EKONOMETRIA ECONOMETRICS 2(52) 2016 ISSN e-issn

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1 EKONOMETRIA ECONOMETRICS (5) 06 ISSN e-issn Marek Walesak Wrocław Unversty of Economcs e-mal: VISUALIZATION OF LINEAR ORDERING RESULTS FOR METRIC DATA WITH THE APPLICATION OF MULTIDIMENSIONAL SCALING WIZUALIZACJA WYNIKÓW PORZĄDKOWANIA LINIOWEGO DLA DANYCH METRYCZNYCH Z WYKORZYSTANIEM SKALOWANIA WIELOWYMIAROWEGO DOI: 0.56/ekt JEL Classfcaton: C38, C430 Summary: The artcle dscusses the two-step research procedure allowng the vsualzaton of lnear orderng results for metrc data. In the frst step, as a result of the applcaton of multdmensonal scalng (see [Borg, Groenen 005; Mar et al. 06]) the vsualzaton of obects n two-dmensonal space s obtaned. In the next step, the lnear orderng of a set of obects s carred out based on the Eucldean dstance from the pattern (deal) obect. The suggested approach expanded the possbltes for the nterpretaton of the lnear orderng results of a set of obects. The artcle apples the concept of soquants and the path of development (the shortest way connectng a pattern and an ant-pattern obect) proposed by [Hellwg 98]. The graphcal presentaton of the lnear orderng results based on ths concept was possble for two varables only. The applcaton of multdmensonal scalng expanded the applcablty of the results of lnear orderng vsualzaton for m varables. The suggested approach s llustrated by an emprcal example wth the applcaton of R envronment scrpt. Keywords: lnear orderng, multdmensonal scalng, dstance measures, composte measures, R envronment. Streszczene: W artykule zaproponowano dwukrokową procedurę badawczą pozwalaącą na wzualzacę wynków porządkowana lnowego. W perwszym kroku w wynku zastosowana skalowana welowymarowego (zob. [Borg, Groenen, 005; Mar n. 06]) otrzymue sę wzualzacę obektów w przestrzen dwuwymarowe. W następnym kroku przeprowadza sę porządkowane lnowe zboru obektów na podstawe odległośc Eukldesa od wzorca rozwou. Zaproponowane podeśce rozszerzyło możlwośc nterpretacyne wynków porządkowana lnowego zboru obektów. W artykule wykorzystano koncepcę zokwant śceżk rozwou (os zboru nakrótsze drog łączące wzorzec antywzorzec rozwou) zaproponowaną w pracy [Hellwg 98]. Grafczna prezentaca wynków porządkowana lnowego w te koncepc możlwa była dla dwóch zmennych. Zastosowane ska-

2 0 Marek Walesak lowana welowymarowego rozszerzyło możlwośc zastosowana wzualzac wynków porządkowana lnowego dla m zmennych. Zaproponowane podeśce zlustrowano przykładem emprycznym z zastosowanem skryptu przygotowanego w środowsku R. Słowa kluczowe: porządkowane lnowe, skalowane welowymarowe, mary odległośc, mary agregatowe, program R.. Introducton The artcle presents the proposal of the applcaton of multdmensonal scalng [Borg, Groenen 005] n lnear orderng of a set of obects based on the pattern of development [Hellwg 968]. A two-step research procedure was suggested, whch allows the vsualzaton of lnear orderng results for metrc data. Frst, followng the applcaton of multdmensonal scalng, the vsualzaton of obects n twodmensonal space s obtaned. Next, the lnear orderng of a set of obects s carred out based on the Eucldean dstance from the pattern of development. The suggested approach s llustrated by an emprcal example. The artcle apples the concept of soquant and the path of development (the axs of the set the shortest way connectng a pattern and an ant-pattern obect ) proposed by [Hellwg 98]. The graphcal presentaton of the lnear orderng results, based on ths concept, was possble for two varables. The applcaton of multdmensonal scalng expanded the applcablty of lnear orderng vsualzaton results for m varables.. The geness of the concept of the pattern of development and measure of development The frst research paper dscussng the concept of the pattern of development and the measure of development n Englsh was presented by Professor Zdzsław Hellwg at the UNESCO conference n Warsaw n 967 [Hellwg 967]. The study was publshed n Englsh n a monograph edted by Z. Gostkowsk [Hellwg 97]. The frst artcle analyzng the pattern of development and the measure of development n Polsh was publshed n the ournal Przegląd Statystyczny (The Statstcal Revew) n 968 [Hellwg 968]. These studes ntroduced the followng terms: stmulants and destmulants, pattern of development, measure of development (dstance from the pattern of development). There are two types of pattern obects: a pattern obect (upper pattern obect, deal obect, upper pole) and an ant-pattern obect (lower pattern obect, ant-deal obect, lower pole). The coordnates of a pattern obect cover the most preferred preference varable (stmulants, destmulants, nomnants) values. The coordnates of an ant-pattern obect cover the least preferred preference varable values.

