Circular(2)-linear regression analysis with iteration order manipulation

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1 Internatonal Journal of Advances n Intellgent Informatcs ISSN: Vol. 3, No., July 7, pp Crcular()-lnear regresson analyss wth teraton order manpulaton Muhamad Irpan Nurhab a,,*, Badaruddn Nurhab b,, Tut Purwanngsh c,3, Mng Foey Teng d,4 a Department of Economcs, STIE MURA, Indonesa b Department of Economcs, IAIN Bengkulu, Indonesa c Department of Statstcs, Unverstas Islam Indonesa, Indonesa d Amercan Unversty of Mddle East, Kuwat rpanmatstat@gmal.com*; badaruddnnurhab@gmal.com; 3 tut.purwanngsh@u.ac.d; 4 mng.teng@unsa.edu.au * correspondng author ARTICLE INFO ABSTRACT Artcle hstory: Receved July 8, 7 Revsed August, 7 Accepted August, 7 Keywords: data crcular crcular() lnear regresson smulaton r-square Data n the form of tme cycle or pont poston to the angle of possblty s no longer sutable to be analyzed usng classcal lnear statstc method because the drecton and the angle nfluence the poston between one data wth other data. Ths paper ams to examne the comparson of Lnear Regresson Analyss wth Crcular Regresson Analyss. The wrtng method used s lterature revew usng smulaton data. Data smulaton and analyss s done wth the help of R program. The results showed that crcular data s better analyzed by Crcular Regresson Analyss rather than Classcal Lnear Regresson Analyss. The use of classcal lnear statstc method s not recommended due to the drecton and the angle nfluence the poston between one data wth other data. Copyrght 7 Internatonal Journal of Advances n Intellgent Informatcs. All rghts reserved. I. Introducton A model of dstrbuton and statstcal technques for analyzng random varables n the form of cycles n nature s Crcular statstcs. Crcular statstcs are used on data whose measurement results are drectons and are usually expressed n angle sze. Ths technque has evolved n several felds of scence where exploraton, modelng and hypothess testng of the drecton and angle data play an mportant role. The presentaton of data n two-dmensonal drectons s not a sngle angle or unt vector because the angular value depends on the choce of startng pont specfed as the angle and the drecton of rotaton. A mathematcan consders the 6 drecton measured from the west as the startng angle and the drecton of rotaton counterclockwse, but the drecton of the same poston s consdered to have a drecton 3 by a geologst measured from the north as the startng angle and rotates clockwse []. Crcular data can be expressed n several ways. The usual way s related to two crcular measurng nstruments, e compass and clock. The observed form s measured usng a compass such as the drecton of the wnd and the drecton of brd movement, ncludng the data measured usng a protractor. Forms of observaton measured by hours may be tme, eg arrval tme (4 hours) of patent n emergency room at a hosptal and number of ncdents n one year or n monthly tme []. Brunsdon and Corcoran [3] use crcular statstcs to see the tmng patterns of crmnal acts n both daly and weekly tmes. Some dstrbutons n crcular statstcs are unform dstrbuton, wrapped dstrbuton, dstrbuton of cardods and dstrbuton of von Mses. One of the most wdely used dstrbutons s the dstrbuton of von Mses. Lke the normal dstrbuton of lnes, the dstrbuton of von Mses has an mportant role n descrptve statstcs and the statstcs of crcular nferences. DOI: W : E : nfo@jan.org

