Kernel Smoothing Function and Choosing Bandwidth for Non-Parametric Regression Methods 1

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1 Ozea Joural of Applied Scieces (), 009 Ozea Joural of Applied Scieces (), 009 ISSN Ozea Publicatio Kerel Smoothig Fuctio ad Choosig Badwidth for No-Parametric Regressio Methods Murat Kayri ad Gürol Zırhlıoğlu Yuzucu Yil Uiversity, Faculty of Educatio, Va/Turkey Abstract: It is possible to get detailed results betwee idepedet ad depedet variables i oparametric regressio methods. By usig o-parametric regressio methods, the estimatio of curve may have complexity. This is a crucial problem to read ad iterpret the regressio curve. To solve this problem, some smoothig fuctios were developed. Kerel fuctio is oe of the smoothig techiques. Kerel fuctio choices the optimal badwidth value (h) to get a readable curve. Also the distributio type is importat for Kerel smoothig fuctio. The Kerel fuctio obtais the differet curves accordig to distributio type (Gaussia, Uiform, Quartic, etc). To show Kerel smoothig process, iteret depedecy levels of itermediate studets were evaluated. The relatio betwee poit (depedet variable) ad daily of iteret use (idepedet variable) were examied ad scatter plot with curve was obtaied. I this study, i terms of differet badwidth values, some curves have bee obtaied ad the differeces of these curves have bee argued. Keywords: curve estimatio, badwidth, o-parametric distributio INTRODUCTION It is kow that regressio model is too crucial process for estimatio of future. Essetially, regressio methods are gathered i three mai topics. These are parametric, o-parametric ad semi-parametric regressio methods (Härdle, Müler, Sperlich & Werwatz, 004; Kaki, 004). I estimatio of curve methods, parametric methods are ot always capable of obtaiig iformatio sufficietly. For all that, it is difficult to get detailed iformatio about populatio with usig o-parametric methods (Härdle ve Tsybakov, 997). Although parametric ad liear equatio (y = f(x)) ca make possible to coclude ay estimatio for each perso, o-liear fuctio ca t coclude regressio equatio for each oe. Geeralized Liear Model (GLM) was developed to estimate parameters robustly because of parametric liear regressio method ca t explai variace sources efficietly. But, GLM is vulerable to explai a o-liear situatio which ca ot iterpret attitude of idividual i distributio curve (Kaki 004; Muller, 00). I case of iterpretig ay oe s situatio i populatio, o-parametric methods ca be used as a alterative to GLM. It must also cosidered that o-parametric methods are ot eough powerful to aalyze plety of discotiuous variables. Ad it will be difficult to get a regressio curve (Pikse, 994). I literature, curse of dimesioality is too crucial problem ad it leads ureadable curve for populatio. Some smoothig methods are used to solve the ureadable curve. By this way, it will be possible to reduce curse of dimesioality. Especially, usig semi-parametric methods ca obtai the most clarity curves with usig partial liear models. Also, it ca rescue the complexity by usig Kerel smoothig method (Pikse, 994; Muller, 00). The other smoothig method is Splie methods which has differet process from Kerel. Kerel smoothig method is so appropriate for distributio types such as Gaussia, uiform ad Quartic. Splie smoothig is preferred for two or more level polyomial cases. Purpose of Kerel ad Splie methods are to cotrol the error of variace. Cotrolig variace error ca make easy the readable of curves. This study was preseted i X.Biostatistic Cogress as a summary oral paper. 49

