Bootstrap Stability Evaluation and Validation of Clusters Based on Agricultural Indicators of EU Countries

Transcription

1 Economc Insghts Trends and Challenges Vol.V(LXVIII) No. / Bootstrap Stablty Evaluaton and Valdaton of Clusters Based on Agrcultural Indcators of EU Countres Crstan Marnou Petroleum-Gas Unversty of Ploeşt, Bd. Bucureşt 39, 00680, Ploeşt, Romana e-mal: Abstract In ths paper we propose a classfcaton of the European Unon countres, accordng to two ndcators: number of agrcultural holdngs (NAH) and utlzed agrcultural area (UAA). The results obtaned are mportant from the perspectve of emphaszng the smlartes and dfferences between European Unon countres n relaton to the specfed ndcators. Snce the clusterng methods usually used for dscoverng a structure n a data set have a tendency to "detect" clusters even n very homogeneous data set, a specal attenton was pad to the valdaton of the obtaned results. For ths reason, n order to valdate the cluster structure obtaned by applyng herarchcal agglomeratve clusterng, three technques were used to estmate the number of clusters, technques whch have led to the same result. In addton to ths, we estmated the stablty degree of the three clusters obtaned. Our results show that one of the clusters has a reasonable stablty and the other two have an exceptonally good stablty. Keywords: herarchcal agglomeratve clusterng; cluster stablty; bootstrap JEL Classfcaton: C38; Q0 Introducton Due to ts agrcultural exceptonal resources and to a common agrcultural polcy (European Commsson, 204) launched snce 962 and ncreasngly based on sustanable management, the European Unon supports today more than 500 mllon consumers and t s prepared to reman an mportant player n ensurng food securty of the whole world. Characterstcs of agrcultural holdngs n the European Unon countres are strongly nfluenced by clmatc factors, geologcal and geographcal factors and also by socal factors. Ths realty s reflected n Eurostat by several ndcators, among whch (Eurostat-Statstcs explaned, 206a): the sze of agrcultural holdngs, the farms labour force, lvestock unts and agrcultural land use. In ths paper we ntend to acheve a classfcaton of members countres of the European Unon based on two agrcultural ndcators, hghlghted by the (Eurostat-Statstcs explaned, 206b): number of agrcultural holdngs (NAH) as a percentage of the total number of EU agrcultural holdngs and utlzed agrcultural area (UAA) as a percentage of total utlzed agrcultural area n UE. The utlzed agrcultural area (UAA) descrbes the area used for farmng and refers to (Eurostat- Statstcs explaned, 206c): arable land, permanent grassland, and crops, other agrcultural land. The classfcaton obtaned n the followng paragraphs creates an overvew of the smlartes and dfferences between EU member states n terms of these two ndcators.

2 66 Crstan Marnou Classfcaton of EU Countres Usng Clusterng Technques The values of the two ndcators (NAH) and (UAA) for the year 203 are presented n Table. Table. Values of NAH and UAA ndcators for year 203 No. Country NAH UAA Crt. Austra Belgum Bulgara Croata Cyprus Czech Republc Denmark Estona Fnland France Germany Greece Hungary Ireland Italy Latva Lthuana Luxembourg Malta Netherlands Poland Portugal Romana Slovaka Slovena Span Sweden Unted Kngdom Source: Eurostat-Statstcs explaned, 206b, Farm_structure_YB206 In order to classfy EU countres accordng to values of NAH and UAA ndcators we appled the herarchcal agglomeratve clusterng method from (Theodords and Koutroumbas, 2009). The dstance between two objects (.e. countres) s the Eucldean dstance and the dstance between two clusters s Ward.D2 (a varant of Ward s method). Accordng to (Shalzy, 2009, p.3) Ward s method says that the dstance between two clusters A and B s how much the sum of squares wll ncrease when we merge them.. Mathematcally, n ths approach, the dstance between clusters A and B s defned by (Shalzy., 2009, p.3): Δ A B 2 ( A, B) = x ma B x ma x mb ma mb () A B A B na B n

3 Bootstrap Stablty Evaluaton and Valdaton of Clusters Based on Agrcultural Indcators of EU 67 where: x are tems that we want to place n clusters; m s the centrod of cluster ; n represents the number of objects from cluster. The dstance Δ ( A, B) between the clusters A and B s nterpreted as the mergng cost of combnng the clusters A and B (Shalzy., 2009, p.3). The prncple of Ward s method s to merge, at every step, the clusters A and B for whch ths cost s mnmal. By applyng the herarchcal agglomeratve clusterng method we obtaned the dendogram shown n Fgure. Fg.. Dendogram from agglomeratve herarchcal clusterng Source: made by the author n R, usng data from Table The Valdaton of the Clusters Structure The dendogram n Fgure suggests the exstence of a structure consstng of two or three clusters. In order to better determne the number of clusters, we addtonally used the suggestons offered by the plots n Fgures 2, 3 and 4. The graph n Fgure 2 (the so-called scree plot (Peeples, M., 20)) s a plot of the wthn group sum of squares dstances aganst the number of clusters. For each cluster ths dstance s computed between the object and the cluster centrod. For the obtaned structure the total sum of

