Transcript
A Fine Grained Heuristic to Capture Web Navigation Patterns Jose´ Borges and Mark Levene Department of Computer Science, University College London, Gower Street, London WC1E 6BT, U.K.
ABSTRACT ! " $#% !#% '& $ ()*+ +," +," -./( + +, 0+,& $ ' & " 1#%& ! #2," !+,& 0#% 3)4%& 3 "
+," '" 5 . +," 3 + " (-56 7#% -
" 8./(/9 (: $*&; + +0 " " +,"6) .. *=>?%@-A'#%&" & " .#%" +,&" )+,& ' 52 . " " +,"61 .. $4%& +$ 5&" & +$ " " +,;:+,"
0
+ (B;)+,& 1 9 . 2 , (++,& % 5 (' "
+," *+, " %C1 & D " , ;. (/+ ( < " . +D#%" +,&.2ED (+,&9,F3" + G &!
" +,&H=E%F G A+. 5 I+,& . < & +," $ 9 + +," ) 5 +,& '+,"
'#%" +,&* " "6+,;) $0, " J ( +,9 " +#%&" &'#% 2 +,& KL/M NO3P2 +,& &$+,& 2
" +,&$Q " !+," . $ " . " ;B#%" +,&)+,& '/- $ 5- .. + + " +2& +,& (/ #% - 58 +," 0
/- - 5 $#%& B+,& $+,9 " +'" '. - (B1. - +' 5- ; ,& +2 #%& 0+,& -+,9 " +2" 2&" &3 +,&" %#% /#% % +% #)& " +," -+,& +D" . +2 $ " + +," '( " 0 &*#%& " *+,& . + 5D " 2" 9 . + ;'
. + (.;-J+D < " .+, " %#%" +,&'&" &' 9 " " +,; *PR+ " 1 . + ." - " ( ()#%&" &!. , +,& %(/" + % +,#% .+,& D +3 9, +D (-" +8 , (/9 " '. < " . 8 +2 + " ($;'+,& -E%F G
" +,&$%C!& 1+,& + " 1 . + '+ / .+,& ' $S 1+,& .& " +," 0 9 , (-+'+,& -E%F G
" +,&T (0 +,& - . + -+ / - -+' +,& /- - 5D + " (1( (/" ; < " . +'#% ' (/+ (!#%" +,&! +,&* 2 (!; +,& +," U ( + (+,& 8 , +8,& #*+,& +35 82 " .+,9 " +D+,& 8/9 2 5 %" (/ ('" 2./ +,& ;'#%" +,&$+,& ( 5 +,& 0+ " !" + " 374%& 5 D;! ++," *+,& 0 0 5 +,& '+ " 1" + " )+,& . ;/+. !( + " $+,& /9 - ($V " +,;1 53 -+ " (/ (W3+,& -V " +,;0 5D " . , ($;. +,&0" +%
+,&* (' " "6+,;
Keywords
'" " 8 "
+," ) ++ 8&; + +- " "6+," X
..
1.
INTRODUCTION
4%& $ " ()" 1" 7+,& 0," S 1 5-+,& 1#% )" ) +$; & +, ,5 . ($" +%" +'+,& -
+2 5D " ' 9 " '( +-P+,&" -+, (1 +," / '($+,& (/" Y' +," .+,& %5 D" $+,; " .+-J(. +2 ('" + +," " 5 . 9 +," 1 1+,& #% 3%Z0 "
+," !" 0
-" 5 . +," +,+, 5+ * -+,& +' . .(/" " + (7[ \]W \^ _,+,& +2" 3#%& 05 #%" $ " +,& - -+,& . + + " *#%&" &1& " #2," 0 (0 (- ," + ." 1J(/" $&" #% ; PR &!(/" +," 7#%&" &)( .#%" +,&!+,&" H (*" " -, (.; ." ," /- % 5 & 8" D#%
-" " 1[ \` _,84%&" 2 &$J ($5/ 2 0+ &" V #%&" &! . 1( +0+$J(1 ++ " !+,& . "
+," ," %4%& % "
+," 1 ," 2+ / 2+,& %5 a 53 V 53 " 25 #% (';'+,& 2 #%&" &0#% %bK,c/d M O/+,& +%& 9 + " &" #% 0 "
+," ! & " X ( + (/" 0 "
+," B & " '" . ) +," + *" )+,& ./ . 5 + " ," 0#% 1," + +'+,& Q 2 (- " +,& ;'" 9 " *" +'+ +," 0+,+, 0 e ();* " (/" ) ( +," 1#%
0[ \f _,*Z0 %," $ 1( +1" . + ()" B$ # 5 . +8" +D" % '" ( 3+
+D5 D " . ;/ (';. + . + ( +/ )gD +, ;! .. " 2 * ;/," $+/ ' " $[ h ] _,W/& #% +,& -+/ %& 2 " " + (1 ;/," 9 " " +,"6 /(/" $ ;. , +,&0 %,.. ;'+ +," +," (*56 V ;* +. 52
