Multivariate statistical methods

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# Multivariate statistical methods - PowerPoint PPT Presentation

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

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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
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,…., n j = 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