GRAPHICAL REPRESENTATIONSOF A
DATA MATRIX
SYSTEM CHARCTERISATION
SYSTEM
Numbers
Measure
CHARACTERISATION
UV,IR,NMR,MS,GC,GC-MS
......................................... .....................
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)