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# Multivariate Data Analysis - PowerPoint PPT Presentation

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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## PowerPoint Slideshow about 'Multivariate Data Analysis' - mizell

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### Multivariate Data Analysis

Principal Component Analysis

• Singular Value Decomposition

• Eigenvector / eigenvalue calculation

• Reduce variables

• Improve projections

• Remove noise

• Find outliers

• Find classes

K

X

I

• Example with 2 variables, 6 objects

• Find best (most informative) direction in space

• Describe direction

• Make projection

x2

x1

x2

x1

Score

Residual

Unit vector

Unit vector

X

K

i

Score vector

I

p

t

X

K

Score vector

I

p

X

K

Score vector

I

p

X = t1p1’ + t2p2’ + ... + tApA’ + E

X=TP’+E

X : properly preprocessed (IxK)

T: Score matrix (IxA)

E: residual matrix (IxK)

ta: score vector

The Wine ExamplePeople magazineWise & Gallagher

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

Mean

20.9900 68.2600 1.7500 75.9000 153.8700

24.9270 38.6718 0.9132 3.2128 110.8182

Standard

Deviation

l1=46%

32%

12%

8%

2%

Component

Czech

Brit

Austral

Mex

USA

Japan

Switz

Italy

France

Russia

Score 1 (46%)

Beer

Life exp.

Heart dis.

Wine

Spirit

### 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%)

Appelkoos

Wavelength, nm

Scree plot

Component number

### What is rank?

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

Gives zero residual

Effective rank = A

Separates model from noise

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 1 (98%)

### ANOVA

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

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

From largest to smallest!

X = TP’ + E

data = model + residual

SStot = SSmod + SSres

R2 = SSmod / SStot = 1 - SSres / SStot

Coefficient of determination (often in %)

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.

Outliers removed

Wavelength, nm

Singular values

l1=81%

16%

3%

Component

Whole fruit

No kernel

Thin slice

Score 2 (16%)

Wavelength, nm

### More nomenclature

Score = Latent Variable

Effective rank = Pseudorank = Model dimensionality = Number of components

SSa = Eigenvalue

Singular value = SSa1/2

• 1. Scale, mean-center data

• 2. Calculate a few components

• 4. Find outliers, groupings, explain

• 5. Remove outliers

• 6. Scale, mean-center data

• 7. Calculate enough components

• 8. Try to detemine pseudorank

• 9. Check score plots

• 11. Check residuals

Residual stdev

2

1

4

0

3

Residual stdev

4

0

1

3

2