GRAPHICAL REPRESENTATIONSOF A

DATA MATRIX

SYSTEM CHARCTERISATION

SYSTEM

Numbers

Measure

CHARACTERISATION

UV,IR,NMR,MS,GC,GC-MS

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SampleInstrument + Computer

Instrumental ProfilesData matrix

Numbers Measure

Information(Graphics)

Latent Projections

Modelling

X

Data matrix

Variable vectors(column vectors)

Object vectors(row vectors)

x’k

xi

DATA MATRIX / DATA TABLE

i j k 1 5 l 3 1 m 8 6

Object/Sample Variable

i j k [ 1 5 ] l [ 3 1 ] m [ 8 6 ]

Object/Sample Variable

Object vectors

i j k 1 5 l 3 1 m 8 6

Object/Sample Variable

Variable vectors

i j k 1 5 l 3 1 m 8 6

Object/Sample Variable

i j k [ 1 5 ] l [ 3 1 ] m [ 8 6 ]

Object/Sample Variable

Object vectors

i j k 1 5 l 3 1 m 8 6

Object/Sample Variable

Variable vectors

i j k 1 5 l 3 1 m 8 6

Object Variable

Column-centreddata matrix

i j k -3 1 l -1 -3 m 4 2

Object Variable

Originaldata matrix

Subtract variable mean, xi=4, xj=4

VARIABLE SPACE

x’l

Shows relationships between objects (angle kl measures similarity).

cos kl = x’k xl/|| x’k || || xl ||

variable i

variable j

x’m

x’k

kl

i j k -3 1 l -1 -3 m 4 2

OBJECT SPACE

object k

object m

object l

xi

xj

ij

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 ||

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 1

variable 2

xk

LV

e1

e2 wa

tka

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)

p2

Object spacepa’ = ta’X/ta’ta

Variable Correlation

Variable spaceta = Xwa

Object Correlation

v2

v1

p1

o1

o3LVV

Object vectors

t3

t2 t1

X

Data matrix

Variable vectors

v1

v2

LV

o2

Score plot axes (w1,w2…)

Loading plotsAxes (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

PCA/SVD wa = pa/||pa||

PLS wa = u’aXa/|| u’aXa ||

MVP wa = ei

MOP wa = xk/||xk||

TP wa = bk/||bk||

METHOD OVERVIEW

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

IS AN INSTRUMENT

TO CREATE ORDER (MODEL)

OUT OF CHAOS (DATA)

LATENT PROJECTION

LATENT VARIABLE MODEL

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’

PCA/PLS (orthogonal scores)

X - XP’

T 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

Visual Interface

Score plot- variable space

Loading plot- object space

Biplot plot - Scores and loadings in one plot!

EXTENDING THE LATENT VARIABLE MODEL

- 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)