Developer Reference

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

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