Principal Components Analysis

1
Principal Components Analysis
2
Correlated Variables
Z2
Z1
3
Uncorrelated Principal Components
P1
P2
4
Linear Transformation
P1
P2
Matrix of Factor Score Coefficients
Z1
Z2
5
Linear Combinations
 
6
Standard Scores
7
Intercorrelated Variables
8
Total Variance of Original Variables
1 1 2
9
Uncorrelated Principal Components
10
Linear Transformation
P1
P2
Matrix of Factor Score Coefficients
Z1
Z2
11
Compute the Factor Score Coefficients

• Matrix of Eigenvalues L
• Matrix of Factor Loadings P
• Factor Scores Coefficients

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1. Compute the Eigenvalues
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Compute Eigenvalues
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Correlation Matrix R
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The Trace of the Correlation Matrix
1 1 2
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Compute Eigenvalues
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Compute Eigenvalues
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Compute Eigenvalues
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Two Eigenvalues

• The first eigenvalue equals 1.3
• The second eigenvalue equals - .7

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Matrix of Eigenvalues L
1.30 .70 2.00
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Eigenvalues

• The original variables have a total variance of
  2.00 (Standardized, 2/2 1).
• The first eigenvalue equals 1.30. The first
  component accounts for a variance of 1.30
  (Standardized, 1.3/2 .65).
• The second eigenvalue equals .70. The second
  component accounts for a variance of .70
  (Standardized, .7/2 .35).

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2. Compute the Factor Loadings
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Compute Eigenvectors
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Compute Eigenvectorsfor the First Eigenvalue
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Compute Eigenvectorsfor the First Eigenvalue
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Compute Eigenvectorsfor the First Eigenvalue
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Eigenvectorsfor the First Eigenvalue
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Compute Eigenvectorsfor the Second Eigenvalue
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Compute Eigenvectorsfor the Second Eigenvalue
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Compute Eigenvectorsfor the Second Eigenvalue
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Eigenvectorsfor the Second Eigenvalue
-V1 V2
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Eigenvector Structure
33
Restrictions on Structural Solution
34
Structural Solution in Squared Form
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Matrix of Factor Loadings P
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Correlations between Variables and Principal
Components
37
P1
P2
Z1
Z2
Component loadings are correlations of original
variables with the components
38
Vector Plot
39
Compute Factor Score Coefficients
40
Compute Factor Score Coefficients
41
Factor Score Coefficients
42
Linear Transformation
P1
P2
Z1
Z2
Matrix of Coefficients
43
Principal Components
44
Uncorrelated Principal Components
45
Correlation Super Matrix
46
Super Matrix of Squared Coefficients
47
Row Sum
1.00
1.00
48
Column Sum
1.30
.70