Fitting Orthogonal Polynomial Model

Transcription

1 Fittig Orthogoal Polyomial Model Mohammad Ehsaul Karim rd Year, Course ASTH, Roll SH Istitute of Statistical Research ad Traiig Uiversity of Dhaka, Dhaka-, Bagladesh

2 We have YEAR values of Z i. For our coveiece of table use, we ceter them aroud YEAR 96. The we will have the followig table- Y Z X = Z Now, from theory, we kow that maximum possible umber of order polyomial that may be fit is oe less tha the umber of distict idepedet values. Here we do ot have ay duplicate values of X. Therefore, we have 9 distict X values ad hece maximum possible order of our polyomial is p = - = 9 = 8. Now whe the order of the fitted polyomial is the maximum order, the model fits the data exactly so that o residual occurs. So, to have some residual so that we may test our hypotheses about the model, we may cosider lowerig our model i some orders. Also, from table use cosideratio, we fit polyomial of 6 th order. Thus, our model for this problem is: Y = 6 j= α ( X ) j j j Ad, uder the Orthogoal Polyomial, the Least Squares estimate of the parameters is give by- αˆ j = Y j ( X ) i j i ( X ) i The data was take from Draper, Smith (98) exercise (Y) i page 9.

3 At this poit, ote that X values are equally spaced ad have same umber of replicates (r = ) each time. Therefore, we may use the table of Pearso, Hartley (958, page ) for =9, which is as follows: λ 5/6 7/ / / The, Similarly, α ˆ = ( ) ˆα ˆα =.975 where =.9979 where = where ( Xi) = 9 Yi( Xi) = 7.7 ( Xi) = 57 Yi( Xi) =.8 ( Xi) = 566 Yi( Xi) = -6.6

4 ˆα ˆα ˆα 5 6 = where = where = -.67 where ˆα = where ( Xi) = 8 Yi( Xi) = ( Xi) = 88 Yi( Xi) = ( Xi) = 898 Yi5( Xi) = -. 6( Xi) = 557 Yi6( Xi) = -7.7 Now, Total SS = Thus, SS( α ˆ j ) = ) = Yi = 7.75 Y ( X ) i j i j i ( X ) Yi( Xi) = ( X ) Similarly, ) =.76 i ( 7.7) 9 ) =.6856 ) = ) = ) =.7 6) =.89 6 = Thus, Residual SS = Total SS - SS( αˆ j ) = =.55 j=

5 Estimatig Parameters Usig SAS We write the followig SAS program to get the estimates: title 'PROC ORTHOREG used with drsm data'; data drsm; iput X Y Y Y Y; Year=X-96; datalies; ; proc iml; use drsm; read all var {X} ito year; read all var {Y} ito Y; /*retur orthogoal polyomial desig matrix of order6 */ x = orpol(year-96,6); /* Labels to use for desig matrix */ order = {"Itercept", "st order", "d order", "rd order", "th order", "5th order", "6th order"}; ds_ssq={9,57,566,8,88,898,557}; x_ew = fuzz(x`#sqrt(ds_ssq))`; /* Fit regressio */ ru regress(x_ew,y,order,,,,); quit; Note that, the code provides estimates of uormalized orthogoal polyomials. Replacig - ds_ssq = {9,57,566,8,88,898,557}; x_ew = fuzz(x`#sqrt(ds_ssq))`; ru regress(x_ew,y,order,,,,); byru regress(x,y,order,,,,);

6 will provide estimates of ormalized Orthoormal Polyomial. This ca also be foud usig S-plus or R commad : orth.fit<-lm(y~poly(x,degree =6)). However, the output of the uormalized orthogoal polyomial codes will be as follows: PROC ORTHOREG used with drsm data :5 NAME B STDB T PROBT Itercept E- st order e-7 d order rd order th order th order th order Covariace of Estimates order COVB Itercept st order d order rd order th order 5th order 6th Itercept. st order. d order 5E-7 rd order 977E-9 th order 9E-9 5th order E-8 6th order 85E- order CORRB Correlatio of Estimates Itercept st order d order rd order th order 5th order 6th Itercept st order d order rd order th order 5th order 6th order Special thaks to Douglas Bates, Dale McLerra, Prof Bria Ripley, Nick Cox for makig a begier like me to uderstad such details.

7 ANOVA table Source of Variatio df SS MS F cal F tab (5%) ˆα ˆα ˆα * ˆα ˆα ˆα * 5 ˆα * 6 Residual Here, R = (Reg SS) / (Total SS) = 5./7.75 = ad Residual df = -k- = 9-6- = As we observed that the terms of the fourth ad lower order (other tha d) accout for most of the variatio i the data, we ca adopt the model as- Y ˆ = αˆ + α ˆ ( X ) ˆ ˆ i + α ( Xi) + α ( Xi) ( X ) i = ( Xi) ( Xi) We ca rearrage the ANOVA table as- New ANOVA table Source of Variatio df SS MS F cal F tab (5%) ˆα ˆα ˆα ˆα Residual Here Residual SS = Total SS - SS( αˆ j ) = = j=.997 ad Residual df = -k- = 9-- = 5.

8 Therefore, the above model is fial ad retaied sice o parameters are rejected at 5% sigificace level. Here R = (Reg SS) / (Total SS) =.977 is a bit less tha the previous oe, but for sake of simplicity of the model ad iterpretatio, we choose this model. Now, i order to obtai the fitted equatio i terms of the origial variables, we have to first substitute for the ad relate these to X I s. The formula for the j Orthogoal Polyomial upto fourth order for equally spaced X s for = 9 is- ( ) = ( ) = λx = X ( ) = λ X = X 9 = X - ( ) 7 = λ X X = X X = 6 ( ) 7 = X ( 9 ) X (9 )(9 9) + 56 = 7 X - 69 X X -77 X Therefore, Z ad X correspods as follows ( ) = : = Z i : where the codig is foud to be X = Z 97 sice there s. Thus the fitted polyomial is Y ˆ = αˆ + α ˆ ( X ) ˆ ˆ i + α ( Xi) + α ( Xi) ( X ) i = ( Xi) ( Xi) 5 = X ( 6 X -77 X) ( 7 X - 69 X + 96)

9 Ad, our Orthogoal Polyomial Model looks like the followig graph (after ormalizig )- Orthogoal Polyomial Fitted 5 5 X Usig Draper, Smith Data Also, for compariso purposes, we also show the origial scatter plot - Scatter Plot Y 5 5 X Usig Draper, Smith Data Dividig each parameters (except the itercept term) by respective Sum of Squares obtaied from table.

10 Checkig the followig Residual plots, we fid them more or less radom, ad coclude that this model that we have fitted is a good oe (ot oly based o R value, but also o validity grouds). Residual Vs. X X Residual Usig Draper Smith Data Residual Vs. Fitted Y Fitted Residual Usig Draper Smith Data