3 Vsualzaton of lnear orderng results for metrc data wth the applcaton... It can be stated, wthout any exaggeraton, that Hellwg s dea ntated an avalanche of proposals for the development of lnear orderng methods. These modfcatons amed at (see [Borys, Strahl, Walesak 990; Pocecha, Zaąc 990]): a) dfferentatng the method for the normalzaton of varable values, b) ntroducng nomnant varables n a set, c) determnng the pattern of development (comparatve base) n a dfferent way, d) applyng varous constructons of the composte measure, e) applyng fuzzy sets n the constructon of the composte measure. Recently the concepts usng fuzzy numbers were developed n lnear orderng based on the pattern of development (see e.g. [Chen 000; Wysock 00; Jefmańsk, Dudek 06]) and takng nto account spatal dependences (see [Antczak 03; Petrzak 04]) and nterval symbolc data (see [Młodak 04]). 3. Lnear orderng for metrc data based on a pattern obect general procedure The general procedure n lnear orderng of the set of obects based on a pattern obect (or an ant-pattern obect) and metrc data takes the followng form: P A X SDN T N d R, () where: P choce of a complex phenomenon the overrdng phenomenon for orderng A set obects, whch s not subect to drect measurement; A choce of obects; X selecton of varables. Collectng data and the constructon of data matrx [ x ] ( x the value of the -th varable on -th obect); SDN dentfyng preferental varables (stmulants, destmulants, nomnants). M varable s a stmulant (see [Hellwg 98, p. 48]), when for every two of S S S S ts observatons x, x k referrng to obects A, A k take x > xk A Ak ( means A obect domnaton over A k obect). M varable s a destmulant (see [Hellwg 98, p. 48]), when for every two of ts observatons D D D D x, x k referrng to obects A, A k take x > xk A Ak ( means A k obect domnaton over A obect). Therefore, M varable represents a unmodal nomnant (see [Borys 984, p. 8]), when for every two of ts observatons x, x referrng to obects A, A ( nom means the nomnal level N N of k -th varable): f x, x nom, then N N k w k N N k N N k k N N k k x > x A A ; f x, x > nom, then x > x A A, T w transformaton of nomnants nto stmulants (requred for an ant-pattern obect only). Transformaton formulas can be found for example n the study by [Walesak 0, p. 8]; N normalzaton of varable values. The revew of methods for the normal-

4 Marek Walesak zaton of varable values s presented n the study [Walesak 04a]; d aggregated measure (composte measure) calculaton for -th obect the applcaton of dstance measures from a pattern obect usng weghts; R orderng of obects n accordance wth d value. Table. Selected dstance measures from a pattern obect for metrc data Name Dstance d Interval Measure of development I [Hellwg 968] Measure of development d d sd d m α z z = II [Hellwg 98] ( ) TOPSIS measure [Hwang, Yoon 98] GDM dstance [Walesak, 00] GDM_TOPSIS TOPSIS measure wth GDM dstance [Walesak, 04b] d d d GDM = m m n α ( z zw )( zw z ) α ( z zl )( zw zl ) = = l= l w, m n m n α ( z zl ) α ( zw zl ) = l= = l= GDM GDM GDM ( ;] [0; ] [0; ] [0; ] [0; ], l =,..., n obect number, =,..., m varable number, z ( zl, z w ) the normalzed value of -th varable for the -th (l-th, w-th) obect, z ( z ) the normalzed -th coordnate of pattern obect m (ant-pattern obect), d ( ) = α z z weghted Eucldean dstance between -th obect and = m pattern obect, d ( ) = α z z = weghted Eucldean dstance between -th obect and ant- n n pattern obect, d = d =, sd = ( d d ) n n, α weght of -th varable ( α [0; ] = m m and α = or α [0; m] and = α = m ), GDM ( GDM ) GDM dstance between = -th obect and ant-pattern obect (pattern obect), z ( z w = z ) for GDM ( GDM w = z ). Source: author s complaton.