$ " ," + !+,& $. +," . $+,& &7 .-" +,;!& . 7+,(; " )( +*" " )+ &9 " V ++ / 256 3 ( +
- 58+,& 2" 5 . +," 1 " " ,&0 'J 24%& -& 5 1+,#%.. " 1 & +-" " .5 D % "
+," 0 ++ D56 : .( + '+,& J+% &$ ( +-" 2. (' +- +," D+ % ( . ( + (. ," ' 5+ ( (.( +" " + &" V ,& " " $ /" +," * - + " 0+ &" V 3 -" 9 / ( -5 2 $[ ` _3 (1[ h h_, 0+,& - (0 & + &" V . ( (!#%&" &! 1 " / (*(/" +, ;* +,& '( + 5 '[ \_,[ h_,[ \i _8 (*[ \ j_, 4%& #% . + ('& " $" D +2 53
" . &0#%" +,& +,& '+ k
8 58, " 5; " $. +2 5D+ &" V +" ( 9 +," 5;' +-#% 0+, " .[ \ W8\ h/Wh ^W3h/W^ _, ![ h_8#% + ( $ #R./( %5 & (/ " !+,& . H 5" " 1 1( +
G @l-mm
U < +," -g% ; " &+ n h ] ] ]poq-rHstu.v.wwxy ;$h ] ] ]
zD . .h/ , .\9D
-f ]
#%& " B+,& $#% *" ../( (B .0 $ .. '#%& + + 2 , (1+.#% $
2 ($/(/+," 1 +&; 9 " 24%& 2 % "
+," 1 ," %" + ($" + +,& %./( /" . ( D+2" (. N Kb N /b K
c d M d Obd /K,
c c/K = ," ;1.>?%@-A% (0( +." " 0+ &" V - (0+ J(1+,& -&" & " "6+,;*+,"
-#%&" &* , (*+'+,& Q - 5 (1 "
+," 7+, " )[ ^ _%#% .& ' + ( ( +,& >?%@: ;/," " "6+," ;$( " ," 0& " +," '+." 9 (/ - +D 58+,"
2#%&
" $ " " +,;$" % 2, " J ('+,& ,& (D4%& %>?%@!./( / " ( D," 2 ( (1+/ %+',.. " .+,& " + +," !#%" +,&*+,& #% ('#%&" &'" D + +," ;' 56 +,&$+-+,& 2#% '," + ( ," (0+.+,& " (/" " (/ D
!+,& $ .& (D;1," 11>?%@k+0 ;/ 0 .
+,& .#% *," + .( ," ' 1 J+-56 k& " 1' ++ -9 ( + (/" $ 5D+,& 2 Q #2," . & " #%&" &0" 2& 9 + " (1;0+,& . +- 5%. + .+, " X ( + (/"
+,& ' . 5 " .1+ *5 #% (!+$ " ( 0 ( +," #% $
2+" +,& -," + +,+, - (/" $+.+,& ," +," G &0 +," 2 ' 5 2 D+ " 0#% *
8+$" $+,& .
'+," . '$ , (D" $+,& %," + %+2" +,/(/ #*
D" . %#%&" & . / $+,& I&" & ;) " ," !48$" $+,& $
$+," . ' $+ ;/. )+,& $," + $ " $ ! 0 + (* +,#% !
5% '+, " 'Pa #:/(/+ 1 ' " !
//(* < 9 , ;$ " $ " +'" +
56 . .+, " %+,& + " ( . + (1/(/+0l #%" 0+,& . '+, " . 0& )+$" ( +," 5;*+,& '. + .. ! " +.#%& . + " + +,& " - ," .+,& +2+,& - + +2 1+,& " + 0 " ( .+,& 2 +,& D& (/>?%@) . %" . + (' D2 #2 9" B+0" . + ;)+ ' (! ;/ 0+,& . Q " (/" 9 " (/ "
+," $&" + ; G &'->?%@)#% (. 2 9 + +," 0 58+,& % Q D/ #% (
53+,& #% $ (' (0 +D % . . ;. " (/ ;/ ($" $ ( 2+" 5 2+,& 2 % 5 ( +, " 2 .#% * .$ (/" +," 7+/ +$ 5 +&!" ) (
-+,& - -. ;' -" + + (0" 3-E%;'& " 0../( D 5 &" %" (/" " (/ D/ #% (
- 58+,& #% $- #% (' -+ (e &
-&" ./( 8#%" +,&0+,& 53&" 2 (35 8" ( +," 5;1 Q 5 (*+, " #%&" &! / #%1+&" $ 1[ h_3#% 2( " (0ED (+,&9,F3" + G &*=E%F G AD
" +,& +0J(! +,& $+, " .& " * J( 0 (), +. -, " J (1+,& ,& (P2 +,& &*,&1
" +,& " ; Y'" +." +'; " ($ ).
1/- $ 5 '" 5-+,& J( " +2+/ #: (0.. D +2 5D ;$,& +% " 5'+,& 1 J( *" 0 +0+/*&" &3IPI
1 9, +$ " " + +,& . " " +,;1 5D" 0+,& .
