Principal component analysis for studying the world security problem

This research is a continuation of the work [1], in which the list of ten most essential global threats to the future of mankind have been presented. The initial data on each threat are taken from the respectable international organizations data bases. Then, we defined the summarized impact of the e...

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Дата:	2009
Автори:	Pomerantseva, T., Boldak, A.
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Мова:	English
Опубліковано:	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України 2009
Назва видання:	Системні дослідження та інформаційні технології
Теми:	Теоретичні та прикладні проблеми і методи системного аналізу
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Цитувати:	Principal component analysis for studying the world security problem / T. Pomerantseva, A. Boldak // Систем. дослідж. та інформ. технології. — 2009. — № 4. — С. 32–46. — Бібліогр.: 9 назв. — англ.

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spelling	irk-123456789-422362013-03-14T03:10:11Z Principal component analysis for studying the world security problem Pomerantseva, T. Boldak, A. Теоретичні та прикладні проблеми і методи системного аналізу This research is a continuation of the work [1], in which the list of ten most essential global threats to the future of mankind have been presented. The initial data on each threat are taken from the respectable international organizations data bases. Then, we defined the summarized impact of the examined ten global threats totality on different countries based on cluster analysis method with the purpose of selecting groups of the countries with “close” performances of summarized threats. By using the Minkovsky type metric the foresight of the future global conflicting has been executed. To facilitate the analysis and make it easier we use the method of Principal Component Analysis (PCA) which allows reduce variables with many properties to several hidden factors. The analysis shows that currently the most considerable threats for most countries are the reduction of energy security, worsening of balance between bio capacity and human demands and the incomes inequality between people and countries. Проведено дослідження національної безпеки різних країн світу з використанням метода головних компонент (Principal Component Analysis) у просторі десяти глобальних загроз. За допомогою обчислення коефіцієнтів кореляції визначено характер залежності між головними чинниками і вихідними загрозами. Визначено три найбільш істотні загрози, які впливають на національну безпеку більшості країн світу: державна нестабільність, дефіцит енергетичних ресурсів і нерівність доходів (Gini Index). Виконано графічну інтерпретацію глобальних загроз і визначено міри залежності між їх основними групами. Проведено исследование национальной безопасности различных стран мира с использованием метода главных компонент (Principal Component Analysis) в пространстве десяти глобальных угроз. С помощью вычисления коэффициентов корреляции определен характер зависимости между главными факторами и исходными угрозами. Проведена кластеризация стран по уровню глобальных угроз. Определены три наиболее существенные угрозы, влияющие на национальную безопасность большинства стран мира: государственная нестабильность, дефицит энергетических ресурсов и неравенство доходов (Gini Index). Выполнена графическая интерпретация глобальных угроз в пространстве трех главных компонент. Проведено исследование факторной структуры угроз и определены степени зависимости между их основными группами. 2009 Article Principal component analysis for studying the world security problem / T. Pomerantseva, A. Boldak // Систем. дослідж. та інформ. технології. — 2009. — № 4. — С. 32–46. — Бібліогр.: 9 назв. — англ. 1681–6048 http://dspace.nbuv.gov.ua/handle/123456789/42236 316.42+330.34+519.711.2 en Системні дослідження та інформаційні технології Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
institution	Digital Library of Periodicals of National Academy of Sciences of Ukraine
collection	DSpace DC
language	English
topic	Теоретичні та прикладні проблеми і методи системного аналізу Теоретичні та прикладні проблеми і методи системного аналізу
spellingShingle	Теоретичні та прикладні проблеми і методи системного аналізу Теоретичні та прикладні проблеми і методи системного аналізу Pomerantseva, T. Boldak, A. Principal component analysis for studying the world security problem Системні дослідження та інформаційні технології
description	This research is a continuation of the work [1], in which the list of ten most essential global threats to the future of mankind have been presented. The initial data on each threat are taken from the respectable international organizations data bases. Then, we defined the summarized impact of the examined ten global threats totality on different countries based on cluster analysis method with the purpose of selecting groups of the countries with “close” performances of summarized threats. By using the Minkovsky type metric the foresight of the future global conflicting has been executed. To facilitate the analysis and make it easier we use the method of Principal Component Analysis (PCA) which allows reduce variables with many properties to several hidden factors. The analysis shows that currently the most considerable threats for most countries are the reduction of energy security, worsening of balance between bio capacity and human demands and the incomes inequality between people and countries.