2 Ozea Joural of Applied Scieces (), 009 I this study, Kerel which is oe of the smoothig methods was ivestigated with its mathematical features. It was also aimed to show Kerel s capabilities o readig ad uderstadig distributio curves. The choicig of badwidth was also uderlied to get optimal distributio. MATERIAL AND METHOD To show Kerel smoothig process, levels of itermediate studets were evaluated. Data set was obtaied for determiig the regressio curve. This data set was cosisted by Guuc & Kayri (009) to develop a Iteret Addictio Scale. The aim of usig this data set is ot idetify the depedecy level. The sigle aim of usig this data set is just to preset how Kerel fuctio is capable of reducig the curve complexity accordig as choosig appropriate badwidth. The relatio betwee poit (depedet variable) ad daily of iteret use (idepedet variable) were examied ad scatter plot with curve was obtaied. I this study, the badwidth (h) value was chose as 0.0, 0.0, 0.50 ad.00. Accordig to these values, the differeces of regressio curves were compared with each other. Also with same badwidth value, Normal, Uiform ad Epaechikov distributio curves were compared. SPSS was used for all aalysis. Kerel Method Kerel fuctio estimates choosig of origi poit a bi grid o curve. To make estimatio, it calculates f(x) fuctio: ^ f h( x),{ Xi [ x h, x + h)} h = () Here, h iterval legth ad [x-h, x+h) are hadled for distributio type (Härdle, Müler, Sperlich & Werwatz, 004). Kerel fuctio has alteratio accordig to distributio type. If distributio is uiform Kerel fuctio will chage. Takig accout of uiform distributio, Kerel fuctio is: I () K(u) = ( u <= ) Here, u=(x-xi)/h ad I is a idetity matrix. I case of examiig equatio, it will be see that ½ weighted additios will be added for each observed values i uiform distributio. Kerel smoothig process for a uiform distributio will carry out a probability desity fuctio. This desity fuctio ca be showed i equatio 3. ^ f X Xi X Xi h( x) = K = I <= h i= h h i= h (3) The differet types of Kerel smoothig fuctios were summarized i Table (Härdle, Müler, Sperlich & Werwatz, 004). I geeral, a probability desity fuctio of Kerel ca be calculated i equatio 4 (Härdle, Müler, Sperlich & Werwatz, 004): Here, Kh ( ) ^ f h x) = K i= ( X Xi) ( (4) = K( / h) h. K ( ) is oe of Kerel fuctio which has bee showed i Table. 50

3 Ozea Joural of Applied Scieces (), 009 Table. Kerel smoothig fuctio related to the distributio types (Härdle, Müler, Sperlich & Werwatz, 004). Kerel Uiform Triagle Epaechikov Quartic (Biweight) Triweight Gaussia Cosie K(u) I ( u <= ) (- u ) I( u <=) ¾(-u )I( u <=) 5/6(-u ) I( u <=) 35/3(-u ) 3 I( u <=) exp u π π π cos u I u <= 4 ( ) Value of h shows the badwidth of curve. Choosig of optimal badwidth is too importat topic for Kerel smoothig parameter (Rice, 984; Sheather & Joes, 99; Hall & Marro, 99). Accordig to badwidth (h) value, the distributio curve shows chageability. Kerel fuctio is capable of obtaiig more readable ad more iterpretable curve if choosig of distributio (Gaussia, Quartic, Uiform) is appropriate. The mai aim of optimal choosig of badwidth is to reduce the mea square error (Doksum, Peterso & Samarov, 000; Kaki, 004). Choosig of optimal badwidth ca be obtaied by two methods. These methods are Plug-i ad Cross-Validatio (Joes, Marro & Sheather, 996). It is kow that commo method of badwidth choosig is Cross-Validatio (Rice & Silverma, 99). The purposes of Cross- Validatio method is to reduce mea square error ad to examie trasformatio of smoothig fuctio i a reliability border. For havig more sesitive examiatio, Geeralized Cross-Validatio was developed. Geeralized Cross-Validatio method was showed i equatio 5. GCV ( h) RRS( h) { tra( h)} = (5) I equatio 5, RRS shows sum of error, show sample size. It is kow that RRS is calculated as equatio 6: RRS = ( Yi m x i )) i= ( (6) I equatio 6, m(x i )shows the smoothig fuctio ad Y i meas the observatio values. RESULTS Readable ad iterpretable regressio curves are obtaied with usig differet badwidth values ad Kerel desity estimatio was showed i Figure. 5