" +,&I" *. " *. . ; 3#%" +,& +,& V +( ( +," 1 53" +D 5 . (' .( 9 . (0+,& * < " + * 5
*+
*, * (7( + '+," . B4%&" ." . +," ;B" + +$#%& )+,& 0 9 ;/+$" $" + +," ;: < " . +," 7#%" +,&7 " 1(/" + ./( 2 J +," !4%& . 5 (!(/ #% . 5+,& 'E%F G (*+$" +,"
+ 0& " +," '#%&" &) #R+0 + !, +$ 5+,& $ 9, +0,&7+,& +%5 0* " : J( D#% . . '+$ +, % +,&*" +-," S ' (1 .
+,&3 4%& % &. + (-& " 3/#%&" &'#% D +,& N /K,c/d N & " +," " . + '" + +," ( " ' &$#%& " +,& . + 5D -" " . + ;*
. + (1;'J+- < 9 " !+, " $#%" +,&)&" &B " " +,; aP + " * . + $" " ( (!#%&" &!. , -+,& .(/" + ' +,#% *+,& . +
G @l-mm
9, + ($" + , (/" 0. < " . % +2 + " (0;$+,& E%F G
" +,&$ C1 ' + $+,& +.+,& '>?%@ ./( 2 , ($+0+,&) ( Z0 / 1./( 3 $[ h_, 2 +, ; 8 % +,& -& . ! ," !+,& . $ 5Z0 / !./( .+1+,& '+,(;* 5 0#% B "
+," a & " \ ! _+,& 1 +,& 0 9 a[ (1$;/+ #%&" &*" (1+'( . +, + .+,& +," " +,;1 5 Z0 / *& " B./( ." ! " ) (/" +," B (! +,&) ;/," *+,& #% 3 U < " . + % , +- + (1#%&" &*,& # +,& +$Z0 / 1./( 2 * 56% +,&*" !+,& . (/" +," 5&++,! V +' (!" B+,& $ (/" +," 7 5-+,& $ + " B+ - V + ( *[ f _8+,& - +,& 2 ... +,&/(
;$5 + " " (/" " ( 6 " $+,& 2 +," 53+,& " D % 9 "
+," 1 ," ('" ![ "_.. +,&/(
;'5 " , " ," $+,& "
+," 1 ++ %& + " ," $ &$ + 2" D + ( 4%& . "
+," B & " 0 5+,& . " + (*;* Z0 / 0./( 8" *#%&" &*+,& -
- ," J (*" +' ( 9 J ($ +
" %P% (0++,& >?%@7./( +,&" 2./( (/ ' +' ;/ 1+,& $ !( +1" B" +.J +' - 5( + " > #% +,& - (0+ &" V - + +," ;0 5635 - .( + / ," $+
2" 0 ( +-" (1>?%@B 9 +," $+,& "
+," * & " 5D. 1 58 #%" +,& ," " ." + + * ((/" +," 3%+,& .>?%@:" -" . + D < 9 + ," - ( " ( ('#%" +,&' Y'" +D
" +,&.2+ ;/ " + "
+," !+, " G +," ah0 +$ 7 " #I 5>?%@2#%&" G +," :^ +0+,& 0 (B& " +," ! (B( + " 1+,& 0 , +$ 5 5 . () < " . +BF3" ; 2" G +," 7f1#% $ " 0 (/" $ .
2.
THE HPG MODEL
PI !J * ," +$" :* 9 1 ( (7 +$ 5.#% B
V +.56 T#%&" &)" +" ," 1+0" 5 . . "
+," ," G " '" +-" . < + (!+,& +$ . " ," +'0#% *," + . -+,& 0 2 "
+," * ," 0" %, ;0( J ( %- V - 5
V +D56 a+,& . 2 2,&$+,& + '+,#%' +," V +- + (0;$. +,& $ # " /+ 8#%& % #T" -$ " 1 . + ![ ! _D+,& . +,& (.5 #k+,& % 2 53 h " & "2" /+ #%&" &. , ( +' \ & "-+ ( (0( " +," 58+,& -
-+," . 2 +,#% $ " + 5 V + G " .+,& 33. ;0 +,& & . ( + ( +,& - . 5D^ ]." /+ 5 #3D48 &" V -+" 5 9 "
+," * ," 56 a '( +' -( " (0" ![ \ \_, PT +," k 5 1 "
+," R ," 1" 0./( (a 0 N KbN/b K c d M d Obd M c /Lc' /N'[ \ h_
+ (*;* N K b N /b K
c d M d Obd( /K,c c/K0= ," ;!>?%@-A.[ h_%#%&" &!" $, ' 5 " " +," 1 $ .. 0[ h/\_,*P >?%@)" %- " " +,"6- .. 2#%&" &'& %- 9,+ 9 . " ' +,#% 1+,& + 58 + " % (0+ " ;/- U &) + " ;/- - , ($+11#%
$ (*$/(/+," B ' , ('+0$ " ! +,#%
$Z0 8+,& ' '+,#%$ ((/" +," +," J" + + ( *#%&" &1 +-+,& + + (0J" ,&!+ + 5%+,& ) "
+," ! ," 24%& " "6+,;1 5D. .. +," '"
" 0;'+,& 2/(/+ 58+,& 2 " "6+," - 58+,& /(/+," (." '" +%( " +," 32C1 2 3+,& %/(/+," #%" +,& ) '" + 5+,9& ($," ( 'Obc/Kb K '/+ L bd O8 ($#% 3+,& /(/+," , (/" '+ " % +,#% .
