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1 Engneerng Grou Trends n Comuter Scence and Informaton Technology DOI htt://dx.do.org/0.735/tcst DOI CC By Eren Bas *, Erol Egroglu and Vedde Rezan Uslu Deartment of Statstcs, Faculty of Arts and Scence, Forecast Research Laboratory, Gresun Unversty, Gresun, 800, Turkey Deartment of Statstcs, Faculty of Arts and Scence, Unversty of Ondokuz Mays, Samsun, 5539, Turkey Dates: Receved: 3 February, 07; Acceted: March, 07; Publshed: 3 March, 07 *Corresondng author: Eren Bas, Gresun Unversty, Faculty of Arts and Scence, Deartment of Statstcs, Gure Camus, Gresun, Turkey, Tel: ; Fax: ; Emal: Keywords: Rdge regresson; Shrnkage arameters; Partcle swarm otmzaton htts:// Research Artcle Shrnkage Parameters for Each Exlanatory Varable Found Va Partcle Swarm Otmzaton n Rdge Regresson Abstract Rdge regresson method s an mroved method when the assumtons of ndeendence of the exlanatory varables cannot be acheved, whch s also called multcollnearty roblem, n regresson analyss. One of the way to elmnate the multcollnearty roblem s to gnore the unbased roerty of. Rdge regresson estmates the regresson coeffcents based n order to decrease the varance of the regresson coeffcents. One of the most mortant roblems n rdge regresson s to decde what the shrnkage arameter (k) value wll be. Ths k value was found to be a sngle value n almost all these studes n the lterature. In ths study, dfferent from those studes, we found dfferent k values corresondng to each dagonal elements of varance-covarance matrx of nstead of a sngle value of k by usng a new algorthm based on artcle swarm otmzaton. To evaluate the erformance of our roosed method, the roosed method s frstly aled to real-lfe data sets and comared wth some other studes suggested n the rdge regresson lterature. Fnally, two dfferent smulaton studes are erformed and the erformance of the roosed method wth dfferent condtons s evaluated by consderng other studes suggested n the rdge regresson lterature.. Introducton The functonal relaton between a deendent varable and more than one ndeendent varable s examned by multle regresson analyss. The urose of the multle regresson analyss s the creaton of the best model that can redct the deendent varable by usng the ndeendent varables. For ths urose, the most common method to create the best model s ordnary least square (OLS) estmates method. In ths method, the sum of error squares to be mnmal s calculated to redct the arameters of the model. There are some vald assumtons for the mlementaton of the multle regresson analyss. These are; the absence of multcollnearty roblem among ndeendent varables, the varance of error term must be constant for all ndeendent varables and the covarance between error term and ndeendent varables must be equal to zero. One of the maor roblems n multle regresson analyss s multcollnearty roblem. If there s a full or hgh degree lnear relatonsh among ndeendent varables, ths stuaton s called as multcollnearty. Besdes, multcollnearty has some mortant effects on OLS estmates of the regresson coeffcents. In the resence of multcollnearty, the OLS of regresson coeffcents have large varance. And also, the regresson coeffcents can be estmated ncorrectly and the standard errors of regresson coeffcents can be found as exaggerated n the resence of multcollnearty. If the regresson coeffcents can be estmated ncorrect, t can be obtaned ncorrect results statstcally. Therefore, rdge regresson method s used to obtan stable coeffcent estmates for the estmaton of the regresson coeffcents. That means, rdge regresson has been suggested to overcome the multcollnearty roblem. In the lterature, t s commonly acceted that f the varance nflaton factors (VIF) values are greater than 0 there s a multcollnearty roblem. Ths s a rule of thumb and ths s not exact nformaton. Smlarly, condton number can be used to determne multcollnearty roblem by usng rule of thumbs. As a result of, determnng of multcollnearty roblem can be realzed by usng some crtera. The two methods most commonly used to determne the effects of multcollnearty roblem are VIF and condton 0 Ctaton: Bas E, Egroglu E, Uslu VR (07) Shrnkage Parameters for Each Exlanatory Varable Found Va Partcle Swarm Otmzaton n Rdge Regresson. Ctaton: Szyszkowcz M (06) Squarng the Crcle Usng Modfed Tartagla Method. Peertechz J Comut Sc Eng (): Trends Comut Sc Inf Technol (): DOI: htt://dx.do.org/0.735/tcst

