PCA Explained: Data Reduction and Interpretation Using Principal Component Analysis

J. Rovan: Multivariate Analysis 9 Principal Component Analysis 1
9 Principal Component Analysis
A principal component analysis (PCA) is concerned with explaining the variance-covariance
structure of a set of variables through a few linear combinations of these variables, called
principal components. Its general objectives are:
• data reduction and
• interpretation.
In general, p principal components are required to reproduce the total system of variability of
the original data set (n measurements on p variables). Fortunatelly, much of this variability can
often be accounted for by a small number of k of principal components. If so, there is (almost)
as much information in the first k components as there is in the original p variables. The k first
principal components can then replace the initial p variables, and the original n p× data set is
reduced to n k× data set consisting of n measurements on k principal components.
An analysis of principal components often reveals relationships that were not previously
suspected and thereby allows interpretations that would not ordinarily result.
Principal components also frequently serve as intermediate steps in much larger investigations,
e.g. as inputs to a multiple regression, cluster analysis, etc.
Example
Suppose one would like to investigate the level of the socio-economic development of some
European countries in the year 1981. An investigation will take into account the following set of
economic, demographic, health, social security and level of living indicators:
• Per capita gross domestic product in $
• Share of agriculture in gross domestic product (%)
• Share of service activities in gross domestic product (%)
• Export/import ratio
• Per capita fuel consumption in kilograms of coal
• Natural change of population (rates per 1000 inhabitants)
• Share of urban population (%)
• Infant mortality per 1000 live birth
• Number of students per 1000 inhabitants
• Number of TV sets per 1000 inhabitants

9.1 Geometry of Principal Component Analysis
Example
Suppose we have a data set of 12 measurements on 2 variables 1X and 2X for 12 randomly
selected units (Sharma, 1966, p. 59). Let us calculate their mean-corrected values.
Table 1
1ix 2ix 1,cix 2,cix
16 8 8 5
12 10 4 7
13 6 5 3
11 2 3 -1
10 8 2 5
9 -1 1 -4
8 4 0 1
7 6 -1 3
5 -3 -3 -6
3 -1 -5 -4
2 -3 -6 -6
0 0 -8 -3
The position of units can be presented with points in the two-dimensional space. The
coordinates of the points are the values of mean-corrected variables 1,cX and 2,cX :
X1C
1086420-2-4-6-8-10
X2C
10
8
6
4
2
0
-2
-4
-6
-8
-10
12
11
10
9
8
7
6
5
4
3
2
1

9.1.1 Identification of Alternative Axes and Forming New Variables
Let *
1,cX be any axis in the two dimensional space that goes through the origin of the two
rectangular axes 1,cX and 2,cX 1. Axis *
1,cX is making an angle of θ degrees with 1,cX . The
perpendicular projections of the units (observations) onto *
1,cX will give the coordinates of the
observations with respect to *
1,cX . These new coordinates are linear combinations of the
coordinates of the points with respect to the original set of axes 1,cX and 2,cX :
*
1,c 1,c 2,ccos sinX X Xθ θ= ⋅ + ⋅
There is one and only one new axis 1, cξ that results in a new variable accounting for the
maximum variance in the data. In our case this axis makes an angle of o
43,261 with 1,cX . The
corresponding equation for computing the values of 1,cξ is
o o
1,c 1,c 2,c 1,c 2,ccos43,261 sin 43,261 0,728 0,685X X X Xξ = ⋅ + ⋅ = + ,
while its values are
1,c 1,c 2,c0,728 0,685i i ix xξ = + , 1,2, ,i n= … .
1 The origin 1, 2,( , ) (0,0)c cx x ′ ′= , i.e. the centroid, is always part of the optimal subspace in the sence of
least squares.
Of course, a one-dimensional space represented by the new axis 1, cξ (in general) does not
account for all the variance of the investigated phenomena, that has been originally presented by
the values of the two variables 1,cX and 2,cX in a two-dimensional space. Therefore, it is
possible to identify a second axis 2, cξ such that the corresponding new variable accounts for the
maximum of the variance that is not accounted for by 1, cξ . Let 2, cξ be the second new axis that
is orthogonal to 1, cξ . Thus, if the angle between 1, cξ and 1,cX is θ then the angle between 2, cξ
and 2,cX will also be θ.
The equation for computing the values of 2,cξ is
o o
2,c 1,c 2,c 1,c 2,csin 43,261 cos43,261 0,685 0,728X X X Xξ = − ⋅ + ⋅ = − + ,
while its values are
2,c 1,c 2,c0,685 0,728i i ix xξ = − + , 1,2, ,i n= … .