-bK,
c Od6bd , N K '/+ L bd O
U < +," -g% ; " &+ n h ] ] ]poq-rHstu.v.wwxy ;$h ] ] ]
zD . .h/ , .\9D
-f\
F H+,& 0 +," a 5- $ "
+," : ," 0#% 0 + " +,& ./- ' 52+," . .$
'#% - V + (D+,& ./- . 5 +," . " +-#% +,& -J+-+ + -" !' ," 38 (1+,& -/- 5 +," . D" +D#% D+,& % +%+ + %" $- ," 3%4%& %/- 2 53+," . 1 V 1 5-+,#%$
' '" B+,& $ ," $ " '+,& /- 5%+," . -+,& . , (/" 0 " *#% +, (C1 ( J -' . + .#%&" &! ++ & +,& .( ," (!#% " &+ 5D'+ + - , (/" 0+.+,& -J+%
- #2 ($" * "
+," ) ," 3 5 ]' ;0+ + 2#%&" &*#% -+,& -J+ " B B +, - ," B& ' " "6+,;B + $+,& BS 1 5 " '" 1.+ +%/(/+," 3 0+,&" - ;$+,& +,"
#%&" &!+ +-#%" +,&*$+ + '+,& +-#% -+,& J+" !$ ," ! " (/ (0;'+,& - .. 0+,& - +,& +, . " 5 \-+,& " "6+,;$ 5-+ +8/(/+," 0" 8 +," 8+-+,& 2 9 8/- - 5D+," . +,& - , (/" 0+ + #% 2 " ," + ( +,&" - +,& ( +," +," !/( . 5D./(/+," !#%" +,&1&" & " "6+,;0 , (+..+ + +,& +2#% % " ," + (1. - 569 + 3!4%& ' . + )+ / 0 ;* $ +,#% B]1 ( \ " (/" 0. +,#% 1+,& +,#%. " -( " ( -P,&3
" -+,& ;/++,& " " +,;*+'+, +,& ./( 5 D+,& 2 &. 5(/" +%+,; D 5 ++ 8" '+,& D "
+," B ," 'F3" ; D+,& . " " +,;! 5%$+, ," +," /(/+," *" 2 ," (0" 1,&1.#% ;'+,& +%" +2" % +," ++,& %56 V ;'#%" +,&$#%&" &0+,& 2 , (/" ' " $#% D+, 9 (
( ) *! + ,- .$ ! + / - $ ! + $
+'
0" 7! ," 3- 0 7+,& 1#% ;!+1
2 ( +,#%" 1+,& #% ;$+.
24%& 5 +,& - " "6+,;1 5 +,& $+, ," +," 0/(/+," $#%" +,& $ )+,& " . 5+,9& ()," ( 3= A h f 3= A \ f ( 3= *-A \ f + +%/(
)
m
^ f
!
"
0.17 0.08
0.25
0.5
A4 A1
0.25
0.35
0.25
0.5 0.6
A5
A2
0.75
A3
0.2 0.4
1
A6
0.04
F3" !\1P \- (
F
0.4
=6A
+' 5-+, " 0 ()+,& 0 , (/" B>?%@k5 +] & "/
F3" .\-=/AD " % $ - 53- +," * 5 2 ," (0F3" .\.=6A% " +,& , (/" 0>?%@/ + ( ;.+, ," +," 1(/" $5 ]+& "/DPB>?%@B" %-5 9,+, #%& *-'" ) ) +,& ' + 52 + " -;/- " 2+,& +D 58+ " 8;/- ) " 2+,& 2" V -+ +%;/- ( " D+,& % +8 5/(/+," $ D4%& D/(/+," ' D 5 +,& >?%@:" 1+,& . " 1" *F3" 'h/24%& 9 +," $ 5 "
+," 1 ," D" $+,& 2 + " %-+ + 5 h f-
2 V +#%& " 3/5 % + + D#% D " ," + ( f-+," . h2 5#%&" &' D+,& DJ+D+ + %" '2 % ," 3D4%& 9 5 5 +] & "2#% 8& 2+,& +8+,& 8 " "6+,;' 5" +8+ +3 9 (/+," *" 3= ) A =+ ] & " f/A h f 7=+ ] & " /h A '! +] & h "/ Z0 D
'#% . " ," + ()f1+," . 2\$ 5#%&" &) .+,&
!
