Mva 06 principal_component_analysis_2010_11

255 views

Published on

Published in: Technology
0 Comments
0 Likes
Statistics
Notes
  • Be the first to comment

  • Be the first to like this

No Downloads
Views
Total views
255
On SlideShare
0
From Embeds
0
Number of Embeds
2
Actions
Shares
0
Downloads
10
Comments
0
Likes
0
Embeds 0
No embeds

No notes for slide

Mva 06 principal_component_analysis_2010_11

  1. 1. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 19 Principal Component AnalysisA principal component analysis (PCA) is concerned with explaining the variance-covariancestructure of a set of variables through a few linear combinations of these variables, calledprincipal components. Its general objectives are:• data reduction and• interpretation.In general, p principal components are required to reproduce the total system of variability ofthe original data set (n measurements on p variables). Fortunatelly, much of this variability canoften 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 firstprincipal components can then replace the initial p variables, and the original n p× data set isreduced to n k× data set consisting of n measurements on k principal components.An analysis of principal components often reveals relationships that were not previouslysuspected 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.J. Rovan: Multivariate Analysis 9 Principal Component Analysis 2ExampleSuppose one would like to investigate the level of the socio-economic development of someEuropean countries in the year 1981. An investigation will take into account the following set ofeconomic, 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
  2. 2. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 39.1 Geometry of Principal Component AnalysisExampleSuppose we have a data set of 12 measurements on 2 variables 1X and 2X for 12 randomlyselected units (Sharma, 1966, p. 59). Let us calculate their mean-corrected values.Table 11ix 2ix 1,cix 2,cix16 8 8 512 10 4 713 6 5 311 2 3 -110 8 2 59 -1 1 -48 4 0 17 6 -1 35 -3 -3 -63 -1 -5 -42 -3 -6 -60 0 -8 -3J. Rovan: Multivariate Analysis 9 Principal Component Analysis 4The position of units can be presented with points in the two-dimensional space. Thecoordinates of the points are the values of mean-corrected variables 1,cX and 2,cX :X1C1086420-2-4-6-8-10X2C1086420-2-4-6-8-10121110987654321
  3. 3. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 59.1.1 Identification of Alternative Axes and Forming New VariablesLet *1,cX be any axis in the two dimensional space that goes through the origin of the tworectangular axes 1,cX and 2,cX 1. Axis *1,cX is making an angle of θ degrees with 1,cX . Theperpendicular projections of the units (observations) onto *1,cX will give the coordinates of theobservations with respect to *1,cX . These new coordinates are linear combinations of thecoordinates 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 themaximum variance in the data. In our case this axis makes an angle of o43,261 with 1,cX . Thecorresponding equation for computing the values of 1,cξ iso o1,c 1,c 2,c 1,c 2,ccos43,261 sin 43,261 0,728 0,685X X X Xξ = ⋅ + ⋅ = + ,while its values are1,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 ofleast squares.J. Rovan: Multivariate Analysis 9 Principal Component Analysis 6Of course, a one-dimensional space represented by the new axis 1, cξ (in general) does notaccount for all the variance of the investigated phenomena, that has been originally presented bythe values of the two variables 1,cX and 2,cX in a two-dimensional space. Therefore, it ispossible to identify a second axis 2, cξ such that the corresponding new variable accounts for themaximum of the variance that is not accounted for by 1, cξ . Let 2, cξ be the second new axis thatis orthogonal to 1, cξ . Thus, if the angle between 1, cξ and 1,cX is θ then the angle between 2, cξand 2,cX will also be θ.The equation for computing the values of 2,cξ iso o2,c 1,c 2,c 1,c 2,csin 43,261 cos43,261 0,685 0,728X X X Xξ = − ⋅ + ⋅ = − + ,while its values are2,c 1,c 2,c0,685 0,728i i ix xξ = − + , 1,2, ,i n= … .