format	Article
author	Pomerantseva, T. Boldak, A.
author_facet	Pomerantseva, T. Boldak, A.
author_sort	Pomerantseva, T.
title	Principal component analysis for studying the world security problem
title_short	Principal component analysis for studying the world security problem
title_full	Principal component analysis for studying the world security problem
title_fullStr	Principal component analysis for studying the world security problem
title_full_unstemmed	Principal component analysis for studying the world security problem
title_sort	principal component analysis for studying the world security problem
publisher	Навчально-науковий комплекс "Інститут прикладного системного аналізу" НТУУ "КПІ" МОН та НАН України
publishDate	2009
topic_facet	Теоретичні та прикладні проблеми і методи системного аналізу
url	http://dspace.nbuv.gov.ua/handle/123456789/42236
citation_txt	Principal component analysis for studying the world security problem / T. Pomerantseva, A. Boldak // Систем. дослідж. та інформ. технології. — 2009. — № 4. — С. 32–46. — Бібліогр.: 9 назв. — англ.
series	Системні дослідження та інформаційні технології
work_keys_str_mv	AT pomerantsevat principalcomponentanalysisforstudyingtheworldsecurityproblem AT boldaka principalcomponentanalysisforstudyingtheworldsecurityproblem
first_indexed	2025-07-04T00:44:01Z
last_indexed	2025-07-04T00:44:01Z
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fulltext	© T. Pomerantseva, A. Boldak, 2009 32 ISSN 1681–6048 System Research & Information Technologies, 2009, № 4 UDC 316.42+330.34+519.711.2 PRINCIPAL COMPONENT ANALYSIS FOR STUDYING THE WORLD SECURITY PROBLEM T. POMERANTSEVA, A. BOLDAK This research is a continuation of the work [1], in which the list of ten most essential global threats to the future of mankind have been presented. The initial data on each threat are taken from the respectable international organizations data bases. Then, we defined the summarized impact of the examined ten global threats totality on dif- ferent countries based on cluster analysis method with the purpose of selecting groups of the countries with “close” performances of summarized threats. By using the Minkovsky type metric the foresight of the future global conflicting has been executed. To facilitate the analysis and make it easier we use the method of Princi- pal Component Analysis (PCA) which allows reduce variables with many properties to several hidden factors. The analysis shows that currently the most considerable threats for most countries are the reduction of energy security, worsening of balance between bio capacity and human demands and the incomes inequality between peo- ple and countries. INTRODUCTION In the work [1] the impact of system world conflicts on sustainable development is studied in the global context. On the basis of data analysis pertaining to the global conflicts taking place from 705 B.C. till now the regularity of their flow is determined. It is shown that the sequence of life cycles of system world conflicts is subordinate to the law of Fibonacci series, and the intensity of these conflicts, depending on a level of technological evolution of a society, builds up under the hyperbolic law. By using the revealed regularities we attempt to foresee the up- coming world conflict, called “the conflict of the XXI century” and analyze its na- ture and principal performances: - durations, main phases of the flow and intensity. The totality of main global threats generating the conflict of the XXI century is given. These global threats are: ES — Energy Security; FB — Footprint and Biocapacity Balance; GINI — Incomes Inequality; GD — Global Diseases; CM — Child Mortality; CP — Corruption Perception; WA — Water Access; GW — Global Warming; SF — State Fragility; ND — Natural Disasters. By the cluster analysis method we define the impact of the above threats on different countries and on twelve large groups of countries (civilizations according to Huntington) combined by common culture features. Assumptions are made as to possible scenarios in the course of the conflict of the XXI century and after its termination. Since it is difficult to analyze the security of this or that country simultane- ously in the space of ten global threats, to make the research more convenient and demonstrative we use the Principal Component Analysis (PCA). This method makes it possible to reduce analysis of many properties to some hidden