Developer Reference

  • 0.9
  • 09/09/2020
  • Public Content
Contents

Summary Statistics Mathematical Notation and Definitions

The following notations are used in the mathematical definitions and the description of the
Intel® 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

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
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

Intel's compilers may or may not optimize to the same degree for non-Intel microprocessors for optimizations that are not unique to Intel microprocessors. These optimizations include SSE2, SSE3, and SSSE3 instruction sets and other optimizations. Intel does not guarantee the availability, functionality, or effectiveness of any optimization on microprocessors not manufactured by Intel. Microprocessor-dependent optimizations in this product are intended for use with Intel microprocessors. Certain optimizations not specific to Intel microarchitecture are reserved for Intel microprocessors. Please refer to the applicable product User and Reference Guides for more information regarding the specific instruction sets covered by this notice.

Notice revision #20110804