NPCoImp {CoImp}

Non-Parametric Copula-Based Imputation Method

Description

Imputation method based on empirical conditional copula functions.

Usage

NPCoImp(X, Psi=seq(0.05,0.45,by=0.05), K=7, method="gower", N=1000, seed=set.seed(7))

Arguments

X

a data matrix with missing values. Missing values should be denoted with NA.

Psi

vector of probabilities to assess the symmetry/asymmetry of the empirical conditional copula (ecc) function and find the best quantile for the imputation (see below for details).

K

the number of data matrix rows more similar to the missing one that are used for the imputation.

method

the distance measure used for the imputation, among Euclidean, Manhattan, Canberra, Gower, and two based on the Kendall-correlation coefficient (see below for details).

N

the number of uniform numbers to numerically construct the empirical conditional copula.

seed

the set.seed for the numerical inversion Monte Carlo method used for constructing the empirical conditional copula.

Details

NPCoImp is a non-parametric imputation method based on the empirical conditional copula function and the numerical inversion Monte Carlo method. To choose the best quantile for the imputation it assesses the (a)symmetry of the empirical conditional copula and it uses the K pseudo-observations more similar to the missing one. The NPCoImp allows the imputation of missing observations according to the multivariate dependence structure of the data generating process without any assumptions on the margins. This method can be used independently from the dimension and the kind (monotone or non monotone) of the missing patterns. Brief description of the approach:

1. numerically reconstruct the empirical conditional copula of the missing observation(s);

2. evaluate the (a)symmetry of the empirical conditional copula around 0.5. Inspired by Patil et al. (2012), we compute C(0.5-Psi|available obs)-( 1 - C(0.5+Psi|available obs)) where Psi in ]0,0.5[ and say that C(.|available obs) is symmetric (negative asymmetric or positive asymmetric) if the above equation is null (positive or negative, respectively);

3. select the quantile of the empirical conditional copula on the basis of its (a)symmetry. Therefore:

   • symmetry: we impute through the median of the empirical conditional copula;

   • negative asymmetry: we impute with a quantile on the left tail of the ecc (see the paper in the references for details);

   • positive asymmetry: we impute with a quantile on the right tail of the ecc (see the paper in the references for details);

4. select the K pseudo-observations closest to the imputed one and the corresponding original observations;

5. impute missing values by replacing them from the average of the original observations derived at the previous step.

Value

An object of S4 class "NPCoImp", which is a list with the following elements:

Imputed.matrix

the imputed data matrix.

Selected.alpha

the selected alpha values. The best quantile for the imputation.

numFlat

the number of possible flat empirical conditional copulas, i.e. when ecc is always zero.

Author(s)

F. Marta L. Di Lascio <[e-mail]>, Aurora Gatto <[e-mail]>

References

Di Lascio, F.M.L, Gatto A. (202x) "A non-parametric conditional copula-based imputation method". Working paper.

Examples

## generate data from a 4-variate Frank copula with different margins

set.seed(21)
n.marg <- 4
theta  <- 5
copula <- frankCopula(theta, dim = n.marg)
mymvdc <- mvdc(copula, c("norm", "gamma", "beta","gamma"), list(list(mean=7, sd=2),
list(shape=3, rate=2), list(shape1=4, shape2=1), list(shape=4, rate=3)))
n      <- 20
x.samp <- copula::rMvdc(n, mymvdc)

# randomly introduce univariate and multivariate missing

perc.mis    <- 0.3
set.seed(11)
miss.row    <- sample(1:n, perc.mis*n, replace=TRUE)
miss.col    <- sample(1:n.marg, perc.mis*n, replace=TRUE)
miss        <- cbind(miss.row,miss.col)
x.samp.miss <- replace(x.samp,miss,NA)
probs <- seq(0.05,0.45,by=0.05)
ndist <- 7
dist.meth <- "gower"  

# impute missing values
NPimp <- NPCoImp(X=x.samp.miss, Psi=probs, K=ndist, method=dist.meth, 
                 N=1000, seed=set.seed(7))

# methods show

show(NPimp)

## Not run: 
## generate data from a 3-variate Clayton copula and introduce missing by
## using the MCAR function and try to impute through a rotated copula

set.seed(11)
n.marg <- 3
theta  <- 5
copula <- claytonCopula(theta, dim = n.marg)
mymvdc <- mvdc(copula, c("beta", "beta", "beta"), list(list(shape1=4, shape2=1),
            list(shape1=.5, shape2=.5), list(shape1=2, shape2=3)))
n      <- 50
x.samp <- copula::rMvdc(n, mymvdc)

# randomly introduce MCAR univariate and multivariate missing

perc.miss <- 0.15
setseed   <- set.seed(13)
x.samp.miss <- MCAR(x.samp, perc.miss, setseed)
x.samp.miss <- x.samp.miss@"db.missing"
probs <- seq(0.05,0.45,by=0.05)
ndist <- 7
dist.meth <- "gower" 

# impute missing values

NPimp2 <- NPCoImp(X=x.samp.miss, Psi=probs, K=ndist, method=dist.meth, N=1000)
# methods show and plot

show(NPimp2)

## End(Not run)