# Multivariate statistical methods - PowerPoint PPT Presentation

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Multivariate statistical methods. Multivariate methods. multivariate dataset – group of n objects, m variables (as a rule n > m, if possible). confirmation vs. eploration analysis confirmation – impact on parameter estimate and hypothesis testing

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Multivariate statistical methods

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## Multivariate statistical methods

### Multivariate methods

• multivariate dataset – group of n objects, m variables (as a rule n>m, if possible).

• confirmation vs. eploration analysis

• confirmation – impact on parameter estimate and hypothesis testing

• exploration – impact on data exploration, finding out of patterns and structure

### Multivariate statistical methods

Unit classification

• Cluster analysis

• Discrimination analysis

Analysis of relations among variables

• Cannonical correlation analysis

• Factor analysis

• Principal component analysis

Methods for analysis of relations among variables

### Principal component analysis

• the oldest and the most used multivariate statistical methods

• standed by Pearson in 1901 and independently from Pearson also by Hotelling in 1933

• principal aims:

• detection of relations among variables

• reduction of variables number and finding of new purposeful variables

### Principal component analysis

• as fundament is linear transformation of original variables into less number of new fictituous variables, so called principal components

• component characteristics:

• are not mutually correlated

• for m original variables is r<=m good dimension, r (best a lot less than m) principal components explain sufficiency variability of original variables

### PCA

• component characteristics:

• method is based on full explanation of total variability

• principal components are ordered according share of explained variance

• the most of variance is explained by first component, the least by last component

### PCA procedure

• starting analysis – exploration of relations among variables (graphs, descriptive statistics)

• exploration of correlation matrix (existence of correlation among original variables – reduction of variables is possible)

• principal component analysis, choice of suitable number of components (usually is enough 70 – 90 % of explained variance)

• interpretation of principal components

### PCA procedure

• PCA is based on

• covariance matrix (the same units of variables, similar variance)

• correlation matrix (standardized data or different units of variables)

### Model of PCA

→ standardized original variable

… weights of principal component

… prin. components in standardized expression

j,k = 1,2, …., p

i = 1,2, …., n- number of units

j = 1,2, …., p- number of variables

### PCA – mathematical model

• original matrix – dataset X (n x m), n objects, m variables

• Z = [zij]standardized matrix X

i = 1,…., nj = 1,…., m

• aim is find out transformation matrix Q, which convert m standardized variables (matrix Z) into m mutual independent component (matrix P)

P = Z . Q

### PCA – mathematical model

• Modification of P = Z . Q→ we get matrix

### PCA – mathematical model

• matrix Λ is matrix of covariance and variance of principal components. With regard to independence of principal components are covariances 0 and matrix Λ is diagonal with variances of principal component on diagonal

• sum of variances standardized variables equals to m.

proportions indicate, how large is the

share of the first, second, … last component on explanation of the total variance of all variables

### PCA – mathematical model

• matrix R is correlation matrix of original variables

where

Diagonal values of matrix Λ are eigenvalues of matrix R, in columns of matrix Q are eigenvectors related to each eigenvalue

### PCA – other notions

• coordinates of nonstandardized principal component are called „score“

• matrix of all score for all objects (n) is called „score matrix“

• scores for objects are in rows

• matrix columns are vectors of score

### PCA – other notions

• share of total variability of each original variable Xi, i = 1, 2,…, m, which is explained by r principals components is called communality of variable Xi.

• is computed as second power of multiple coefficient of correlation → r2

### PCA – graphical visualisation

• Cattel´s graph → scree plot

• tool for determination of number of principal components

### PCA – graphical visualization

• graph of coefficients of correlation (1st and 2nd principal component)

### PCA – graphical visualization

• Graph of component score