4 Ozea Joural of Applied Scieces (), 009 h = 0.0 h = 0.0 3,00 5,00,50 4,00 3,00,50 depedecy level h = 0.50 h =.00 (Normal) 4,00,50 3,00,50 Figure. Kerel desity estimatio for h = 0.0, 0.0 ad h = 0.50 ad h =.00 I case of scrutiizig Figure, if h value has a low badwidth value, the curve is ureadable ad the regressio curve is so roughly. I the evet of Icreasig h value, the distributio has more readable ad smooth. But choosig iappropriate high badwidth value, the variace of populatio will be hidde. Because of all these reasos, Kerel desity fuctio smooth at a optimal value of badwidth. Cotrast to high level value of h, the small value ca produce curse of dimesioality. So that choosig of badwidth value is cosidered too crucial. Cosiderig distributio type, Cross-Validatio algorithm ca determie optimal badwidth value. Accordig to distributio type such as Gaussia, Uiform, Epaechikov ad Quartic, it will be obtaied differet regressio curve. Each distributio has differet curve ad mathematical equatio which has bee showed i Table. I this study, It was also proved that Uiform, Normal ad Epaechikov distributio types have differet curves. Ad this result has bee showed i Figure. 5

5 Ozea Joural of Applied Scieces (), 009 h =.00 (Normal) h =.00 (Uiform),50,50,50,50 h =.00 (Uiform),50,50 Figure. Normal, Epaichikov ad Uiform curves with same badwidth (h =.00). Although badwidth value is same for each distributio the regressio curves are obtaied differet. I researches, the type of distributio must be cosidered ad it must be determied before aalyzig or gettig ay curve. DISCUSSION Although parametric regressio methods are isufficiet for parameter estimatio o-parametric methods are detailed iformatio about populatio uecessarily. So that regressio curves which belog to oparametric methods are too roughly ad it makes it difficult the readability ad iterpretability of curves (Pikse, 994; Müler, 00).This problem is called as curse of dimesioality. Solvig this problem depeds o some smoothig methods which ehace the readability of curves (Härdle, Müler, Sperlich & Werwatz, 004). Besides, determiig distributio type of populatio ad choosig of badwidth must be take accout of process. It must be cosidered that smoothig fuctio varies accordig to distributio type. Kerel fuctio is oe of the smoothig methods. Kerel fuctio is capable of creatig optimal curves accordig to appropriate distributio types. I this study, curse dimesioality was focused ad Kerel fuctio was scrutiized to fix this problem. REFERENCES Doksum, K., Peterso, D.& Samarov, A. (000). O Variable Badwidth Selectio i Local Polyomial Regressio. Joural of the Royal Statistical Society, Series B, 6:

6 Ozea Joural of Applied Scieces (), 009 Härdle, W., Müler, M., Sperlich, S. & Werwatz, A. (004). Noparametric ad Semiparametric Models. Spriger Series i Statistics. Spriger press, Germay. Härdle, w. & Tsybakov, A.B. (997). Local Polyomial Estimators of the Volatility Fuctio i Noparametric Auto-regressio, Joural of Ecoometrics 8(): 3-4. Hall, P. & Marro, J.S. (99).Local Miima i Cross-Validatio Fuctios. Joural of the Royal Statistical Society, Series B 53: Joes, M.C., Marro, J.S. & Sheather, S.J. (996). Progress i Data-Based Badwidth Selectio for Kerel Desity Estimatio. Computatioal Statistics (3): Kaki, B. (004). Yarı Parametrik Regresyo Yötemii Hayvacılıkta Kullaımı. Yüzücü Yıl Üiversitesi Fe Bilimleri Estitüsü, Yayılamamış Yüksek Lisas Tezi, VAN. Muller, M. (00). Estimatio ad Testig i Geeralized Partial Liear m-a Comparative Study, Statistics ad Computig : Pikse, C.A.P. (994). Noparametric ad Semiparametric Estimatio ad Testig. Upublished doctorate thesis, Lodo School of Ecoomics ad Politicial Sciece, Lodo. Rice, J. (984). Badwidth Choice of Noparametric Regressio. The Aals of Statistics, (4): Rice, J. & Silverma, B.W. (99). Estimatig the Mea ad Covariace Structure Noparametrically Whe the Data are Curves. Joural of the royal Statistical society, Series B, 53(): Sheather, S.J. & Joes, M.C. (99). A Reliable Data-Based Badwidth Selectio Method for Kerel Desity. Joural of the Royal Statistical Society, Series B 53: