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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 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 methods1
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
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 analysis1
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
  • 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
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 procedure1
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
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
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 model1
PCA – mathematical model
  • Modification of P = Z . Q→ we get matrix
pca mathematical model2
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 model3
PCA – mathematical model
  • matrix R is correlation matrix of original variables


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

pca other notions
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 notions1
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
PCA – graphical visualisation
  • Cattel´s graph → scree plot
  • tool for determination of number of principal components
pca graphical visualization
PCA – graphical visualization
  • graph of coefficients of correlation (1st and 2nd principal component)
pca graphical visualization1
PCA – graphical visualization
  • Graph of component score