Spatial Modelling of Peak Frequencies of Brain Signals

Transcription

1 Malaysan Journal of Mathematcal Scences 3(1): 13-6 (9) Spatal Modellng of Peak Frequences of Bran Sgnals 1 Mahendran Shtan, Hernando Ombao, 1 Kok We Lng 1 Department of Mathematcs, Faculty of Scence, and Appled and Computatonal Statstcs Laboratory, Insttute for Mathematcal Research, Unverst Putra Malaysa Center for Statstcal Scences, Brown Unversty, Rhode Island, USA E-mal: 1 mahen698@gmal.com, ombao@stat.brown.edu ABSTRACT Spatal modellng of varous phenomena has been undertaken n many dversfed felds. In ths project, we concentrate on the modellng of the peak frequences of bran sgnals and the objectve s to ft and llustrate spatal regresson wth Smultaneous Autoregressve (SAR) covarance structure. We found that the peak frequences can be modelled approprately as, Y = β + β + β xx 1 1x1x + ε, wth a smultaneous autoregressve correlaton structure. INTRODUCTION Spatal modellng of varous phenomena has been undertaken n many dversfed felds. For nstance, spatal modellng of ranfall (Smth, 1994), spatal regresson of relatve humdty (Mahendran Shtan, 4), trend surface analyss for agrcultural land value data n Iowa (Clff and Ord, 1981) and forest landscape patterns (Jn-Png, Guo and Yang, Xao, 1999), etc. Whenever we deal wth spatal data, t s vtal to be thoughtful of spatal correlaton amongst the neghbourng stes and ths feature has to be taken nto consderaton n the modellng process. In ths project, we concentrate on the modellng of the peak frequences of bran sgnals and the objectve s to ft and llustrate spatal regresson wth Smultaneous Autoregressve (SAR) covarance structure. In secton regresson wth smultaneous autoregressve errors s brefly descrbed and the methodology s n secton 3. The results are presented n secton 4 and fnally the conclusons are drawn n secton 5.

2 Mahendran Shtan, Hernando Ombao & Kok We Lng REGRESSION WITH SIMULTANEOUS AUTOREGRESSIVE (SAR) ERRORS In ths study we ft regresson model wth Smultaneous Autoregressve (SAR) covarance structure and hence we brefly dscuss the SAR model. The SAR model was frst proposed by Whttle n 1954 where a gven set of observatons observed on a lattce are modelled as functons of the neghbourng stes. That s gven a set of observatons, say { X }, the obsevatons are modelled as follows, n { ε } X = g X +, = 1,,..., n, (1) where { j } j j j = 1 j g s a sequence of constants, { } errors wth E ( ε ) =0 and Var ( ε ) = σ ε s a sequence uncorrelated. A detaled account of the SAR model can be found n Clff and Ord, A class of models that ncorporates correlaton reflectng the spatal structure s of the form, Y = µ + ε, where Y s the random varable at ste, µ s the mean at ste whch s modelled n terms of the covarates and ε s the random error terms. Further, we could allow ε to be a functon of the neghbourng stes as, n ε = g jεj + δ, =1,,..., n, () j = 1 j where { g j } s a sequence of constants, { δ } s a sequence uncorrelated errors wth E ( δ ) = 0 and Var ( δ ) = σ. Ths s what we call as Regresson wth Smultaneous Autoregressve (SAR) Errors. Ths model can be wrtten n matrx forms as, ε = Gε + δ, where T T the vector ε ( ε, ε,..., ε ), vector δ ( δ, δ,..., δ ), = 1 n = 1 n 14 Malaysan Journal of Mathematcal Scences

