Learn a projection method for statistics and applies it

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Description

project is a generic function with two methods. If the first argument is a parameter name, project.character defines a projection function from several statistics to an output statistic predicting this parameter. project.default produces a vector of projected statistics using such a projection. project is particulary useful to reduce a large number of summary statistics to a vector of projected summary statistics, with as many elements as parameters to infer. This dimension reduction can substantially speed up subsequent computations. The concept implemented in project is to fit a parameter to the various statistics available, using machine-learning or mixed-model prediction methods. All such methods can be seen as nonlinear projection to a one-dimensional space. project.character is an interface that allows different projection methods to be used, provided they return an object of a class that has a defined predict method with a newdata argument (as expected, see predict).

Usage

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project(x,...)

## S3 method for building the projection 
## S3 method for class 'character'
project(x, stats, data, 
             trainingsize= if (method=="REML") 
               {Infusion.getOption("projTrainingSize")} else {NULL},
             knotnbr= if (method %in% c("REML","GCV"))  {
                        Infusion.getOption("projKnotNbr")
                      } else {floor(1000*log2(length(stats)+1))},
             method="REML",methodArgs=list(),verbose=TRUE,...)

## S3 method for applying the projection
## Default S3 method:
project(x, projectors,...)

Arguments

x

The name of the parameter to be predicted, or a vector/matrix/list of matrices of summary statistics.

stats

Statistics from which the predictor is to be predicted

data

A list of simulated empirical distributions, as produced by add_simulation, or a data frame with all required variables.

trainingsize

For REML only: size of random sample of realizations from the data from which the smoothing parameters are estimated.

knotnbr

Size of random sample of realizations from the data from which the predictor is built given the smoothing parameters.

method

character string: "REML", "GCV", or the name of a suitable projection function. The latter may be defined in another package, e.g. "randomForest", or predefined by Infusion (function "nnetwrap"), or defined by the user. See Details for predefined functions and for defining new ones.

methodArgs

A list of arguments for the projection method. One may not need to provide arguments in the following cases, where project kindly (tries to) assign values to the required arguments if they are absent from methodArgs:

If "REML" or "GCV" methods are used (in which case methodArgs is completely ignored); or

if the projection method uses formula and data arguments (in particular if the formula is of the form response ~ var1 + var2 + ...; otherwise the formula should be provided through methodArgs). This works for example for methods based on nnet; or

if the projection method uses x and y arguments. This works for example for randomForest (though not with the generic function method="randomForest", but only with the internal function method="randomForest:::randomForest.default").

projectors

A list with elements of the form <name>=<project result>, where the <name> must differ from any name of x. <project result> may indeed be the return object of a project call.

verbose

Whether to print some information or not. In particular, TRUE, true-vs.-predicted diagnostic plots will be drawn if any of the following methods have been used: "REML", "GCV", or a call to caret::train.

...

further arguments passed to or from other methods (currently not used).

Details

Prediction can be based on a linear mixed model (LMM) with autocorrelated random effects, internally calling the corrHLfit function with formula <parameter> ~ 1+ Matern(1|<stat1>+...+<statn>). This approach allows in principle to produce arbitrarily complex predictors (given sufficient input) and avoids overfitting in the same way as restricted likelihood methods avoids overfitting in LMM. REML methods are then used by default to estimate the smoothing parameters. However, faster methods may be required, and method "neuralNet" interfaces a neural network approach.

The data may involve hundreds of thousands of realizations of the summary statistic, and REML fitting is already slow for much smaller data sets, which is why faster alternative methods may be worth considering, and why random subset(s) of the data may be considered at various steps. The default size of these subsets aim to ensure that the computations can be performed in reasonable time.

For REML, the trainingsize and knotnbr arguments determine respectively the size of the subset used to estimate the smoothing parameters and the size of the subset defining the predictor given the smooothing parameters.

If method="GCV", a generalized cross-validation procedure (Golub et al. 1979) is used to estimate the smoothing parameters. This is faster but still slow, so a random subset of size knotnbr is still used to estimate the smoothing parameters and generate the predictor.

Alternatively, various machine-learning methods can be used (see e.g. Hastie et al., 2009, for an introduction). A random subset of size knotnbr is again used, with a larger default value bearing the assumption that these methods are faster. method="neuralNet" interfaces a neural network method. It calls the train function from the caret package.

In principle, any object suitable for prediction could be used as one of the projectors. That is, if predictions of a parameter can be performed using an object MyProjector of class MyProjectorClass, MyProjector could be used in place of a project result if predict.MyProjectorClass(object,newdata,...) is defined. However, if the learning method that generated the projector used a formula-data syntax, then its predict method is likely to request names for its newdata, that need to be provided through attr(MyProjector,"stats") (these names cannot be assumed to be in the newdata when predict is called through optim).

Value

project.character returns an object of class returned by the method (methods "REML" and "GCV" will call corrHLfit which return an object of class spaMM) project.default returns an object of the same class and structure as the input x, containing the projected statistics inferred from the input summary statistics.

References

Golub, G. H., Heath, M. and Wahba, G. (1979) Generalized Cross-Validation as a method for choosing a good ridge parameter. Technometrics 21: 215-223.

T. Hastie, R. Tibshirani, and J. Friedman. The Elements of Statistical Learning: Data Mining, Inference, and Prediction. Springer, New York, 2nd edition, 2009.

Examples

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##########
if (Infusion.getOption("example_maxtime")>250) {
## Transform normal random deviates rnorm(,mu,sd)
## so that the mean of transformed sample is not sufficient for mu,
## and that variance of transformed sample is not sufficient for sd,
blurred <- function(mu,s2,sample.size) {
  s <- rnorm(n=sample.size,mean=mu,sd=sqrt(s2))
  s <- exp(s/4)
  return(c(mean=mean(s),var=var(s)))
}

set.seed(123)
dSobs <- blurred(mu=4,s2=1,sample.size=20) ## stands for the actual data to be analyzed

## Sampling design as in canonical example 
parsp <- init_grid(lower=c(mu=2.8,s2=0.4,sample.size=20),
                      upper=c(mu=5.2,s2=2.4,sample.size=20))
# simulate distributions
dsimuls <- add_simulation(,Simulate="blurred", par.grid=parsp) 

## Use projection to construct better summary statistics for each each parameter 
mufit <- project("mu",stats=c("mean","var"),data=dsimuls)
s2fit <- project("s2",stats=c("mean","var"),data=dsimuls)

## plots
mapMM(mufit,map.asp=1,
  plot.title=title(main="prediction of normal mean",xlab="exp mean",ylab="exp var"))
mapMM(s2fit,map.asp=1,
  plot.title=title(main="prediction of normal var",xlab="exp mean",ylab="exp var"))

## apply projections on simulated statistics
corrSobs <- project(dSobs,projectors=list("MEAN"=mufit,"VAR"=s2fit))
corrSimuls <- project(dsimuls,projectors=list("MEAN"=mufit,"VAR"=s2fit))

## Analyze 'projected' data as any data (cf canonical example)
densb <- infer_logLs(corrSimuls,stat.obs=corrSobs) 
} else data(densb)
#########
if (Infusion.getOption("example_maxtime")>10) {
slik <- infer_surface(densb) ## infer a log-likelihood surface
slik <- MSL(slik) ## find the maximum of the log-likelihood surface
}
if (Infusion.getOption("example_maxtime")>500) {
slik <- refine(slik,10) ## refine iteratively
}