G @l-mm
" '
# %$ &
# %$
)
+, ," +," /(
%
J 3/(
] h" ] " ] j " ] h " ] f ] h ] " ] h " ] ! ] f
*
] h" ] f ] h" \ ]
* * *
F3" h D4%& %&; + +% " "6+,"6 .. , (/9 " '+.+,& - +," ! 5D+, " 2" 1+,& - -5 ]+& "/
4%& 0+,"
$
+ ();!+,& 0 .. $ , (B+*+,& . "
+," :+, " 2 ()+,& $ " T" '+0" ( +," 5;)+,& $,9 +D 58+,& +,"
#%&" &0 , ($+-+,& 2 +,& +D + & + " %+,& 8 8 & " %#%& " ," +," -+,& D," + %=F #a 1#% +,& .+ .-+, " % (1+," '" + &
6; A Pa+, " D" " ( (*" *+,& . .. Q
." 5%" +-( " 9 +," 7 " " +,;7" $ *+,& Lb /d b6 a4%& 0
+,9 " +" (* 52+,#%$(/" +," +.+,& ,& ( / #%& =] \ A2" +,& $OL /Kb8+,& ,& (* ( =] \ A2+,& /'N N-+,& ,& (!4%& ." +," +," B &" (B+,& ', +." +,& +%" +%" 2+,& 5 + 58+,& -+,9 " +2 , ," .5 %"
+2+,& +,"
#%& 2J+2( " +," *+ '& % #B " 9 " +,; / , (/" .+-, +% 5+,& %&; + +D;/+ : ; " ," + ( + +,& +-+,& ' 1 *+ *+ +/(/+," #%& B" '+,& 0/- 0 5 .. 0+ + W+,& 5 +,& " " +,"6 5D+,& -/(/+," - 5D.-&1. - 9 ( 2+,& 0+,& 58+,& +, ," +," -/(/+," -,
+," '+,& + +,& ', +. 0,& (!+ / '" +1 ++,& '/- . 5 + + .+,& % < " . +D + (" '+,&" D#% #% 8J (.+,& , +2 + \ 3D4%& J( -" %+,& 5 + 2 58+,& +,9 " +. , ," 05 " * +.+,"
.#%& $( " +," + " -+, ," +," ./(/+," #%" +,&*. 8 " " +,"6 ," " " +,;!+$+,& .+ ( (* /" +," ! '" " 1+,& .,9 +D+,& ,& (.& D+,& D 5 +," +," +,& 2" " -5 8 +-+,& 2+,"
%#%&" &'+ +D +D+ + 8+,& +D %56 V +, ;" +,& "
+," * ," 2Z0 +,& - J( +,& ,& ( & +,& 58" $ +2+,& -+,"
-#%" +,&$ #7 " " +,; +,& +D ('+,& , +%+ +D4%& 2 2 53+,& , +% ( J( -+,& ,& (2 " +,& % 2 +, %+,& 2/- +,& 2
+,&$ ('+,& 2 " " +,;0 53+,& 2+, " 2+- %" ( ($" +,& 9, +
10 3 2 #.4 4 5
*7
2 $.7
0.25
0.25
0.4
)
'6 7
0.25
0.11
)
2'5
=/A S
)
] h" ] ] i ] \ \ ] \ j ]
^ " ] ] f
0
43 "
\ h
)
C1 0 + 1+,& +$+,& 0 (7./( . / $+,& 1 ,+," +,& +.+,& $& " 0 5-+,& ' +
$+1 " ," +'( ($ ;! +,& $ +
$" )+,& ' Q #2 ' (!+,& +.+,& $>?%@ , (+Z0 / '& " 3 +D5 #22+,& +%+,& $ +,"
'" B+,& 0 .. Q
* , (/" )+*+, " +,& +%#% 2 +%+, ($ %#%& ;.- DP2 +,& &0+,&" ,+," 1" . #%& +2 +, ," 8" +%" ;' " " +,"
+*./( ;B+,& 0 +' 0 ," 0," 1+,& 0 " 9 " +,;$ 58 '+, " " '5 #% (0 D+," . %" 1 +, ;0+,& . '#% ;$" #R (3" ) ((/" +," B+,& (/" +," )" & ( Z0 #%& 7* $" '" , +," B*#% B
1 (B( 9 " (/" !#%&" &! " )+$5 #k +. ;15 #a" 5- ;* 5 +,&
U < +," -g% ; " &+ n h ] ] ]poq-rHstu.v.wwxy ;$h ] ] ]
zD . .h/ , .\9D
-f/h
25
7[ h_#% ' + ()+,& ' , +$ 5-+,& $ < " . +$#%" +,&B
" +,& #%&" &BJ($ .. 0+,"
0#%" +,&B " 9 " +,;1 '+,& ., " J (!+,9 " +$P2 +,& &!+,& '
" +,&$ #%&" &*" $./(/" J (* (+,&9J+ &33" - ;1 Y'" +-" + & %+,& (/ #% ' 53 + +," ;$ +," .- ;
/9 D 5 35 D. D 5+,& %+,9 " +D ( " . ((/" +," 3 +,& . . '+/$,& +.#%& !+,& .+,9 " +." $+$ C!&" $F3" $h$,& #2-+,& . " +," ) 52+,& /- . 5% #%" +,&*+,& J( '+,& ,& (*F3" 'f',& #2-+,& " +," 5 +,&'+,& D
-=( P 8A (. < " -k=Z 8A %
+,& #%" +,&$+,& 2 . +,& ,& ( ' ' 2 $+,& +8" $ ( 2+- 9 + " 0 ' +,& +,& ,& (0& + +2#%" +,&1 #: (!+,& +- ('+00 ;0
'/- ' 52 *P2 +,& & " +" ," 1+$ !+,& ' " B$
' 9, +.;1+,& "
+,&) (* " " +,;/2 -;* . ' +,& ." + " D" ) ( +." ( +," 5;1+,& - + 3+,& . " +," B 5D.