factors determining these properties. In this case the security of a country may be presented in a simplified form not by all ten global threats, but some most significant factors. Principal component analysis for studying the world security problem Системні дослідження та інформаційні технології, 2009, № 4 33 APPLICATION OF THE PRINCIPAL COMPONENT METHOD FOR THE ANALYSIS OF THE IMPACT OF GLOBAL THREATS TOTALITY ON SUSTAINABLE DEVELOPMENT The example of sustainable development global simulation [2] presents global threats and degree of their impact on different countries. Let us format table 1 in the form of the initial data matrix, m NX , 106=N , 10=m , in such a way that its lines NiX i ,1, = correspond to the analyzed countries, and the columns jX , mj ,1= contain the values of threats (indicators) kPX , 10,,1 == mmk . Then, for each country there will be the corresponding vector 〉〈= m iiii xxxX ,,, 21 … of threats values (the upper index corresponds to the threat’s ordinal number). The purpose of the given study conducted with application of the principal component method is finding out and interpreting latent common factors with si- multaneous goal to minimize both their number and the degree of dependence iPX on their specific residual random components. Suppose that each threat iPX is a result of impact m′ of hypothetical and one characteristic factor [3]: ∑ ′ = += m j ij i ji eFqPX 1 , mi ,1= , where i jq — factor loadings; jF — factors to be defined; ie — characteristic factor for the i-th initial feature representing inde- pendent random value with zero mathematical expectation and finite variance. The expression for iPX may be presented in matrix form: EVQX Tm N += , where (1) 1. V — matrix of factor scores; Q — matrix of factor loadings; E — ma- trix of residuals. Searching of principal components is reduced to finding the matrix decom- position m NX in the form (Lindsay I. Smith, 2002): ETPX Tm N += , where T — matrix of scores with dimension )( mmmN ≤′′× . Each line of this matrix is a projection of data vector m iX on m′ of principal components. Number of lines — N corresponds to the number of vectors of the initial data. Number of columns or number of principal components vectors selected for projection is equal 'm . P — loadings matrix of dimension mm ×′ , where m′ — number of lines (data space dimension); m — number of columns (number of vectors of principal compo- nents selected for projection); E — matrix of residuals. Matrix of scores assigns a set of vectors mjNitT j ii ′==〉〈= ,1,,1, , deter- mining projectors of vectors mjNiX j i ,1,,1, == in the principal components space (number of components is equal mm ≤′ ). Matrix of loadings assigns the mapping of the initial space basis in principal components space. The principal component method allows find such mapping 'mFm RR ⎯→⎯ , that mm ≤′ and min2 →∑ ∑i j ije for all possible T and P [3]. Defining principal components is connected with calculation of eigenvectors of the covariance matrix [3, 4], defined as: mjmiPXPXccC jiijij ,1,,1)),,(cov:( ==== , (2) T. Pomerantseva, A. Boldak ISSN 1681–6048 System Research & Information Technologies, 2009, № 4 34 where 1 )()( ),(cov 1 − −− = ∑ = N XxXx PXPX N k jj k ii k ji — covariance of parameters iPX and jPX . For selection of sufficient number mm ≤′ of principal components a cumu- lative variance is often used [5]: mi m D i j j i ,1,1 == ∑ = λ , (3) where mjj ,1, =λ — eigenvalues of covariance matrix C are used. Preliminary analysis of principal components is given in Table 1. T a b l e 1 . Analysis of principal components Value Eigenvalues Total variance, % Comulative Eigenvalues Comulative, % 1 5,065629 50,65629 5,065629 50,65629 2 1,331475 13,31475 6,397103 63,97103 3 1,065071 10,65071 7,462175 74,62175 We shall define the sufficient number of principal components by using the “slide rocks” criterion suggested by [6]. “Slide rocks” is a geological term to define rock debris accumulated in the lower part of a rocky slope. Using this anal- ogy it is possible to show graphically (Fig. 1) the eigenvalues presented in table 1. It is necessary to find such a place in the plot where a decrease of eigenvalues left to right is maximally slow. It is supposed that to the right from this point only “factorial slide rocks” are located. In accordance with this criterion only 2 or 3 factors may be left. As seen from the above presented data it is sufficient