GRAPHICAL REPRESENTATIONS OF A DATA MATRIX

GRAPHICAL REPRESENTATIONSOF ADATA MATRIX

SYSTEM CHARCTERISATION SYSTEM Measure Numbers

CHARACTERISATION Sample Instrument + Computer UV,IR,NMR, MS,GC,GC-MS Instrumental Profiles Data matrix ..................... .................... . ....................

Numbers Measure Latent Projections Information (Graphics) Modelling

Data matrix X x’k xi Object vectors (row vectors) Variable vectors (column vectors)

Object/Sample Variable i j k 1 5 l 3 1 m 8 6 DATA MATRIX / DATA TABLE

Variable Object/Sample i j k [ 1 5 ] l [ 3 1 ] m [ 8 6 ] Object vectors

Variable Object/Sample i j k 15 l 31 m 86 Variable vectors

Object/Sample Variable Variable Object/Sample i j k 1 5 l 3 1 m 8 6 i j k [ 1 5 ] l [ 3 1 ] m [ 8 6 ] Variable Object/Sample i j k 15 l 31 m 86 Object vectors Variable vectors

Subtract variable mean, xi=4, xj=4 Variable Object i j k 1 5 l 3 1 m 8 6 Original data matrix Variable Object i j k -3 1 l -1 -3 m 4 2 Column-centred data matrix

variable j x’m x’k variable i kl VARIABLE SPACE i j k-3 1 l-1 -3 m 4 2 x’l Shows relationships between objects (angle kl measures similarity). cos  kl = x’k xl/|| x’k || || xl ||

object m xi ij object l xj object k OBJECT SPACE i j k-3 1 l-1 -3 m 4 2 Shows relationships (correlation/covariance) between variables (correlation structure) The angle ij represents the correlation between variable i and j. cos ij = x’i xj/|| x’i || || xj ||

Object space shows common variation in a suite of variables! common variation underlying factor!

VARIABLE SPACE AND OBJECT SPACE CONTAIN TOGETHER ALL AVAILABLE INFORMATION IN A DATA MATRIX

WHAT TO DO IF THE NUMBER OF VARIABLES IS GREATER THAN 2-3? PROJECT ONTO LATENT VARIABLES (LV)!

variable 2 xk LV tka e2 wa e1 variable 1 PROJECTING ONTO LATENT VARIABLES Projection (in variable space) of object vector xk (object k) on latent variable wa : tka = x’kwa , k=1,2,..,N (score)

X Data matrix Variable vectors Object vectors o3 v1 v2 LVV LV t3 p1 p2 o2 v1 v2 t2 t1 o1 Variable space ta = Xwa Object Correlation Object space pa’ = ta’X/ta’ta Variable Correlation Score plot axes (w1,w2…) Loading plots Axes (t1/||t1||,t2/||t2||…) BIPLOT LATENT VARIABLE PROJECTIONS

Successive orthogonal projections (SOP) i) Select wa ii) Project objects (sample, experiment) on wa: ta = Xawa iii) Project variable vectors on t: p’a = t’aXa/t’ata iv) Remove the latent-variable a from preditor space, i.r. substitute Xa with xa - tap’a. Repeat i) - iv) for a= 1,2,..A, where A is the dimension of the model

METHOD OVERVIEW PCA/SVD wa = pa/||pa|| PLS wa = u’aXa/|| u’aXa || MVP wa = ei MOP wa = xk/||xk|| TP wa = bk/||bk||

Decomposition Properties/Criteria Principal Components (PCA) Maximum variance Partial Least Squares (PLS) Relevant components Rotated (target) “Real” factors Marker Projections (MOP/MVP) “Real” factors METHOD OVERVIEW

LATENT PROJECTION IS AN INSTRUMENT TO CREATE ORDER (MODEL) OUT OF CHAOS (DATA)

X = UG1/2P’ + E T U orthonormal matrix of score vectors, {ua} G diagonal matrix, ga = t’ata P’ loading matrix BIPLOT (SVD, PLS, orthogonal rotations,...) Scores: UG1/2 Loadings: G1/2P’ LATENT VARIABLE MODEL

X - X T P’ E = + Centred Data Scores Loadings Residuals Scores - projection of the object vectors (in variable space)(scores - samples) Loadings - projection of the variable vectors (in object space) shows the variables correlation structure PCA/PLS (orthogonal scores)

Visual Interface Score plot- variable space Loading plot- object space Biplot plot - Scores and loadings in one plot!

- introduce interactions and squared terms in the variables (non-additive model) Horst (1968) Personality: measurements of dimensions Clementi et al. (1988), Kvalheim (1988) - introduce interactions and squared terms in the latent variables McDonal (1967) Nonlinear factor analysis Wold, Kettanch-Wold (1988), Vogt (1988) - introduce new sets of measurements, new data matrices systematic method for induction Kvalheim (1988) EXTENDING THE LATENT VARIABLE MODEL

GRAPHICAL REPRESENTATIONS OF A DATA MATRIX