   f1 <- matrix(0,n,n)
   for (i in 1:n) for (j in 1:n) 
       { f1[i,j] <- -exp(4*x1[i])/(1+exp(4*x1[i]))+2*sin(pi*x2[j])}
   f <- as.vector(t(f1))
   y <- f+rnorm(length(f))*.2
   x11 <-  matrix(0,n,n)
   x11[,1:n] <- x1
   x11 <- as.vector(t(x11))
   x22 <- rep(x2,n)
   dat <- list(x1=x11,x2=x22,y=y)
## fit model  and plot ...
   b <- scam(y~s(x1,x2,k=c(10,10),bs=c("tesmd1","ps")),data=dat,optimizer="efs")
   summary(b)
   par(mfrow=c(2,2),mar=c(4,4,2,2))
   plot(b,se=TRUE)
   plot(b,pers=TRUE,theta = 30, phi = 40)
   plot(y,b$fitted.values,xlab="Simulated data",ylab="Fitted data",pch=".",cex=3)

## example with random effect smoother...
   set.seed(2)
   n <- 200
   x1 <- runif(n)*6-3
   f1 <- 3*exp(-x1^2) # unconstrained term
   x2 <- runif(n)*4-1;
   f2 <- exp(4*x2)/(1+exp(4*x2)) # increasing smooth
   f <- f1+f2
   a <- factor(sample(1:10,200,replace=TRUE))   
   Xa <- model.matrix(~a-1)    # random main effects
   y <- f + Xa%*%rnorm(length(levels(a)))*.5 + rnorm(n)*.4    
   dat <- data.frame(x1=x1,x2=x2,y=y,a=a)
   ## fit model and plot...
   b <- scam(y~s(x1,bs="cr")+s(x2,bs="mpi")+s(a,bs="re"), data=dat)
   summary(b)
   scam.check(b)
   plot(b,pages=1,shade=TRUE)

## example with AR1 errors...
set.seed(8)
n <- 500
x1 <- runif(n)*6-3
f1 <- 3*exp(-x1^2) # unconstrained term
x2 <- runif(n)*4-1;
f2 <- exp(4*x2)/(1+exp(4*x2)) # increasing smooth
f <- f1+f2
e <- rnorm(n,0,sd=2)
for (i in 2:n) e[i] <- .6*e[i-1] + e[i]
y <- f + e
dat <- data.frame(x1=x1,x2=x2,y=y)  
b <- scam(y~s(x1,bs="cr")+s(x2,k=25,bs="mpi"),
            data=dat, AR1.rho=.6, optimizer="efs")
b
## Raw residuals still show correlation...
acf(residuals(b)) 
## But standardized are now fine...
x11()
acf(b$std.rsd)
 
## End(Not run)

scam documentation built on Aug. 10, 2021, 5:06 p.m.