conformal.multidim.split {conformalInference.multi}

Split conformal prediction intervals with Multivariate Response

Description

Compute prediction intervals using split conformal inference with multivariate response.

Usage

conformal.multidim.split(
  x,
  y,
  x0,
  train.fun,
  predict.fun,
  alpha = 0.1,
  split = NULL,
  seed = FALSE,
  randomized = FALSE,
  seed.rand = FALSE,
  verbose = FALSE,
  rho = 0.5,
  score = "l2",
  s.type = "st-dev",
  mad.train.fun = NULL,
  mad.predict.fun = NULL
)

Arguments

x

The feature variables, a matrix n x p.

y

The matrix of multivariate responses (dimension n x q)

x0

The new points to evaluate, a matrix of dimension 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, and y: matrix of responses.

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.

split

Indices that define the data-split to be used (i.e., the indices define the first half of the data-split, on which the model is trained). Default is NULL, in which case the split is chosen randomly.

seed

Integer to be passed to set.seed before defining the random data-split to be used. Default is FALSE, which effectively sets no seed. If both split and seed are passed, the former takes priority and the latter is ignored.

randomized

Should the randomized approach be used? Default is FALSE.

seed.rand

The seed for the randomized version. Default is FALSE.

verbose

Should intermediate progress be printed out? Default is FALSE.

rho

Split proportion between training and calibration set. Default is 0.5.

score

The non-conformity measure. It can either be "max", "l2", "mahalanobis". The default is "l2".

s.type

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

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.

Details

If the two mad functions are provided they take precedence over the s.type parameter, and they force a local scoring via the mad function predicted values.

Value

A list with the following components: x0,pred,k_s,s.type,s,alpha,randomized,tau, average_width,lo,up. In particular pred, lo, up are the matrices of dimension n0 x q, k_s is a scalar, s.type is a string, s is a vector of length q, alpha is a scalar between 0 and 1, randomized is a logical value, tau is a scalar between 0 and 1,and average_width is a positive scalar.

References

The s_regression and the "max" score are taken from "Conformal Prediction Bands for Multivariate Functional Data" by Diquigiovanni, Fontana, Vantini (2021).

See Also

conformal.multidim.full

Examples

n=50
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()

final.point = conformal.multidim.split(x,y,x0, fun$train.fun, fun$predict.fun,
                             alpha=0.1,
                                split=NULL, seed=FALSE, randomized=FALSE,seed.rand=FALSE,
                                verbose=FALSE, rho=0.5,score ="l2",s.type="st-dev")

ppp2<-plot_multidim(final.point)