HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION

Transcription

1 HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR E-mal: ghapor@umedumy Abstract Ths paper presents the hypothess testng of parameters for ordnary lnear crcular regresson model assumng the crcular random error dstrbuted as von Msses dstrbuton The man nterests are n testng of the ntercept and slope parameter of the regresson lne As an llustraton ths hypothess testng wll be used n analyzng the wnd and wave drecton data recorded by two dfferent technques whch are HF radar system and anchored wave buoy Key words: crcular random varable contnuous lnear varables regresson model hypothess testng von Mses dstrbuton 1 Introducton A crcular random varable s a varable whch takes values on the crcumference of a crcle e the angle s n the range ( π ) radans or ( 36 ) Ths random varable must be analysed by technques dfferng from those approprate for the usual Eucldean type varables because the crcumference s a bounded closed space for whch the concept of orgn s arbtrary or undefned A contnuous lnear varable s a random varable wth realsatons on the straght lne whch may be analysed by usual technques Ordnary lnear crcular regresson model s appled when we wsh to determne the relatonshp between a sngle crcular explanatory varable X and a crcular response varable Y As an example s n the relatonshp of the measurements of wnd and wave drectons measured by an anchored buoy (Y) and radar (X) as shown n fgures below Sova (1995) Fgure below gves a smple (or even smplstc) scatter plot for each data set In both cases the observatons (drecton n radans) has been measured smultaneously by an anchored buoy (Y) and by radar (X) Although measurements are n radans we wll often use the more famlar degrees n nformal dscusson of the data In ths case we arbtrarly chose varate Y as the anchored buoy and varate X as radar We would antcpate an deal model y = x as approprate for the data The scatter plots of one drecton aganst the other (as measured n radans antclockwse from North) gve a cluster of ponts along the X=Y dagonal and then a few n the top left and bottom rght corners (359 o aganst 1 o etc) If we thnk of these data as arsng from an ordnary lnear regresson model we would Pak j stat oper res VolII No 6 pp79-86

2 Abdul Ghapor Hussn regard those ponts at the top left and bottom rght as outlers but n relaton to an ordnary lnear crcular regresson model ths s not so because the measurements are on the crcle or crcumference not a straght lne Snce 1 o s only o from 359 o the pont (1 o 359 o ) on the smple scatter plot should not really be far from the deal model y = x In ths respect the smple scatter plot s msleadng but t llustrate that an ordnary lnear regresson model whch gnore the crcularty of the data s equally msleadng Perhaps such scatter plots should be drawn on a torus whch mantans the wrappng of the measurements scales Ths shows the chef problem wth ordnary lnear regresson when appled to crcular varables and below we wll propose an ordnary lnear crcular regresson model whch s more suted to ths form of data Hussn (4) 6 5 Anchored buoy Radar (a) Wnd drecton data 6 5 Anchored buoy Radar (b) Wave drecton data Fg 1: Scatter plot for wnd and wave drecton data (n radans) 8 Pak j stat oper res VolII No 6 pp79-86

3 Hypothess testng of parameters for ordnary lnear crcular regresson The Model and Parameters Estmaton Suppose for any crcular observatons ( x1 y1 )( xn yn) of crcular varables X and Y we propose a model of y = α + βx + ε (mod π ) (1) where ε s a crcular random error havng a von Mses dstrbuton wth mean crcular and concentraton parameter κ An applcaton when both X and Y are crcular for example n modellng a relatonshp for calbraton between two nstruments for wnd drecton requres β 1 Parameters α and β may be estmate by maxmum lkelhood estmaton Based on the von Mses densty functon the log lkelhood functon for model (1) s gven by log L( α β κ; x1 xn y1 yn) = nlog( π) nlog I ( κ) + κ cos( y α βx ) We dfferentate log L wth respect to α β and κ to get α β and κ whch are gven by 1 tan S > C > C 1 α = tan + π C < () C 1 tan + π S < C > C β β + 1 x sn( y α β x ) x cos( y α β x ) (3) Ths expresson for βˆ may be solved teratvely gven some sutable ntal guesses at the estmate We can then update α and β and proceed teratvely Ths teraton procedure wll contnue untl the convergence crteron satsfed Further the estmate of parameter concentraton s gven by κ = cos( α A β ) y x (4) n 1 1 Pak j stat oper res VolII No 6 pp

4 Abdul Ghapor Hussn We can now proceed to obtan useful approxmatons usng the results of Dobson (1978) who gves varous smple approxmatons for the functon A (rato of the modfed Bessel functons for the frst knd of order one and the frst knd of order zero) for the von Mses concentraton parameter κ For example for 65 < w < 1 the approxmaton s A 1 9 w w w ( ) 81 ( w) Thus by usng maxmum lkelhood estmaton we have shown that αβ and κ can be estmated teratvely Snce both the x and y are measurements of the same quantty unty would be logcal ntal estmate of β and so a possble ntal estmate for teraton s β = 1 n () and (3) 3 The Hypothess Testng of Parameters Ths secton s concerned wth testng hypotheses on α and β e testng the hypothess H :β = (modπ ) vs H 1 :β (modπ ) for the ordnary lnear crcular regresson of the form y = α + βx + ε (mod π ) = 1 n usng observatons ( x1 y1 ) ( xn yn) and assumng that ε are VM( κ ) It follows drectly that the observatons y are VM( α + βx κ ) In partcular our nterest s n testng the hypotheses that H : β = 1 aganst H : β 1 and H 1 : α = (mod π ) aganst H : α (mod π ) In a practcal applcaton 1 α = and β = 1 mples an exact relatonshp between the two varables For large κ Marda (197) has shown that κs has a χ degrees of freedom where { cos( ) } S = n y α β x dstrbuton wth n Further f S and S are the values of S under H and H 1 S1 S( = αβ ) and S = S( α 1) say then for large κ 1 respectvely e κs 1 ~ χ n - 8 Pak j stat oper res VolII No 6 pp79-86