The following conclusions can be made from the above figure and the statistical measures:
• the perpendicular projections of the points onto the original axes give the values of the
original variables 1,cX and 2,cX , and the perpendicular projections of the points onto the
new axes give the values for the new variables 1, cξ and 2, cξ . The new axes and the
corresponding variables are called principal components and the values of the new variables
are called principal component scores. Each of the new variables are linear combinations of
the original variables and remain mean-corrected.
• The total variance of the principal components is the same as the total variance of the
original variables.The variance accounted for by the first principal component is greater than
the variance accounted for by any one of the original variables.
The geometrical illustration of principal component analysis can be easily extended to more
than two variables. An n p× data set now consists of p variables and each unit (observation)
can be represented as a point in a p-dimensional space with respect to the p new axes – principal
components. The projections of points on principal components are called principal component
scores.
If a substantial amount of the total variance in the data set is accounted for by a few first
principal components, than we can use these principal components for further analysis or for
interpretations instead of the original variables. This would result in a substantial data reduction
– an n k× data set ( k p ) of principal component scores is sufficient for further analysis.
Hence, principal component analysis is commonly referred to as a data-reduction technique.
9.2 Analytical Approach
Let us form the following p linear combinations:
1 11 1 12 2 1
2 21 1 22 2 2
1 1 2 2
p p
p p
p p p pp p
w X w X w X
w X w X w X
w X w X w X
ξ
ξ
ξ
= + + +
= + + +
= + + +
…
…
…
where 1 2, , , pξ ξ ξ… are the p principal components and jkw ( , 1,2, , )j k p= … is the weight of the
k-th variable for the j-th principal component.

The principal component weights are estimated in such a way that:
1. The first principal component, 1ξ , accounts for the maximum variance in the data, the
second principal component, 2ξ , accounts for the maximum variance that has not been
accounted for by the first principal component, and so on
2. For each principal component, the sum of squares of its weights should be equal to 1
2
1
1
p
jk
k
w
=
=∑ , 1,2, ,j p= …
3. Sum of the products of the corresponding weights of two principal components should be
equal to 0
1
0
p
jk j k
k
w w ′′
=
=∑ , j j′′≠
The last condition ensures that principal components are ortogonal to each other.
How do we obtain the weights such that the above listed conditions are satisfied? We are
dealing with an optimization problem, usually based on covariance or correlation matrix. We
need to calculate eigen vectors, that define principal component weights, and eigenvalues that
represent variances of principal components.
9.3 Issues Relating to the Use of Principal Component Analysis
9.3.1 Effect of Type of Data on Principal Component Analysis
Principal component analysis can either be done on raw or mean-corrected data on one hand or
on standardised data on the other. Each data set could give a different solution depending upon
the extent to which the variances of the variables differ.
In case of raw or mean-corrected data, the basis for principal component analysis is covariance
matrix. The influence of an individual variable on principal components is determined by the
magnitude of its variance. The higher the variance of the variable, the stronger the effect of a
variable on principal components.
In case of standardized data, the basis for principal component analysis is correlation matrix. All
the variances are equal to 1 and therefore they all have the same influence on principal
components.
In cases for which there is a reason to believe that the variances of the variables do indicate the
importance of given variable and the units of measure are commensurable, the raw or the mean-
corrected data should be used. In all other cases standardised data are preferable alternative.

