robust pca in stata
Post on 27-Jan-2016
86 Views
Preview:
DESCRIPTION
TRANSCRIPT
Robust PCA in StataRobust PCA in Stata
Vincenzo Verardi (vverardi@fundp.ac.be)
FUNDP (Namur) and ULB (Brussels), BelgiumFNRS Associate Researcher
PCA, transforms a set of correlated variables into a smaller set of uncorrelated variables (principal components).
For p random variables X1,…,Xp. the goal of PCA is to construct a new set of p axes in the directions of greatest variability.
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
X1
X2
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
X1
X2
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
X1
X2
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
X1
X2
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
Hence, for the first principal component, the goal is to find a linear transformation Y=1 X1+2 X2+..+ p Xp (= TX) such that tha variance of Y (=Var(TX) =T ) is maximal
The direction of is given by the eigenvector correponding to the largest eigenvalue of matrix Σ
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The second vector (orthogonal to the first), is the one that has the second highest variance. This corresponds to the eigenvector associated to the second largest eigenvalue
And so on …
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The new variables (PCs) have a variance equal to their corresponding eigenvalue
Var(Yi)= i for all i=1…p
The relative variance explained by each PC is given by i / i
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
How many PC should be considered?
Sufficient number of PCs to have a cumulative variance explained that is at least 60-70% of the total
Kaiser criterion: keep PCs with eigenvalues >1
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
PCA is based on the classical covariance matrix which is sensitive to outliers … Illustration:
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
PCA is based on the classical covariance matrix which is sensitive to outliers … Illustration:
. set obs 1000
. drawnorm x1-x3, corr(C)
. matrix list C
c1 c2 c3r1 1r2 .7 1r3 .6 .5 1
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
x3 -0.0148 0.5216 1.0000 x2 0.0005 1.0000 x1 1.0000 x1 x2 x3
(obs=1000). corr x1 x2 x3
(100 real changes made). replace x1=100 in 1/100
x3 0.6162 0.5216 1.0000 x2 0.7097 1.0000 x1 1.0000 x1 x2 x3
(obs=1000). corr x1 x2 x3
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
x3 -0.0148 0.5216 1.0000 x2 0.0005 1.0000 x1 1.0000 x1 x2 x3
(obs=1000). corr x1 x2 x3
(100 real changes made). replace x1=100 in 1/100
x3 0.6162 0.5216 1.0000 x2 0.7097 1.0000 x1 1.0000 x1 x2 x3
(obs=1000). corr x1 x2 x3
x3 -0.0148 0.5216 1.0000 x2 0.0005 1.0000 x1 1.0000 x1 x2 x3
(obs=1000). corr x1 x2 x3
(100 real changes made). replace x1=100 in 1/100
x3 0.6162 0.5216 1.0000 x2 0.7097 1.0000 x1 1.0000 x1 x2 x3
(obs=1000). corr x1 x2 x3
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
This drawback can be easily solved by basing the PCA on a robust estimation of the covariance (correlation) matrix.
A well suited method for this is MCD that considers all subsets containing h% of the observations (generally 50%) and estimates Σ and µ on the data of the subset associated with the smallest covariance matrix determinant.
Intuition …
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The generalized variance proposed by Wilks (1932), is a one-dimensional measure of multidimensional scatter. It is defined as .
In the 2x2 case it is easy to see the underlying idea:
det( )GV
22 2 2
2 and det( )x xyx y xy
xy y
Raw bivariate spread
Spread due to covariations
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
Remember, MCD considers all subsets containing 50% of the observations …
However, if N=200, the number of subsets to consider would be:
Solution: use subsampling algorithms …
582009.0549×10 ...
100
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The implemented algorithm:
Rousseeuw and Van Driessen (1999)
1.P-subset
2.Concentration (sorting distances)
3.Estimation of robust ΣMCD
4.Estimation of robust PCA
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
Consider a number of subsets containing (p+1) points (where p is the number of variables) sufficiently large to be sure that at least one of the subsets does not contain outliers.
