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GRAPHICAL REPRESENTATIONSOF ADATA MATRIX: GRAPHICAL REPRESENTATIONS OF A DATA MATRIX

Slide2: SYSTEM CHARCTERISATION SYSTEM Numbers Measure

Slide3: CHARACTERISATION UV,IR,NMR, MS,GC,GC-MS ..................... .................... . .................... Sample Instrument + Computer Instrumental Profiles Data matrix

Slide4: Numbers Measure Information (Graphics) Latent Projections Modelling

Slide5: X Data matrix Variable vectors (column vectors) Object vectors (row vectors) x’k xi

Slide6: DATA MATRIX / DATA TABLE

Slide10: 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

Slide11: 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

Slide12: 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 ||

Slide13: Object space shows common variation in a suite of variables! common variation underlying factor!

Slide14: VARIABLE SPACE AND OBJECT SPACE CONTAIN TOGETHER ALL AVAILABLE INFORMATION IN A DATA MATRIX

Slide15: WHAT TO DO IF THE NUMBER OF VARIABLES IS GREATER THAN 2-3? PROJECT ONTO LATENT VARIABLES (LV)!

Slide16: 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)

Slide17: LATENT VARIABLE PROJECTIONS

Slide18: 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

Slide19: PCA/SVD wa = pa/||pa|| PLS wa = u’aXa/|| u’aXa || MVP wa = ei MOP wa = xk/||xk|| TP wa = bk/||bk|| METHOD OVERVIEW

Slide20: METHOD OVERVIEW

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

Slide24: LATENT VARIABLE MODEL

Slide25: PCA/PLS (orthogonal scores)

Slide26: Visual Interface Score plot - variable space Loading plot - object space Biplot plot - Scores and loadings in one plot!

Slide27: EXTENDING THE LATENT VARIABLE MODEL