bam: Generalized additive models for very large datasets

Description

Fits a generalized additive model (GAM) to a very large data set, the term `GAM' being taken to include any quadratically penalized GLM. The degree of smoothness of model terms is estimated as part of fitting. In use the function is much like gam, except that the numerical methods are designed for datasets containing upwards of several tens of thousands of data. The advantage of bam is much lower memory footprint than gam, but it can also be much faster, for large datasets. bam can also compute on a cluster set up by the parallel package.

Usage

bam(formula,family=gaussian(),data=list(),weights=NULL,subset=NULL,
    na.action=na.omit, offset=NULL,method="fREML",control=list(),
    scale=0,gamma=1,knots=NULL,sp=NULL,min.sp=NULL,paraPen=NULL,
    chunk.size=10000,rho=0,AR.start=NULL,sparse=FALSE,cluster=NULL,gc.level=1,
    use.chol=FALSE,samfrac=1,...)

Arguments

formula

A GAM formula (see formula.gam and also gam.models). This is exactly like the formula for a GLM except that smooth terms, s and te

family

This is a family object specifying the distribution and link to use in fitting etc. See glm and family for more details. A negative binomial family is provided: s

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.

weights

prior weights on the data.

subset

an optional vector specifying a subset of observations to be used in the fitting process.

na.action

a function which indicates what should happen when the data contain `NA's. The default is set by the `na.action' setting of `options', and is `na.fail' if that is unset. The ``factory-fresh'' default is `na.omit'.

offset

Can be 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 and glm.

method

The smoothing parameter estimation method. "GCV.Cp" to use GCV for unknown scale parameter and Mallows' Cp/UBRE/AIC for known scale. "GACV.Cp" is equivalent, but using GACV in place of GCV. "REML" for REML estimatio

control

A list of fit control parameters to replace defaults returned by gam.control. Any control parameters not supplied stay at their default values.

scale

If this is positive then it is taken as the known scale parameter. Negative signals that the scale paraemter is unknown. 0 signals that the scale parameter is 1 for Poisson and binomial and unknown otherwise. Note that (RE)ML methods can only work with

gamma

It is sometimes useful to inflate the model degrees of freedom in the GCV or UBRE/AIC score by a constant multiplier. This allows such a multiplier to be supplied.

knots

this is an optional list containing user specified knot values to be used for basis construction. For most bases the user simply supplies the knots to be used, which must match up with the k value supplied (note that the number of knots is n

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. Negative elements indicate that the parameter should be estimated, and hence a mixture of fixed

min.sp

Lower bounds can be supplied for the smoothing parameters. Note that if this option is used then the smoothing parameters full.sp, in the returned object, will need to be added to what is supplied here to get the smoothing parameters actua

paraPen

optional list specifying any penalties to be applied to parametric model terms. gam.models explains more.

chunk.size

The model matrix is created in chunks of this size, rather than ever being formed whole.

rho

An AR1 error model can be used for the residuals (based on dataframe order), of Gaussian-identity link models. This is the AR1 correlation parameter.

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 then there are no breaks in AR1 correlaion.

sparse

If all smooths are P-splines and all tensor products are of the form te(...,bs="ps",np=FALSE) then in principle computation could be made faster using sparse matrix methods, and you could set this to TRUE. In practice the speed

cluster

bam can compute the computationally dominant QR decomposition in parallel using parLapply from the parallel package, if it is supplied with a cluster on which to do this (a cluster he

gc.level

to keep the memory footprint down, it helps to call the garbage collector often, but this takes a substatial amount of time. Setting this to zero means that garbage collection only happens when R decides it should. Setting to 2 gives frequent garbage col

use.chol

By default bam uses a very stable QR update approach to obtaining the QR decomposition of the model matrix. For well conditioned models an alternative accumulates the crossproduct of the model matrix and then finds its Choleski decomposition,

samfrac

For very large sample size Generalized additive models the number of iterations needed for the model fit can be reduced by first fitting a model to a random sample of the data, and using the results to supply starting values. This initial fit is run with

...

further arguments for passing on e.g. to gam.fit (such as mustart).

Value

An object of class "gam" as described in gamObject.

WARNINGS

The routine will be slow if the default "tp" basis is used.

You must have more unique combinations of covariates than the model has total parameters. (Total parameters is sum of basis dimensions plus sum of non-spline terms less the number of spline terms).

This routine is less stable than `gam' for the same dataset.

The negbin family is only supported for the *known theta* case.

concept

Varying coefficient model
Functional linear model
Penalized GLM
Generalized Additive Model
Penalized regression
Spline smoothing
Penalized regression spline
Generalized Cross Validation
Smoothing parameter selection
tensor product smoothing
thin plate spline
P-spline
Generalized ridge regression

Details

bam operates by first setting up the basis characteristics for the smooths, using a representative subsample of the data. Then the model matrix is constructed in blocks using predict.gam. For each block the factor R, from the QR decomposition of the whole model matrix is updated, along with Q'y. and the sum of squares of y. At the end of block processing, fitting takes place, without the need to ever form the whole model matrix.