. 9 +. " " +$+,& 0
" +,&$Q 0 " " +,;B+0)" 7. " B. . ; Z0 3+,& -( + . -+," . - ( (1+'+ -.
9, +- * ' ' " " + +," )#%& *+,& ;/+-" " + +," ;! 5 " 1 +," 0+ +#%" +,&! 2(/" 569 5 +-./( 2 J +," 14%& . (1+$+,& +,(;* 5+,& '& " +," $ + (!" )+,& . +. +," )#%&" & #2-+' + ..&" &* " " +,;*, +- 5%+,& 9 +#%&" . " $ '+' +, 8 +,&1" +-," S . (1+,& -
+,& 58" +%
3.
FINE-GRAINED HEURISTIC
3.1 Iterative Deepening Heuristic 4%& " ( $ &" (*+,&" -& " +," ." +." . +. Bd6bN K,c/bd, N /NN 'N d . &B= -," ; m-A#%& " !+,& 9, +-" -" 9 . + ;'" +2+," 3" +D& 2+,& ( ," (0& + " +," 248 +,& +$ +BOb d c/K,
c N bN K'" ' " ( (:#%&" &7. 9 , -+,& (/" + . +,#% 1+,& - 9, + " $" +- (1+,& 'c /d c M%KL/M N ,ON b64%& '. < " . % 9, +" +,& ' + 5% " (/ (';-+,& 2E%F G
" +,&$#%&" &." ( 2 +, " %#%" +,& " "6+,;1 +,& +,9 " + ($" 2 (0 2 5 5 +,& & " +," Q , +'4%& . ;/+., " J +,& . $ 5% Ob d Kd6bN Kd /#%&" &1#% 2 2.+,& ,& (05 +,& -+ 9
G @l-mm
n=1000 n=5000 n=10000
20 15 10 5
0 0.1
0.3
0.5 Confidence
0.7
0.9
F3" 0^+0gD& + " +," $ 5+,& ' 9, +." (/ (B#%" +,&)+,& E%F G
" +,&$
BFS - synthetic data 7 MRL, n=10000 ARL, n=10000 MRL, n=5000 ARL, n=5000
6 5 Length
BFS - synthetic data 30
Num. rules (x1000)
" , ;7 " ," + ()
$' +.&" $( " ," 3Rg% ," ( %5 21 . #2," * B " 1, . / +'+,& +.& +'( " ( .#%&" &1 " *+.5 #a#%&" " , +," 0.
.#%" +,& , " -' !>2" .( " ," 7#%" ;)( (). $ +,& 0V " +,;) 5+,& $' '+,& . $+,& B B -+,& '
& 1 " ," + (B 5 & ( B+,& 1 +,& '& (+,& 0 " 9 " +,;0 52&/ ," 0' " 1" - +- + ;0" ( ( +' 5%+,& #2," '&" + ; %4%& 5 './( 8+,& +% (-+,& +,#% ./( - 5D & " " ( . ++ - + +," ! 5%+,& Q 2 "
+," * ++ 4%&/" ![ h_8#% . / - 5D+,&
+$[ j_,D#%& \ D( + " '+,& ' . . ;$#%& 0 "
+," 0+,& #% 3" ; " 0+,& +#%& 1 " ,9 " +," *$
$ ;*+,& " , ;) " ," + (!
" +,& . " !+,& . #%" 2&/ $+'5 #a + *
&1 5D+,& -+ + 2 , (1+.' V 5
" ,9 " + (1 (/" 1+$$+, ( ' * +,#% !+,& ../( % ;1 ( " + < " +,; 8," .#%& 1+,& ( - 5%+,& ./( 8" *(/ '" +'," S . , ();!+,& 0/- $ 5+ + 74%& ( " ," ! 58#%&" & +.&/ $ ," + ($;$+ +," +," + &" V 3,&0 ( + " (0" *[ i _,
4 3 2 1 0 0.1
0.3
0.5 Confidence
0.7
0.9
F3" 0f+0gD& + " +," $ 5+,& ' 9, +." (/ (B#%" +,&)+,& E%F G
" +,&$
" . + D4%& % &5 8 D+ 8#%& .+,& % 2 5 +,& + " ' . + , -+,& - . 5D+,& + "
" + " 3E%;' ++," '+,& 5D+,& + " $" + " *+,& ;/+-" " ! +, % +,& -/- . (0
+,&! 52+,& ' + " (W2+,& ' 9, +. + " (!" .0 ', +. 5 +,& -. < " . D + 4%& m*& " +," #% 8 35 #2/ 0N M /K,c/bd bK,NND" 3" + 56 R+,& -+ +2+ + - ('" 1+,& 2J+%" + +," * ;$+,& -+, " #%" +,&)
+,&B 0 $ < (7> " 1" +$+,& 0+, $#%" +,& ( +,&1 -+,& - 5D+,& -+ " ' . + " 2 + ( (B + (B
" +0+,& 0+ " !" + " 3-, " J (B; +,& - ;/+ 58+,& -+ " $" + " 0" 2. +2+,& < +," + +,& #%" -+,& ( +,&0 53+,& +, 2" D" . + ($;. (. +,& %+, " D#%" +,&
+,&'+,#% % < (D4%& %+ "
. + 8" 8+,& 9, + (. ( (
" '#%" +,&.+,& + " 0" + " 3W8+,& -/ - +," / +," %+,& + "
" + " 1" 2. + C!& !0+, " D" - " 1 < (! ;0 +,9 " !56 " + + + + -" + (*+'( + " " 5D" +2" )c d OOd M N-N c + Od $ - )d
c d OOd M N N
c Od 8P:+, " % < ," *" (/9 " ," *" 5.#%" (7+*!+, " -#%" +,&B " " +,;: 1+,&
U < +," -g% ; " &+ n h ] ] ]poq-rHstu.v.wwxy ;$h ] ] ]
zD . .h/ , .\9D
-f ^
+,9 " +- ($" (" ," 6 ." 5D +%Z0 '+, " 3" %c' d c MD#%& 1" +& -$ (" ," $ < ," ' (1$+, " D" - c /d /c/bN2#%& $" +%& +% +2 (" ," ' < ," 3
unexplored link
= A'+ ($5 $+,& 1, 5+,& 0 " "6+," * 5. +, " " #%& / + *+,& +0] = A \ ," '+,& .,I 52+,& " " +,"6 ' 52$ + 52+, " 3 +-" 9 (/" ',9,+, " " $ ' < +," 1+, /#%&" &' , ( +00+/& +," 0+, 3" . #% ;/ . . V +*\ ! ((/" 9 +," 3 = A = A2," .#%&" +,&1 +2" ( '+,& . < " . +, " .#%" +,&!