to use three first prin- cipal components (the eigenvalues corresponding to them are indicated in red) to represent the data variability higher than 74 %. Fig. 1. Defining principal components by using “slide rocks” criterion 5,0656 1,3315 1,0651 0,8481 0,5083 0,4811 0,3091 0,2162 0,0996 0,0754 1 2 3 4 5 6 7 8 9 1 0 0 ,0 0 ,5 1 ,0 1 ,5 2 ,0 2 ,5 3 ,0 3 ,5 4 ,0 4 ,5 5 ,0 Number of Eigenvalues V al ue Principal component analysis for studying the world security problem Системні дослідження та інформаційні технології, 2009, № 4 35 Definition of factor loadings. Now let us analyze principal components and consider solving a problem with three factors. For this we consider correlations between threats and factors (or “new” variables) which are calculated by the for- mula [7]: X xX x X xX x r N i ll i N i kk i ll i N i k i lk, ∑∑ ∑ == = −− −− = 1 2 1 2 1 1 )()( )()( , (4) where lk,r — correlation coefficient of parameters lX and kX ; kXX ,1 — average values of parameters lX and kX ; N x X N i l il ∑ == 1 ; N x X N i k ik ∑ == 1 . The correlation coefficient itself does not have informal interpretation. How- ever, its square called the coefficient of determination shows to what extent varia- tions of dependent characteristics may be explained by variations of an independ- ent one. It is thought that correlation coefficients which by their module are more that 0.7 indicate a strong connection (in this case coefficients of determination > 50%, i.e. one characterististics determines the other more than by half. Correlation coefficients which by their module are less that 0.7, but more than 0.5 indicate that connection is average (in this case the coefficients of determination are less than 50%, but more than 25%). At last, correlation coefficients which by their module are less than 0.5 indicate a weak connection (here the coefficients of de- termination are less than 25 %). Table 2 shows the values of correlation coeffi- cients between principal factors and initial threats. The coefficients corresponding to strong connections are indicated in red. From Table 2 it is seen that the first factor to greater extent correlates with threats than the second and third factors. It should be expected, since, as it has been mentioned above, factors are defined sequentially and contain less and less total variance. T a b l e 2 . Correlation coefficients between principal factors and initial threats Variable Factor 1 Factor 2 Factor 3 ES 0,208964 0,817502 0,342974 FB – 0,855800 0,412124 0,053021 GINI – 0,355499 0,105301 – 0,716591 CP – 0,856876 0,248258 – 0,003646 NA – 0,809616 – 0,315140 0,210144 GW 0,723432 – 0,392527 – 0,006533 CM – 0,844045 – 0,267343 – 0,024123 ND – 0,326707 – 0,285766 0,615743 SF – 0,899250 – 0,086816 – 0,005283 GD – 0,788874 – 0,080839 – 0,084617 Expl. Var 5,065629 1,331475 1,065071 Prp. Totl 0,506563 0,133147 0,106507 Interpretation of factor structure. It is convenient to carry out interpreta- tion of factors (principal components) by using a diagram where threats are shown as vectors the coordinates of which correspond to factor loadings (Fig. 2). T. Pomerantseva, A. Boldak ISSN 1681–6048 System Research & Information Technologies, 2009, № 4 36 In accordance with maximum factor loadings threats may be divided into three categories (red, blue and green coulours). The first group of threats includes: FB, CP, SF, GD, NA, CM, GW. As seen in fig. 2 these threats are in the plane of the first and second factors. It means that for more detail analysis it is advisable to show them in the projection on this plane (Fig. 3). As seen from Fig. 3 the pairs of vectors SF-GD, FB-GW are practically colinear, which indicates their high degree of dependence. It is interesting that we study only two factors, then the pair of vectors CP-GINI may be considered as colinear. It should be also noted that the vector ES is orthogonial to FB (GW). GW ES ND GINI GD NA FB CP SF CM Fig. 2. Interpretation of threats in coordinates of principal components Fig. 3. Projection of threats on the plane of the first and second factors p p ES FB GINI CP NA GW CM ND SF GD -1 ,0 -0,8 -0,6 -0,4 -0,2 0 ,0 0 ,2 0 ,4 0 ,6 0 ,8 -0 ,6 -0 ,4 -0 ,2 0 ,0 0 ,2 0 ,4 0 ,6 0 ,8 1 ,0 Fa ct or 2 Factor 1 Principal component analysis for studying the world security problem Системні дослідження та інформаційні технології, 2009, № 4 37 It means that: • between level of energy security (ES), balance of biological