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

Data structure for PCA, data matrix Rows are objects, ”samples” Columns are variables

Scatter plots, vectors Vector x=( x 1,x 2,…x K ) Can be plotted. If several vectors are plotted it is called a scatter plot

X=(x1,x2,x3) x1 x2 x3

Principal component analysis Data Matrix X Variables Objects PCA Scores plot Loadings plot Other results

X1 X2 X3 X PC 1 PC 2

Model X=TP T + 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 components % 100% Cumulative plot (in % or absolute units)

123 Number of components Explained variance 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 X new =X old /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 types Difficult 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)

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.