Summary Statistics Mathematical Notation and
Definitions
The following notations are used in the mathematical definitions and the description of the Summary Statistics functions.
Intel® oneAPI Math Kernel Library
Matrix and Weights of Observations
For a random
,...,
), this manual denotes
the following:
p
-dimensional
vector
ξ
= (ξ
1
,...,
ξ
i
ξ
p
- (is the result ofX)=(ix)ijj=1..nnindependent observations for thei-th componentξof the vectoriξ.
- The two-dimensional arrayis the matrix of observations.X=(x)ijnxp
- The column[of the matrixX]=(jx)iji=1..pXis thej-th observation of the random vectorξ.
Each observation
, where
[
is assigned a
non-negative weight
X
]j
w
j
- The vector(is a vector of weights corresponding tow)jj=1..nnobservations of the random vectorξ.
- is the accumulated weight corresponding to observationsX.
Vector of sample means


for all
i
= 1, ...,
p
.
Vector of sample partial sums


for all
i
= 1, ...,
p
.
Vector of sample variances



for all
i
= 1, ...,
p
.
Vector of sample raw/algebraic moments of
k-th order,
k≥ 1
k
-th order,
k
≥
1

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

for all
i
= 1, ...,
p
.
Vector of sample central moments of the
third and the fourth order



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

for all
i
= 1, ...,
p
.
Vector of sample excess kurtosis
values


for all
i
= 1, ...,
p
.
Vector of sample skewness values


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


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


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


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


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




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




for all
i
= 1, ...,
p
.
Vector of sample quantile values
For a positive integer number
is the
if
}
and
}
, where
q
and
k
belonging to
the interval [0,
q
-1], point
z
i
k
-th
q
quantile of
the random variable
ξ
i
P
{ξ
i
≤
z
i
≥
β
P
{ξ
i
≤
z
i
≥
1 -
β
- Pis the probability measure.
- β=k/nis 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
+1f
(y
i
,j
+2y
i
,j
+1for 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
. One observation
belongs to one group only. The group mean and variance-covariance matrices are
calculated similarly to the formulas above:
[ = ()
belongs to the
group
X
]j
x
ij
i
=1..p
r
if
j
∈
G
r



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:


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


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


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
( is defined as
P
(X
) =(p
ij
X
))k
xk

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
- kis the dimension ofY.
- p(ijX) are elements of the partial variance-covariance matrix.
Sorted dataset
Matrix
),
in which the
is obtained as a result
of sorting in ascending order the row
( =
() in the original matrix of observations.
Y
=
(y
ij
pxn
i
-th row
(Y
)i
X
)i
x
ij
j
= 1..n