5 Vsualzaton of lnear orderng results for metrc data wth the applcaton... 3 Table presents the chosen dstance measures from the pattern of development characterzed by the normalzed varablty nterval. The subect lterature also offers other dstance measures from the pattern (deal) obect (see e.g. [Grabńsk 984; Pawełek 008]). 4. Research procedure allowng the vsualzaton of lnear orderng results for the set of obects for metrc data The research procedure allowng the vsualzaton of lnear orderng results for a set of obects covers the followng steps:. The choce of a complex phenomenon n lnear orderng whch s not subect to an mmedate measurement (e.g. the economc development level of countres worldwde, toursm attractveness level of countes).. Determnng the set of obects and the set of varables substantvely related to the analyzed complex phenomenon. The varables used to descrbe the obects are measured on metrc scale. Followng data collecton a data matrx s constructed [ x ] n xm ( x -th varable value for -th obect; =,, n obect number, =,, m varable number). 3. Among the varables the followng preferental varables are dstngushed: stmulants, destmulants, nomnants. Nomnants are transformed nto stmulants. 4. A pattern obect (upper pole of development) and an ant-pattern obect (lower pole of development) are added to the set of obects and result n a data matrx [ x ] nxm ( n= n ). 5. If the varables descrbng obects are measured on an nterval or rato scale, they should be comparable usng normalzaton (see [Walesak, 04a]) and receve a normalzed data matrx [ z ] nxm. 6. The dstances between obects are calculated and arranged n a dstance matrx [ δ k ]. The followng dstance measures can be appled n ths case (measures takng nto account weghts of varables) e.g. cty-block, Eucldean, GDM (see [Walesak 0, pp. 3 4]). Multdmensonal scalng: f : dk dk s carred out. Multdmensonal scalng s the method representng the dstance matrx between the obects n m-dmensonal space [ ] k δ nto the dstance matrx between the obects n q-dmensonal space [ ] (q < m) for the purposes of the graphcal vsualzaton of relatons occurrng between the analyzed obects and to specfy (nterpret) the content of q dmensons. The dmensons cannot be observed drectly. They represent latent type of varables, whch allow explanng the smlartes and dfferences between the analyzed obects. Due to the possblty for the graphcal presentaton of lnear orderng results, q equals. The teratve procedure n the smacof algorthm was presented n the study [Borg, d k

6 4 Marek Walesak Groenen 005, pp ]. Fnally, the data matrx n two-dmensonal space [ v ] nx s obtaned. 7. The graphcal presentaton and the nterpretaton of the results n a twodmensonal (multdmensonal scalng results) and one-dmensonal space (lnear orderng results): n the fgure a straght lne connects the ponts determnng a pattern and an antpattern obect n the so-called axs of the set n a two-dmensonal space (multdmensonal scalng results). Isoquants of development are determned based on a pattern obect, e.g. dvdng the set axs nto four parts allows determnng four soquants. The obects between soquants present a smlar development level. The same development level can be acheved by the obects placed n dfferent ponts on the same soquant of development (due to a dfferent confguraton of varable values). Such a presentaton of the results expands the nterpretaton of the lnear orderng results; normalzed d dstances of -th obect from the pattern of development are calculated n accordance wth the formula (cf. [Hellwg 98, p. 6]): d = ( v v ) = ( v v ) =, d [0; ], () where: ( v v ) Eucldean dstance between -th obect and pattern ob- = ect (deal pont co-ordnates), ( v v ) Eucldean dstance between pattern obect and ant- = pattern obect (ant-deal pont co-ordnates). The obects of the study are ordered by the growng values of dstance measure (). The lnear orderng results are graphcally presented n the fgure. 5. Emprcal results The emprcal study uses the statstcal data presented n the artcle [Gryszel, Walesak 04], referrng to the attractveness level of 9 Lower Slesan countes. The evaluaton of the tourstc attractveness of Lower Slesan countes was performed usng 6 metrc varables (measured on a rato scale):