capacity of the Earth and people’s needs (FB) and CO2 emissions(GW) the dependence is inconsiderable; • balance between biological capacity of the Earth and people’s needs(FB) and CO2 emissions (GW) has negative correlation; • level of state fragility (SF)) is closely connected with level of global dis- eases vulnerability(GD); • corruption perception index (CP) is closely connected with level inequal- ity between people and countries (GINI) in the context determined by the first and second factors. The most significant global threats are defined by using factor loadings of the initial list of threats. For this it is necessary to select such factors which have maximum loading by absolute value on the first, second and third factors. This choice ensured the definition of maximum impact of initial threats under condi- tion of their maximum independence on the aggregated indicator (Minkovsky norm) of these threats (Fig. 4). In accordance with the indicated approach such threats are SF, ES, GINI, (Fig. 4) i.e. the most significant threats in descending order are state fragility, global decrease of energy security and growing inequality between people and countries. Clustering of countries by the level of global threats and the correspond- ing graphic interpretation is done in the plane of the first and second factors. For this purpose we cluster countries by the degree of their remoteness from threats (Minkovsky norm) using the clustering method of K-averages. As seen from Fig. 5 the isolines which assign the Minlovsky norm approxi- mation are practically orthogonal to the first factor axis. It gives the ground to state that the first factor values mostly determine the countries’ remoteness from global threats (Fig. 5). 0,209 0,8558 0,3555 0,8569 0,8096 0,7234 0,844 0,3267 0,8993 0,7889 0,8175 0,4121 0,1053 0,2483 0,3151 0,3925 0,26730,2858 0,08680,0808 0,343 0,053 0,7166 0,0036 0,2101 0,00650,0241 0,6157 0,0053 0,0846 ES FB GINI CP NA GW CM ND SF GD 11 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 Fig. 4. Definition of most significant global threats T. Pomerantseva, A. Boldak ISSN 1681–6048 System Research & Information Technologies, 2009, № 4 38 C a n a d a S w e d e n N o rw a y A u s tr a lia F in la n d N e w Z e a la n d D e n m a rk S w itz e rl a n d N e th e rl a n d s A u s tr ia L u xe m b o u rg Ja p a n Ir e la n d F ra n ce G e rm a n y P o rt u g a l S lo ve n ia B e lg iu m U ru g u a y U n ite d K in g d o m S p a in C o s ta -R ic a Ita ly H u n g a ry L a tv ia G re e ce C h ile S lo va ki a L ith u a n ia P o la n d C ze ch R e p u b lic U S A B u lg a ri a A lb a n ia E s to n ia C ro a tia A rg e n tin a B e la ru s Is ra e l T h a ila n d M e xi co Ja m a ic a Jo rd a n M a la ys ia T u n is ia P a n a m a B o s n ia a n d H e rz e g o vi n a V ie tn a m B ra zi l U kr a in e C o lu m b ia K o re a R e p u b lic E l S a lv a d o r M o ld o va R o m a n ia S ri -L a n ka T u rk e y E cu a d o r E g yp t G u a te m a la D o m in ic a n R e p u b lic N a m ib ia R u s s ia M o ro cc o P e ru P h ili p p in e s Ir a n In d o n e s ia G e o rg ia N ic a ra g u a T ri n id a d a n d T o b a g o B o ts w a n a C h in a P a ki s ta n In d ia K yr g yz s ta nN e p a l V e n e zu e la A lg e ri a B o liv ia A rm e n ia M o n g o lia H o n d u ra s A ze rb a ija n U zb e ki s ta n G h a n a S e n e g a l B a n g la d e s h K a za kh s ta n T a n za n ia T a jik is ta n Iv o ry C o a s t Y e m e nB e n in S o u th A fr ic a K e n ya Z im b a b w e C a m e ro o n C a m b o d ia Z a m b ia H a iti T u rk m e n is ta n N ig e ri a E th io p ia M o za m b iq u e -2 ,5 -2 ,0 -1 ,5 -1 ,0 -0 ,5 0 ,0 0 ,5 1 ,0 1 ,5 2 ,0 2 ,5 F a c to r 1 -2-1012345 Factor 2 C lu s te r 1 2 3 4 5 5 Fi g. 5 . I nt er pr et at io n of g lo ba l t hr ea ts in th e pl an e o f t he fi rs t a nd se co nd fa ct or s Principal component analysis for studying the world security problem Системні дослідження та інформаційні технології, 2009, № 4 39 RESEARCHING THE DEPENDENCE OF COUNTRIES’ NATIONAL SECURITY ON PARTICULAR THREATS BY USING MODIFIED METHOD OF WEIGHTED LOCAL CORRELATION Let us consider that the quantitative value of Minkovsky norm for this or that country is an estimate of its national security level. We define the level of Minkovsky norm dependence on initial threats by calculating the corresponding correlation coefficients (Table 3). T a b l e 3 . Correlation