Developer Reference

  • 2020.2
  • 07/15/2020
  • Public Content
Contents

Summary Statistics Mathematical Notation and Definitions

The following notations are used in the mathematical definitions and the description of the
Intel® MKL
Summary Statistics functions.

Matrix and Weights of Observations

For a random
p
-dimensional vector
ξ
= (
ξ
1
,...,
ξ
i
,...,
ξ
p
), this manual denotes the following:
  • (
    X
    )
    i
    =(
    x
    ij
    )
    j
    =1..
    n
    is the result of
    n
    independent observations for the
    i
    -th component
    ξ
    i
    of the vector
    ξ
    .
  • The two-dimensional array
    X
    =(
    x
    ij
    )
    n
    x
    p
    is the matrix of observations.
  • The column
    [
    X
    ]
    j
    =(
    x
    ij
    )
    i
    =1..
    p
    of the matrix
    X
    is the
    j
    -th observation of the random vector
    ξ
    .
Each observation
[
X
]
j
is assigned a non-negative weight
w
j
, where
  • The vector
    (
    w
    j
    )
    j
    =1..
    n
    is a vector of weights corresponding to
    n
    observations of the random vector
    ξ
    .
  • is the accumulated weight corresponding to observations
    X
    .

Vector of sample means

with
for all
i
= 1, ...,
p
.

Vector of sample partial sums

with
for all
i
= 1, ...,
p
.

Vector of sample variances

with
,
for all
i
= 1, ...,
p
.

Vector of sample raw/algebraic moments of
k
-th order,
k
1

with
for all
i
= 1, ...,
p
.

Vector of sample raw/algebraic partial sums of
k
-th order,
k
=
2, 3, 4 (raw/algebraic partial sums of squares/cubes/fourth powers)

with
for all
i
= 1, ...,
p
.

Vector of sample central moments of the third and the fourth order

with
,
for all
i
= 1, ...,
p
and
k
= 3, 4.

Vector of sample central partial sums of
k
-th order,
k
=
2, 3, 4 (central partial sums of squares/cubes/fourth powers)

with
for all
i
= 1, ...,
p
.

Vector of sample excess kurtosis values

with
for all
i
= 1, ...,
p
.

Vector of sample skewness values

with