PCA for analysis of complex multivariate data

Interpretation of large data tables by PCA

• In industry, research and finance the amount of data is often very large

• Little information is available a priori

• There is a need for methods based on few assumptions and which can give a simple and easily understandable overview

– Overall broad interpretation

– Ideas for further analyses

– Generating hypotheses

• PCA is such a method!!!!

PCA used for

• Interpretation

• Pre-processing for regression

• Classification

• SPC

• Noise reduction

• Pre-processing for other statistical analyses

Examples of use in industry

• Process monitoring

• Sensory analysis (tasting etc.)– Product development and quality control

• Rheological measurements

• Process prediction

• Spectroscopy (NIR and other)

Examples of use outside industry

• Psychology

• Food science

• Information retrieval systems

• Consumer studies, marketing

PCA

1. Compresses the information– Finds the directions with most variability– Projects the information down on these dimensions

2. Presents the information in simple plots– Scores plot

• Projection of data onto subspace

– Loadings plot• Plot of relation between original variables and subspace

dimensions

NKN

K

xx

x

xxx

X

..

.

.

.

.

.

.

..

1

21

11211

Data structure for PCA, data matrix

Rows are objects, ”samples”Columns are variables

Scatter plots, vectors

• Vector x=( x1,x2,…xK)

• Can be plotted. If several vectors are plotted it is called a scatter plot

X=(x1,x2,x3)

x1x2

x3

Principal component analysis

DataMatrix

X

Variables

Objects

PCAScores plot

Loadings plot

Other results

X1X2

X3

X

PC 1

PC 2

Model

X=TPT + E

The matrix X is modelled as components (systematic effects) plus residuals, E (noise)

PCA model

The main plots

• Scores plot– For interpreting relations among samples

• Loadings plot– For interpreting relations among variables

• Explained variance plot

PC1

PC2

Scores plot/projection (T)

t1

t2

70%

25%

x1 pc1

pc2

Loadings plot

x2

x3

Loadings plots

• Usually 2-dimensional

• For spectroscopy and other continuous measurements, 1-dimensional plots are used.

Guidelines for how to interpret the plots

• Variables which are close have high correlation

• Samples which are close are similar

• Variables on opposite side of origin have negative correlation

• Objects on the right are dominated by variables to the right and so on….

Variance pr. component

• Sum of the variances of the original x-variables is equal to the sum of the variances of the scores.

• We can talk about variance pr. component and explained variance (in %) pr. component

• Can be presented in a cumulative way (or not)

Explained variance

No. of components1 2 3

50%

100%

Cumulative plot (in % or absolute units)

1 2 3

Number of components

Explained variance

0.5

1.0

Non-cumulative plot (in % or absolute units)

Bar plots can also be used

Sensory analysis of sausages Goals of the analysis

• Investigate the possibility of using dairy ingredients in sausages– Type and concentration– Focus on sensory properties

• Investigate the interaction of diary ingredients with other ingredients and process parameters

• Characterise the differences among the dairy ingredients used in sausages

Sensory analysis of sausages• Factorial design in 4 variables

– 5 dairy ingredients• Na caseinate• Na caseinate (high viscosity)• Skim milk• Whey protein• Demineralised whey powder

– 3 concentration levels• 1%, 3% and 5%

– 2 starch levels• 2% and 4%

– 2 cooking temperatures• 76 and 82 degrees C.

Published: Baardseth et al, J. Food Science.

Variables/attributes used• Graininess• Stickiness• Firmness• Juiciness• Fatness• Elasticity• Colour hue• Colour intensity• Whiteness• Meat taste• Off-taste• Rancidity• Smokiness

70%

  

Loadings and scores

Scores split up according to ingredient on next slide

Above average

Below average

Demineralised whey powder

Na caseinate

Na caseinate (high viscosity)

Skim milk

Whey protein

Can also be done using colours

We have got information about

• Which samples that are similar• Which variables that are similar or very different• Which samples that are characterised by which

variables• Which design variables that are most important for

variation• Differences among the ingredients

Pre-processing

• If variables are in very different units, it may be advantageous to standardise the variables prior to PCA

• Xnew=Xold/std(X) for each variable

• Be aware of noise!! Can be tested by ANOVA or replicates.

Standard deviations

Viscosity

pH

Water content

Temp

Variables of different typesDifficult to compare

Pre-processing

• In spectroscopy usually not done

• Very important if measurements from different instruments are used together

Outlier detection

• Outliers may always be present

• Influence the solution

• New information?

• Important to detect them

Tools for outlier detection

• Residuals =

– Plot residuals pr. object

– Compute sum of squared residuals pr. object

• Leverage, distance to mean within space

(Mahalanobis distance)

'ˆˆ TPXXXE

e

”normal samples”

PCA plane

Leverage point

x1 x2

x3

Validation

• Plots, how natural is the solution: Relate to knowledge and design.

• Steep increase of explained variance

• Can also use cross-validation– Leave out one sample and test on the rest. Repeat for all samples.

Compute explained prediction variance.