  4. 4. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 7The 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 theoriginal variables 1,cX and 2,cX , and the perpendicular projections of the points onto thenew axes give the values for the new variables 1, cξ and 2, cξ . The new axes and thecorresponding variables are called principal components and the values of the new variablesare called principal component scores. Each of the new variables are linear combinations ofthe original variables and remain mean-corrected.• The total variance of the principal components is the same as the total variance of theoriginal variables.The variance accounted for by the first principal component is greater thanthe variance accounted for by any one of the original variables.J. Rovan: Multivariate Analysis 9 Principal Component Analysis 8The geometrical illustration of principal component analysis can be easily extended to morethan 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 – principalcomponents. The projections of points on principal components are called principal componentscores.If a substantial amount of the total variance in the data set is accounted for by a few firstprincipal components, than we can use these principal components for further analysis or forinterpretations 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 ApproachLet us form the following p linear combinations:1 11 1 12 2 12 21 1 22 2 21 1 2 2p pp pp p p pp pw X w X w Xw X w X w Xw X w X w Xξξξ= + + += + + += + + +………where 1 2, , , pξ ξ ξ… are the p principal components and jkw ( , 1,2, , )j k p= … is the weight of thek-th variable for the j-th principal component.
  5. 5. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 9The principal component weights are estimated in such a way that:1. The first principal component, 1ξ , accounts for the maximum variance in the data, thesecond principal component, 2ξ , accounts for the maximum variance that has not beenaccounted for by the first principal component, and so on2. For each principal component, the sum of squares of its weights should be equal to 1211pjkkw==∑ , 1,2, ,j p= …3. Sum of the products of the corresponding weights of two principal components should beequal to 010pjk j kkw 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 aredealing with an optimization problem, usually based on covariance or correlation matrix. Weneed to calculate eigen vectors, that define principal component weights, and eigenvalues thatrepresent variances of principal components.J. Rovan: Multivariate Analysis 9 Principal Component Analysis 109.3 Issues Relating to the Use of Principal Component Analysis9.3.1 Effect of Type of Data on Principal Component AnalysisPrincipal component analysis can either be done on raw or mean-corrected data on one hand oron standardised data on the other. Each data set could give a different solution depending uponthe extent to which the variances of the variables differ.In case of raw or mean-corrected data, the basis for principal component analysis is covariancematrix. The influence of an individual variable on principal components is determined by themagnitude of its variance. The higher the variance of the variable, the stronger the effect of avariable on principal components.In case of standardized data, the basis for principal component analysis is correlation matrix. Allthe variances are equal to 1 and therefore they all have the same influence on principalcomponents.In cases for which there is a reason to believe that the variances of the variables do indicate theimportance 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.