9.3.2 Is Principal Component Analysis the Appropriate Technique
The use of principal component analysis is appropriate at least in two cases:
• if principal components have meaningful interpretation, what is particularly important for
their further use in other statistical techniques and/or
• if the objective is to reduce the number of variables in the data set to a few principal
components without a substantial loss of information.
Principal component analysis is most appropriate if the variables are interrelated, for only then
will it be possible to reduce a number of variables to a manageble few without much loss of
information.
Many statistical tests are available for determining if the variables are significantly correlated
among themselves. For standardised data we can use Bartlett's test, but we should keep in mind
that it is very sensitive on the sample size:
0H : =P I , 1H : ≠P I
2 1
6
( 1) (2 5) lnn pχ = − − − +⎡ ⎤⎣ ⎦ R
2
( )/ 2m p p= −
9.3.3 Number of Principal Components to Extract
We suggest the use of the following two empirical rules :
1. Kaiser's rule
In the case of standardised data, retain only those components whose eigenvalues (variances)
are greater than 1.
, s
2
1jj ξλ σ= ≥
The rationale for this rule is that for standardised data the amount of variance extracted by
each component should, at minimum, be equal to the variance of at least one variable.
2. Scree plot (Cattell, 1966)
Plot the percentage of variance (or the eigenvalue) accounted for by each of principal
components (on vertical axis) against the ordinal number of the components (on horizontal
axis) and look for an elbow.
However, no one rule is best under all circumstances. One should take into consideration the
purpose of the study, the type of data, and the trade-off between parsimony and the amount of
variation in the data that the researcher is willing to sacrifice in order to achieve parsimony.
Lastly, and more importantly, one should determine the interpretability of the principal
components in deciding upon how many principal components should be retained (Sharma,
1996, p. 79)

9.3.4 Interpreting Principal Components
Since principal components are linear combinations of the original variables, one can use
loadings (simple correlations between the original variables and principal components) for
interpreting the principal components. The higher the loading of a variable, the more influence it
has in the formation of the principal component score and vice versa. Traditionally, a loading of
0.5 or above is used as the cutoff point.
9.3.5 Use of Principal Component Scores
The principal component scores can be plotted for further interpreting the results. Based on
visual examination of the plot, clusters can be defined.
Principal component scores can also be used as input variables for further analysing the data
using other multivariate techniques such as cluster analysis, multiple regression, and
discriminant analysis. The advantage of using principal component scores is that they are not
correlated and the problem of multicollinearity is avoided. Unfortunatelly, a new problem can
arise due to the inability to meaningfully interpret the principal components.
Example (the level of the socio-economic development of some European countries - continued)
GET
FILE='F:Predmeti EFMagistrski studijMultivariate Analysis (IMB)Priprava prosojnic6_predavanjePCA.sav'.
EXECUTE .
LIST .
List
country gdp agric service expimp energy growth urban infmort student tv
Austria 8725 4,4 55,7 ,753 4160,00 ,0 54 14,7 15,7 290
Belgium 9702 2,1 61,2 ,896 6037,00 ,1 72 11,1 20,3 296
Bulgaria 4150 16,9 25,4 1,136 5678,00 ,7 64 19,8 13,2 200
Czechslovakia 5820 8,4 16,9 ,983 6482,00 ,7 63 15,8 12,5 252
Denmark 10874 4,8 66,7 ,898 5225,00 ,3 84 8,8 20,5 361
Finland 10028 8,2 56,2 ,987 5135,00 ,3 62 7,6 17,4 318
France 12214 4,2 60,1 ,840 4351,00 ,4 78 10,1 19,0 299
Greece 3887 15,5 56,7 ,477 2137,00 1,1 62 18,7 12,4 151
Italy 6085 6,4 50,7 ,826 3318,00 ,5 69 15,3 19,1 232
Yugoslavia 2620 13,3 34,8 ,694 2049,00 ,9 42 34,0 20,0 195
Hungary 4180 14,3 26,8 ,954 3850,00 ,4 54 23,7 9,9 251
GDR (East Germany) 7180 9,1 22,1 ,893 7408,00 -,2 77 13,0 23,0 344
Netherlands 9760 4,0 63,0 1,040 6183,00 ,7 76 8,7 23,2 298
Norway 13522 4,5 55,1 1,150 6434,00 ,4 53 8,8 18,5 294
Poland 3900 15,3 20,6 ,856 5590,00 ,9 57 21,3 16,9 218
Portugal 2370 13,0 41,0 ,423 1097,00 1,1 31 39,0 8,6 126
Romania 1904 11,0 25,0 ,904 4593,00 1,0 50 31,6 8,6 166
Spain 5678 8,0 55,0 ,632 2530,00 1,1 74 15,0 17,7 267
Sweden 13326 3,1 65,5 ,991 5296,00 ,3 87 7,5 23,9 375
Switzerland 15069 6,1 55,0 ,881 3708,00 -,3 58 10,0 12,6 320
United Kingdom 9358 1,9 63,5 1,003 4835,00 ,0 91 12,8 13,6 336
Sowiet Union 4550 15,1 23,5 1,115 5598,00 ,9 62 25,6 19,1 307
FRG (West Germany) 11135 2,2 49,9 1,074 5727,00 -,2 85 13,5 18,0 343
Number of cases read: 23 Number of cases listed: 23