2 8 Internatonal Journal of Advances n Intellgent Informatcs ISSN: Vol. 3, No., July 7, pp. 7-6 The study of ths research ams to examne the comparson between Lnear Regresson Analyss and Crcular Regresson Analyss. More specfc ths model wll be brng to Crcular Regresson()- Lnear Order. Ths paper wll gve new nformaton about the possbltes to have a better predcton when manpulate the order of teraton. It works as the new approach for mprovng value of R. II. Lterature Revew A. Data and Crcular statstcs Crcular data s the data of measurng result that the values always repeat perodcally. The value wll be found agan after meetng a full perod. The defnton of crcular varable tself s data n the frst and the last scale whch meets each other [4]. The crcular data s dvded nto two types, drectoncrcular data and tme-crcular data [4]. Crcular statstcs s a dstrbuton model and statstcal technque to analyze random varable of the cycle n the nature. Crcular statstcs s used on data whch has drecton-measurement output and s usually expressed n angular sze. Ths technque has developed n some branches of scence n whch exploraton, modelng, and hypothess tral from drectonal data and angle have crucal role. Von Mses dstrbuton s the normal spread of crcular wth dsperson usng (). g( ;, ) I ( ) e cos( ) The method used to evaluate Von Mses dsperson s QQ-plot by fndng Z usng () then Z s arranged based on the mnmum grade to the maxmum grade untl Z... Z n and after that make plot usng (3). Z sn,,...,n sn q, Z,..., sn q n, Z n If the data follows Von Mses dstrbuton, plot wll follow the straght lne (,) n declvty 45 [5]. Data could be easly analyzed f t s llustrated on a graph. Accordng to [5], the representaton of crcular data on a graph s very mportant n the analyss of crcular data. To analyze crcular data, two trgonometry functons used as the foundaton are snus and cousns. Both two functons are utlzed to poston the data. Those functons are used to harmonze coordnate systems. Jammalamadaka and Sengupta [] state the drectonal poston could be determned by polar coordnate or Cartesan coordnate. In Cartesan coordnate, P pont s stated as value (X,) or value (r,θ) on polar coordnate by whch r s the dstance of P pont from the center pont O. Polar coordnate can be converted to Cartesan coordnate by usng trgonometry (4). The relaton between cartesan coordnate and polar coordnate shown n Fg 3. x r cos, y r sn In crcular analyss, the concerned thng s drecton, not vector quantty. Consequently these vectors are changed to unt vector whch s a vector that has length unt wth r =. Every drecton has a connecton wth P pont n the crcumference of a crcle. Conversely, ths pont n the crcle crcumference could be named as an angle. If P pont s stuated n the crcle crcumference, the change of polar coordnate and Cartesan coordnate usng (5). (,αα (x cos α,y sn α) Muhamad Irpan Nurhab et.al (Crcular()-lnear regresson analyss wth teraton order manpulaton)

3 ISSN: Internatonal Journal of Advances n Intellgent Informatcs 9 Vol. 3, No., July 7, pp. 7-6 The average drecton of the crcular sample data s obtaned by calculatng the vector resultant of unt vectors from each sample. The drecton of vector resultant shows the average way of data sample, and the average length of resultant from each sample descrbes the concentraton of data aganst the average drecton. For example, there are samples α, α,, α n wth n crcular observaton stated n angle. Known the transformaton from polar coordnate to Cartesan coordnate for each observaton usng (6). (, ) ( x cos, y sn ),,,..., n The result s the resultant vector from vector unt by summng up for each component usng (7). R = n cos, n sn = (C, S) (7) ( wth R s calculated usng (8), ) R R C S, R n R stands for the length of resultant vector R that s calculated usng (9), R R, R n where R stands for the average length of resultant vector and also shows the concentraton measure from data aganst average drecton. The drecton of vector resultant R s the drecton of crcular mean that s symbolzed by and defned usng (), cos α = C R, sn α = S R For more explct t s gven nverse quadrant-specfc from tangent usng (), arctan S / C f / f S arctan* arctans / C f C arctans / C f undefned f C., S C, S C C, S C, S () If all dot angles show the same drecton, so that data s concentrated and R s close wth n. On the other hand, f data spreads n all crcle, therefore t s not concentrated and R verges on []. On [4] defned the mode of crcular sample s V = - R. As smaller the value of crcular mode, as concentrated the data nto a certan pont. The value of V s on nterval [,]. B. Smple Lnear Regresson (SLR) Smple lnear regresson dfferent wth multple lnear regresson, the dfference s n number of explanatory varable. Data used n ths model are scalar, mentoned as y as the dependent varable and X as explanatory varable. SLR develop a model between and X [6]. SLR model have many practcal uses. There are two broad categores whch commonly used by data analyst: Muhamad Irpan Nurhab et.al (Crcular()-lnear regresson analyss wth teraton order manpulaton)