  6. 6. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 119.3.2 Is Principal Component Analysis the Appropriate TechniqueThe use of principal component analysis is appropriate at least in two cases:• if principal components have meaningful interpretation, what is particularly important fortheir 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 principalcomponents without a substantial loss of information.Principal component analysis is most appropriate if the variables are interrelated, for only thenwill it be possible to reduce a number of variables to a manageble few without much loss ofinformation.Many statistical tests are available for determining if the variables are significantly correlatedamong themselves. For standardised data we can use Bartletts test, but we should keep in mindthat it is very sensitive on the sample size:0H : =P I , 1H : ≠P I2 16( 1) (2 5) lnn pχ = − − − +⎡ ⎤⎣ ⎦ R2( )/ 2m p p= −J. Rovan: Multivariate Analysis 9 Principal Component Analysis 129.3.3 Number of Principal Components to ExtractWe suggest the use of the following two empirical rules :1. Kaisers ruleIn the case of standardised data, retain only those components whose eigenvalues (variances)are greater than 1., s21jj ξλ σ= ≥The rationale for this rule is that for standardised data the amount of variance extracted byeach 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 principalcomponents (on vertical axis) against the ordinal number of the components (on horizontalaxis) and look for an elbow.However, no one rule is best under all circumstances. One should take into consideration thepurpose of the study, the type of data, and the trade-off between parsimony and the amount ofvariation 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 principalcomponents in deciding upon how many principal components should be retained (Sharma,1996, p. 79)
  7. 7. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 139.3.4 Interpreting Principal ComponentsSince principal components are linear combinations of the original variables, one can useloadings (simple correlations between the original variables and principal components) forinterpreting the principal components. The higher the loading of a variable, the more influence ithas in the formation of the principal component score and vice versa. Traditionally, a loading of0.5 or above is used as the cutoff point.9.3.5 Use of Principal Component ScoresThe principal component scores can be plotted for further interpreting the results. Based onvisual examination of the plot, clusters can be defined.Principal component scores can also be used as input variables for further analysing the datausing other multivariate techniques such as cluster analysis, multiple regression, anddiscriminant analysis. The advantage of using principal component scores is that they are notcorrelated and the problem of multicollinearity is avoided. Unfortunatelly, a new problem canarise due to the inability to meaningfully interpret the principal components.J. Rovan: Multivariate Analysis 9 Principal Component Analysis 14Example (the level of the socio-economic development of some European countries - continued)GETFILE=F:Predmeti EFMagistrski studijMultivariate Analysis (IMB)Priprava prosojnic6_predavanjePCA.sav.EXECUTE .LIST .Listcountry gdp agric service expimp energy growth urban infmort student tvAustria 8725 4,4 55,7 ,753 4160,00 ,0 54 14,7 15,7 290Belgium 9702 2,1 61,2 ,896 6037,00 ,1 72 11,1 20,3 296Bulgaria 4150 16,9 25,4 1,136 5678,00 ,7 64 19,8 13,2 200Czechslovakia 5820 8,4 16,9 ,983 6482,00 ,7 63 15,8 12,5 252Denmark 10874 4,8 66,7 ,898 5225,00 ,3 84 8,8 20,5 361Finland 10028 8,2 56,2 ,987 5135,00 ,3 62 7,6 17,4 318France 12214 4,2 60,1 ,840 4351,00 ,4 78 10,1 19,0 299Greece 3887 15,5 56,7 ,477 2137,00 1,1 62 18,7 12,4 151Italy 6085 6,4 50,7 ,826 3318,00 ,5 69 15,3 19,1 232Yugoslavia 2620 13,3 34,8 ,694 2049,00 ,9 42 34,0 20,0 195Hungary 