3 Spatal Modellng of Peak Frequences of Bran Sgnals ( ) ε~ MVN 0,Σ, δ~ MVN( 0, σ I) and the matrx G s gven as follows, 0 g1 g13 g1 n g1 0 g3 gn G = g 31 g3 0 g3n. g n1 gn 0 Snce gj are constants that need estmaton and there are too many of them to be estmated, some smplfcaton can be made by allowng G = ρw, where ρ s an unknown constant that can be estmated for a gven data set and 0 w1 w13 w1 n w1 0 w3 w n W = w 31 w3 0 w3 n, w n1 wn 0 s a matrx of known weghts. The covarance matrx, Σ would then be gven as σ ρ 1 T ( I W) ( I ρw ) 1 for the SAR model. An applcaton of regresson model wth SAR covarance structure has been appled to the Sudden Infant Death Syndrome (SIDS) data set for North Carolna Countes (see Kaluzny, et. al., 1998). METHODOLOGY In ths secton the data set used n ths study and the model fttng are descrbed. Data Set The prmary data set conssted of the event-related optcal (EROS) sgnals observed over tme at 81 spatal locatons (9 9 grd) over the cortcal surface of the bran. The center of the bran surface s referenced by the co-ordnate (0, 0). The x and y axes each stretch from 4 to 4. Each tme seres had a length of n = 15. Malaysan Journal of Mathematcal Scences 15

4 Mahendran Shtan, Hernando Ombao & Kok We Lng A perodogram of the tme seres at each spatal locaton was then obtaned (see Brockwell and Davs, for detals). Thereafter, we produced a smoothed perodogram at each spatal locaton over the cortcal surface of the bran. The frequency at whch the peak of the smoothed perodogram occurred was noted at each spatal locaton. Fgure 1 shows three dmensonal plots of the peak frequency values over the cortcal surface of the bran. The plot clearly suggest fttng a regresson surface and n ths paper we model the peak frequences as functons of the locaton co-ordnates together wth a spatally correlated error structure. Rotaton Angle to a axs : 65 Angle to x axs : 40 Dstance : 5 Rotaton Angle to a axs : 65 Angle to x axs : 75 Dstance : 5 Rotaton Angle to a axs : 65 Angle to x axs : 45 Dstance : 5 Fgure 1: 3D plot over the cortcal surface 16 Malaysan Journal of Mathematcal Scences

5 Spatal Modellng of Peak Frequences of Bran Sgnals Model Fttng To apply the method descrbed n secton and to obtan the weghts, the researcher frst needs to ascertan or defne whch are the neghbourng stes and then work out the weghts. For ths study the neghbours for a gven spatal locaton has been defned as all spatal locatons located wthn a unt n scale from the pont of nterest. The weghts, wj =1f pont and j are neghbours and w j =0, otherwse. The neghbours of the eghty one spatal locatons consdered n ths study are lsted out n Table 1. Varous models of ncreasng complexty as dscussed n the results secton (see Secton 3), were ftted to the data set and the modellng process was done usng S-plus Spatal Statstcs Module (Kaluzny, et. al., 1998). To evaluate between competng models, the test statstc (Cresse, 1993) used n ths study s, n p r U = ( Lp Lp + r ) χ ( r), n (3) where n s the number of data ponts, p s the number of parameters estmated, r s the addtonal number of parameters estmated, L s the negatve log lkelhood for the smaller model and L p+ r s the negatve log lkelhood for the larger model. The log lkelhood functon for the SAR model s gven by n n 1 T T log( π) log( σ ) + log I ρw ε ( I ρw )( I ρw) ε. (4) σ To determne whether any of the coeffcents of the covarates were sgnfcant or not, we used the Lkelhood Rato Test gven as log λ~ χ ( k ) (see Maddala, 1989), where p Maxmum of Lkelhood under restrctons λ = (5) Maxmum of Lkelhood wthout restrctons Malaysan Journal of Mathematcal Scences 17