4 Internatonal Journal of Advances n Intellgent Informatcs ISSN: Vol. 3, No., July 7, pp. 7-6 a. For predcton purposes, the X varables as nput varables to the SLR. as the response varables usually need to be predct at the next perod f tme seres data, and next object as cross-secton data. The model wll be good f the R value bgger than other b. To know the strength of relatonshp/ nfluence from X to varables. The bgger SLR coeffcent s representng the bgger nfluence of X too. Fg. llustrates the relatonshp between data and X data, t s show us that X and have postve correlaton n the pcture. At the pcture there are three tems we have to know, the frst s the observatons as data shown as red, t consst of X and data. Second s regresson lne, ths lne estmate whch poston s the best for represent the relatonshp between X and, the thrd s error. The error symbolzed as the dstance between red and regresson lne [7]. Fg.. Illustraton of how error n lnear regresson Lnear Regresson can be defned as n () [6]. X The attrbute for equaton () consst of, X,,, and. s dependent varable for object, Independent varable (predctor) varable for object, X represent the lnear X relaton between and X, and s a mean of when X= (-ntercept), for the slope n mean of when X ncreases by measurement and are random error terms. C. Crcular Regresson The regresson formula for crcular data s dvded nto three optons [8], they are :. Crcular regresson-lnear: the regresson analyss wth ndependent varable s crcular varable and dependent varable s lnear varable.. Lnear regresson-crcular: the regresson analyss wth ndependent varable s lnear varable and dependent varable s crcular varable 3. Crcular regresson-crcular: the regresson analyss wth both ndependent and dependent varable are crcular varable. Crcular regresson model()-lnear between lnear varable and ndependent varables crcular α can be wrtten usng (3) [9], []. E( X ) A A cos( ) A cos( ) for example, and k A k cos k Bk A k cos k B, t can be wrtten as E( X ) A B cos B sn B cos B sn the formula of crcular regresson model()-lnear can be mentoned as Muhamad Irpan Nurhab et.al (Crcular()-lnear regresson analyss wth teraton order manpulaton)

5 ISSN: Internatonal Journal of Advances n Intellgent Informatcs Vol. 3, No., July 7, pp. 7-6 A B cos B sn B cos B sn The applcaton for crcular model are many, one of the proof s studed by Lnder and Wllander [].the study show about examnaton of causes for reluctance. They assume on a hypothess-testng framework of busness model nnovaton, and show the sgnfcant roles of crcular busness models whch mply sgnfcant challenges to proactve uncertanty reducton for the entrepreneur. Moreover, the study show that many product servce system varants that facltate return flow control n crcular busness models further aggravate the potental negatve effects of faled uncertanty reducton because of ncreased captal commtments. The other study s about crcular model appled n nonparametrc data, studed by D Marzo et al. []. Guerrero and Solar [3] appled crcular data wth Gaussan process. The specal research dd by Km and Sengupta [] about crcular model wth nversed approach and the another research came from Pers and Km [4] whch restrctng nference of Crcular - Lnear and Lnear - Crcular Regresson Model. D. Regresson Coeffcent Assesment The regresson coeffcent A, B, B, B, B can be expected by usng The Smallest Quadrate Method. Ths method chooses ts parameter value so the value of Error Sum of Squares (SSE) s mnmum. The soluton of ths equaton s the smallest quadrate