conformal.multidim.full {conformalInference.multi}

Full Conformal Prediction Regions, Multivariate Response

Description

Compute prediction intervals using full conformal inference with multivariate response

Usage

conformal.multidim.full(
  x,
  y,
  x0,
  train.fun,
  predict.fun,
  alpha = 0.1,
  mad.train.fun = NULL,
  mad.predict.fun = NULL,
  score = "l2",
  s.type = "st-dev",
  num.grid.pts.dim = 100,
  grid.factor = 1.25,
  verbose = FALSE
)

Arguments

x

Matrix of features, of dimension (say) n x p.

y

Matrix of responses, of length (say) n X q.

x0

Matrix of features, each row being a point at which we want to form a prediction interval, of dimension (say) n0 x p.

train.fun

A function to perform model training, i.e., to produce an estimator of E(Y|X), the conditional expectation of the response variable Y given features X. Its input arguments should be x: matrix of features, y: vector of responses, and out: the output produced by a previous call to train.fun, at the same features x. The function train.fun may (optionally) leverage this returned output for efficiency purposes. See details below.

predict.fun

A function to perform prediction for the (mean of the) responses at new feature values. Its input arguments should be out: output produced by train.fun, and newx: feature values at which we want to make predictions.

alpha

Miscoverage level for the prediction intervals, i.e., intervals with coverage 1-alpha are formed. Default for alpha is 0.1.

mad.train.fun

A function to perform training on the absolute residuals i.e., to produce an estimator of E(R|X) where R is the absolute residual R = |Y - m(X)|, and m denotes the estimator produced by train.fun. This is used to scale the conformal score, to produce a prediction interval with varying local width. The input arguments to mad.train.fun should be x: matrix of features, y: vector of absolute residuals, and out: the output produced by a previous call to mad.train.fun, at the same features x. The function mad.train.fun may (optionally) leverage this returned output for efficiency purposes. See details below. The default for mad.train.fun is NULL, which means that no training is done on the absolute residuals, and the usual (unscaled) conformal score is used. Note that if mad.train.fun is non-NULL, then so must be mad.predict.fun (next).

mad.predict.fun

A function to perform prediction for the (mean of the) absolute residuals at new feature values. Its input arguments should be out: output produced by mad.train.fun, and newx: feature values at which we want to make predictions. The default for mad.predict.fun is NULL, which means that no local scaling is done for the conformal score, i.e., the usual (unscaled) conformal score is used.

score

Method to compute nonconformity measure in the multivariate regime. The user can choose between squared l^2 norm of the residual, mahalanobis depth of the residual, the max norm of the residual.

s.type

The type of modulation function. Currently we have 3 options: "identity","st-dev". Default is "st-dev"

num.grid.pts.dim

Number of grid points per dimension used when forming the conformal intervals (each num.grid.pts.dim^q points is a trial point). Default is 100.

grid.factor

Expansion factor used to define the grid for the conformal intervals, i.e., the grid points are taken to be equally spaced in between -grid.factor x max(abs(y)) and grid.factor x max(abs(y)). Default is 1.25. In this case (and with exchangeable data, thus unity weights) the restriction of the trial values to this range costs at most 1/(n+1) in coverage. See details below.

verbose

Should intermediate progress be printed out? Default is FALSE.

Details

Due to eventual computational overload the function is restricted to a bivariate y.

This function is based on the package future.apply to perform parallelisation.

If the data (training and test) are assumed to be exchangeable, the basic assumption underlying conformal prediction, then the probability that a new response value will lie outside of (-max(abs(y)), max(abs(y))), where y is the vector of training responses, is 1/(n+1). Thus the restriction of the trials values to (-grid.factor x max(abs(y)), grid.factor x max(abs(y))), for all choices grid.factor >= 1, will lead to a loss in coverage of at most 1/(n+1). This was also noted in "Trimmed Conformal Prediction for High-Dimensional Models" by Chen, Wang, Ha, Barber (2016) <arXiv:1611.09933> (who use this basic fact as motivation for proposing more refined trimming methods).

Value

A list with the following components: pred, valid_points. The first is a matrix of dimension n0 x q, while the second is a list of length n0, containing in each position a matrix of varying number of rows (depending on which points where accepted by the method) and with a number of columns equal to q + 1. Indeed, valid_points contains the selected points on the y-grid as well as the p-values.

See Also

conformal.multidim.split

Examples

n=25
p=4
q=2


mu=rep(0,p)
x = mvtnorm::rmvnorm(n, mu)
beta<-sapply(1:q, function(k) c(mvtnorm::rmvnorm(1,mu)))
y = x%*%beta + t(mvtnorm::rmvnorm(q,1:n))
x0=x[ceiling(0.9*n):n,]
y0=y[ceiling(0.9*n):n,]

n0<-nrow(y0)
q<-ncol(y)


fun=mean_multi()

#################################### FULL CONFORMAL

final.full=conformal.multidim.full(x, y, x0, fun$train.fun,
                                fun$predict.fun, score="l2",
                                num.grid.pts.dim=5, grid.factor=1.25,
                                verbose=FALSE)

ppp<-plot_multidim(final.full)