+,&! . . V + " ( +,& . (/" ( + $+, " .#%&" &! J '= (1+,& 5 .& &" & D " "6+,;AD 5+,& " D" (" ," 6 - < ," D" ( ( " P2 = A = A2," . ;0+, " 8" " - 2 J<. 58-. < " . 8+, " " 8F3" ; 5
&
6
5
4 & m7& " +," " + +," ;0 + % 5 ( /\ h &&& +," " %
//(' &0 < " . +," * 5 DF % " 1#% . , & #: " 56 ;$+,& . ' 5%+,& + " . + /#%&" &'" %( J (' D+,& % . " " 9 " " +,;$ 5+%+ < %4%& + " . . + 2" % + ( #%" +,&1. 9,+ $ / & (* ('" ( J (0
A9 A4
7 7
6 + -+,& +2+,& - . 58+,& + " ' . + 2" ( J (0+
+,& ,k 53+,& 2 " "6+," - 58+,& (" ," . < ," 5+,& % (/" ( + 2+, " D4%& D < +," $+ 8#%& .+,& D+ 9 " . + D" %,Y'" +, ;'. W#% 2 ;+,& +%/- %" ,Y'" +, ;1. 3" 5D" +%" 2 +,& 1 . ( J ( ] (1+,& 'Ob d Kd6bN Kd 8P2+2+,& - (0 5D+,& < +," +,& - 9, +2" " 1; ," +,&1+,& . < " . D (0+,& (/" ( + .+, " - - ," ( (0 #%& 1+,& -+ " $" 9 + " )" '. +*F3" ; 2 + $+,& + *" ' ! ' ( +," . + - 5D+,& /- - 5D+, " +,& +2 1+," 3 " (
$.0
F3" ("D +8 . 2 5 < +," $+, %+2 " 5; +,& D +3" +,/(/ (" +,&" 8 +," 3 -+,& 8J %% " 8 5
G @l-mm
0.5; 0.5
S
0.1; 0.02
0.9; 0.225
A16
A 10
0.1; 0.025
A17 0.6; 0.24
0.5; 0.5 0.8; 0.4
A2
0.9; 0.22
A 11
A5 0.4; 0.16
A 12
0.1; 0.02
A18
0.2; 0.1 0.9; 0.09
A 13
A6
0.1; 0.01
A 14 k=2
F3" %" U / < 58 0 < +," *+, + " (0#%" +,&0+,& h ( ]+& \'"/ m7& " +," 5 (
0
/- D +%+ " ' +%+,& 2 " " +,;0 53+,& % " 5 #% ($;$+,& " " +,;1 5D+,& -+, " 8 " " $" 1+ + ) F h8 ( +] & '\ ", 5 - 5 " $+,& - / & (8+,& +8 53 (/" ( + +, " D" / ,8C!& ' ' 9 + 0 / & (1" 5 . (056 k+,& - +-+ + 5%+,& 9 (/" ( + .+, " " +2" " J (*+,& +2 +,& 3 ( 3 " (" ," ' < ," 8+,& 5 " .. < " . D+, " Z0 +,& D / & ('" ( +," J 2 D ." (" ," < ," B 5 D ( D - ! (" ," 0 < ," ) 5 +,& $ . 0+, " %" '1 (/" ( + 1+, " BF3" ; 2 +,& ( 8 " ( +," J (! (" ," $ < ," . 5 3. " '+,& + " 2 '. (/" ( + .+, "
0
%
%
% % %
+
%
h0 $ +,& $+, " '#%" +,&!