2 ^ number methods. The dagonal elements of Var are called as VIF and are gven by the Equaton. VIF R,, () In ths Equaton, R s the determnaton coeffcent obtaned from the multle regresson of X on the remanng regressor varables n the model. It can be sad that there s a multcollnearty roblem among the relevant ndeendent varables f these VIF values ncrease (VIF values 0). And also, f VIF values are ncreased, the degree of the multcollnearty ncreases wth the ncrease of VIF values. Condton number method s another method to determne the multcollnearty roblem whch s based on the egenvalues of X'X matrx. The formula of the condton number () was gven n Equaton. () mn ' In ths Equaton, shows the egenvalues of XX. the relatonsh between condton number and multcollnearty s gven n Table. In summary, the determnng of multcollnearty roblem can be done by followng two rules of thumbs. The frst one s that f VIF values are greater than 0 multcollnearty s hgh. The second one s checkng condton number as gven n Table. In addton, another roblem n rdge regresson s fndng otmal basng arameter (k) value. Ths k value s a very small constant determned by the researcher []. Several methods were roosed for fndng t n the lterature. These methods have been roosed n the studes of [-]. And also, there are many methods n the lterature for rdge regresson [3-9]. And also, [30] roosed some new methods that take care of the skewed egenvalues of the matrx of exlanatory varables. [3] Proosed an teratve aroach to mnmze the mean squared error n rdge regresson. [3] Proosed new rdge arameters for rdge regresson. [33] Proosed an otmal estmaton for the rdge regresson arameter. [34,35] Proosed some new estmators for estmatng the rdge arameter. Ths k value was found to be a sngle value n almost all these studes n the lterature. But n ths study, we found dfferent k values corresondng to each dagonal elements of varancecovarance matrx of nstead of a sngle value of k by usng a new algorthm based on artcle swarm otmzaton. The rest art of the aer can be outlned as below: Table : Condton number and ts effects. Condton Number Multcollnearty <00 There s no serous multcollnearty 00 < <000 Strong multcollnearty >000 Severe multcollnearty exst n the data The second secton of the aer s about rdge regresson. The methodology of the aer s gven n Secton 3. The mlementaton of our roosed method s gven n Secton 4. Two dfferent smulaton studes are erformed under the ttle of smulaton study and fnally, dscussons are resented n Secton 6. Rdge regresson Rdge regresson s a remedy used n the resence of multcollnearty roblem and t was frstly roosed by []. Rdge regresson method has two mortant advantages accordng to OLS method. One of them s to solve the multcollnearty roblem and the other one s to decrease the mean square error (MSE). The soluton technque of rdge regresson s smlar wth OLS. Besdes, the dfference between rdge regresson and OLS s the k value. Ths k value s also called as based arameter or shrnkage arameter and t takes values between 0 and. Ths k value s added to the dagonal elements of the correlaton matrx and thus based regresson coeffcents are obtaned. The OLS estmates of regresson coeffcents and rdge estmates of regresson coeffcents are shown n the Equatons 3 and 4 resectvely. X ' X X ' Y ˆ (3) ˆ R X' X ki X' Y (4) As noted above, rdge regresson s a based regresson method. The roof of ths stuaton s shown n Equaton 5. ˆ R X' X ki X' Y (5) X' X ki ( X' X)ˆ Zˆ ˆ EZ ˆ Z E R It s clearly seen that rdge estmates of regresson coeffcents ˆ R are based estmates. One of the most mortant onts to be consdered n the rdge regresson s the k value. There are many methods roosed n the lterature to fnd the otmal k value. Rdge trace s one of these methods. Rdge trace s a lot of the elements of the rdge estmator versus k usually n the nterval (0, ) []. The other methods n the lterature used to fnd the otmal k value were gven n the Equatons 6-4, resectvely. k k ^ [] (6) ^ ^ k ' ^ ^ [4] (7) ^ ^ ^ ^ / / [36] (8) 03 Ctaton: Bas E, Egroglu E, Uslu VR (07) Shrnkage Parameters for Each Exlanatory Varable Found Va Partcle Swarm Otmzaton n Rdge Regresson. Trends Comut Sc Inf Technol (): DOI: htt://dx.do.org/0.735/tcst

