Cloud Class

public class Cloud

Public Class Cloud

public ref class Cloud

type Cloud = class end

Inheritance

Object    Cloud

Points and variables

Let us consider points in , and assume that a weighting scheme has been imposed to them to control their contributions to the statistical characteristics of their ensemble: that is, a relative weight is assigned to point , with . Such set of weighted points can thus be represented by means of the pair , referred to as a weighted multidimensional structure where



is the matrix whose -th row is , and is the weighting scheme expressed as a sequence:

Given a basis of , a structure can be represented by a cloud, say , which can be formally defined as the triplet , where



is the coordinates matrix w.r.t. of the points in , i.e., its -th row, , represents the coordinates of point .

Given a Cloud instance representing , you can inspect its Basis, , the Coordinates of the points , and their Weights, .

The mean point of is the vector whose coordinates are given by:



Such coordinates are returned by method Mean. The variance of , say , is returned by method Variance and defined as follows:



where is the distance induced by basis .

The columns of represent the active variables observed at the points w.r.t. basis . The covariance matrix of such variables can be defined as follows



where



Such matrix can be inspected by calling method Covariance.

A variable which is not active is said supplementary. Supplementary variables have as many rows as the number of points in the cloud, while the number of columns is the number of supplementary variables. Their variances can be computed, which means that the weighting scheme of the cloud will be applied.

Cloud modifications

A cloud can be re-based, so that the cloud coordinates are updated to be referable to a new basis. Clouds can also be centered so as to have zero mean coordinates, or standardized by subtracting to each variable its mean and then dividing the difference by its standard deviation.

Reference