coefficients between Minkovsky norm and global threats Varuable ES FB GINI CP NA GW CM ND SF GD Minkovsky norm – 0,16 0,80 0,31 0,82 0,83 – 0,54 0,83 0,40 0,89 0,78 The calculated correlation coefficients show a high degree of dependence of Minkovsky norm on initial threats, but at the same time do not answer the ques- tion what risks the countries are running from the point of view of their approach- ing various threats. The reason is the averaging of correlation coefficients on the entire data sample. For detailed analysis of global threats the countries may face, it is necessary to localize the sample on which correlation is estimated. It is natural to assume that this sample should include “alike” countries the degree of similarity of which may be estimated as, for example, a Euclidean distance in the space of threats. The second assumption is connected with the idea that the closer is a country to the point in which the correlation is analyzed; the higher is the degree of the coun- try’s indicators impact on the correlation coefficient. In accordance with the above assumptions we define the weighted mean [8] as: ∑ ∑ = i i i ii w xw WXm ),( , (5) where X — data sample; W — weighted function. If we define W , as function depending on distance, for example, ),(),( txdetxW λ−= , (6) in which: ),( txd — distance between points nRtx ∈, , and λ — distribution parameter and substitute in (5), then we get the expression for calculating the weighted localized mean in point t for sample X : Xx e xe tXm i i xtd i i xtd i i ∈= ∑ ∑ − − ,),( ),( ),( λ λ . (7) Similarly, we can define the weighted localized covariation: T. Pomerantseva, A. Boldak ISSN 1681–6048 System Research & Information Technologies, 2009, № 4 40 ∑ ∑ − − −− = i xtd i ii xtd i i e tYmytXmxe tYX ),( ),( )),())(,(( ),,(cov λ λ . (8) And we define the weighted localized correlation (WLC): ),,(cov),,(cov ),,(cov),,(corr tYYtXX tYXtYX = . (9) The distribution parameter of weights λ may be chosen in such a way that it is possible to restrict the impact area of point’s located at large distances. For ex- ample, we assume that points located at mean distance from the point where WLC is calculated have the weight equal S (distribution scale). I.e. se tdmt =− )()(λ , then )( )ln()( tdm st =λ , (10) where )( tdm — mean distance from the sample points to point t . Examples of weights distribution for different values of mean distance and distribution scale are given in Figs. 6, 7. With distribution scale equal 1, WLC coincides with Pearson product- moment correlation coefficient. As seen from (10), the weights distribution pa- rameter is calculated for each point t , which is a sample point. And for each new point the mean distance value is calculated )( tdm anew. Hence, the suggested method of estimating threats local dependence is adaptive. The interpretation of WLC values is presented in Table 4. 0,000000 0,100000 0,200000 0,300000 0,400000 0,500000 0,600000 0,700000 0,800000 0,900000 1,000000 -0,1 0,0 0,1 0,2 0,3 0,4 0,5 0,6 0,7 0,8 0,9 1,0 1,1 s=0,1 s=0,01 s=0,001 1 — 2 — 3 — 1 2 3 Fig. 6. Weights distribution for mean distance equal 0,5 Principal component analysis for studying the world security problem Системні дослідження та інформаційні технології, 2009, № 4 41 T a b l e 4 . Interpretation of values of weighted localized correlation (WLC) Value of WLC Behavior of global threats under study Interpretation [– 1.0, – 0.5) High degree of negative corre- lation (more than 25 %). The growth of one threat is connected with reduction of the other With a decrease of a particular threat the general remoteness from the totality of global threats considerably decreases. The studied threat has low (as compared to oth- ers) contribution to the general remoteness from global threats [– 0.5, – 0.3) Mean degree of negative cor- relation (9–25%). The growth of one threat is connected with reduction of the other With a decrease of a particular threat the general remoteness from the totality of global threats considerably decreases at the mean degree [– 0.3, 0.3] Low degree of correlation (less than 9%) It is possible to speak about an inconsiderable dependence of the degree of remoteness from the totality of global threats on the studied threat (0.3, 0.5] Mean degree of positive corre- lation (9 – 25 %). The growth of one threat is connected with the growth of other With a decrease of the particular threat the gen- eral