3 k ^ ^ ^ n ^ k 0, ^ ^ ' nvif k s ^ ^ ^ [4](9) [9] (0) [37] () ^ ^ ^ k ^ k ^,,,, k ^ [5]() ^ [6] (3) / ^ ^ 4 ^ ^ 4 ^ ^ ^ / /4 6 / / [38] (4) In ths aer, for the urose of comarng the results we ust consder the methods of whch a bref ntroducton s gven as below. [] Suggested another method for fndng k value whch s gven n Equaton 5 ˆ k (5) ˆ ' ˆ In ths Equaton ˆ and ˆ are the OLS estmates. Ths method s called as fxed ont rdge regresson method (FPRRM). [39] Introduced an teratve method for fndng the otmal k value. In ths method k s calculated n Equaton 6; ˆ t ' t ˆ t k ˆ In ths Equaton, ˆ t and ˆ t (6) are the corresondng resdual mean square and the estmate vector of regresson coeffcents at (t-)th teraton, resectvely. Ths method s called as teratve rdge regresson method (IRRM). And also, the generalzed rdge regresson estmator of Hoerl and Kennard [, 40] s gven n [4] by followng Equatons 7-0. Let and Q be the matrces of egenvalues and egenvectors of X ' X. In the orthogonal verson of the classcal lnear regresson model: Z XQ ˆ Z', k, k,, k, k 0, Q', y K dag then ~ K ˆ Q (7) ~ Is the generalzed rdge estmator of. Hoerl and Kennard [, 40], have shown that the values of k whch mnmze the MSE of regresson coeffcent are gven by k (8) And the estmaton of k values can be obtaned by usng Equaton 9. ˆ ˆ k ˆ (9) In [4], other estmaton formulas for otmum shrnkage arameters are gven below. kˆ ˆ n kˆ ˆ (0) Methodology Fndng the otmal k value s an mortant roblem n rdge regresson. The k values recommended n the lterature were gven n the revous secton. And also, there are some heurstc methods such as genetc algorthms to fnd the otmal k value n the lterature roosed by [8, ]. And also, [] have found the k value by usng artcle swarm otmzaton (PSO). In all these methods suggested n the lterature, ths k value was found as a sngle value. But n ths study, we found dfferent k values corresondng to each exlanatory varable nstead of a sngle value of k by usng an algorthm based on artcle swarm otmzaton. And also, ths aer s the mrovement form of the study of []. The obectve functon of the aer was created by consderng both mean absolute ercentage error () crteron and VIF values at the same tme. The am of the obectve functon s to fnd the otmal k values by fndng the VIF values less than 0 and (sum of square errors) mnmum, at the same tme. And also, we add a arameter ( k) to the second art of the obectve functon. Ths arameter can be called as enalty arameter. If the VIF value corresonds to any exlanatory varable s bgger than 0 the value of the obectve functon s ncreased. Ths s an effect of the enalty arameter. Ths s an undesrable result. The otmzaton roblem n the roosed method can be gven n Equaton. Ctaton: Bas E, Egroglu E, Uslu VR (07) Shrnkage Parameters for Each Exlanatory Varable Found Va Partcle Swarm Otmzaton n Rdge Regresson. Trends Comut Sc Inf Technol (): DOI: htt://dx.do.org/0.735/tcst

4 Obectve functon: k, k,, k mn k, k,, k ( k, k,, k ) () wth subect to: 0 k, k,, k,,, where k, k,, k k k be defned n Equatons and 3 resectvely. k k and,,, k can ^ n y y,,, k () n y k, k,, k 0 VIF 0,,,, VIF otherwse ( shows the number of exlanatory varables.) (3) The otmzaton roblem defned as n () was solved by usng PSO n the roosed method. PSO s a oular artfcal ntellgence technque and t was frstly roosed by [4]. The algorthm of the roosed method s gven below. Algorthm Ste. The arameters such as n, c, c etc., are determned. These arameters are as follows: n: artcle number of swarm c : Cogntve coeffcent c : Socal coeffcent nterval t: Maxmum teraton number w: Inerta weght Ste. Generate random ntal ostons and veloctes. The