assessment, such as Aˆ, Bˆ ˆ ˆ ˆ, B,, B p, B p [8], [5] [7]. If The regresson model of Crcular()-Lnear can be wrtten n matrx form as (6). Z wth n a b b ; b b b b m m n (7) Z = [ cosα cosα cosα n snα snα snα n cos mα cos mα cos mα n sn mα sn mα sn mα n cos α cos α cos α n sn α sn α sn α n cos mα cos mα cos mα n sn mα sn mα sn mα n ] (8) where s the observaton vector n sze (nx), Z s matrx n sze (nx(+4m)), β s regresson coeffcent vector n sze ((+4m)x), and ε = error random vector n sze (nx). Then, t needs to search the smallest quadrate assessment vector ˆ that can mnmze the functon of error quadrate L usng (9). X Z ' X Z L ' X ' X ' Z' X ' Z' Z So, the assessment vector of β usng (), Muhamad Irpan Nurhab et.al (Crcular()-lnear regresson analyss wth teraton order manpulaton)

6 Internatonal Journal of Advances n Intellgent Informatcs ISSN: Vol. 3, No., July 7, pp. 7-6 Z ' Z Z' ˆ The Error sum of squares (SSE) s calculated usng () by substtutng (9) to () SSE ' ˆ' Z' E. The reducton of Error Sum of Squares (SSE) The mportant thng to defne the order n polynomal regresson s by reducng SSE when m s ncreased. The decson s taken on the degree of trgonometry polynomal (m+) by addng columns. To determne whether or not we take degree (m+), frstly we should calculate the reducton of SSE usng (). If reducton s obvously great n number, we decde to put degree (m+) n [8]. III. Methods The data used n ths research are smulaton data and secondary data. Independent varable γ and δ smulaton data s obtaned by usng rvm (6,,) n software R.3.. []. The procedures needed to reach the purpose of ths research are : Frst step: creatng a descrptve analyss about crcular statstcs for each varable γ and δ. Graphcal representaton of crcular data for each varable γ and δ by usng transmt dagram and rose dagram Compatblty graph of Von Mses dstrbuton The average of crcular and lnear drecton for each γ varable and δ varable The vector length of crcular average on each γ varable and δ varable usng The data mode on crcular statstcs and lnear statstcs for each γ varable and δ varable Second step: multple lnear regresson analyss and crcular regresson()-lnear for γ varable and δ varable as ndependent varable aganst lnear varable as dependent varable. The regresson equaton of crcular() lnear for order m= usng () A B cos B sn B cos B sn Thrd step : determnaton of order m from crcular regresson()-polynomal lnear usng (3). A B B cos B sn B IV. Result and Dscusson m sn B cosm B m cos sn m A. Descrptve statstcs smulaton data γ crcular varable and δ crcular varable It was shown on the average way of γ n table wth crcular statstcs about 35,73. Whle for the average way of γ wth lnear statstcs was around 3,47. On Table t can also be seen the resultant length was about 3,7 and the average length of resultant was,5 that ndcated a bg concentraton value of data to the average drecton of γ crcular varable. The mode value on statstcs crcular data was,49 whch showed the small data dsperson. However, the mode mark on lnear statstcs was about 7738,73 whch proved the bg data dsperson. The average drecton of δ varable wth crcular statstcs was,97. In the meantme by usng lnear statstcs, the average way of δ was 74,7. It was also drawn from δ varable that the resultant length was 8,75 and the average length of resultant was,48. It demonstrated the small concentraton of data to the average drecton on statstcal crcular δ varable. The mode value on crcular statstcs was,5 that ndcated ts small data dstrbuton. On the other hand, the mode value on lnear statstcs was 76,5 that showed the great data dsperson. Muhamad Irpan Nurhab et.al (Crcular()-lnear regresson analyss wth teraton order manpulaton)

