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 = (ξ1,..., ξi,..., ξp), this manual denotes the following:

  • (X)i=(xij)j=1..n is the result of n independent observations for the i-th component ξi of the vector ξ.

  • The two-dimensional array X=(xij)n x p is the matrix of observations.

  • The column [X]j=(xij)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 wj , where

  • The vector (wj)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 = (yij)pxn, in which the i-th row (Y)i = (yij)j=1..n is obtained as a result of sorting in the ascending order of row (X)i = (xij)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 zi is the k-th q quantile of the random variable ξi if P{ξizi} β and P{ξizi} 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) = (Q1(X,β), ..., Qp(X,β))

where

(Qi(X,β) = yi,j+1 + f(yi,j+2 - yi,j+1)

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

Variance-covariance matrix

C(X) = (cij(X))p x p

where

,

Cross-product matrix (matrix of cross-products and sums of squares)

CP(X) = (cpij(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 = (xij)i=1..p belongs to the group r if jGr. 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) =(pij(X))kxk 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.

  • pij(X) are elements of the partial variance-covariance matrix.

Sorted dataset

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

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