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GRAPHICAL REPRESENTATIONSOF ADATA MATRIX: 

GRAPHICAL REPRESENTATIONS OF A DATA MATRIX

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SYSTEM CHARCTERISATION SYSTEM Numbers Measure

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

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Numbers Measure Information (Graphics) Latent Projections Modelling

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

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DATA MATRIX / DATA TABLE

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i j k 1 5 l 3 1 m 8 6 Object Variable Column-centred data matrix i j k -3 1 l -1 -3 m 4 2 Object Variable Original data matrix

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

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

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

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

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

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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)

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LATENT VARIABLE PROJECTIONS

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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

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

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METHOD OVERVIEW

LATENT PROJECTION: 

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

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LATENT VARIABLE MODEL

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PCA/PLS (orthogonal scores)

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

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EXTENDING THE LATENT VARIABLE MODEL