Calculate the covariance matrix on each subset and keep the one with the smallest determinant
Do some fine tuning to get closer to the global solution
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The minimal number of subsets we need to have a probability (Pr) of having at least one clean if % of outliers corrupt the dataset can be easily derived:
log(1 Pr)*
log(1 (1 ) )
Pr 1 1 1
p
Np
N
Contamination: %
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The minimal number of subsets we need to have a probability (Pr) of having at least one clean if % of outliers corrupt the dataset can be easily derived:
log(1 Pr)*
log(1 (1 ) )
Pr 1 1 1
p
Np
N
Will be the probability that one random point in the dataset is not an outlier
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The minimal number of subsets we need to have a probability (Pr) of having at least one clean if % of outliers corrupt the dataset can be easily derived:
log(1 Pr)*
log(1 (1 ) )
Pr 1 1 1
p
Np
N
Will be the probability that none of the p random points in a p-subset is an outlier
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The minimal number of subsets we need to have a probability (Pr) of having at least one clean if % of outliers corrupt the dataset can be easily derived:
log(1 Pr)*
log(1 (1 ) )
Pr 1 1 1
p
Np
N
Will be the probability that at least one of the p random points in a p-subset is an outlier
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The minimal number of subsets we need to have a probability (Pr) of having at least one clean if % of outliers corrupt the dataset can be easily derived:
log(1 Pr)*
log(1 (1 ) )
Pr 1 1 1
p
Np
N
Will be the probability that there is at least one outlier in each of the N p-subsets considered (i.e. that all p-subsets are corrupt)
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The minimal number of subsets we need to have a probability (Pr) of having at least one clean if % of outliers corrupt the dataset can be easily derived:
log(1 Pr)*
log(1 (1 ) )
Pr 1 1 1
p
Np
N
Will be the probability that there is at least one clean p-subset among the N considered
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The minimal number of subsets we need to have a probability (Pr) of having at least one clean if % of outliers corrupt the dataset can be easily derived:
log(1 Pr)*
log(1 (1 ) )
Pr 1 1 1
p
Np
N
Rearranging we have:
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The preliminary p-subset step allowed to estimate a preliminary Σ* and μ*
Calculate Mahalanobis distances using Σ* and μ* for all individuals
Mahalanobis distances, are defined as
.
MD are distributed as for Gaussian
data.
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
1( ) ( )'i iMD x x 2p
The preliminary p-subset step allowed to estimate a preliminary Σ* and μ*
Calculate Mahalanobis distances using Σ* and μ* for all individuals
Sort individuals according to Mahalanobis distances and re-estimate Σ* and μ* using the first 50% observations
Repeat the previous step till convergence
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
In Stata, Hadi’s method is available to estimate a robust Covariance matrix
Unfortunately it is not very robust
The reason for this is simple, it relies on a non-robust preliminary estimation of the covariance matrix
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
1. Compute a variant of MD
2. Sort individuals according to . Use the subset with the first p+1 points to re-estimate μ and Σ.
3. Compute MD and sort the data.
4. Check if the first point out of the subset is an outlier. If not, add this point to the subset and repeat steps 3 and 4. Otherwise stop.
1( ) ( )'i MED i MEDMD x x
MDIntroduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
clearset obs 1000local b=sqrt(invchi2(5,0.95))drawnorm x1-x5 ereplace x1=invnorm(uniform())+5 in 1/100mcd x*, outliergen RD=Robust_distancehadimvo x*, gen(a b) p(0.5)scatter RD b, xline(`b') yline(`b')
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
02
46
8R
obu
st d
ista
nce
0 1 2 3 4 5Hadi distance (p=.5)
Hadi
Fast-MCD
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
