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

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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


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
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

    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
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


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