On the generalization of von Mises distributions on the circle

Information Science and Applied Mathematics, Vol. 16, 2008, B.I.I.S., Senshu University 2 ZKHUHˆPˆNDQG%HVVHO,>N@GHQRWHVWKHPRGLILHG%HVVHOIXQFWLRQRIWKHILUVWNLQGDQGRUGHUZKLFKLV GHILQHGE\ %HVVHO,# N' ccccccccccc S à S ÆN&RV#T'Å T VHH*UDGVKWH\QDQG5\]KLN>@,WKDVSRZHUVHULHVH[SDQVLRQ %HVVHO,# N' Å Q  ˆ _ ccccccccccccccccc +Q/ -N cccccc  1 Q  VHH7LWFKPDUVK>@ 7KHSDUDPHWHUPLVWKHPHDQGLUHFWLRQDQGWKHSDUDPHWHUNLVNQRZQDVWKHFRQFHQWUDWLRQSDUDPHWHU([DPSOHVRIWKH VKDSHRIWKHGLVWULEXWLRQDUHJLYHQLQ)LJXUH7KHGLVWULEXWLRQLVXQLPRGDODQGLVV\PPHWULFDODERXWT P  )LJXUH'HQVLW\RIWKHYRQ0LVHVGLVWULEXWLRQ0NIRUP  DQGN EROGSODLQGDVKHG ,WLVVHHQWKDWZKHQN WKHYRQ0LVHVGLVWULEXWLRQ0PNLVWKHXQLIRUPGLVWULEXWLRQDQGWKDW0PNEHFRPHV FRQFHQWUDWHGDWWKHSRLQWT PDVN‘ˆ$OVRLIDUDQGRPYDULDEOH4EHORQJVWRWKHYRQ0LVHVGLVWULEXWLRQ0PNWKH QRUPDOL]HGYDULDEOH4PrNFDQEHDSSUR[LPDWHGE\WKHVWDQGDUGQRUPDOGLVWULEXWLRQ1DVN‘ˆ )RUVPDOONWKHYRQ0LVHVGLVWULEXWLRQ0PNLVDSSUR[LPDWHGE\WKHFDUGLRLGGLVWULEXWLRQZLWKWKHVDPHPHDQP GLUHFWLRQ &DUGLRLGGLVWULEXWLRQ&PU

I+T P U/ cccccccccccccccccccccccccccccccc  U &RV+T  P/ccccccccccccccccccccccccccccccc

 S  s  † U † s 

Information Science and Applied Mathematics, Vol. 16, 2008, B.I.I.S., Senshu University  F+T P U/ cccccc S Æ Q ˆ ˆ ccccccccccccccccccccccccccccccccDcccccccccccccccccccccccccccc D+T  P  SQ/!  ccccccccccc S  Å_{Q } ˆ UQ_{&RV Q+T  P/!}  ccccccccccc S   U ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc   U_{U &RV+T  P/}  )LJXUH'HQVLW\RIWKH:UDSSHG&DXFK\GLVWULEXWLRQ:&PU IRUP DQGU EROGSODLQGDVKHG

On the generalization of von Mises distributions on the circle _

7KHSUREDELOLW\GHQVLW\IXQFWLRQRIWKHDQJXODUFHQWUDO*DXVVLDQGLVWULEXWLRQLV

STTG 

ccccccccccccccccccccccccccccccccccccccccccc

_{SE &RV +TG/}

r

E

 S†TS  ZKHUHEDQGS†GS VHH0DUGLDDQG-XSS>@DQGDOVR-RKQVRQ.RW]DQG%DODNULVKQDQ>@ :HKHUHFRQVLGHUGLIIHUHQWZD\VIRUREWDLQLQJWKHGLVWULEXWLRQVRQWKHFLUFOHDQGZHEHJLQE\VHWWLQJXSWKHDSSURSULDWH ELYDULDWH1RUPDOGLVWULEXWLRQ I+[ [ P P V U/  ccccccccccccccccccccccccccccccccccccccccccccccccccc  S Vr_{  U} H[S%cccccccccccccccccccccccccccccccccccccccc +  U_{/ V} +[[ P [+U P P/   U PP P [+U [ P U P//) IRUˆ  [ ˆ DQG  ˆ  [ ˆ ZKHUHUV!ˆ  P ˆ DQG  ˆ  P ˆ VHH-RKQVRQ.RW]DQG%DODNULVKQDQ>@  )LJXUH7KHGHQVLW\IXQFWLRQRI

I

+[



 [



 P



 P



 V U

/

 ZLWKWKHSDUDPHWHUV^V‘U‘

P

_‘S

P

_‘S` >@:HILUVWFRQVLGHUWKHZHOONQRZQWUDQVIRUPDWLRQWRSRODUFRRUGLQDWHV  [‘U &RV#T' [‘U 6LQ#T'