4180 14,3 26,8 ,954 3850,00 ,4 54 23,7 9,9 251GDR (East Germany) 7180 9,1 22,1 ,893 7408,00 -,2 77 13,0 23,0 344Netherlands 9760 4,0 63,0 1,040 6183,00 ,7 76 8,7 23,2 298Norway 13522 4,5 55,1 1,150 6434,00 ,4 53 8,8 18,5 294Poland 3900 15,3 20,6 ,856 5590,00 ,9 57 21,3 16,9 218Portugal 2370 13,0 41,0 ,423 1097,00 1,1 31 39,0 8,6 126Romania 1904 11,0 25,0 ,904 4593,00 1,0 50 31,6 8,6 166Spain 5678 8,0 55,0 ,632 2530,00 1,1 74 15,0 17,7 267Sweden 13326 3,1 65,5 ,991 5296,00 ,3 87 7,5 23,9 375Switzerland 15069 6,1 55,0 ,881 3708,00 -,3 58 10,0 12,6 320United Kingdom 9358 1,9 63,5 1,003 4835,00 ,0 91 12,8 13,6 336Sowiet Union 4550 15,1 23,5 1,115 5598,00 ,9 62 25,6 19,1 307FRG (West Germany) 11135 2,2 49,9 1,074 5727,00 -,2 85 13,5 18,0 343Number of cases read: 23 Number of cases listed: 23
  8. 8. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 15FACTOR/VARIABLES gdp agric service expimp energy growth urban infmort student tv/MISSING LISTWISE /ANALYSIS gdp agric service expimp energy growth urban infmort student 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 AnalysisF:Predmeti EFMagistrski studijMultivariate Analysis (IMB)Priprava prosojnic6_predavanjePCA.savDescriptive Statistics7653,78 3941,192 238,339 4,9609 2345,670 17,0293 23,88722 ,190763 234670,4783 1613,28267 23,483 ,4448 2365,43 14,981 2316,800 8,8066 2316,683 4,5156 23271,26 69,072 23Per capita gross domestic product in $Share of agriculture in gross domestic product (%)Share of services activities in gross domestic product (%)Export/import ratioPer capita fuel consumption in kilograms of coalNatural change of population (rates per 1000 inhabitants)Share of urban population (%)Infant mortality per 1000 live birthNumber of students per 1000 inhabitantsNumber of TV sets per 1000 inhabitantsMean Std. Deviation Analysis NJ. Rovan: Multivariate Analysis 9 Principal Component Analysis 16Correlation Matrixa1,000 -,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 ,001Per capita gross domestic product in $Share of agriculture in gross domestic produShare of services activities in gross domesticExport/import ratioPer capita fuel consumption in kilograms of cNatural change of population (rates per 1000Share of urban population (%)Infant mortality per 1000 live birthNumber of students per 1000 inhabitantsNumber of TV sets per 1000 inhabitantsPer capita gross domestic product in $Share of agriculture in gross domestic produShare of services activities in gross domesticExport/import ratioPer capita fuel consumption in kilograms of cNatural change of population (rates per 1000Share of urban population (%)Infant mortality per 1000 live birthNumber of students per 1000 inhabitantsNumber of TV sets per 1000 inhabitantsCorrelationSig. (1-tailePer capitagrossdomesticproduct in $Share ofagriculturein grossdomesticproduct (%)Share ofservicesactivities ingrossdomesticproduct (%)Export/importratioPer capita fueconsumptionin kilogramsof coalNaturalchange ofpopulation(rates per1000inhabitants)Share ofurbanpopulation(%)nfant mortalityper 1000 livebirthNumber ofstudents per1000inhabitantsNumber of TVsets per 1000inhabitantsDeterminant = 2,926E-05a.KMO and Bartletts Test,769186,16645,000Kaiser-Meyer-Olkin Measure of SamplingAdequacy.Approx. Chi-SquaredfSig.Bartletts Test ofSphericity
  9. 9. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 17Communalities1,000 1,0001,000 1,0001,000 1,0001,000 1,0001,000 1,0001,000 1,0001,000 1,0001,000 1,0001,000 1,0001,000 1,000Per capita gross domestic product in $Share of agriculture in gross domestic product (%)Share of services activities in gross domestic product (%)Export/import ratioPer capita fuel consumption in kilograms of coalNatural change of population (rates per 1000 inhabitants)Share of urban population (%)Infant mortality per 1000 live birthNumber of students per 1000 inhabitantsNumber of TV sets per 1000 inhabitantsInitial