6 Mahendran Shtan, Hernando Ombao & Kok We Lng TABLE 1: Peak frequences and the neghbours of 81 spatal locatons on the bran Row Spatal Locaton X 1 (x-axs) Y (y-axs) Peak frequences Neghbours 1 (-4, -4) , 10 (-4, -3) ,3, 3 (-4,-) ,4,1 4 (-4,-1) ,5,13 5 (-4, 0) ,6,14 6 (-4, 1) ,7,15 7 (-4, ) ,8,16 8 (-4, 3) ,9,17 9 (-4, 4) ,18 10 (-3, -4) ,,19 (-3, -3) ,10,1,0 1 (-3, -) ,,13,1 13 (-3, -1) ,1,14, 14 (-3, 0) ,13,15,3 15 (-3, 1) ,14,16,4 16 (-3, ) ,15,17,5 17 (-3, 3) ,16,18,6, 18 (-3, 4) ,17,7 19 (-, -4) ,0,8 0 (-, -3) ,19,1,9 1 (-, -) ,0,, (-, -1) ,1,3,31 3 (-, 0) ,,4,3 4 (-, 1) ,3,5,33 5 (-, ) ,4,6,34 6 (-, 3) ,5,7,35 7 (-, 4) ,6,36 8 (-1, -4) ,9,37 9 (-1, -3) ,8,,38 (-1, -) ,9,31,39 31 (-1, -1) ,,3,40 3 (-1, 0) ,31,33,41 33 (-1, 1) ,3,34,4 34 (-1, ) ,33,35,43 35 (-1, 3) ,34,36,44 36 (-1, 4) ,35,45 37 (0, -4) ,38,46 38 (0, -3) ,37,39,47 39 (0, -) ,38,40,48 40 (0, -1) ,39,41,49 41 (0, 0) ,40,4,50 4 (0, 1) ,41,43,51 43 (0, ) ,4,44,5 44 (0, 3) ,43,45,53 45 (0, 4) ,44,54 46 (1, -4) ,47,55 47 (1, -3) ,46,48,56 48 (1, -) ,74,79,57 49 (1, -1) ,48,50,58 18 Malaysan Journal of Mathematcal Scences

7 Spatal Modellng of Peak Frequences of Bran Sgnals TABLE 1(contnued): Peak frequences and the neghbours of 81 spatal locatons on the bran Row Spatal X 1 Y Peak Locaton (x-axs) (y-axs) frequences Neghbours 50 (1, 0) ,49,51,59 51 (1, 1) ,50,5,60 5 (1, ) ,51,53,61 53 (1, 3) ,5,54,6 54 (1, 4) ,53,63 55 (, -4) ,56,64 56 (, -3) ,55,57,65 57 (, -) ,56,58,66 58 (, -1) ,57,59,67 59 (, 0) ,58,60,68 60 (, 1) ,59,61,69 61 (, ) ,60,6,70 6 (, 3) ,61,63,71 63 (, 4) ,6,7 64 (3, -4) ,65,73 65 (3, -3) ,64,66,74 66 (3, -) ,65,67,75 67 (3, -1) ,66,68,76 68 (3, 0) ,67,69,77 69 (3, 1) ,68,70,78 70 (3, ) ,69,71,79 71 (3, 3) ,70,7,80 7 (3, 4) ,71,81 73 (4, -4) ,74 74 (4, -3) ,73,75 75 (4, -) ,74,76 76 (4, -1) ,75,77 77 (4, 0) ,76,78 78 (4, 1) ,77,79 79 (4, ) ,78,80 80 (4, 3) ,79,81 81 (4, 4) ,80 RESULTS In ths secton the results of our study are presented. We have seen that Fgure 1 clearly suggests fttng a regresson surface. However data values of the neghbourng ponts are lkely to be correlated. As such tests for spatal correlaton were conducted usng the Moran and Geary Statstc. The Moran spatal correlaton was found to be 0.59 wth a standard error of The computed z statstc value was and had a p-value of The Geary spatal correlaton value was wth a standard error of The computed z statstc value was and a p-value of Malaysan Journal of Mathematcal Scences 19

