scar {scar} | R Documentation |
Compute the maximum likelihood estimator of the generalised additive regression with shape constraints
Description
This function uses the active set algorithm to compute the maximum likelihood estimator (mle) of the generalised additive regression with shape constraints. Each component function of the additive predictors is assumed to belong to one of the nine possible shape restrictions. The estimator's value at the data points is unique.
The output is an object of class scar
which contains all the information
needed to plot the estimator using the plot
method, or
to evaluate it using the predict
method.
Usage
scar(x, y, shape = rep("l", d), family = gaussian(),
weights = rep(1, length(y)), epsilon = 1e-08)
Arguments
x |
Observed covariates in |
y |
Observed responses, in the form of a numeric |
shape |
A vector that specifies the shape restrictions for each component function,
in the form of a string vector of length
|
family |
A description of the error distribution and link function to be used in the model. This can be a character string naming a family function, a family function or the result of a call to a family function. Currently only the following five common exponential families are allowed: Gaussian, Binomial, Poisson, and Gamma. By default the canonical link function is used. |
weights |
An optional vector of prior weights to be used when maximising the
likelihood. It is a numeric vector of length |
epsilon |
Positive convergence tolerance epsilon when performing the
iteratively reweighted least squares (IRLS) method at each iteration of
the active set algorithm. See |
Details
For i = 1,\ldots,n
, let X_i
be the d
-dimensional
covariates, Y_i
be the corresponding one-dimensional response and
w_i
be its weight. The generalised additive model can be written as
g(\mu) = f(x),
where x=(x_1,\ldots,x_d)^T
,
g
is a known link function and f
is an additive function (to be estimated).
Assume the canonical link function is used here, then the maximum likelihood estimator
of the generalised additive model based on observations
(X_1,Y_1), \ldots, (X_n,Y_n)
is the function that maximises
\frac{1}{n} \sum_{i=1}^n w_i \{Y_i f(X_i) - B(f(X_i))\}
subject to the restrictions that for every j = 1,\ldots,d
,
the j
-th additive component of f
satisfies the constraint indicated by the
j
-th element of shape
. Here B(.)
is the log-partition function of
the specified exponential family distribution, and w_i
are the weights. For i.i.d. data,
w_i
should be 1
for each i
.
To make each component of f
identifiable, we write
f(x) = \sum_{j=1}^d f_j(x_j) + c
and let f_j(0) = 0
for every j = 1,\ldots,d
.
In case zero is outside the range of the j
-th observed covariate,
for the sake of convenience, we set f_j
to be zero at the sample mean of
the j
-th predictor.
This problem can then be re-written as a concave optimisation problem, and our function uses the active set algorithm to find out the maximum likelihood estimator. A general introduction can be found in Nocedal and Wright (2006). A detailed description of our algorithm can be found in Chen and Samworth (2016). See also Groeneboom, Jongbloed and Wellner (2008) for some theoretical supports.
Value
An object of class scar
, with the following components:
x |
Covariates copied from input. |
y |
Response copied from input. |
shape |
Shape vector copied from input. |
weights |
The vector of weights copied from input. |
family |
The exponential family copied from input. |
componentfit |
Value of the fitted component function at each observed
covariate, in the form of an |
constant |
The estimated value of the constant |
deviance |
Up to a constant, minus twice the maximised log-likelihood.
Where applicable, the constant is chosen to make the saturated
model to have zero deviance. See also |
nulldeviance |
The deviance for the null model. |
iter |
Total number of iterations of the active set algorithm |
.
Note
We acknowledge that glm.fit
from the R package
stats is called to perform the method of iterated reweighted least squares
(IRLS) in our routine. It is possible to speed up the implementation considerably
by simply suppressing all the run-time checks there.
If all the component functions are linear, then it is prefered to call directly
the function glm
.
For the one-dimensional covariate, see the pool adjacent violators algorithm (PAVA) of Robertson, Wright and Dykstra (1998) and the support reduction method of Groeneboom, Jongbloed and Wellner (2008).
A different approach to tackle this problem is to use splines. See the R package
scam
. We stress here that our approach is free
of tuning parameters while scam
is not, which
can be viewed as a major difference.
To estimate the generalised additive regression function without any shape
restrictions, see Wood (2004) and Hastie and Tibshirani (1990).
Their corresponding R implementations are mgcv
and gam
.
Author(s)
Yining Chen and Richard Samworth
References
Chen, Y. and Samworth, R. J. (2016). Generalized additive and index models with shape constraints. Journal of the Royal Statistical Society: Series B, 78, 729-754.
Groeneboom, P., Jongbloed, G. and Wellner, J.A. (2008). The support reduction algorithm for computing non-parametric function estimates in mixture models. Scandinavian Journal of Statistics, 35, 385-399.
Hastie, T. and Tibshirani, R. (1990). Generalized Additive Models. Chapman and Hall, London.
Meyer, M. C. (2013). Semi-parametric additive constrained regression. Journal of nonparametric statistics, 25, 715-743.
Nocedal, J., and Wright, S. J. (2006). Numerical Optimization, 2nd edition. Springer, New York.
Robertson, T., Wright, F. T. and Dykstra, R. L. (1988). Order Restricted Statistical Inference. Wiley, New York.
Venables, W. N. and Ripley, B. D. (2002). Modern Applied Statistics with S. Springer, New York.
Wood, S.N. (2004). Stable and efficient multiple smoothing parameter estimation for generalized additive models. Journal of American Statistical Association, 99, 673-686.
See Also
plot.scar
, predict.scar
, scair
,
scam
, mgcv
,
gam
, glm
Examples
## An example in the Poission additive regression setting:
## Define the additive function f on the scale of the predictors
f<-function(x){
return(1*abs(x[,1]) + 2*(x[,2])^2 + 3*(abs(x[,3]))^3)
}
## Simulate the covariates and the responses
## covariates are drawn uniformly from [-1,1]^3
set.seed(0)
d = 3
n = 500
x = matrix(runif(n*d)*2-1,nrow=n,ncol=d)
rpoisson <- function(m){rpois(1,exp(m))}
y = sapply(f(x),rpoisson)
## All the components are convex so one can use scar
shape=c("cvx","cvx","cvx")
object = scar(x,y,shape=shape, family=poisson())
## Plot each component of the estimatied additive function
plot(object)
## Evaluate the estimated additive function at 10^4 random points
## drawing from the interior of the support
testx = matrix((runif(10000*d)*1.96-0.98),ncol=d)
testf = predict(object,testx)
## and calculate the (estimated) absolute prediction error
mean(abs(testf-f(testx)))