Mathematical Notation and Definitions

Matrix and Weights of Observations

For a random p-dimensional vector ξ = (ξ₁,..., 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)_{p x n} 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 variances

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

Vector of sample raw/algebraic moments of k-th order, k ≥ 1

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 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 Y = (y_ij)_pxn, 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

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 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) = (Q₁(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

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:

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 P(X) =(p_ij(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

• k is the dimension of Y.

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