Contents

# Summary Statistics Mathematical Notation and Definitions

The following notations are used in the mathematical definitions and the description of the
Intel® oneAPI Math Kernel Library
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 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 for all
i
= 1, ...,
p
.

## Vector of sample variation coefficients with for all
i
= 1, ...,
p
.

## Matrix of order statistics

Matrix
Y
= (
y
ij
)
p
x
n
, in which the
i
-th row
(
Y
)
i
= (
y
ij
)
j
=1..
n
is obtained as a result of sorting in the ascending order of row
(
X
)
i
= (
x
ij
)
j
=1..
n
in the original matrix of observations.

## Vector of sample minimum values , where for all
i
= 1, ...,
p
.

## Vector of sample maximum values , where for all
i
= 1, ...,
p
.

## Vector of sample median values , where for all
i
= 1, ...,
p
.

## Vector of sample median absolute deviations , where with , for all
i
= 1, ...,
p
.

## Vector of sample mean absolute deviations , where with , for all
i
= 1, ...,
p
.

## Vector of sample quantile values

For a positive integer number
q
and
k
belonging to the interval [0,
q
-1], point
z
i
is the
k
-th
q
quantile of the random variable
ξ
i
if
P
{
ξ
i
z
i
}
β
and
P
{
ξ
i
z
i
}
1 -
β
, where
• P
is the probability measure.
• β
=
k
/
n
is the quantile order.
The calculation of quantiles is as follows:
j
= [(
n
-1)
β
] and
f
= {(
n
-1)
β
} as integer and fractional parts of the number (
n
-1)
β
, respectively, and the vector of sample quantile values is
Q
(
X
,
β
) = (
Q
1
(
X
,
β
), ...,
Q
p
(
X
,
β
))
where
(
Q
i
(
X
,
β
) =
y
i
,
j
+1
+
f
(
y
i
,
j
+2
-
y
i
,
j
+1
)
for all
i
= 1, ...,
p
.

## Variance-covariance matrix

C
(
X
) = (
c
ij
(
X
))
p
x
p
where , ## Cross-product matrix (matrix of cross-products and sums of squares)

CP
(
X
) = (
cp
ij
(
X
))
p
x
p
where ## Pooled and group variance-covariance matrices

The set
N
= {1, ...,
n
} is partitioned into non-intersecting subsets The observation
[
X
]
j
= (
x
ij
)
i
=1..
p
belongs to the group
r
if
j
G
r
. One observation belongs to one group only. The group mean and variance-covariance matrices are calculated similarly to the formulas above: with , for all
i
= 1, ...,
p
, where , for all
i
= 1, ...,
p
and
j
= 1, ...,
p
.
A pooled variance-covariance matrix and a pooled mean are computed as weighted mean over group covariance matrices and group means, correspondingly: with for all
i
= 1, ...,
p
, , for all
i
= 1, ...,
p
and
j
= 1, ...,
p
.

## Correlation matrix , where for all
i
= 1, ...,
p
and
j
= 1, ...,
p
.

## Partial variance-covariance matrix

For a random vector
ξ
partitioned into two components
Z
and
Y
, a variance-covariance matrix
C
describes the structure of dependencies in the vector
ξ
: .
The partial covariance matrix
P
(
X
) =(
p
ij
(
X
))
k
x
k
is defined as .
where
k
is the dimension of
Y
.

## Partial correlation matrix

The following is a partial correlation matrix for all
i
= 1, ...,
k
and
j
= 1, ...,
k
: , where where
• k
is the dimension of
Y
.
• p
ij
(
X
) are elements of the partial variance-covariance matrix.

## Sorted dataset

Matrix
Y
= (
y
ij
)
pxn
, in which the
i
-th row (
Y
)
i
is obtained as a result of sorting in ascending order the row (
X
)
i
= (
x
ij
)
j
= 1..
n
in the original matrix of observations.

#### Product and Performance Information

1

Performance varies by use, configuration and other factors. Learn more at www.Intel.com/PerformanceIndex.