ExtractionExtraction Method: Principal Component Analysis.Total Variance Explained5,879 58,787 58,787 5,879 58,787 58,7871,751 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 98,8706,090E-02 ,609 99,479 6,090E-02 ,609 99,4795,207E-02 ,521 100,000 5,207E-02 ,521 100,000Component12345678910Total % of Variance Cumulative % Total % of Variance Cumulative %Initial Eigenvalues Extraction Sums of Squared LoadingsExtraction Method: Principal Component Analysis.J. Rovan: Multivariate Analysis 9 Principal Component Analysis 18Component 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 -,054Per capita gross domestic product in $Share of agriculture in gross domestic product (%)Share of services activities in gross domesticproduct (%)Export/import ratioPer capita fuel consumption in kilograms of coalNatural change of population (rates per 1000inhabitants)Share of urban population (%)Infant mortality per 1000 live birthNumber of students per 1000 inhabitantsNumber of TV sets per 1000 inhabitants1 2 3 4 5 6 7 8 9 10ComponentExtraction Method: Principal Component Analysis.10 components extracted.a.
  10. 10. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 19Component 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 -1,041Per capita gross domestic product in $Share of agriculture in gross domestic product (%)Share of services activities in gross domesticproduct (%)Export/import ratioPer capita fuel consumption in kilograms of coalNatural change of population (rates per 1000inhabitants)Share of urban population (%)Infant mortality per 1000 live birthNumber of students per 1000 inhabitantsNumber of TV sets per 1000 inhabitants1 2 3 4 5 6 7 8 9 10ComponentExtraction Method: Principal Component Analysis.Component Scores.Component Score Covariance Matrix1,000 ,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 1,000Component123456789101 2 3 4 5 6 7 8 9 10Extraction Method: Principal Component Analysis.Component Scores.J. Rovan: Multivariate Analysis 9 Principal Component Analysis 20COMPUTE pc1 = fac1_1*2.4246 .EXECUTE .COMPUTE pc2 = fac2_1*1.3234 .EXECUTE .LIST VARIABLES=country fac1_1 fac2_1 pc1 pc2 .ListF:Predmeti EFMagistrski studijMultivariate Analysis (IMB)Priprava prosojnic6_predavanjePCA.savcountry FAC1_1 FAC2_1 pc1 pc2Austria ,20385 -,83959 ,49 -1,11Belgium ,85869 -,31095 2,08 -,41Bulgaria -,68262 1,67025 -1,66 2,21Czechslovakia -,28816 1,39670 -,70 1,85Denmark 1,05110 -,54403 2,55 -,72Finland ,54720 -,01549 1,33 -,02France ,70860 -,84492 1,72 -1,12Greece -1,28526 -1,51094 -3,12 -2,00Italy -,07291 -,65342 -,18 -,86Yugoslavia -1,39868 -,42705 -3,39 -,57Hungary -,84721 ,73613 -2,05 ,97GDR (East Germany) ,69647 1,43444 1,69 1,90Netherlands ,87914 ,04262 2,13 ,06Norway ,78937 ,41629 1,91 ,55Poland -,83951 1,15341 -2,04 1,53Portugal -2,24731 -1,48964 -5,45 -1,97Romania -1,40470 ,75353 -3,41 1,00Spain -,33848 -1,29162 -,82 -1,71Sweden 1,40417 -,42425 3,40 -,56Switzerland ,62967 -,80592 1,53 -1,07United Kingdom ,93875 -,42776 2,28 -,57Sowiet Union -,43132 1,70467 -1,05 2,26FRG (West Germany) 1,12915 ,27755 2,74 ,37Number of cases read: 23 Number of cases listed: 23
  11. 11. J. Rovan: Multivariate Analysis 9 Principal Component Analysis 21DESCRIPTIVESVARIABLES=fac1_1 fac2_1 pc1 pc2/STATISTICS=MEAN STDDEV MIN MAX .DescriptivesF:Predmeti EFMagistrski studijMultivariate Analysis (IMB)Priprava prosojnic6_predavanjePCA.savDescriptive Statistics23 -2,24731 1,40417 ,0000000 1,0000000023 -1,51094 1,70467 ,0000000 1,0000000023 -5,45 3,40 ,0000 2,4246023 -2,00 2,26 ,0000 1,3234023REGR factor score 1 for analysis 1REGR factor score 2 for analysis 1PC1PC2Valid N (listwise)N Minimum Maximum Mean Std. DeviationJ. Rovan: Multivariate Analysis 9 Principal Component Analysis 22GRAPH/SCATTERPLOT(BIVAR)=pc1 WITH pc2 BY country (NAME)/MISSING=LISTWISE .GraphF:Predmeti EFMagistrski studijMultivariate Analysis (IMB)Priprava prosojnic6_predavanjePCA.sav

×