7 ISSN: Internatonal Journal of Advances n Intellgent Informatcs 3 Vol. 3, No., July 7, pp. 7-6 Table. descrptve statscts of smulaton data from γ crcular varable and δ crcular varable Varable γ varable δ varable Number of observaton 6 6 Crcular average way 35,73,97 Lnear average way 3,47 74,7 The length of resultant 3,7 8,75 The average length of resultant,5,48 Crcular mode,49,5 Lnear mode 7738,73 76,5 B. The Compatblty graph of Von Mses dstrbuton smulaton data on crcular varable (γ) and crcular varable (δ) The compatblty result of Von Mses dstrbuton that was done wth Von Mses Q-Q plot on α varable and δ varable can be seen n Fg., as n Q-Q plot, data for γ varable and δ varable demonstrated the data dsperson was followng the straght lne (,) n declvty 45 o, consequently t can be sad that data of γ varable and δ varable was comng after the normal crcular dstrbuton or von Mses. (a) (b) Fg.. Compatblty graph of Von Mses dstrbuton smulaton data (a) Q-Q plot graph n γ varable (b) Q- Q plot graph n δ varable C. Representatve Graph smulaton data crcular varable (γ) and crcular varable (δ) The transmt and rose dagram n Fg. 3 and 4 llustrated that the red-straght lne was the average drecton of crcular statstcs from γ varable whch was 35,73 meanng γ varable wth crcular statstcs had nclnaton toward the north, and the black-dash lne was the average way of lnear statstcs from γ varable whch was 3,47 that meant γ varable wth lnear statstcs had southward tendency. Fg. 3. The transmt dagram γ varable (b) The rose dagram γ varable Muhamad Irpan Nurhab et.al (Crcular()-lnear regresson analyss wth teraton order manpulaton)

8 4 Internatonal Journal of Advances n Intellgent Informatcs ISSN: Vol. 3, No., July 7, pp. 7-6 It proves the countng dfference of the average drecton on data between crcular statstcs at data dstrbuton and lnear statstcs keepng away from data dstrbuton. Fg. 4. The transmt dagram δ varable Fg 4 on transmt dagram and rose dagram were seen the red-straght lne was the average drecton of crcular statstcs from δ varable about,97 whch meant δ varable wth crcular statstcs had northward nclnaton and the black-dash lne was the average way of lnear statstcs from δ varable about 74,7 meanng δ varable by usng lnear statstcs had southward nclnaton. It ndcates the calculatng dfference about the average drecton between crcular statstcs on the data dstrbuton and lnear statstcs sheerng away from the data dstrbuton. D. Multple lnear regresson and Crcular regresson()-lnear at smulaton data to analyze the nfluence of γ crcular varable and δ crcular varable aganst varable In Table, the value of determnaton coeffcent on multple lnear regresson was around 33,4% whch t meant around 33,4% the varety of varable can be explaned by γ and δ varable n a lnear correlaton, and the rest was nfluenced by other factors. However, the grade of determnaton coeffcent on crcular regresson()-lnear n Table was about 95,% for order and 95,3% for order meanng about more than 95,% dversty of varable can be elucdated by γ and δ crcular varable, and the rest was by other factors. From the result, we can see that crcular regresson()- lnear had much better output than multple lnear regresson to know the nfluence of γ and δ crcular varable aganst lnear varable. Table. Multple Lnear Regresson and Crcular Regresson()-lnear on smulaton data to see the nfluence of γ and β crcular varable to lnear varable. Type SSE R P-value Model Multple Lnear Regresson 6,959 33,4%, =,5 -,39 -,46 + ε Crcular Regresson()-Lnear 4,557 95,%, ˆ =, +,97 cos +,4 sn + Order, cos +, sn + ε Crcular Regresson()-Lnear 4,395 95,3%, Order P-value on multple lnear regresson was,, so wth the error possblty α =, that P-value (,) < α (,). It can be descrbed that the model of multple lnear regresson can be used sgnfcantly to see the nfluence of γ and δ varable to the average of varable wth credence degree 9%. In crcular regresson()-lnear wth error degree α =,, the P-value (,) < α (,). It means the model of crcular regresson()-lnear order and was sgnfcantly used to know the nfluence of γ and δ crcular varable to the average of lnear varable wth degree of credence 9%. One of the ways to determne the best model s by usng reducton method of SSE. If the value of SSE order SSE order = 4,557-4,395 =,6, t ndcates that the decrease of SSE s very small so the model of crcular regresson()-lnear order s better than order. Therefore, the best model used to see the nfluence of γ and δ crcular varable toward varable on smulaton data was ˆ =, +,97 cos +,4 sn +, cos +, sn + ε. ˆ Muhamad Irpan Nurhab et.al (Crcular()-lnear regresson analyss wth teraton order manpulaton)