remoteness from global threats increases at the mean degree (0.5, 1.0] High degree of positive corre- lation. The growth of threat is con- nected with the growth of other (more than by 25%) With a decrease of the particular threat the gen- eral remoteness from the totality of global threats considerably increases. The studied threat considerably influences the general remoteness from the totality of global threats Figs. 8–10 present the plotted values of weighted localized correlation (WLC) between Minkovsky norm and most significant threats, respectively: SF, ES и GINI. Fig. 7. Weights characteristics for scale distribution equal 0,1 0,000000 0,100000 0,200000 0,300000 0,400000 0,500000 0,600000 0,700000 0,800000 0,900000 1,000000 -0,2 0,0 0,2 0,4 0,6 0,8 1,0 1,2 mean(d)=0.2 mean(d)=0.5 mean(d)=0.7 3 2 1 1 — 2 — 3 — T. Pomerantseva, A. Boldak ISSN 1681–6048 System Research & Information Technologies, 2009, № 4 42 0, 0 0, 2 0, 4 0, 6 0, 8 1, 0 1, 2 C lu st er 1 C lu st er 2 C lu st er 3 C lu st er 4 C lu st er 5 Canada Norway Finland Denmark Netherlands Luxembourg Ireland Germany Slovenia Uruguay Spain Italy Latvia Chile Lithuania Czech Republic Bulgaria Estonia Argentina Israel Mexico Jordan Tunisia nia and Herzegovina Brazil Columbia El Salvador Romania Turkey Egypt Dominican Republic Russia Peru Iran Georgia Trinidad and Tobago China India Nepal Algeria Armenia Honduras Uzbekistan Senegal Kazakhstan Tajikistan Yemen South Africa Zimbabwe Cambodia Haiti Nigeria Mozambique 6 Bosnia and Herzegovina M in ko vs ky n or m S F W LC b et w ee n M in ko vs ky n or m - S F W LC m ea n = .5 72 68 39 80 17 04 81 W LC m ea n - V ar ia nc e W LC m ea n + Va ria nc e H ig he st N eg at ive C or re la tio n L ow es t N eg at iv e C or re la tio n L ow es t P os iti ve C or re la tio n H ig he st P os itiv e C or re la tio n Fi g. 8 . V al ue s o f W LC b et w ee n M in ko vs ky n or m a nd st at e fr ag ili ty (S F) Principal component analysis for studying the world security problem Системні дослідження та інформаційні технології, 2009, № 4 43 Canada Norway Finland Denmark Netherlands Luxembourg Ireland Germany Slovenia Uruguay Spain Italy Latvia Chile Lithuania Czech Republic Bulgaria Estonia Argentina Israel Mexico Jordan Tunisia nia and Herzegovina Brazil Columbia El Salvador Romania Turkey Egypt Dominican Republic Russia Peru Iran Georgia Trinidad and Tobago China India Nepal Algeria Armenia Honduras Uzbekistan Senegal Kazakhstan Tajikistan Yemen South Africa Zimbabwe Cambodia Haiti Nigeria Mozambique 6 Bosnia and Herzegovina Fi g. 9 . V al ue s o f W LC b et w ee n M in ko vs ky n or m a nd e ne rg y se cu rit y (E S) T. Pomerantseva, A. Boldak ISSN 1681–6048 System Research & Information Technologies, 2009, № 4 44 Fi g. 9 . V al ue s o f W LC b et w ee n M in ko vs ky n or m a nd e ne rg y se cu rit y (E S) Canada Norway Finland Denmark Netherlands Luxembourg Ireland Germany Slovenia Uruguay Spain Italy Latvia Chile Lithuania Czech Republic Bulgaria Estonia Argentina Israel Mexico Jordan Tunisia nia and Herzegovina Brazil Columbia El Salvador Romania Turkey Egypt Dominican Republic Russia Peru Iran Georgia Trinidad and Tobago China India Nepal Algeria Armenia Honduras Uzbekistan Senegal Kazakhstan Tajikistan Yemen South Africa Zimbabwe Cambodia Haiti Nigeria Mozambique 6 Bosnia and Herzegovina Fi g. 1 0. V al ue s o f W LC b et w ee n M in ko vs ky n or m a nd p op ul at io n in eq ua lit y (G in i) Principal component analysis for studying the world security problem Системні дослідження та інформаційні технології, 2009, № 4 45 As seen from Fig. 8 the level of state fragility (SF) for most countries has considerable impact on their national security. As to the impact of energy security on the level of national security (Fig. 9), the following groups of countries may be identified [9]: • A group of countries with high level of ES and high values of Minkovsky norm (Canada, Sweden, Norway, Australia) for which energy security considera- bly influences their national security. • An adjacent group (Finland, New Zealand, Denmark, Switzerland, Neth- erlands, Austria, Luxembourg, Japan), for which a mean level of dependence be- tween energy security and Minkovsky norm is observed. • A group of countries for which this dependence is weak. • A group of countries with mean level of national security (Belarus, Israel, Thailand, Mexico, Jamaica, Jordan, Malaysia, Tunisia, Panama, Bosnia