,WLVNQRZQWKDW5 ;;UHSUHVHQWVWKHGLVWDQFHRI+; ;/IURPWKHRULJLQZKLOH4 DUFWDQ+;s ;/UHSUH

VHQWVWKHDQJOHRI+; ;/ZLWKUHVSHFWWRWKH;D[LV7KHQ5 U ! DQG4 T ±T  S  T  S 7KHMRLQWGLVWULEX

WLRQRI5DQG4LVJLYHQE\WKHWUDQVIRUPDWLRQPHWKRGDQGWKHGHVLUHGMRLQWGHQVLW\LV

Information Science and Applied Mathematics, Vol. 16, 2008, B.I.I.S., Senshu University 6 J+U T  O P V U/ U ccccccccccccccccccccccccccccccccccccccccccccccccc  S r  U_V H[SL N MMM Ucccccccccccccccccccccccccccccccc O U O &RV#T  P'  UccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccU6LQ# T'  O U +O 6LQ# P'   U 6LQ#T  P'/cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc +  U_{/ V} \ ^ ]]] ZKHUHUˆSTSV!O!UDQGS†PS )LJXUH7KHGHQVLW\IXQFWLRQRIJUTOPVU IRU^V‘U‘O‘P‘`

7KHQWKHFRQGLWLRQDOGHQVLW\IXQFWLRQIRUWKHYDULDEOH4 TJLYHQ5 UFDQEHJLYHQE\ JF+T  O P V U U/  ccccc - H[SLNMMM U _{ O}_{ U O &RV#T  P'  U}_{U6LQ# T'  O U +O 6LQ# P'   U 6LQ#T  P'/} cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc +  U_{/ V} \ ^ ]]] ZKHUH

- Ã_SSÆccccccccccccccccccccccccccccccccUO U O &RV#TP'U U 6LQ# T'O U +O 6LQ# P' U 6LQ#TP'/cccccccccccccccccccccccccccccccc+U/ Vccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc_{Å T}

     

Information Science and Applied Mathematics, Vol. 16, 2008, B.I.I.S., Senshu University 8 )LJXUH7KHMRLQWGHQVLW\IXQFWLRQRIJDEOPVUKF ZLWKWKHSDUDPHWHUV^V‘F‘U‘P‘SK‘S` 7KHILHWXUHRIWKLVGLVWULEXWLRQLVWKDWLWKDV]HURSUREDELOLW\DWD WKLVLVEHFDXVHWKHGHQVLW\KDVWKHIDFWRU

7DQ D

 ,IZHFRQVLGHUWKH]HURPHDQVSHFLDOFDVHK‘ZHFDQGHULYHWKHIROORZLQJFRQGLWLRQDOGLVWULEXWLRQV 7KHFRQGLWLRQDOGLVWULEXWLRQRIWKHYDULDEOHDIURPJDEOPVUFLV FRQG FÆ

ccccccccccccccccccccccccccccccccF+U 6LQ# E'/ 7DQ#D'_{+U/ V}cccccccccccccccccccccccccccccc

6HF#D'_{+  U 6LQ# E'/ 7DQ#D'}

ccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

+  U_{/ V} 

      D

)LJXUH   7KH GHQVLW\ RI WKH FRQGLWLRQDO GLVWULEXWLRQ RI D IRU E ‘ Ss  F ‘  V ‘  DQG U ‘   +EROG SODLQ GDVKHG/

7KHFRQGLWLRQDOGHQVLW\RIWKHYDULDEOHEIRUWKHMRLQWGHQVLW\JDEOPVUFLV

FRQGJ Æ

ccccccccccccccccccccccccccccccccF U &RV#E' 6LQ#E' 7DQ#D'_{+U/ V}ccccccccccccccccccccccccccccc

On the generalization of von Mises distributions on the circle _ F_U7DQ#D' ccccccccccccccccccccccccccccc_+U_{/ V} DQG U   )LJXUH7KHGHQVLW\IXQFWLRQRIFRQGLWLRQDOGLVWULEXWLRQRIE IRUD‘SF‘V‘U‘^`EROGSODLQGDVKHG >@8VLQJWKHWUDQVIRUPDWLRQIRUD[LDODQJXODUGLVWULEXWLRQIRUIL[HGFRQVWDQWF! [‘F 7DQ#D' [‘F 7DQ#E' 

ZH KDYH D MRLQW GLVWULEXWLRQ ZLWK D GHQVLW\ IXQFWLRQ

On the generalization of von Mises distributions on the circle ₁₁ )LJXUH7KHGHQVLW\IXQFWLRQRIFRQGLWLRQDOGLVWULEXWLRQRIDIRU V ‘  U ‘   P ‘ Ss   K ‘ S s   E ‘ S s  F ‘ rs    r +EROG SODLQ GDVKHG/  )URPWKHVHJUDSKVLWLVVHHQWKDWWKHSDUDPHWHUF!RIWKHGHQVLW\IXQFWLRQLVLPSRUWDQWVLQFHWKHJUDSKLVFRQFDYHLI FDQGDOVRLWLVFRQYH[LIF!

