Definitions

and two sets, a function:

Subset g image :

Subset inverse g image:

is injective

is surjective

is bijective is injective and surjective

For a given , let's define the function :

will be called the prediction function in subsequent sections.

For a given normed space vector on corpse and , let' define the binary relation :

(nearest neighborhood order)

For a given normed space vector on corpse and , let' define the binary relation :

(nearest neighborhood equality)

k-NN multiple imputation

For a given , let's define the finite data set .

Lemme 1

For , is a complete order on .

Proof.

Reflexivity:

Antisymmetry:

Transitivity:

Completeness: order on finite set is complete (trivial proof to be done).

k-NN

For , is a fully ordered finite set.

For , let's define:

the ordered finite set of the nearest neighbors of in

the k-th nearest neighbor of in

.

Imputation on $2$ features data set

Let's define a two features data set where is known and can be missing at random.

Unique missing value imputation

a partially missing value, as is known, to impute in .

For , we consider the image of the k nearest neighbors of in as a sample of the possible imputed elements for .

(discuss conditions on )

Impute with the mean:

Impute with random sampling:

Several missing values imputation

For , the finite set of partially missing values, describe how are built for imputed value.

Multiple imputation

For , describe how are built the imputed dataset using k-NN imputation with random sampling.

Imputation on $N \in \mathbb{N}$ features data set

Given , let's define a features finite data set.