and Her- zegovina, Vietnam, Brazil, Ukraine, Columbia, Korea Republic), for which there exist threats more serious than energy security. • A group of countries with low level of national security (Kenya, Zim- babwe, Cameroon, Cambodia, Zambia, Haiti, Turkmenistan, Nigeria), for which both energy security and other threats are equally important. • A group of most problem countries (Ethiopia, Mozambique), where the level of energy security at least extent determines the level of national security. As to the impact of population inequality on national security (fig.10) it is possible to identify a group of countries (Canada, Sweden, Norway, Australia, Finland, New Zealand, Denmark, Switzerland, Netherlands, Austria, Luxem- bourg, Japan, Ireland, France, Germany, Portugal, Slovenia, Belgium), for which a mean positive correlation between this threat and Minkovsky norm is observed. For the rest of countries this correlation is insignificant. CONCLUSIONS 1. Since it is very complicated to analyze security of this or that country si- multaneously in the space of ten global threats the principal component analysis (PCA) was used. This method allowed reducing ten global threats influencing the general level of national security (in the sense of Minkovsky norm) to three hidden factors determining this characteristic. The application of this approach allowed considerably facilitate research of national security, reducing it to the analysis in the space of three determining factors. 2. By using this method a comprehensive study of national security of different countries was carried out in the space of three determining factors. Factor loadings were defined by calculating coefficients of correlation between principal factors and initial threats. Clustering of countries was made according to the level of global threats, and three most significant threats were defined influencing national security of most countries: state fragility (SF), energy security (ES) and people’s inequality (Gini). Graphic interpretation of global threats was done in the space of three principal components. The factor structure of threats was studied, and the degrees of dependence between main groups were defined. T. Pomerantseva, A. Boldak ISSN 1681–6048 System Research & Information Technologies, 2009, № 4 46 3. The method of weighted localized correlation was modified, which al- lowed carry out research of the dependence of national security level (Minkovsky norm) on particular global threats. By using this method the dependence between Minkovsky norm and most significant threats were analyzed in detail, in particular, state fragility (SF), energy security (ES) and people’s inequality (Gini). Recommendations were made for different countries regarding strengthening their national security. REFERENCES 1. Zgurovsky M.Z., Gvishiani A.G. Sustainable development global simulation. Report 2008. (http://wdc.org.ua/en/node/357) Kiev.:Polytechnika, 2008. — 363 p. 2. System Analysis and Decisions, The example of sustainable development global simulation, 2009, from Word Wide Web: http://systemdecisions.com/index.php? option=com_content&view=article&id=30&Itemid=27. 3. Lindsay I Smith. A tutorial on Principal Components Analysis, 2002, URL: http://www.cs.otago.ac.nz/cosc453/student_tutorials/principal_components. pdf. 4. Strang Gilbert. Linear algebra and its applications, Thomson, Brooks/Cole, Belmont, CA, ISBN 0-030-10567-6, 2006. 5. Jambu M. Exploratory and multivariate data analysis. Academic Press., 1991. 6. Cattel R.B. The screen test for the number of factors. Multivariate Behavioral Re- search, 1, 245–276, 1966. 7. Harman H.H. Modern factor analysis. Chicago: University of Chicago Press, 1966. 8. A MATLAB Toolbox for computing Weighted Correlation Coefficients, 2008, http://www.mathworks.com/matlabcentral/fileexchange/20846. 9. Pomerantseva T., Boldak A. Human Security Analysis Under Impact of the Totality of Global Threats on Sustainability. 5-th International EURO-Mini Conference “Knowledge-Based Technologies and OR Methodologies for Strategic Decisions of Sustainable Development” (KORSD-2009), Vilnius, Lithuania, September 30 – October 3, 2009. — P. 164–169. Received 09.07.2009 From the Editorial Board: the article corresponds completely to submitted manuscript.

Principal component analysis for studying the world security problem

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