6RPHVFDOHPL[WXUHVRIWKHGLVWULEXWLRQVRQWKHFLUFOH

,QWKLVVHFWLRQZHFRQVLGHUWKHH[WHQVLRQVRIWKHJLYHQGLVWULEXWLRQVE\XVLQJWKHVFDOHPL[WXUHZLWKVHYHUDONQRZQ PL[LQJIXQFWLRQVRIWKHVFDOH V_{RIWKHPXOWLDQJXODUGLVWULEXWLRQVREWDLQHGLQWKHVHFWLRQ+HUHZHHPSOR\WKHLQYHUVH} *DPPDGLVWULEXWLRQIRUWKHPL[LQJGLVWULEXWLRQIXQFWLRQDQGDSSO\LWWRWKHFRQGLWLRQDOGHQVLW\IXQFWLRQV6LQFHWKH PL[WXUHGLVWULEXWLRQVHOGRPKDVDQH[SOLFLWIRUPZHVWXG\WKHJUDSKLFDOIHDWXUHVRIWKHGHQVLW\IXQFWLRQRIWKHPL[WXUH RIGLVWULEXWLRQ 7KHLQYHUVH*DPPDGLVWULEXWLRQKDVDSUREDELOLW\GHQVLW\IXQFWLRQVHH*UDGVKWH\QDQG5\]KLN>@ PI+[/ [+D/Æ  ccccccccc_{E [} cccccccccccccccccccccccccccccccccccccc *+D/ED 

IRU[ !  SDUDPHWHUV  D ! +VKDSH/ E ! +VFDOH/ >@FDVHIRUWKHMRLQWGLVWULEXWLRQJRIYDULDEOHVEDQGU

,QVHFWLRQ>@XVLQJWKHSRODUWUDQVIRUPDWLRQZHKDYHREWDLQHGWKHMRLQWGLVWULEXWLRQJRIWKHYDULDEOHVEDQGU

J Æ

FU F U &RV#EP'U U 6LQ# E'F U +F 6LQ# P' U 6LQ#EP'/

Information Science and Applied Mathematics, Vol. 16, 2008, B.I.I.S., Senshu University 12 P[J_{+U E/ Ã}  ˆ J PI+[/Å[  ccccccccccccccccccccccccccccccccc S r  U  D_{D E U} +  U_{/ s +  E +F}_U_{/   U}_{ E F U &RV#E  P' } E U_{U6LQ# E'  E F U +F 6LQ# P'   U 6LQ#E  P'//}D_

IRUU  ˆ E S S 

On the generalization of von Mises distributions on the circle _1 ,QVHFWLRQ>@XVLQJWKHGLIIHUHQWSRODUWUDQVIRUPDWLRQZHKDYHREWDLQHGWKHMRLQWGLVWULEXWLRQJRIWKHDQJXODU YDULDEOHVDDQGE J cccccccccccccccccccccccccccccccc ccccccccccccccc  S r  U_V H[S+'/ F _6HF#D'_6HF#E'_ ZKHUH ' cccccccccccccccccccccccccccccccccccccccccccc +  U_{/ V}

F_+7DQ#D'_7DQ#E'_{+U &RV#P'  6LQ#P'/ 7DQ#E' 7DQ#K' }

+  U 6LQ# P'/ 7DQ#K'_{ 7DQ#D' +U 7DQ#E'  +&RV#P'  U 6LQ#P'/ 7DQ#K'// } 7KHMRLQWGHQVLW\IXQFWLRQRIWKHPL[WXUHGLVWULEXWLRQRIJLVJLYHQDV P[J+D E/  cccccccccccccccccccccccccccccccccccccc  S r  U L N MMMMDEF_6HF#D'_6HF#E'L N MM ccccccccccccccccccccccccccccccccccc

+  U_/+E F+7DQ#D'7DQ#E'+U &RV#P'  6LQ#P'/ 7DQ#E' 7DQ#K' 

On the generalization of von Mises distributions on the circle _1

IRUSDSF!

,WVKRXOGEHQRWDEOHWKDWWKLVPDUJLQDOGLVWULEXWLRQLVHTXLYDOHQWWRWKHZUDSSHG&DXFK\GLVWULEXWLRQ:&PUIRUT

DJLYHQLQVHFWLRQLIZHVHWP U +  F_/IRUF_{7KHUHIRUHWKHGLVWULEXWLRQZLWKWKHGHQVLW\PWLVD}

JHQHUDOL]DWLRQRIWKHZUDSSHG&DXFK\GLVWULEXWLRQ:&PUIRUT DZLWKSDUDPHWHUF!

)LJXUH   7KH GHQVLW\ RI WKH PDUJLQDO GLVWULEXWLRQ RI D ZLWK Q  IRU F ‘ s    +EROG SODLQ GDVKHG/

,IZHVHWQ LQP[JQDELWVPDUJLQDOGHQVLW\RIDLVJLYHQE\  r F &RV#D'

cccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccccc

+  F_{+  F}_{/ &RV# D'/}s IRU  Ss   D  S s  F ! 

)LJXUH   7KH GHQVLW\ RI WKH PDUJLQDO GLVWULEXWLRQ RI D ZLWK Q  IRU F ‘ s    +EROG SODLQ GDVKHG/