x3 0.5471 0.8145 0.1931 0 x2 0.5815 -0.5358 0.6123 0 x1 0.6021 -0.2227 -0.7667 0 Variable Comp1 Comp2 Comp3 Unexplained
Principal components (eigenvectors)
Comp3 .26595 . 0.0886 1.0000 Comp2 .471721 .205771 0.1572 0.9114 Comp1 2.26233 1.79061 0.7541 0.7541 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 3 Number of comp. = 3Principal components/correlation Number of obs = 1000
. pca x1-x3
. drawnorm x1-x3, corr(C)
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
1
.7 1
.6 .5 1
C
x3 0.5471 0.8145 0.1931 0 x2 0.5815 -0.5358 0.6123 0 x1 0.6021 -0.2227 -0.7667 0 Variable Comp1 Comp2 Comp3 Unexplained
Principal components (eigenvectors)
Comp3 .26595 . 0.0886 1.0000 Comp2 .471721 .205771 0.1572 0.9114 Comp1 2.26233 1.79061 0.7541 0.7541 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 3 Number of comp. = 3Principal components/correlation Number of obs = 1000
. pca x1-x3
. drawnorm x1-x3, corr(C)
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
1
.7 1
.6 .5 1
C
x3 0.7073 -0.0143 0.7068 0 x2 0.7064 0.0512 -0.7059 0 x1 -0.0261 0.9986 0.0463 0 Variable Comp1 Comp2 Comp3 Unexplained
Principal components (eigenvectors)
Comp3 .487058 . 0.1624 1.0000 Comp2 1.00075 .513695 0.3336 0.8376 Comp1 1.51219 .511435 0.5041 0.5041 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 3 Number of comp. = 3Principal components/correlation Number of obs = 1000
. pca x1-x3
(100 real changes made). replace x1=100 in 1/100
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
x3 0.7073 -0.0143 0.7068 0 x2 0.7064 0.0512 -0.7059 0 x1 -0.0261 0.9986 0.0463 0 Variable Comp1 Comp2 Comp3 Unexplained
Principal components (eigenvectors)
Comp3 .487058 . 0.1624 1.0000 Comp2 1.00075 .513695 0.3336 0.8376 Comp1 1.51219 .511435 0.5041 0.5041 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 3 Number of comp. = 3Principal components/correlation Number of obs = 1000
. pca x1-x3
(100 real changes made). replace x1=100 in 1/100
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
x3 0.7073 -0.0143 0.7068 0 x2 0.7064 0.0512 -0.7059 0 x1 -0.0261 0.9986 0.0463 0 Variable Comp1 Comp2 Comp3 Unexplained
Principal components (eigenvectors)
Comp3 .487058 . 0.1624 1.0000 Comp2 1.00075 .513695 0.3336 0.8376 Comp1 1.51219 .511435 0.5041 0.5041 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 3 Number of comp. = 3Principal components/correlation Number of obs = 1000
. pca x1-x3
(100 real changes made). replace x1=100 in 1/100
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
x3 0.5564 0.7581 0.3402 0 x2 0.5701 -0.6462 0.5074 0 x1 0.6045 -0.0883 -0.7917 0 Variable Comp1 Comp2 Comp3 Unexplained
Principal components (eigenvectors)
Comp3 .27952 . 0.0932 1.0000 Comp2 .473402 .193882 0.1578 0.9068 Comp1 2.24708 1.77368 0.7490 0.7490 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 3 Number of comp. = 3Principal components/correlation Number of obs = 1000
. pcamat covRMCD, n(1000)
The number of subsamples to check is 20. mcd x*
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
x3 0.5564 0.7581 0.3402 0 x2 0.5701 -0.6462 0.5074 0 x1 0.6045 -0.0883 -0.7917 0 Variable Comp1 Comp2 Comp3 Unexplained
Principal components (eigenvectors)
Comp3 .27952 . 0.0932 1.0000 Comp2 .473402 .193882 0.1578 0.9068 Comp1 2.24708 1.77368 0.7490 0.7490 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 3 Number of comp. = 3Principal components/correlation Number of obs = 1000
. pcamat covRMCD, n(1000)
The number of subsamples to check is 20. mcd x*
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
x3 0.5564 0.7581 0.3402 0 x2 0.5701 -0.6462 0.5074 0 x1 0.6045 -0.0883 -0.7917 0 Variable Comp1 Comp2 Comp3 Unexplained
Principal components (eigenvectors)
Comp3 .27952 . 0.0932 1.0000 Comp2 .473402 .193882 0.1578 0.9068 Comp1 2.24708 1.77368 0.7490 0.7490 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 3 Number of comp. = 3Principal components/correlation Number of obs = 1000
. pcamat covRMCD, n(1000)
The number of subsamples to check is 20. mcd x*
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
QUESTION: Can a single indicator accurately sum up research excellence?