7 Vsualzaton of lnear orderng results for metrc data wth the applcaton... 5 x beds n hotels per km of a county area, x number of nghts spent daly by resdent toursts (Poles) per 000 nhabtants of a county, x 3 number of nghts spent daly by foregn toursts per 000 nhabtants of a county, x 4 gas polluton emsson n tons per km of a county area, x 5 number of crmnal offences and crmes aganst lfe and health per 000 nhabtants of a county, x 6 number of property crmes per 000 nhabtants of a county, x 7 number of hstorcal buldngs per 00 km of a county area, x 8 % of a county forest cover, x 9 % share of legally protected areas wthn a county area, x 0 number of events as well as cultural and tourst ventures n a county, x number of natural monuments calculated per km of a county area, x number of tourst economc enttes per 000 nhabtants of a county (natural and legal persons), x 3 expendture of muncpaltes and countes on toursm, culture and natonal hertage protecton as well as physcal culture per nhabtant of a county n PLN, x 4 cnema goers per 000 nhabtants of a county, x 5 museum vstors per 000 nhabtants of a county, x 6 number of constructon permts (hotels and accommodaton buldngs, commercal and servce buldngs, transport and communcaton buldngs, cvl and water engneerng constructons) ssued n a county n 0-0 per km of a county area. The statstcal data were collected n 0 and come from the Local Data Bank of the Central Statstcal Offce of Poland, the data for x 7 varable only were obtaned from the Regonal Conservaton Offcer. R envronment scrpt was used n ths artcle and prepared n accordance wth the research procedure dscussed n secton 4, whch appled the followng methodology: x 4, x 5 and x 6 varables take the form of destmulants, x 9 s a nomnant (50% level was adopted as the optmal one). The other varables represent stmulants, whereas x 9 nomnant was transformed nto a stmulant. a pattern and an ant-pattern obect were added to the set of 9 countes. Therefore, the data matrx covers 3 obects descrbed by 6 varables. due to the fact that all varables are metrcal, the normalzaton of varable values was performed usng the followng method (see [Walesak 04a; Walesak, Dudek 06]):

8 6 Marek Walesak z = n x = med ( x ) med, (3) where: x ( z ) value (normalzed value) of -th varable for -th obect, med = med( x ) medan for -th varable. dstance matrx [ δ k ] between obects was calculated usng GDM, for whch the where: same weghts were used (see [Walesak 0, p. 47]): m m n a a b a a b k k l kl = = l= l k, m n m n a al a bkl = l= = l= δ, k = δ k GDM dstance measure, kl,, =,, n obect number, =,, m varable number, ap = z z p for p = k, l, b = z z for r =, l kr k r z ( zk, z l ) normalzed -th (k-th, l-th) observaton for -th varable, m α weght of -th varable ( α [0; m] and α = m or α [0; ] = m and α = ). = The multdmensonal scalng of 3 obects was carred out (9 Lower Slesan countes plus the pattern and the ant-pattern obect) n terms of tourst attractveness level usng the smacofsym functon of the smacof package [Mar et al. 06], as a result of whch the confguraton of 3 obects (ponts) n a twodmensonal space was obtaned. Fgure llustrates the graphcal presentaton of the multdmensonal scalng results for 3 obects. The pattern obect (30) and the ant-pattern obect (3) were connected by a straght lne and the so-called axs of the set was obtaned. Four soquants of development were determned by dvdng the axs nto four equal parts. The dstances of each obect (county) from a pattern obect were calculated n accordance wth formula (). Countes were ordered by the growng measure values () and the next four classes of smlar countes, regardng ther tourst attractveness, were dstngushed. The orderng of 3 obects referrng to 9 countes, (4)

9 Vsualzaton of lnear orderng results for metrc data wth the applcaton... 7 the pattern obect (30) and the ant-pattern obect (3) regardng tourst attractveness, presented by the growng measure values () are as follows (see Table ). Fgure. Graphcal presentaton of multdmensonal scalng results n a two-dmensonal space of 3 obects contanng 9 countes, pattern obect (30) and ant-pattern obect (3) referrng to the Lower Slesan countes tourst attractveness Source: author s complaton usng R program. Table. The orderng of 3 obects regardng tourst attractveness Obect Name Dstance 3 30 Pattern Jelenogórsk Kłodzk Jelena Góra Wrocław

10 8 Marek Walesak Table, cont. 3 7 Wałbrzysk Śwdnck Polkowck Legnca Bolesławeck Lubańsk Oleśnck Lubńsk Trzebnck Ząbkowck Lwóweck Jaworsk Dzerżonowsk Mlck Górowsk Legnck Głogowsk Wołowsk Kamennogórsk Zgorzeleck Strzelńsk Oławsk Średzk Wrocławsk Złotorysk Ant-pattern Source: author s complaton usng R program. The graphcal results of the lnear orderng for 3 obects coverng 9 countes, the pattern obect (30) and the ant-pattern obect (3), n terms of tourst attractveness, presented by the growng measure values () are presented n Fgure. Such a form of presentng the results allows for: the presentaton of countes orderng n terms of ther tourst attractveness n accordance wth measure () values and n the form of graphcal presentaton n Fgure,