ntal ostons and veloctes are generated by unform dstrbuton wth (0,) arameters. Each artcle has veloctes u to the number of exlanatory varables and each artcle has ostons u to the number of exlanatory varables whch reresents k, k,, k values. x Reresents the oston of tm artcle m at teraton t and v t reresents the velocty of the m artcle m at teraton t. Ste 3. The ftness functon was defned as n () and the ftness values of the artcles are calculated. Ste 4. Pbest and Gbest artcles gven n (4) and (5), resectvely, are determned accordng to ftness values. Pbest ( m), m,,, n (4) t Gbest t m ( g) (5) Pbest s constructed by the best results obtaned n the related ostons at teraton t. Gbest s the best result n the swarm at teraton t. Ste 5. New veloctes and ostons of the artcles are calculated by usng the Equatons gven n (6) and (7). t wv m crand t vm (6) t t t t Pbestm xmcrandgbest xm x x v (7) t t t m m m Where from U (0,). rand and rand are random numbers generated Ste 6. Ste 3 to Ste 6 s reeated untl t<t. Ste 7. The otmal k, k,, k values are obtaned as Gbest. Imlementaton The roosed algorthm was aled to two dfferent and well known data sets n order to nvestgate of the roosed method. These two data sets named Imort Data and Longley Data were used to evaluate the erformance of the roosed method. Imort data was analyzed by [43]. The varables of Imort Data are; morts (IMPORT-Y), domestc roducton (DOPROD-X), stock formaton (STOCK-X) and domestc consumton (CONSUM-X3), all measured n bllons of French francs for the years 949 through 959. Both Imort data and Longley data were solved by usng fxed ont method ([]), teratve method ([39]), [] s method and the algorthm roosed n ths aer. In the roosed algorthm, PSO arameters were chosen as n 30, w 0.9, c c and t 00. In the teratve rdge method the stong crtera were chosen as 0 6. The results of each method were resented n Tables and 3, resectvely. As we can see from Table, our roosed method has mnmum and values. And also there s no multcollnearty roblem when Imort Data solved by our roosed method. But, there s a multcollnearty roblem when Imort Data solved by FPRRM and IRRM methods because of the VIF values of these methods are bgger than 0. Although, other methods can gve smaller and values they do not stll solve the multcollnearty roblem. Because t s clearly seen that some VIF values of these methods are greater than 0. As we can see from Table 3, our roosed method has mnmum value when comared wth other methods. But value of our roosed method s not the smallest one. The value of OLS s smaller than our roosed methods. But, t s clearly seen that the OLS method has multcollnearty roblem when Longley Data solved by ths method. But our roosed method has no multcollnearty roblem. As a result, fndng k values for each exlanatory varable gves better results than fndng a sngle k value. And also, our roosed has no multcollnearty roblem. Smulaton study Two dfferent smulaton studes are erformed n ths secton of the aer n order to show the erformance of the Ctaton: Bas E, Egroglu E, Uslu VR (07) Shrnkage Parameters for Each Exlanatory Varable Found Va Partcle Swarm Otmzaton n Rdge Regresson. Trends Comut Sc Inf Technol (): DOI: htt://dx.do.org/0.735/tcst

5 roosed method n dfferent levels of multcollnearty and standard devaton of error term and the suerorty of the roosed method when comared wth other methods. The Frst Smulaton Study: In ths smulaton study, the roosed method was comared wth rdge regresson methods gven n [,,39] by a smulaton study. The number of observatons (n) was taken as 00, 500 and 000; the standard devaton of error term ( ) was taken as 0.0 and and comarsons were made for the total 6 cases. For each case, 000 data set ncludng multcollnearty roblem was created. The frst three ndeendent varables were generated from standard normal dstrbuton as gven n Equaton 8. X ~ N 0,,,3 (8) The last two ndeendent varables were generated by usng Equaton 9. Thus, t s rovded to arse multcollnearty roblem for the data set by rovdng a hgh correlaton between ndeendent varables X and X, 4 X and X 5. X U 0,0 