FACTOR
/VARIABLES gdp agric service expimp energy growth urban infmort student tv
/MISSING LISTWISE /ANALYSIS gdp agric service expimp energy growth urban infmort st
udent tv
/PRINT UNIVARIATE INITIAL CORRELATION SIG DET KMO EXTRACTION FSCORE
/PLOT EIGEN
/CRITERIA FACTORS(10) ITERATE(25)
/EXTRACTION PC
/ROTATION NOROTATE
/SAVE REG(ALL)
/METHOD=CORRELATION .
_
- - - - - - - - - - - - F A C T O R A N A L Y S I S - - - - - - - - - - - -
Factor Analysis
F:Predmeti EFMagistrski studijMultivariate Analysis (IMB)Priprava prosojnic6_predavanjePCA.sav
Descriptive Statistics
7653,78 3941,192 23
8,339 4,9609 23
45,670 17,0293 23
,88722 ,190763 23
4670,4783 1613,28267 23
,483 ,4448 23
65,43 14,981 23
16,800 8,8066 23
16,683 4,5156 23
271,26 69,072 23
Per capita gross domestic product in $
Share of agriculture in gross domestic product (%)
Share of services activities in gross domestic product (%)
Export/import ratio
Per capita fuel consumption in kilograms of coal
Natural change of population (rates per 1000 inhabitants)
Share of urban population (%)
Infant mortality per 1000 live birth
Number of students per 1000 inhabitants
Number of TV sets per 1000 inhabitants
Mean Std. Deviation Analysis N
Correlation Matrixa
1,000 -,801 ,686 ,410 ,389 -,728 ,542 -,842 ,460 ,799
-,801 1,000 -,723 -,261 -,314 ,655 -,611 ,704 -,447 -,704
,686 -,723 1,000 -,103 -,151 -,332 ,465 -,606 ,359 ,438
,410 -,261 -,103 1,000 ,817 -,409 ,406 -,445 ,290 ,573
,389 -,314 -,151 ,817 1,000 -,449 ,479 -,544 ,437 ,595
-,728 ,655 -,332 -,409 -,449 1,000 -,476 ,622 -,278 -,751
,542 -,611 ,465 ,406 ,479 -,476 1,000 -,735 ,554 ,744
-,842 ,704 -,606 -,445 -,544 ,622 -,735 1,000 -,549 -,784
,460 -,447 ,359 ,290 ,437 -,278 ,554 -,549 1,000 ,635
,799 -,704 ,438 ,573 ,595 -,751 ,744 -,784 ,635 1,000
,000 ,000 ,026 ,033 ,000 ,004 ,000 ,014 ,000
,000 ,000 ,115 ,072 ,000 ,001 ,000 ,016 ,000
,000 ,000 ,319 ,245 ,061 ,013 ,001 ,046 ,018
,026 ,115 ,319 ,000 ,026 ,027 ,017 ,090 ,002
,033 ,072 ,245 ,000 ,016 ,010 ,004 ,019 ,001
,000 ,000 ,061 ,026 ,016 ,011 ,001 ,100 ,000
,004 ,001 ,013 ,027 ,010 ,011 ,000 ,003 ,000
,000 ,000 ,001 ,017 ,004 ,001 ,000 ,003 ,000
,014 ,016 ,046 ,090 ,019 ,100 ,003 ,003 ,001
,000 ,000 ,018 ,002 ,001 ,000 ,000 ,000 ,001
Share of agriculture in gross domestic produ
Share of services activities in gross domestic
Export/import ratio
Per capita fuel consumption in kilograms of c
Natural change of population (rates per 1000
Share of agriculture in gross domestic produ
Export/import ratio
Per capita fuel consumption in kilograms of c
Correlation
Sig. (1-taile
Per capita
gross
domestic
product in $
Share of
agriculture
in gross
domestic
product (%)
Share of
services
activities in
gross
domestic
product (%)
Export/import
ratio
Per capita fue
consumption
in kilograms
of coal
Natural
change of
population
(rates per
1000
inhabitants)
Share of
urban
population
(%)
nfant mortality
per 1000 live
birth
Number of
students per
1000
inhabitants
Number of TV
sets per 1000
inhabitants
Determinant = 2,926E-05a.
KMO and Bartlett's Test
,769
186,166
45
,000
Kaiser-Meyer-Olkin Measure of Sampling
Adequacy.
Approx. Chi-Square
df
Sig.
Bartlett's Test of
Sphericity