GOAL: Determine the underlying factors measured by the variables used in the Shanghai ranking
Principal component analysis
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
Alumni: Alumni recipients of the Nobel prize or the Fields Medal;
Award: Current faculty Nobel laureates and Fields Medal winners;
HiCi : Highly cited researchers
N&S: Articles published in Nature and Science;
PUB: Articles in the Science Citation Index-expanded, and the Social Science Citation Index;
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
scoreonpub 0.3767 0.6409 0.5726 0.3453 0.0161 0 scoreonns 0.5008 0.1280 -0.3848 -0.1104 -0.7567 0 scoreonhici 0.4829 0.2651 -0.4261 -0.3417 0.6310 0 scoreonaward 0.4405 -0.5202 -0.1339 0.6991 0.1696 0 scoreonalu~i 0.4244 -0.4816 0.5697 -0.5129 -0.0155 0 Variable Comp1 Comp2 Comp3 Comp4 Comp5 Unexplained
Principal components (eigenvectors)
Comp5 .118665 . 0.0237 1.0000 Comp4 .189033 .0703686 0.0378 0.9763 Comp3 .414444 .225411 0.0829 0.9385 Comp2 .872601 .458157 0.1745 0.8556 Comp1 3.40526 2.53266 0.6811 0.6811 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 5 Number of comp. = 5Principal components/correlation Number of obs = 150
. pca scoreonalumni scoreonaward scoreonhici scoreonns scoreonpub
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
scoreonpub 0.3767 0.6409 0.5726 0.3453 0.0161 0 scoreonns 0.5008 0.1280 -0.3848 -0.1104 -0.7567 0 scoreonhici 0.4829 0.2651 -0.4261 -0.3417 0.6310 0 scoreonaward 0.4405 -0.5202 -0.1339 0.6991 0.1696 0 scoreonalu~i 0.4244 -0.4816 0.5697 -0.5129 -0.0155 0 Variable Comp1 Comp2 Comp3 Comp4 Comp5 Unexplained
Principal components (eigenvectors)
Comp5 .118665 . 0.0237 1.0000 Comp4 .189033 .0703686 0.0378 0.9763 Comp3 .414444 .225411 0.0829 0.9385 Comp2 .872601 .458157 0.1745 0.8556 Comp1 3.40526 2.53266 0.6811 0.6811 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 5 Number of comp. = 5Principal components/correlation Number of obs = 150
. pca scoreonalumni scoreonaward scoreonhici scoreonns scoreonpub
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
scoreonpub 0.3767 0.6409 0.5726 0.3453 0.0161 0 scoreonns 0.5008 0.1280 -0.3848 -0.1104 -0.7567 0 scoreonhici 0.4829 0.2651 -0.4261 -0.3417 0.6310 0 scoreonaward 0.4405 -0.5202 -0.1339 0.6991 0.1696 0 scoreonalu~i 0.4244 -0.4816 0.5697 -0.5129 -0.0155 0 Variable Comp1 Comp2 Comp3 Comp4 Comp5 Unexplained
Principal components (eigenvectors)
Comp5 .118665 . 0.0237 1.0000 Comp4 .189033 .0703686 0.0378 0.9763 Comp3 .414444 .225411 0.0829 0.9385 Comp2 .872601 .458157 0.1745 0.8556 Comp1 3.40526 2.53266 0.6811 0.6811 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 5 Number of comp. = 5Principal components/correlation Number of obs = 150
. pca scoreonalumni scoreonaward scoreonhici scoreonns scoreonpub
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
The first component accounts for 68% of the inertia and is given by:Φ1=0.42Al.+0.44Aw.+0.48HiCi+0.50NS+0.38PUB
Variable Corr. (Φ1,Xi)
Alumni 0.78
Awards 0.81
HiCi 0.89
N&S 0.92
PUB 0.70
Total score 0.99
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
scoreonpub 0.3948 0.1690 0.8682 -0.1233 0.2158 0 scoreonns 0.3178 0.6537 -0.1712 -0.3163 -0.5851 0 scoreonhici 0.5322 0.3220 -0.3983 0.3494 0.5765 0 scoreonaward -0.5128 0.4375 -0.0544 -0.5293 0.5123 0 scoreonalu~i -0.4437 0.4991 0.2350 0.6946 -0.1277 0 Variable Comp1 Comp2 Comp3 Comp4 Comp5 Unexplained
Principal components (eigenvectors)