U 5,0 X N 0,7 4,5 (9) The observatons of deendent varable were obtaned usng Equaton 30. So, all the coeffcents n the regresson model are taken as. Y X N(0, ) (30) For each data generated n each case, VIF,, and values are calculated by usng roosed method, the studes [,, 39]. The formula of s gven n Equaton 3. n ^ ( y y ) (3) The most mortant ndcator for the comarson of methods s that VIF and would be small. The methods [] and [39] do not guarantee the soluton of multcollnearty roblem as seen n the numercal examles. The method [] and roosed method guarantee that all VIF values are smaller than 0. Therefore, t s sutable to comare the roosed method wth [] method n terms of and crtera. The results of medan and nter quartle range (IQR) values were gven between Tables 4-9. When all tables are examned, t s clearly seen that VIF and values of roosed method s lower than the other methods n all cases. However, t s seen that the roosed method roduces lower values comared to others deste roducng hgher values. Ths s because the obectve functon of the roosed method may be deendng to the. The Second Smulaton Study: A second smulaton study was erformed n the aer accordng to dfferent levels of multcollnearty roblem and standard devaton of error term. The regressors were generated by usng Equatons 3-36 gven by [44]. w ~ N 0, ;,,, n ;,,,6 (3) x w w,,, n ;,,3 / ( ),6; (33) x w,,,, n ; 4,5 (34) Table : The comarson of VIF values, and obtaned from OLS, FPRRM, IRRM, [] and roosed method for Imort Data. OLS (k=0) [] (k=0.006) [39] (k=0.004) [] k=(0.0090) Proosed Rdge Method k values obtaned from Proosed Rdge Method Varable S.C. VIF S.C. VIF S.C. VIF S.C. VIF S.C. VIF X k X k 0 X k S.C.: Standardzed Coeffcents. Table 3: The VIF Values, and values obtaned from OLS, FPRRM, IRRM, [] and roosed method for Longley Data. OLS (k=0) [] (k=0.0003) [39] (k=0.0006) [] (k=0.07) Proosed Rdge Method Varable S.C. VIF S.C. VIF S.C. VIF S.C. VIF S.C. VIF k values obtaned from Proosed Rdge Method X k X k X k X k 4 0 X k X k S.C.: Standardzed Coeffcents. Ctaton: Bas E, Egroglu E, Uslu VR (07) Shrnkage Parameters for Each Exlanatory Varable Found Va Partcle Swarm Otmzaton n Rdge Regresson. Trends Comut Sc Inf Technol (): DOI: htt://dx.do.org/0.735/tcst

6 e ~ N 0, ;,,, n (35) 5 y x, e ;,,, n (36) Where w, ndeendent standard normal are seudorandom numbers and s theoretcal correlaton between any two exlanatory varables. Smulaton study was conducted for a total of 8 cases for Table 8: Smulaton results for n=000, 0.0 Method [39] [] [] Proosed Method Medan IQR Medan IQR VIF IQR Medan Medan IQR Table 4: Smulaton results for n=00, 0.0 Method [39] [] [] Proosed Method Medan IQR Medan IQR VIF IQR Medan Medan IQR Table 9: Smulaton results for n=000, 0.0 Method [39] [] [] Proosed Method Medan IQR Medan IQR Medan VIF IQR Medan IQR Table 5: Smulaton results for n=00, 0.0 Method [39] [] [] Proosed Method Medan IQR Medan IQR Medan VIF IQR Medan IQR Table 6: Smulaton results for n=500, 0.0 Method [39] [] [] Proosed Method Medan IQR Medan IQR VIF IQR Medan Medan IQR Table 7: Smulaton results for n=500, 0.0 Method [39] [] [] Proosed Method Medan IQR Medan IQR Medan VIF IQR Medan IQR samle sze s 00, ( n 00) devaton of error term ( 0.0, 0.,,5 of multle connectons ( 0.99, ) (Tables 0-7)., standard devaton of the standard ) and dfferent degrees It s clearly seen that n the tables of the smulaton Study, VIF and values of the roosed method do not change sgnfcantly when standard devaton of error term values are changed. VIF And values of the roosed method are ncreased dramatcally when multcollnearty s ncreased. And also there s no a hardly ever change to be seen n the values of the roosed method wth the reasonable standard devaton of error term values 0.0, 0. or there s a decrease to be seen n the values of the roosed method when multcollnearty s ncreased. In ths smulaton study, dfferent levels of standard devaton of error term are also emloyed. As a result of ths smulaton study t s clearly seen that when standard devaton of error term value s greater than and > the model has very bg devaton