11 Vsualzaton of lnear orderng results for metrc data wth the applcaton... 9 Fgure. Graphcal presentaton of lnear orderng of 3 obects contanng 9 countes, pattern obect (30) and ant-pattern obect (3) referrng to the Lower Slesan countes tourst attractveness by the growng measure values () Source: author s complaton usng R program. dstngushng the classes of countes (countes between soquants) presentng the smlar level of tourst attractveness (see Fgure ), dentfyng countes characterzed by a smlar level of tourst attractveness, but dfferent regardng ther locaton on the soquant of development (see Fgure ). For example, Zgorzeleck County (7) and Strzelńsk County () have a smlar level of tourstc attractveness, but a dfferent locaton on the soquant of development. A smlar stuaton occurs for Polkowck County (3) and Legnca County (8). Therefore these countes acheved a smlar level of development, however they are characterzed by qute dfferent confguratons of varable values. 6. Fnal remarks The artcle presents the proposal of a research procedure allowng the vsualzaton of lnear orderng results for the set of obects by applyng multdmensonal scalng for ths purpose.

12 0 Marek Walesak The concept of soquants and the path of development suggested n the study [Hellwg 98], allow for the graphcal presentaton of lnear orderng results for two varables only. The applcaton of multdmensonal scalng extends the possbltes of vsualzng lnear orderng results for m varables. Followng such a soluton, the nterpretaton of lnear orderng results was expanded. The proposed approach was llustrated by an emprcal example usng R envronment scrpt [R Development Core Team 06]. It should be borne n mnd that the applcaton of multdmensonal scalng results n a partal loss of nformaton about the obects. A set of obects s ntally presented n the space of m varables. As a result of multdmensonal scalng applcaton, the graphcal presentaton of obects n a two-dmensonal space s obtaned. In the smacofsym functon of the smacof package STRESS- Kruskal s ft measure s used [Borg, Groenen 005, pp ]. Bblography Antczak E., 03, Przestrzenny taksonomczny mernk rozwou [Spatal taxonomc measure of development], Wadomośc Statystyczne, 7, pp Borg I., Groenen P.J.F., 005, Modern Multdmensonal Scalng. Theory and Applcatons, nd Edton, Sprnger ScenceBusness Meda, New York. Borys T., 984, Kategora akośc w statystyczne analze porównawcze [Category of Qualty n Statstcal Comparatve Analyss], Prace Naukowe Akadem Ekonomczne we Wrocławu, nr 84, Sera: Monografe Opracowana nr 3, Wydawnctwo Akadem Ekonomczne we Wrocławu, Wrocław. Borys T., Strahl D., Walesak M., 990, Wkład ośrodka wrocławskego w rozwó teor zastosowań metod taksonomcznych [The contrbuton of the center of Wroclaw n the development of the theory and applcaton of taxonomc methods], [n:] Pocecha J. (red.), Taksonoma teora zastosowana, Wydawnctwo Akadem Ekonomczne w Krakowe, Kraków, pp. 3. Chen C.T., 000, Extensons of the TOPSIS for group decson-makng under fuzzy envronment, Fuzzy Sets and Systems, 4(), pp. 9. Grabńsk T., 984, Welowymarowa analza porównawcza w badanach dynamk zawsk ekonomcznych [Multvarate Comparatve Analyss n Research Over the Dynamcs of Economc Phenomena], Zeszyty Naukowe Akadem Ekonomczne w Krakowe, Sera specalna: Monografe nr 6, Wydawnctwo Akadem Ekonomczne w Krakowe, Kraków. Gryszel P., Walesak M., 04, Zastosowane uogólnone mary odległośc GDM w ocene atrakcynośc turystyczne powatów Dolnego Śląska [The applcaton of the general dstance measure (GDM) n the evaluaton of Lower Slesan dstrcts attractveness], Fola Turstca, nr 3, pp Hellwg Z., 967, Procedure of Evaluatng Hgh-Level Manpower Data and Typology of Countres by Means of the Taxonomc Method, COM/WS/9, Warsaw, 9 December, 967 (unpublshed UNESCO workng paper). Hellwg Z., 968, Zastosowane metody taksonomczne do typologcznego podzału kraów ze względu na pozom ch rozwou strukturę wykwalfkowanych kadr [Procedure of evaluatng hgh level manpower data and typology of countres by means of the taxonomc method], Przegląd Statystyczny, tom 5, z. 4, pp Hellwg Z., 97, Procedure of Evaluatng Hgh-Level Manpower Data and Typology of Countres by Means of the Taxonomc Method, [n:] Gostkowsk Z. (ed.), Towards a system of Human Re-

