1' Introduction to multiway analysis

1
1. Introduction to multiway analysis

• Quimiometria Teórica e Aplicada
• Instituto de Química - UNICAMP

2
Why build models of chemical data?

• Data exploration
• e.g. find important sources of variation in
  complex environmental samples
• Compound identification and calibration in
  mixtures
• e.g. identification and quantification of
  pollutants in river water
• Statistical process control
• e.g. detect disturbances in product quality

• Models are useful approximations of reality
• first-principles models are based on
  chemical/physical knowledge do they fit well
  with the measured data?
• empirical models (e.g. PCA, PLS) are purely
  mathematical do they have a chemical meaning?

3
Multiway data

• Multiway data is becoming more common in
  chemistry. Examples are

• Chromatography
• sample number ? elution time ? wavelength

• On-line analysis
• experiment number ? time ? wavelength/temperature/
  pressure

• Tandem mass spectroscopy (MS-MS)
• sample number ? parent ion mass ? daughter ion
  mass

• Image analysis
• experiment number ? time ? x-position ?
  y-position

4
Multiway data an example

• Batch process data

time
time
batch
process variable
process variable
One batch A series
of batches X (J ? K)
X (I ? J ? K)
5
Multiway modelling

• The PARAFAC (or CANDECOMP) and Tucker models
  were developed by psychometricians 30 years ago,
  but are especially useful in chemistry, because
  chemical data often has a multilinear structure.

• PARAFAC and Tucker are different generalizations
  of PCA for higher-order data.

• There also exist generalizations of PLS for
  higher-order data, e.g. N-PLS.

6
Two-way modelling

• Two-way data can be modelled using bilinear
  models

PT


X
E
T
time
process variable
(1) X TPT E
PCA
7
Multiway models - PARAFAC

• Multiway data can be modelled using multilinear
  models, such as the PARAFAC model...

CT


BT
X
E
batch
A
time
process variable
8
Multiway models - Tucker

• ...or the Tucker model

CT


G
BT
E
X
batch
time
A
process variable
9
Unfolding

• Another option is to matricize (or unfold) the
  data and use standard two-way methods

X
X1
...
XI
I
I
K
XI?JK
JK
J

• Can also unfold along other modes XJ?KI and
  XK?IJ

• But if a multiway structure exists in the data,
  multiway methods have some important advantages!!

10
Advantages of multiway

• Multiway models need fewer model parameters to
  describe the data, e.g. a three-component model
  of X (30 ? 800 ? 200) uses
• 540090 parameters for unfold-PCA
• 3090 parameters PARAFAC
• PARAFAC is more parsimonious than unfold-PCA.

?

• Multiway models use one set of loadings for each
  mode results are much easier to plot and
  understand.

?
11
Disadvantages of multiway

• PARAFAC and Tucker models are usually calculated
  using a technique called alternating least
  squares (ALS).
• This is sometimes slow...

?
...and sometimes gives convergence problems if an
inappropriate model is used.
?
?
?
?
?
12
Conclusions

• PARAFAC and Tucker are both generalizations of
  the PCA model for multiway data.
• PARAFAC and Tucker models use fewer parameters
  and are easier to interpret than unfold-PCA.
• Models can be calculated in MATLAB using N-way
  Toolbox (or PLS_Toolbox)