from lnear regresson model because values are obtaned about 60 and ths value s not sutable. And also, t s clearly seen that n the tables of the smulaton study, the redcton erformance of the roosed s affected qute negatvely when standard devaton of error term s ncreased. Dscusson There are some vald assumtons to create a model n multle regresson analyss. One of them s that t should not be multcollnearty roblem among ndeendent varables. Rdge regresson method s often used n the lterature when there s a multcollnearty roblem among ndeendent varables. But, rdge regresson has also some roblems. One of the most mortant roblems n rdge regresson s to decde what 07 Ctaton: Bas E, Egroglu E, Uslu VR (07) Shrnkage Parameters for Each Exlanatory Varable Found Va Partcle Swarm Otmzaton n Rdge Regresson. Trends Comut Sc Inf Technol (): DOI: htt://dx.do.org/0.735/tcst

7 the shrnkage arameter (k) value wll be. There are many studes n the lterature to fnd the otmal k value. In these studes, ths k value was found to be a sngle value. But n ths study, we found dfferent k values corresondng to each exlanatory varable nstead of a sngle value of k by usng a new algorthm based on artcle swarm otmzaton. And also, the roosed method was suorted by two smulaton studes. Besdes, t s an mortant novelty for rdge regresson lterature. Table 0: Smulaton results for n=00, 0.99, 0.0 Method [39] [] [] Proosed Method Medan IQR Medan IQR VIF IQR.E-0.E-0.06E-0.3E-0 Medan 3.70E E E E-0 Medan IQR Table : Smulaton results for n=00, 0.99, 0. Method [39] [] [] Proosed Method Medan IQR Medan IQR VIF IQR Medan Medan IQR Table : Smulaton results for n=00, 0.99, Method [39] [] [] Proosed Method Medan IQR Medan IQR Medan VIF IQR Medan IQR Table 4: Smulaton results for n=00, 0.99, 5 Method [39] [] [] Proosed Method Medan IQR Medan IQR Medan 3.90E E E+0.96E+0 VIF IQR.36E+0.36E+0.76E E+0 Medan IQR Table 5: Smulaton results for n=00, 0.99, 5 Method [39] [] [] Proosed Method Medan IQR Medan IQR VIF IQR Medan Medan IQR Table 6: Smulaton results for n=00, 0.99, 5 Method [39] [] [] Proosed Method Medan IQR Medan IQR Medan VIF IQR Medan IQR Table 7: Smulaton results for n=00, 0.99, 5 Method [39] [] [] Proosed Method Medan IQR Medan IQR Medan 6.79E VIF IQR 3.59E Medan IQR Table 3: Smulaton results for n=00, 0.99, 5 Method [39] [] [] Proosed Method Medan IQR Medan IQR Medan 8.9E VIF IQR 5.3E Medan IQR In the future studes, dfferent artfcal ntellgence otmzaton technques can be used to fnd these k values for each exlanatory varable. References. Hoerl AE, Kennard RW (970) Rdge regresson: based estmaton for nonorthogonal roblems. Technometrcs : Lnk: htts://goo.gl/5zv56t. Hoerl AE, Kennard RW, Baldwn KF (975) Rdge regresson: some smulatons. Communcatons n Statstcs 4: Lnk: htts://goo.gl/qggp3l 08 Ctaton: Bas E, Egroglu E, Uslu VR (07) Shrnkage Parameters for Each Exlanatory Varable Found Va Partcle Swarm Otmzaton n Rdge Regresson. Trends Comut Sc Inf Technol (): DOI: htt://dx.do.org/0.735/tcst

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9 to Nonorthogoral Problems. Technometrcs : Lnk: htts://goo.gl/hknemy 4. Frnguett L (999) A generalzed rdge regresson estmator and ts fnte samle roertes. Communcatons n Statstcs-Theory and Methods 8: 7-9. Lnk: htts://goo.gl/vtzhb 4. Kennedy J, Eberhart R (995) Partcle swarm otmzaton. In Proceedngs of IEEE Internatonal Conference on Neural Networks, Pscataway, NJ, USA, IEEE Press Chatteree S, Had (006) A Regresson Analyss by Examle. John Wley and Sons. Lnk: htts://goo.gl/dx6qn 44. Gbbons DG (98) A smulaton study of some rdge estmators. Journal of the Amercan Statstcal Assocaton 76:3 39. Lnk: htts://goo.gl/xmzqjs Coyrght: 07 Bas E, et al. Ths s an oen-access artcle dstrbuted under the terms of the Creatve Commons Attrbuton Lcense, whch ermts unrestrcted use, dstrbuton, and reroducton n any medum, rovded the orgnal author and source are credted. Ctaton: Bas E, Egroglu E, Uslu VR (07) Shrnkage Parameters for Each Exlanatory Varable Found Va Partcle Swarm Otmzaton n Rdge Regresson. Trends Comut Sc Inf Technol (): DOI: htt://dx.do.org/0.735/tcst