Communalities
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
1,000 1,000
Share of services activities in gross domestic product (%)
Export/import ratio
Natural change of population (rates per 1000 inhabitants)
Initial Extraction
Extraction Method: Principal Component Analysis.
Total Variance Explained
5,879 58,787 58,787 5,879 58,787 58,787
1,751 17,514 76,302 1,751 17,514 76,302
,830 8,305 84,607 ,830 8,305 84,607
,437 4,367 88,973 ,437 4,367 88,973
,399 3,995 92,968 ,399 3,995 92,968
,260 2,603 95,570 ,260 2,603 95,570
,224 2,237 97,808 ,224 2,237 97,808
,106 1,062 98,870 ,106 1,062 98,870
6,090E-02 ,609 99,479 6,090E-02 ,609 99,479
5,207E-02 ,521 100,000 5,207E-02 ,521 100,000
Component
1
2
3
4
5
6
7
8
9
10
Total % of Variance Cumulative % Total % of Variance Cumulative %
Initial Eigenvalues Extraction Sums of Squared Loadings
Component Matrixa
,890 -,220 -,226 -,074 ,226 -,118 ,047 ,094 -,091 ,136
-,831 ,341 ,131 -,007 -,024 -,374 ,161 -,061 ,053 ,056
,589 -,745 ,060 ,135 ,186 ,027 ,076 -,124 ,137 ,024
,572 ,705 -,117 ,153 ,242 ,113 ,236 -,094 -,040 -,044
,623 ,719 ,031 ,029 ,075 ,031 -,259 ,003 ,112 ,077
-,764 -,024 ,486 ,261 ,290 ,049 ,006 ,160 -,002 ,000
,795 ,005 ,296 ,367 -,368 ,018 ,051 -,019 -,050 ,065
-,909 ,077 -,035 -,159 -,098 ,297 ,166 ,007 ,039 ,122
,651 ,035 ,647 -,382 ,052 ,022 ,012 -,073 -,042 -,001
,931 ,098 ,005 -,122 -,133 -,029 ,193 ,195 ,106 -,054
product (%)
Export/import ratio
inhabitants)
1 2 3 4 5 6 7 8 9 10
Component
10 components extracted.a.