4 68 Crstan Marnou squares of each cluster s nterpreted as a measure of overall error. In general, by representng graphcally the overall error value aganst to the number of clusters of the structure, we obtan a plot whch s smlar to that shown n Fgure 2,.e. a plot whch rapdly decreases to a pont from whch the decrease slows down. In ths pont the plot makes an elbow. The rght number of clusters s ndcated by the number of clusters whch correspond to ths elbow of the plot: a greater number of clusters than ths number does not produce a sgnfcant decrease n the value of global error. As the plot n Fgure 2 suggests ths number s 3. Fg. 2. The plot of the sum of squared error aganst number of clusters (scree plot) Source: made by the author n R, usng data from Table The second method used s based on gap statstc (Tbshran, Walther and Haste, 200). The Gap statstc plot from Fgure 3 s the plot of the functon values defned by the relaton (3), from (Tbshran, Walther and Haste, 200, p. 42), aganst the number of clusters k. Accordng to Tbshran, Walther and Haste (200, p. 45) the estmated number of clusters s the lowest k for whch Gap [ k] > Gap[ k + ] sk+, (2) where: Gap[k] s the value of the functon n pont k s s the standard error n pont k + k + In a less accurate nterpretaton, but more ntutve, the estmated number of clusters s the lowest k from whch the ncrease of the functon Gap slows near the pont ( k, Gap( k)). Because the vertcal bars n the Fgure 3 represent the values of the standard errors s k n the ponts k, we observe mmedately that the estmated number of clusters s 3.

5 Bootstrap Stablty Evaluaton and Valdaton of Clusters Based on Agrcultural Indcators of EU 69 Fg. 3. The gap statstc plot Source: made by the author n R usng functons defned by Chen (200) In order to evaluate the qualty of the structure based on three clusters t s also useful to plot the average slhouette wdth aganst number of clusters. The results are shown n Table 2. Table 2. Average slhouette wdths based on the number of clusters Number of clusters Average slhouette wdth Source: made by the author wth results obtaned n R We fnd that the maxmum of these values, named Slhouette Coeffcent (SC) s reached for k =3, whch renforces the dea that the number of clusters s estmated correctly (Stuyf, Hubert and Rousseeuw, p. 0). In Fgure 4 we show the slhouette plot of the structure found for the number of clusters k = 3. Fg. 4. Slhouette plot of the obtaned structure. Source: made by the author n R The obtaned average slhouette wdth (0.69) shows that a reasonable structure has been found (Stuyf, Hubert and Rousseeuw, p. 0).

6 70 Crstan Marnou The Stablty of the Clusters Structure In Henng s (2006, p.2) opnon stablty means that a meanngful vald cluster shouldn t dsappear easly f the data set s changed n a non essental way. Also n (2006, p.2), Henng ndcates more practcal modaltes n order to evaluate the stablty of the clusters. In ths paper we used hs algorthm, mplemented by the functon clusboot from R. In ths algorthm, the evaluaton of the clusters stablty s done by usng Jaccard s coeffcent. The change of ntal data n a non essental way s realzed by usng the bootstrap method (Efron and Tbshran, 993). In order to easly understand the algorthm, let us denote by x =,2, the lne vector whose components are gven by the ndcators values NAH and UAA from lne of Table. A * * * randomzed verson of the ntal data set, whch we denote by x, x 2,... x 28 s a random sample of sze 28 drawn wth replacement from the ntal populaton x, x2,... x28. The Jaccard s coeffcent of two sets A and B s defned n (Jaccard, 92) by: A B J ( A, B) =, f sets A and B are nonempty and by J ( A, B) =, f sets A and B are A B empty, where A B s the ntersecton of sets A and B, A B s the reunon of sets A and B, A represents the cardnal of the set A. In prncple, the method proposed by Henng can be descrbed by the followng algorthm:. For the ntal data set x, x2,... x28 estmate the number of clusters and choose a clusterng method; 2. Determne the component of the obtaned clusters C through the chosen method * * 3. Determne B (for example B = 00 ) bootstrap versons x, x 2,... x 28 of the ntal data set x, x2,... x28 4. For each bootstrap verson b {,2,... B} : 4. Determne the component of the clusters by the chosen method at pont b 4.2 For each ntal cluster C calculate J max, as beng the maxmum of Jaccard s coeffcents calculated for C and the obtaned clusters n current bootstrap verson b 5. Calculate the stablty ndex of each cluster C as beng 6. Stop Stab = B B b= * J max b