In the generalized case, the same trick is used with the weighted model matrix and weighted pseudodata, at each step of the PIRLS. Smoothness selection is performed on the working model at each stage (performance oriented iteration), to maintain the small memory footprint. This is trivial to justify in the case of GCV or Cp/UBRE/AIC based model selection, and for REML/ML is justified via the asymptotic multivariate normality of Q'z where z is the IRLS pseudodata.

Note that POI is not as stable as the default nested iteration used with gam, but that for very large, information rich, datasets, this is unlikely to matter much.

Note also that it is possible to spend most of the computational time on basis evaluation, if an expensive basis is used. In practice this means that the default "tp" basis should be avoided: almost any other basis (e.g. "cr" or "ps") can be used in the 1D case, and tensor product smooths (te) are typically much less costly in the multi-dimensional case.

If cluster is provided as a cluster set up using makeCluster (or makeForkCluster) from the parallel package, then the rate limiting QR decomposition of the model matrix is performed in parallel using this cluster. Note that the speed ups are often not that great. On a multi-core machine it is usually best to set the cluster size to the number of physical cores, which is often less than what is reported by detectCores. Using more than the number of physical cores can result in no speed up at all (or even a slow down). Note that a highly parallel BLAS may negate all advantage from using a cluster of cores. Computing in parallel of course requires more memory than computing in series. See examples.

If the argument sparse=TRUE then QR updating is replaced by an alternative scheme, in which the model matrix is stored whole as a sparse matrix. This only makes sense if all smooths are P-splines and all tensor products are of the form te(...,bs="ps",np=FALSE), but no check is made. The computations are then based on the Choleski decomposition of the crossproduct of the sparse model matrix. Although this crossproduct is nearly dense, sparsity should make its formation efficient, which is useful as it is the leading order term in the operations count. However there is no benefit in using sparse methods to form the Choleski decomposition, given that the crossproduct is dense. In practice the sparse matrix handling overheads mean that modest or no speed ups are produced by this approach, while the computation is less stable than the default, and the memory footprint often higher (but please let the author know if you find an example where the speedup is really worthwhile).

References

http://www.maths.bath.ac.uk/~sw283/

Examples

Run this code

library(mgcv)
## Some examples are marked 'Not run' purely to keep 
## checking load on CRAN down. Sample sizes are small for 
## the same reason.
set.seed(3)
dat <- gamSim(1,n=15000,dist="normal",scale=20)
bs <- "cr";k <- 20
b <- bam(y ~ s(x0,bs=bs,k=k)+s(x1,bs=bs,k=k)+s(x2,bs=bs,k=k)+
           s(x3,bs=bs,k=k),data=dat)
summary(b)
plot(b,pages=1,rug=FALSE)  ## plot smooths, but not rug
plot(b,pages=1,rug=FALSE,seWithMean=TRUE) ## `with intercept' CIs

ba <- bam(y ~ s(x0,bs=bs,k=k)+s(x1,bs=bs,k=k)+s(x2,bs=bs,k=k)+
            s(x3,bs=bs,k=k),data=dat,method="GCV.Cp") ## use GCV
summary(ba)

## A Poisson example...

k <- 15
dat <- gamSim(1,n=11000,dist="poisson",scale=.1)
system.time(b1 <- bam(y ~ s(x0,bs=bs,k=k)+s(x1,bs=bs,k=k)+s(x2,bs=bs,k=k),
            data=dat,family=poisson()))
b1

## repeat on a cluster (need much larger n to be worthwhile!)
require(parallel)  
nc <- 2   ## cluster size, set for example portability
if (detectCores()>1) { ## no point otherwise
  cl <- makeCluster(nc) 
  ## could also use makeForkCluster, but read warnings first!
} else cl <- NULL
  
system.time(b2 <- bam(y ~ s(x0,bs=bs,k=k)+s(x1,bs=bs,k=k)+s(x2,bs=bs,k=k)
            ,data=dat,family=poisson(),cluster=cl))

## ... first call has startup overheads, repeat shows speed up...

system.time(b2 <- bam(y ~ s(x0,bs=bs,k=k)+s(x1,bs=bs,k=k)+s(x2,bs=bs,k=k)
            ,data=dat,family=poisson(),cluster=cl))

fv <- predict(b2,cluster=cl) ## parallel prediction

if (!is.null(cl)) stopCluster(cl)
b2

## Sparse smoother example...
dat <- gamSim(1,n=10000,dist="poisson",scale=.1)
system.time( b3 <- bam(y ~ te(x0,x1,bs="ps",k=10,np=FALSE)+
             s(x2,bs="ps",k=30)+s(x3,bs="ps",k=30),data=dat,
             method="REML",family=poisson(),sparse=TRUE))
b3

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