Component Score Coefficient Matrix
,151 -,126 -,272 -,170 ,565 -,454 ,211 ,889 -1,492 2,603
-,141 ,195 ,158 -,015 -,059 -1,436 ,718 -,574 ,873 1,082
,100 -,425 ,073 ,309 ,465 ,104 ,338 -1,170 2,243 ,459
,097 ,402 -,141 ,350 ,605 ,432 1,054 -,887 -,660 -,849
,106 ,411 ,038 ,066 ,187 ,120 -1,157 ,030 1,833 1,475
-,130 -,014 ,585 ,597 ,726 ,187 ,025 1,505 -,032 ,007
,135 ,003 ,356 ,840 -,921 ,069 ,228 -,179 -,826 1,256
-,155 ,044 -,042 -,365 -,246 1,142 ,740 ,065 ,633 2,343
,111 ,020 ,779 -,875 ,131 ,085 ,053 -,687 -,692 -,025
,158 ,056 ,006 -,279 -,334 -,111 ,862 1,834 1,743 -1,041
product (%)
Export/import ratio
inhabitants)
1 2 3 4 5 6 7 8 9 10
Component
Component Scores.
Component Score Covariance Matrix
1,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000
,000 1,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000
,000 ,000 1,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000
,000 ,000 ,000 1,000 ,000 ,000 ,000 ,000 ,000 ,000
,000 ,000 ,000 ,000 1,000 ,000 ,000 ,000 ,000 ,000
,000 ,000 ,000 ,000 ,000 1,000 ,000 ,000 ,000 ,000
,000 ,000 ,000 ,000 ,000 ,000 1,000 ,000 ,000 ,000
,000 ,000 ,000 ,000 ,000 ,000 ,000 1,000 ,000 ,000
,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000 1,000 ,000
,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000 ,000 1,000
Component
1
2
3
4
5
6
7
8
9
10
1 2 3 4 5 6 7 8 9 10
Component Scores.
COMPUTE pc1 = fac1_1*2.4246 .
EXECUTE .
COMPUTE pc2 = fac2_1*1.3234 .
EXECUTE .
LIST VARIABLES=country fac1_1 fac2_1 pc1 pc2 .
List
country FAC1_1 FAC2_1 pc1 pc2
Austria ,20385 -,83959 ,49 -1,11
Belgium ,85869 -,31095 2,08 -,41
Bulgaria -,68262 1,67025 -1,66 2,21
Czechslovakia -,28816 1,39670 -,70 1,85
Denmark 1,05110 -,54403 2,55 -,72
Finland ,54720 -,01549 1,33 -,02
France ,70860 -,84492 1,72 -1,12
Greece -1,28526 -1,51094 -3,12 -2,00
Italy -,07291 -,65342 -,18 -,86
Yugoslavia -1,39868 -,42705 -3,39 -,57
Hungary -,84721 ,73613 -2,05 ,97
GDR (East Germany) ,69647 1,43444 1,69 1,90
Netherlands ,87914 ,04262 2,13 ,06
Norway ,78937 ,41629 1,91 ,55
Poland -,83951 1,15341 -2,04 1,53
Portugal -2,24731 -1,48964 -5,45 -1,97
Romania -1,40470 ,75353 -3,41 1,00
Spain -,33848 -1,29162 -,82 -1,71
Sweden 1,40417 -,42425 3,40 -,56
Switzerland ,62967 -,80592 1,53 -1,07
United Kingdom ,93875 -,42776 2,28 -,57
Sowiet Union -,43132 1,70467 -1,05 2,26
FRG (West Germany) 1,12915 ,27755 2,74 ,37
Number of cases read: 23 Number of cases listed: 23

DESCRIPTIVES
VARIABLES=fac1_1 fac2_1 pc1 pc2
/STATISTICS=MEAN STDDEV MIN MAX .
Descriptives
Descriptive Statistics
23 -2,24731 1,40417 ,0000000 1,00000000
23 -1,51094 1,70467 ,0000000 1,00000000
23 -5,45 3,40 ,0000 2,42460
23 -2,00 2,26 ,0000 1,32340
23
REGR factor score 1 for analysis 1
REGR factor score 2 for analysis 1
PC1
PC2
Valid N (listwise)
N Minimum Maximum Mean Std. Deviation
GRAPH
/SCATTERPLOT(BIVAR)=pc1 WITH pc2 BY country (NAME)
/MISSING=LISTWISE .
Graph

PCA Explained: Data Reduction and Interpretation Using Principal Component Analysis

Recomendados

Recomendados

Más contenido relacionado

La actualidad más candente

La actualidad más candente (19)

Destacado

Destacado (14)

Similar a PCA Explained: Data Reduction and Interpretation Using Principal Component Analysis

Similar a PCA Explained: Data Reduction and Interpretation Using Principal Component Analysis (20)

Último

Último (20)

PCA Explained: Data Reduction and Interpretation Using Principal Component Analysis