7 Bootstrap Stablty Evaluaton and Valdaton of Clusters Based on Agrcultural Indcators of EU 7 From the defnton of the Jaccard s coeffcent we notce that t takes values between 0 and and so 0 <= Stab <=. The closer to s the stablty value Stab, the more stable s the cluster C. Based on ths observaton we fnd that the values of stablty ndexes presented n Table 3 reveal very good values for clusters C and C2 and an reasonable value for the cluster C3. Table 3. Stablty ndexes for the found clusters Cluster C C2 C3 Stablty ndex Source: made by the author wth results obtaned n R Hgh values of stablty ndexes for clusters C and C2 ndcate they are real clusters,.e. not an artfcal creaton of the clusterng algorthm. Cluster C3, wth a reasonable stablty ndex (0.66) does not seem to be a cluster of confdence. He appeared most lkely due to the exceptonal values that Romana s ndcators NAH and UAA have: Romana s the country that has the largest number of land propertes of all EU countres and ranks sxth n terms of utlzed agrcultural area. Results and Dscussons The structure of the three clusters estmated based on the dendogram from the Fgure has been valdated by three methods: scree plot, gap statstc and the method based on the use of slhouettes. In addton, the stablty of the obtaned clusters was analysed usng the method proposed n (Efron and Tbshran, 993), whch s based on the use of bootstrap technque. The three obtaned clusters (hghlghted n dendogram from Fgure by enclosng each one n a rectangle) have the followng composton: o Cluster : Austra, Belgum, Bulgara, Croata, Cyprus, Czech Republc, Denmark, Estona, Fnland, Greece, Hungary, Ireland, Latva, Lthuana, Luxembourg, Malta, Netherlands, Portugal, Slovaka, Slovena, Sweden. o Cluster 2: France, Germany, Italy, Poland, Span, Unted Kngdom. o Cluster 3: Romana. Countres that belong to cluster 2 are among the top seven countres n EU n terms of utlzed agrcultural area (UAA). Together they own 63.9% of the total utlzed agrcultural area n EU and 40.% of the number of agrcultural holdngs. Countres that belong to cluster own 28.6% of total utlzed agrcultural area n EU and 26.4% of the number of agrcultural holdngs. Romana (cluster 3) has a partcular stuaton: t owns 7.5% n terms of utlzed agrcultural area (UAA), rankng 6th n the EU and t has a thrd of EU agrcultural holdngs (33.5%), the largest of the EU countres. It may be observed that the number of agrcultural holdngs n Romana s 7.% hgher than the number of agrcultural propertes for the countres n cluster 2 although the utlzed agrcultural area n Romana s almost four tmes smaller.

8 72 Crstan Marnou Conclusons In ths paper we classfy EU countres n relaton to two ndcators: number of agrcultural holdngs as a percentage of the total number of agrcultural propertes (NAH) n EU and utlzed agrcultural area (UAA) as a percentage of the total utlzed agrcultural area n EU. The method used was herarchcal agglomeratve clusterng based on Ward s dstance and led to fndng a three cluster structure. The number of clusters was valdated by usng three graphcal methods (scree plot, gap statstc and slhouette plot), and we concluded that a reasonable structure has been found. We also studed the stablty of the obtaned clusters usng an algorthm based on the bootstrap method. Whle clusters and 2 have exceptonally good stablty ndexes (0.98 and respectvely 0.86) cluster 3 has a reasonable stablty ndex (0.66). The 0.66 value of stablty ndex of cluster 3 reflects the specal stuaton of ts only member, Romana: a country wth a large agrcultural area (sxth country n the EU), but wth the hghest number of agrcultural holdngs. References. Chen, E., 200, Gap statstc avalable at [ accessed on ]. 2. Efron, B. and Tbshran, R., 993. An ntroducton to the bootstrap, Chapman& Hall, Inc, New York, U.S.A. 3. European Commsson, 204, The European Unon explaned :Agrculture, [onlne] avalable at [ accessed on ]. 4. Eurostat-Statstcs explaned, 206a, Farm structure statstcs.[onlne] avalable at [ accessed on ] 5. Eurostat-Statstcs explaned, 206b, Farm_structure_YB206, [onlne] avalable at 8&clent=frefox-b&gws_rd=cr&e=uSJVV7OuL8bwUqTsvdAH [ accessed on ] 6. Eurostat-Statstcs explaned, 206c, Glossary:Agrcultural area (AA) [onlne] avalable at [ accessed on ] 7. Henng, C., Clusterr-wse assesment of cluster stablty, Research Report no. 27, Department of Statstcal Scence, Unversty College London, December 2006, avalable at [ accessed on ]. 8. Jaccard, P., 92. The dstrbuton of the flora n the alpne zone, The new phytologst, vol XI, no. 2 February 92, avalable at [ accessed on ]. 9. Peeples,M., 20 R Scrpt for K-Means Cluster Analyss [onlne] avalable at [ accessed on ]. 0. Shalzy, C., 2009 Dstances between Clusterng, Herarchcal Clusterng Statstcs : Data Mnng course avalable at [ accessed on ].. Stuyf, A., Hubert, M. and Rousseeuw, J., Clusterng n an Object-Orented Envronment. Journal of Statstcal Software, vol., number, (997) avalable at [ accessed on ]. 2. Theodords S. and Koutroumbas K., Pattern Recognton, Academc Press Elsever 3. Tbshran, R., Walther, G. and Haste, T., 200. Estmatng the number of clusters n a data set va the gap statstc, Journal of the Royal Statstcal Socety B, 63, Part 2, pp avalable at [ accessed on ].