Multivariate data analysis
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Multivariate Data Analysis. Principal Component Analysis. Principal Component Analysis (PCA). Singular Value Decomposition Eigenvector / eigenvalue calculation. Data Matrix (IxK). Reduce variables Improve projections Remove noise Find outliers Find classes. K. X. I. PCA.

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Multivariate data analysis

Multivariate Data Analysis

Principal Component Analysis


Principal component analysis pca
Principal Component Analysis (PCA)

  • Singular Value Decomposition

  • Eigenvector / eigenvalue calculation


Data matrix ixk
Data Matrix (IxK)

  • Reduce variables

  • Improve projections

  • Remove noise

  • Find outliers

  • Find classes

K

X

I


PCA

  • Example with 2 variables, 6 objects

  • Find best (most informative) direction in space

  • Describe direction

  • Make projection


x2

x1


x2

x1



1st PC

Score

Residual


1st PC

Loading p2

Unit vector

Loading p1


1st PC

Unit vector

Loading p2 = sin (a)

Loading p1 = cos(a)


t

X

K

i

Score vector

I

p

Loading vector


k

t

X

K

Score vector

I

p

Loading vector


t

X

K

Score vector

I

p

Loading vector


X = t1p1’ + t2p2’ + ... + tApA’ + E

X=TP’+E

X : properly preprocessed (IxK)

T: Score matrix (IxA)

P: loading matrix (KxA)

E: residual matrix (IxK)

ta: score vector

pa: loading vector


The wine example people magazine wise gallagher
The Wine ExamplePeople magazineWise & Gallagher


Wine Beer Spirit LifeEx HeartD

France

Italy

Switz

Austra

Brit

U.S.A.

Russia

Czech

Japan

Mexico

63.5000 40.1000 2.5000 78.0000 61.1000

58.0000 25.1000 0.9000 78.0000 94.1000

46.0000 65.0000 1.7000 78.0000 106.4000

15.7000 102.1000 1.2000 78.0000 173.0000

12.2000 100.0000 1.5000 77.0000 199.7000

8.9000 87.8000 2.0000 76.0000 176.0000

2.7000 17.1000 3.8000 69.0000 373.6000

1.7000 140.0000 1.0000 73.0000 283.7000

1.0000 55.0000 2.1000 79.0000 34.7000

0.2000 50.4000 0.8000 73.0000 36.4000


Beer Wine Spirit LifeEx HeartD

Mean

20.9900 68.2600 1.7500 75.9000 153.8700

24.9270 38.6718 0.9132 3.2128 110.8182

Standard

Deviation


Singular value

l1=46%

32%

12%

8%

2%

Component


Score 2 (32%)

Czech

Brit

Austral

Mex

USA

Japan

Switz

Italy

France

Russia

Score 1 (46%)


Loading 2

Beer

Life exp.

Heart dis.

Wine

Spirit

Loading 1


Conclusions

Conclusions

Scores = positions of objects in multivariate space

Loadings = importance of original variables for new directions

Try to explain a large enough portion of X (46+32 = 78%)


The apricot example
The Apricot Example

Manley & Geladi


Pseudoabsorbance

Appelkoos

Wavelength, nm


Singular value

Scree plot

Component number


What is rank

What is rank?

Mathematical rank = max(min(I,K))

Gives zero residual

Effective rank = A

Separates model from noise


Anova

ANOVA

SS

SS%

SS%cum

Comp#

1

2

3

4

5

6

7

8

9

10

68.8269

1.2843

0.0463

0.0045

0.0007

0.0003

0.0002

0.0001

0.0000

0.0000

98.10

1.83

0.07

0.01

0.00

0.00

0.00

0.00

0.00

0.00

98.10

99.93

100

Total

70.1634

100


Score 2 (2%)

Score 1 (98%)


Anova1

ANOVA

SStot = l1 + l2 + l3 +...+ l(I or K)

SStot = SS1 + SS2 + SS3 +...+ SS(I or K)

From largest to smallest!


Anova2
ANOVA

X = TP’ + E

data = model + residual

SStot = SSmod + SSres

R2 = SSmod / SStot = 1 - SSres / SStot

Coefficient of determination (often in %)


Examples
Examples

Wines R2 = SSmod = 78% SSres = 22%2 Comp.

Apricots 1 R2 = SSmod = 99.93% SSres = 0.07%

2 Comp.

Apricots 2 R2 = SSmod = 100% SSres = ±0.0%

3 Comp.


Absorbance

Outliers removed

Wavelength, nm


No outliers

Singular values

l1=81%

16%

3%

Component


Score 3 (3%)

Whole fruit

No kernel

Thin slice

Score 2 (16%)


Loading 23

Wavelength, nm


Loading 3

Loading 2


More nomenclature

More nomenclature

Score = Latent Variable

Loading vector = Eigenvector

Effective rank = Pseudorank = Model dimensionality = Number of components

SSa = Eigenvalue

Singular value = SSa1/2


An analysis sequence
An analysis sequence

  • 1. Scale, mean-center data

  • 2. Calculate a few components

  • 3. Check scores, loadings

  • 4. Find outliers, groupings, explain

  • 5. Remove outliers


An analysis sequence1
An analysis sequence

  • 6. Scale, mean-center data

  • 7. Calculate enough components

  • 8. Try to detemine pseudorank

  • 9. Check score plots

  • 10. Check loading plots

  • 11. Check residuals


Wines

Residual stdev

2

1

4

0

3


Wines

Residual stdev

4

0

1

3

2


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