Comp5 .326847 . 0.0654 1.0000 Comp4 .409133 .0822867 0.0818 0.9346 Comp3 .835928 .426794 0.1672 0.8528 Comp2 1.46006 .624132 0.2920 0.6856 Comp1 1.96803 .507974 0.3936 0.3936 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 5 Number of comp. = 5Principal components/correlation Number of obs = 150
. pcamat covMCD, n(150) corr
The number of subsamples to check is 20. mcd scoreonalumni scoreonaward scoreonhici scoreonns scoreonpub, raw
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
scoreonpub 0.3948 0.1690 0.8682 -0.1233 0.2158 0 scoreonns 0.3178 0.6537 -0.1712 -0.3163 -0.5851 0 scoreonhici 0.5322 0.3220 -0.3983 0.3494 0.5765 0 scoreonaward -0.5128 0.4375 -0.0544 -0.5293 0.5123 0 scoreonalu~i -0.4437 0.4991 0.2350 0.6946 -0.1277 0 Variable Comp1 Comp2 Comp3 Comp4 Comp5 Unexplained
Principal components (eigenvectors)
Comp5 .326847 . 0.0654 1.0000 Comp4 .409133 .0822867 0.0818 0.9346 Comp3 .835928 .426794 0.1672 0.8528 Comp2 1.46006 .624132 0.2920 0.6856 Comp1 1.96803 .507974 0.3936 0.3936 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 5 Number of comp. = 5Principal components/correlation Number of obs = 150
. pcamat covMCD, n(150) corr
The number of subsamples to check is 20. mcd scoreonalumni scoreonaward scoreonhici scoreonns scoreonpub, raw
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
scoreonpub 0.3948 0.1690 0.8682 -0.1233 0.2158 0 scoreonns 0.3178 0.6537 -0.1712 -0.3163 -0.5851 0 scoreonhici 0.5322 0.3220 -0.3983 0.3494 0.5765 0 scoreonaward -0.5128 0.4375 -0.0544 -0.5293 0.5123 0 scoreonalu~i -0.4437 0.4991 0.2350 0.6946 -0.1277 0 Variable Comp1 Comp2 Comp3 Comp4 Comp5 Unexplained
Principal components (eigenvectors)
Comp5 .326847 . 0.0654 1.0000 Comp4 .409133 .0822867 0.0818 0.9346 Comp3 .835928 .426794 0.1672 0.8528 Comp2 1.46006 .624132 0.2920 0.6856 Comp1 1.96803 .507974 0.3936 0.3936 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 5 Number of comp. = 5Principal components/correlation Number of obs = 150
. pcamat covMCD, n(150) corr
The number of subsamples to check is 20. mcd scoreonalumni scoreonaward scoreonhici scoreonns scoreonpub, raw
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
scoreonpub 0.3948 0.1690 0.8682 -0.1233 0.2158 0 scoreonns 0.3178 0.6537 -0.1712 -0.3163 -0.5851 0 scoreonhici 0.5322 0.3220 -0.3983 0.3494 0.5765 0 scoreonaward -0.5128 0.4375 -0.0544 -0.5293 0.5123 0 scoreonalu~i -0.4437 0.4991 0.2350 0.6946 -0.1277 0 Variable Comp1 Comp2 Comp3 Comp4 Comp5 Unexplained
Principal components (eigenvectors)
Comp5 .326847 . 0.0654 1.0000 Comp4 .409133 .0822867 0.0818 0.9346 Comp3 .835928 .426794 0.1672 0.8528 Comp2 1.46006 .624132 0.2920 0.6856 Comp1 1.96803 .507974 0.3936 0.3936 Component Eigenvalue Difference Proportion Cumulative
Rotation: (unrotated = principal) Rho = 1.0000 Trace = 5 Number of comp. = 5Principal components/correlation Number of obs = 150
. pcamat covMCD, n(150) corr
The number of subsamples to check is 20. mcd scoreonalumni scoreonaward scoreonhici scoreonns scoreonpub, raw
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
Two underlying factors are uncovered:Φ1 explains 38% of inertia and Φ2 explains 28% of inertia
Variable Corr. (Φ1,∙) Corr. (Φ2,∙)
Alumni -0.05 0.78
Awards -0.01 0.83
HiCi 0.74 0.88
N&S 0.63 0.95
PUB 0.72 0.63
Total score 0.99 0.47
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
Classical PCA could be heavily distorted by the presence of outliers.
A robustified version of PCA could be obtained either by relying on a robust covariance matrix or by removing multivariate outliers identified through a robust identification method.
Introduction
Robust
Covariance
Matrix
Robust PCA
Application
Conclusion
top related