8 Mahendran Shtan, Hernando Ombao & Kok We Lng Both these tests ndcate that the observatons were sgnfcantly spatally correlated due to the extremely small p-value thereby rejectng the null hypothess of no spatal correlaton. Hence, ths prompted us to ft varous models of ncreasng complexty wth spatally correlated error structure. Let Y represent the peak frequency value recorded at pont, x1 be the coordnate of x-axs and x be the coordnate of the y-axs over the cortcal surface of the bran. The eghteen (18) models consdered n ths study were, Y = β + ε (Model 1), Y = β + β x 3 + ε (Model ) 1, Y = β + β x + β x + ε (Model 3) , Y = β + β x 3 + β xx + ε (Model 4) 1 1 1, Y = β + β xx + β x 3 + ε (Model 5) 1 1, Y = β + β xx + β x x + ε (Model 6) 1 1 1, Y = β + β x + β xx + β x + ε (Model 7) , Y = β + β x + β x 3 + β xx + ε (Model 8) , Y = β + β xx + β x + β x 3 + ε (Model 9) , Y = β + β xx + β xx + β x 3 + ε (Model 10) , Y = β + β xx + β x x + β x 3 + ε (Model ) , Y = β + β x + β xx + β x x + ε (Model 1) , Y = β + β x + β xx + β x x + ε (Model 13) , Y = β + β x + β x + β xx + β x + ε (Model 14) , 0 Malaysan Journal of Mathematcal Scences

9 Spatal Modellng of Peak Frequences of Bran Sgnals Y = β + β x + β x + β xx + β x x + ε (Model 15) , Y = β + β x + β xx + β x x + β x 3 + ε (Model 16) , Y = β + β xx + β xx + β x x + β x 3 + ε (Model 17) , Y = β + β x + β x + β xx + β x + β x 3 + ε (Model 18) , The parameter estmates of our ftted models are contaned n Table. Usng equaton (3), the test statstcu were computed for the varous models consdered n ths study and are also tabulated n Table together wth the p values. From Table, we notce that the estmated parameter coeffcents take on a wde varety of values both postve and negatve. For every model consdered n ths study, the estmate for σ was found to be 0.4. The estmated value for ρ s n the range of to The log lkelhood remans n the vcnty of 43.1 to and U does not exceed However, the most crucal thng that needs to be observed n Table s the p value, whch ranges from to The p value ndcates whether or not a partcular model dffers from the null model (Model 1) sgnfcantly. Clearly then a smaller p value would assst us n the selecton of a model. Of all the models consdered n ths study, two models namely model 6 and had the smallest p values of and hence they were sgnfcant at the 0.05 level. However snce model 6 s the smpler model of the two models, t would seem reasonable to choose model 6 over model. The sgnfcance of the coeffcents of the co-varates were establshed by the Lkelhood Rato Test whch gave the value, x = wth degree of freedom and p value of To test for the sgnfcance of ρ the Lkelhood Rato Test gave a value, x = wth 1 degree of freedom and p value of Ths s very hghly sgnfcant at the 0.1 level. Malaysan Journal of Mathematcal Scences 1

10 Mahendran Shtan, Hernando Ombao & Kok We Lng TABLE : Results of ftted models Model 1 Model Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Model 10 Model Estmated parameter coeffcents ˆ 0.17 ˆ σ ˆ ρ Log Lkelhood U p- value β = ˆ β = 0.17 ˆ β = ˆ β = 0.45 ˆ β = 0.7 ˆ β = 0.3 ˆ β = 0.17 ˆ β = 0.1 ˆ β = ˆ β = 0.16 ˆ β = 0.5 ˆ β = 0.1 ˆ β = 0.16 ˆ β = 0.5 ˆ β = ˆ β = 0. ˆ β = ˆ β = 0.4 ˆ β = ˆ β = 0.43 ˆ β = 0. 0 ˆ β = 0.1 ˆ β = ˆ β = 0.8 ˆ β = 0.5 ˆ β = ˆ β = 0.1 ˆ β = 0.16 ˆ β = 0.5 ˆ β1 = 0.1 ˆ β = 0.1 ˆ β = 0.16 ˆ β = 0.5 ˆ β1 = 0.1 ˆ β = Malaysan Journal of Mathematcal Scences

