1. GRAPHICAL REPRESENTATIONS OF A DATA MATRIX

2. SYSTEM CHARCTERISATION
Measure
Numbers

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

4. Numbers
Measure
Latent Projections
Information
(Graphics)
Modelling

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

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

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

8. i j k l m
Object/Sample
Variable
Variable vectors

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

10. Subtract variable mean, xi=4, xj=4
Object
Variable
i j k l m
Original
data matrix
Subtract variable mean, xi=4, xj=4
Object
Variable
i j k l m
Column-centred
data matrix

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

12. OBJECT SPACE xi i j k -3 1 l -1 -3 m 4 2 xj
object k
object m
object l xi xj
 ij
i j
k -3 1 l
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 ||

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

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

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

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

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

18. 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

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

20. 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

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

24. 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’

25. PCA/PLS (orthogonal scores)
X - X P’ 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

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

27. 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)