## 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