11 Spatal Modellng of Peak Frequences of Bran Sgnals TABLE : Results of ftted models (contnued) Model 1 Model 13 Model 14 Model 15 Model 16 Model 17 Model 18 Estmated parameter coeffcents ˆ β = 0.16 ˆ β = ˆ β = 0.5 ˆ β = ˆ β = 0.16 ˆ β = ˆ β = 0.5 ˆ β = ˆ β = 0.7 ˆ β = ˆ β = ˆ β = 0.5 ˆ β = ˆ β = 0.16 ˆ β = ˆ β = ˆ β = 0.5 ˆ β = ˆ β = 0.15 ˆ β = ˆ β = 0.5 ˆ β1 = 0.1 ˆ β = 0.1 ˆ β = 0.16 ˆ β = 0.5 ˆ β1 = 0.1 ˆ β1 = 0.1 ˆ β = 0.1 ˆ β = 0.3 ˆ β = ˆ β = ˆ β = 0.5 ˆ β = ˆ β = 0.1 ˆ σ ˆ ρ Log Lkelhood U p- value Malaysan Journal of Mathematcal Scences 3

12 Mahendran Shtan, Hernando Ombao & Kok We Lng Some dagnostcs plots were also obtaned for the resduals of Model 6 and n Fgure the hstogram of the resduals s shown. In Fgure 3 the normal probablty plot s shown and n Fgure 4 the ftted values versus the resduals s shown. Fgure : Hstograms of the resduals Fgure 3: Normal probablty plot of the resduals Fgure 4: Ftted values vs. Resduals 4 Malaysan Journal of Mathematcal Scences

13 Spatal Modellng of Peak Frequences of Bran Sgnals It s clear from Fgures and 3 that the resduals are approxmately normally dstrbuted. Plot of the ftted values aganst the resduals also ndcate that Model 6 s an approprate one. Other models besdes models 6 and, that were sgnfcant at the 0.05 level were models 5, 1, 13, 16 and 17. The remanng models can be safely dscarded. CONCLUSION The objectve of ths research was to ft and llustrate spatal regresson modellng that takes accounts of spatal correlaton amongst ts neghbors. It has been found that the model Y = β + β + β xx 1 1x1x + ε1 (Model 6) s an approprate one, n the sense that t has the smallest p value when compared wth the null model (Model 1). The coeffcents of the covarates were also found to be sgnfcant. The parameter ρ was hghly sgnfcant at 0.1 level explanng the mportance of takng the spatal correlaton between neghborng ponts nto consderaton n the modellng process. The usefulness of ths model s that t would help us to estmate the peak frequences at locatons where no observatons were recorded and would also lead to an understandng of the phenomenona. Dfferent neghborhood structures and weghts can also be attempted n any further study. Alternatvely, further research can be done to ft spatal regresson models wth ether Condtonal Autoregressve (CAR) errors or Movng Average (MA) errors and to make comparsons wth the proposed model n ths research. REFERENCES Brockwell, P. J. and Davs, R. A.. Introducton to Tme Seres and Forecastng, Second Edton, Sprnger, New York. Clff, A. D. and Ord, J. K Spatal Processes: Models and Applcatons, Pon, London. Cresse, N. A. C Statstcs for Spatal Data, Wley, New York. Malaysan Journal of Mathematcal Scences 5

14 Mahendran Shtan, Hernando Ombao & Kok We Lng Jn Png, Guo and Yang, Xao Trend Surface Analyss of Forest Landscape Pattern n Guandshan Forest Regon of Shanx, Chna, Journal of Envronmental Scences. Kaluzny, S. P., Vega, S. C., Cardoso T. P. and Shelly, A. A S+SpatalStats User's Manual for Wndows and UNIX, Sprnger, New York. Maddala, G. S Introducton to Econometrcs, Macmllan Publshng Company, New York. Mahendran Shtan. 4. Trend Surface Analyss wth Smultaneous Autoregressve (SAR) Errors for Annual Mean Relatve Humdty n Pennsular Malaysa, Proceedngs of Ecologcal and Envronmental Modellng (ECOMOD 4), September 4, Penang, Malaysa. Smth, R. L Spatal Modellng of Ranfall Data, Statstcs for the Envronment : Water Related Issues, Wley, New York. 6 Malaysan Journal of Mathematcal Scences