5 Hypothess testng of parameters for ordnary lnear crcular regresson whch gves κ( S1 S ) ~ χ1 Further κs 1 and κ ( S1 S ) may be assumed to be ndependently dstrbuted Consequently for testng H aganst H 1 the followng F-statstc may be used ( n ) S S F 1 n- = S ( ) 1 In ths secton we wll use the above result to test the hypotheses H :β = 1 aganst H 1 :β 1 and H : α = (mod π ) aganst H 1 :α (mod π ) n order to make a comparson between the ftted model and the deal model of y = x Suppose that ( α 1 ) and ( αβ ) are the maxmum lkelhood estmates of the parameters under H :β = 1 and H 1 :β 1 respectvely Under 1 H :β = 1 the log lkelhood functon s gven by log L = nlog( π) nlog I ( κ) + κ cos( y α x ) Dfferentatng log L wth respect to α gves log L α = κ sn( y α x) Settng ths equal to zero and smplfyng we get Hence tan α = S = C sn( y x ) cos( y x ) say α 1 tan S > C > C 1 = tan π + C < C 1 tan π + S < C > C Therefore ( ) α and S1 = ( n cos( y α β x ) ) S = n cos( y x ) Pak j stat oper res VolII No 6 pp

6 Abdul Ghapor Hussn and the test statstc s gven by F 1 n- = ( ) ( n ) S S 1 S 1 The same technque can be used to test the hypothess that H :α = (modπ) aganst H 1 : α (mod π ) Suppose that ( β ) and ( α β) are the maxmum lkelhood estmates of the parameters under H :α = (modπ) and H 1 : α (mod π ) respectvely Under H :α = (modπ) the log lkelhood functon s gven by log L = nlog( π) nlog I ( κ) + κ cos( y β x ) Dfferentatng log L wth respect to β gves log L β = κ x cos( y βx) Ths may be solved teratvely for β and t can be shown that β = β + x sn( y β x) x cos( y β x ) where β s an ntal estmate of β Hence { } β and S1 = { n cos( y x } ) S = n cos( y x ) α β We wll use the same F-statstc as above to test the hypothess for ntercept α 4 Applcaton and Concluson In ths secton we wll apply the results of the hypothess testng and wll use the wnd and wave drecton data as our examples Suppose varable X s the observaton (drectons) measured by radar and varable Y s the observaton (drectons) measured by anchored buoy The scatter plots of the data set n Fgure 1 ndcate that there s generally a lnear relatonshp between the drectons taken by radar and anchored buoy wth β 1 We assume that each y observaton on Y and x observaton on X can be descrbed by the model y = α + βx+ ε (mod π) 84 Pak j stat oper res VolII No 6 pp79-86

7 Hypothess testng of parameters for ordnary lnear crcular regresson where ε represents departures from the straght lne mplct n the model also called crcular random errors and are dstrbuted as a von Mses dstrbuton wth crcular mean and concentraton parameter κ e ε ~ VM( κ ) Estmates for the wnd drecton data are gven n Table 1 together wth ther approxmate standard errors Parameter Estmate St error α x1 - β x1 - κ Table 1: Parameter estmates for wnd drecton data Testng the null hypothess that α s equal to mod( π ) gave an F-statstc equal to 575 whch we compare wth F(117) The null hypothess s rejected at the 5% sgnfcance level ndcatng a non-zero α s requred However testng the null hypothess that β equals 1 gves F=17 and we draw the concluson that the null hypothess can not be rejected at the 5% sgnfcance level Table gve the estmates of wave drecton data together wth ther approxmate standard errors Parameter Estmate St error α β x1 - κ Table : Parameter estmates for wave drecton data Testng the null hypothess that α s equal to mod( π ) gves F = 1815 and the null hypothess that β s equal to 1 gves F = 77 We compare these F values wth F(1 76) and we draw the concluson that both null hypotheses can not be rejected at 5% sgnfcance level Pak j stat oper res VolII No 6 pp

8 Abdul Ghapor Hussn References 1 Dobson A J Appled Statstcs 7 (1978) pp Marda K V Statstcs of Drectonal Data Academc Press London (197) 3 Hussn AG Feller N R J and Eleanor E C Journal of Appled Scence & Technology 8 Nos 1 & (4) pp Sova M S The samplng varablty and the valdaton of hgh frequency radar measurements of the sea surface PhD Thess School of Mathematcs and Statstcs Unversty of Sheffeld (1995) 86 Pak j stat oper res VolII No 6 pp79-86