9 ISSN: Internatonal Journal of Advances n Intellgent Informatcs 5 Vol. 3, No., July 7, pp. 7-6 Fg. 5 llustrates the graph of predcton at multple lnear regresson s less close on the real value of so creates a hgh error. Fg 5 shows the graph of predcton on crcular regresson()- lnear s very close wth the real grade. In concluson, the crcular regresson()-lnear possess better result than multple lnear regresson on smulaton data smulaton data Error predcton smulaton data Fg. 5. The comparatve graph of predcton smulaton data, smulaton data, and error at multple lnear regresson V. Concluson Dagnoss data before dong the regresson analyss s an early stage should be done to determne the approprate type of regresson. The type of data that s dmensonless drecton (the drecton of the wnd, the drecton of navgaton, the drecton of the clouds) and tme (day, month, year, tme) s a crcular knd of data. Data were analyzed usng a crcular multple lnear regresson produces less regresson model, when compared wth the regresson model generated by the crcular regresson () -lnear. Acknowledgements We thank for all partes who support ths research. Wth the collaboraton of all partes, ths research could reach ts purpose. References [] S. R. Jammalamadaka and A. Sengupta, Topcs n crcular statstcs. Rver Edge, N.J: World Scentfc,. [] P. E. Jupp and K. V Marda, A general correlaton coeffcent for drectonal data and related regresson problems, Bometrka, vol. 67, no., pp , 98. [3] C. Brunsdon and J. Corcoran, Usng crcular statstcs to analyse tme patterns n crme ncdence, Comput. Envron. Urban Syst., vol. 3, no. 3, pp. 3 39, 6. [4] M. I. Nurhab, A. Kurna, and I. M. Sumertajaya, Crcular Crcular Lnear Regresson Analyss of Order m n Crcular Varable α and β aganst Lnear Varable (). [5] N. I. Fsher, Statstcal Analyss of Crcular Data, 3rd ed. New ork: Cambrdge Unversty Press, 995. [6] D. A. Freedman, Statstcal models: theory and practce. cambrdge unversty press, 9. [7] M. Krzywnsk and N. Altman, Ponts of Sgnfcance: Multple lnear regresson, Nat. Methods, vol., no., pp. 3 4, 5. [8] Jammalamadaka, S. Rao and. R. Sarma, Statstcal Theory and Data Analyss II: Proceedngs of the Second Pacfc Area Statstcal Conference, nd ed. North Holland: Elsever Scence Ltd, 988. [9] A. H. Abuzad, I. B. Mohamed, and A. G. Hussn, Procedures for outler detecton n crcular tme seres models, Envron. Ecol. Stat., vol., no. 4, pp , Dec. 4. [] S. Km and A. SenGupta, Inverse Crcular--Lnear/Lnear--Crcular Regresson, Commun. Stat. Methods, vol. 44, no., pp , 5. Muhamad Irpan Nurhab et.al (Crcular()-lnear regresson analyss wth teraton order manpulaton)

10 6 Internatonal Journal of Advances n Intellgent Informatcs ISSN: Vol. 3, No., July 7, pp. 7-6 [] M. Lnder and M. Wllander, Crcular busness model nnovaton: nherent uncertantes, Bus. Strateg. Envron., vol. 6, no., pp. 8 96, 7. [] M. D Marzo, A. Panzera, and C. C. Taylor, Nonparametrc crcular quantle regresson, J. Stat. Plan. Inference, vol. 7, pp. 4, 6. [3] P. Guerrero and J. R. del Solar, Crcular Regresson Based on Gaussan Processes, n Pattern Recognton (ICPR), 4 nd Internatonal Conference on, 4, pp [4] T. Pers and S. Km, Restrcted Inference n Crcular-Lnear and Lnear-Crcular Regresson, Sr Lankan J. Appl. Stat., vol. 7, no., 6. [5] M. B. Mles, A. M. Huberman, and J. Saldana, Qualtatve data analyss. Sage, 3. [6] M. Olvera Pérez, R. M. Crujeras Casas, and A. Rodr \guez Casal, NPCrc: An R package for nonparametrc crcular methods, 4. [7] A. Rambl, A. H. M. Abuzad, I. Bn Mohamed, and A. G. Hussn, Procedure for Detectng Outlers n a Crcular Regresson Model, PLoS One, vol., no. 4, p. e5374, 6. Muhamad Irpan Nurhab et.al (Crcular()-lnear regresson analyss wth teraton order manpulaton)