13 Vsualzaton of lnear orderng results for metrc data wth the applcaton... sources Indcators for Less Developed Countres, Papers Prepared for UNESCO Research Proect, Ossolneum, The Polsh Academy of Scences Press, Wrocław, pp Hellwg Z., 98, Welowymarowa analza porównawcza e zastosowane w badanach welocechowych obektów gospodarczych [Multvarate Comparatve Analyss and Applcatons n Research of Multfeature Economc Obects], [n:] W. Welfe (ed.), Metody modele ekonomcznomatematyczne w doskonalenu zarządzana gospodarką socalstyczną, PWE, Warszawa, pp Hwang C.L., Yoon K., 98, Multple Attrbute Decson Makng Methods and Applcatons. A State-of-the-Art Survey, Sprnger-Verlag, New York. Jefmańsk B., Dudek A., 06, Syntetyczna mara rozwou Hellwga dla trókątnych lczb rozmytych [Hellwg s Measure of Development for Trangular Fuzzy Numbers], [n:] Appenzeller D. (ed.), Matematyka nformatyka na usługach ekonom. Wybrane problemy modelowana prognozowana zawsk gospodarczych, Wydawnctwo Unwersytetu Ekonomcznego w Poznanu, Poznań, pp Mar P., De Leeuw J., Borg I., Groenen P.J.F., 06, smacof: Multdmensonal Scalng. R package verson.8-3, URL Młodak A., 04, On the constructon of an aggregated measure of the development of nterval data, Computatonal Statstcs, vol. 9, October, Issue 5, pp Pawełek B., 008, Metody normalzac zmennych w badanach porównawczych złożonych zawsk ekonomcznych [Normalzaton of Varables Methods n Comparatve Research on Complex Economc Phenomena], Wydawnctwo Unwersytetu Ekonomcznego w Krakowe, Kraków. Petrzak M.B., 04, Taksonomczny mernk rozwou (TMR) z uwzględnenem zależnośc przestrzennych [Taxonomc measure of development (TMD) wth the ncluson of spatal dependence], Przegląd Statystyczny, tom 6, z., pp Pocecha J., Zaąc K., 990, Wkład ośrodka krakowskego w rozwó teor zastosowań metod taksonomcznych [The Contrbuton of the Center of Kraków n the Development of the Theory and Applcaton of Taxonomc Methods], [n:] Pocecha J. (ed.), Taksonoma teora zastosowana, Wydawnctwo Akadem Ekonomczne w Krakowe, Kraków, pp R Development Core Team, 06, R: A Language and Envronment for Statstcal Computng, R Foundaton for Statstcal Computng, Venna, URL Walesak M., 00, Propozyca uogólnone mary odległośc w statystyczne analze welowymarowe [The Proposal of the Generalsed Dstance Measure n Multvarate Statstcal Analyss], [n:] Paradysz J. (ed.), Statystyka regonalna w służbe samorządu lokalnego bznesu, Internetowa Ofcyna Wydawncza, Centrum Statystyk Regonalne, Akadema Ekonomczna w Poznanu, Poznań, pp. 5. Walesak M., 0, Uogólnona mara odległośc GDM w statystyczne analze welowymarowe z wykorzystanem programu R [The Generalzed Dstance Measure GDM n Multvarate Statstcal Analyss wth R], Wydawnctwo Unwersytetu Ekonomcznego we Wrocławu, Wrocław. Walesak M., 04a, Przegląd formuł normalzac wartośc zmennych oraz ch własnośc w statystyczne analze welowymarowe [Data normalzaton n multvarate data analyss. An overvew and propertes], Przegląd Statystyczny, tom 6, z. 4, pp Walesak M., 04b, Wzmacnane skal pomaru w statystyczne analze welowymarowe [Renforcng measurement scale for ordnal data n multvarate statstcal analyss], Taksonoma, Prace Naukowe Unwersytetu Ekonomcznego we Wrocławu, nr 37, pp Walesak M., Dudek A., 06, clustersm: Searchng for Optmal Clusterng Procedure for a Data Set. R package verson 0.45-, URL Wysock F., 00, Metody taksonomczne w rozpoznawanu typów ekonomcznych rolnctwa obszarów weskch [The Methods of Taxonomy Recognton of Economc Types n Agrculture and Rural Areas], Wydawnctwo Unwersytetu Przyrodnczego w Poznanu, Poznań.