+,&) ' ' V + ," J (*#% - 0 + +,& 5D+,& + " ' 9 + D48-+,& +2 + = A3 +2+,& " " +,;0 58+,& + 5%. < " . 2 (1 (/" ( + '+, " #%& -
+,&*" - V 8+'h/ ($& 2+,& -
A += % A & =+% A & =+
A ]+& h"%&8]+& h"%&8]+& f ]+& ` * ((/" +," 3 = h A" +,& " " +,;! 52+,&
+,&! 5
PT ++ * +," . + *5 7 ! + (a#%" +,& 9,+ / & (8#%& " 5 0" 2
. & = A = AW & #% +,& - . 5D 9,+ 0 / & (!#% ($ (1+''( 9 -" $ 5 .
7 '7
A15 0.9; 0.20
0.5; 0.25
A1
A = A& =
A8
0.5; 0.125 0.5; 0.25
0.5; 0.125
A3
inadmissible expansion
A7
admissible expansion
+ ('5 '+,& $ .. Q
*#%" +,& )+,& 0 V +,9 " + ( ( + 2" + (/" " +,; 4%&
" 8 + " (.;-+,& %E%F G
" +,&R (" ( % ;. < 9 " . +, " -,9,+, " " 8" " " +,6;'( J ('," 2" +8" %2 J< 58. < " . 8+, " %C1 2 + . +,& - +%( +,&0 58+,& < 9 +," *+, . , ($;.+,& /- 58 " ( +,& . +- 52+, " -" !+,& . < +," B+, 3 &* 5%
+,&* . V 2+,& * (!#%" +,&* " " +,;) $+,& '+,9 " + 4%&
" ( +,&1+,& -. < " . D ($ (/" ( + +, " 56 < +," !+, 2Z0 #% - + . 9 ( " '( +," 0+,& -
+,&! 5D'+, " % ( = A +,& 2 " " +,;0 53+, " " $+,& 2
2-
+,& @2" = A%" + (1#%" +,&1' 9,+ / & (1+,& +D" % 5 . ($+" ( +," 5;' +,&0+,& (" ," (1" (" ," 6 0 < ," B (* V +, ;*+,& .. < " . ($ (/" ( + .+, " 2" D4%& 2 " " +,; = AD 9 , (D+2+,& D,7 5+,& 8 " " +,"6 % 5+,& D. < " . +, " (0+,& " (" ," 6 ' < ," - 5D+,& - (/" ( + .+, "
0
trail
=
.. < " . D+, " 3 V D+,& 1h/ ($" 2 " 1;'+,& ,k 5D+,& - " " +,"6 58 +,&0+,& . < " . D+, " (0+,& " (" ," ' < ," 58+,& - (/" ( + .+, "
=
h A
+= A& += % A
&
]+& h" !]+& ]/h"
4%& 5 +,& - 5D+,& -+ " ' . + "
=
A =
A
]+& ` *]+& h j"
]+& h j"/
]+& / ! h"
#%&" &1 , (-+.+,& -, 58+,& - " " +,"6 5D 8 (/9 " ," * < ," 0 5-+,& $ (/" ( + 1+, " 7F3" ; ," +,&
.#%" +,&1+,9 " + ]+& \'"-"
0 % % +
%%
+ ( = A ]+& h" &!]+& h&!]+& h h! & ]+& \! ]+& i ^+,& 5 #% - 0 " 5;$+,& +
=
U < +," -g% ; " &+ n h ] ] ]poq-rHstu.v.wwxy ;$h ] ] ]
A
]+& `
=
A +] & i ^ =
h A
&
]+& h j"
zD . .h/ , .\9D
-f f
3.2
Synthetic Data Experiments
3.3
C1 $& 0 (/+ (B1 +' 5- < " . +$#%" +,&B+,& m 9 &) (! +,& &B+,& .& " +," $" . Y'" +.+,& '+ "
" + " )" +J $ &3!F3" !1,& #2+,& $ " +," 5%+,& .+ " $ . + #%" +,&*+,& . < +," )( +,&3 (0F3" 'j.,& #2+,& - " +," ! 5D+,& - +
. 58 + " ()#%" +,&)+,& ' < +," 7( +,&3 * +,&!J " +" ,& #%7+,& +$ . +0i ] 5.+,& 1+ " ! . + 0 " 9 +," */5 h'. " 0+,& +2" +" -(/" Y' ++' +, +,& .," S ' 52+,& " (/ (1 9, +#%" +,&!+,& .+ " 0" + " .5 + " 53+,& + " ." + " $" D +% +D+ ] & " h+,& 2+, " . < () ;1+," ( \' (!#% ' + " ! .+,& Bh ] 5D+,& - " 5%+,& -+ " 0" + " 1" + +2+ ] & f i'+,& -J < +," *( +,&$" h- ($ -+.i ] a 58+,& 2 " (/ (4%& , + - ;$ ," + +5 2(/" +- 9 . '," S . .#% . ) ';*+,& ' B 5+,& ' " ." )+,& +%48 . -+,&" 2 #% ( " ( (1+-" +,& +" !0#% ;0#%&" &!#% (* (!+$. ." " W+,&" + &" V " 2( " (0" 1+,& +2 +," 3
ID (BF=3, conf=0.1) 1.2 n=1000 n=5000 n=10000
Stopping parameter
1 0.8 0.6 0.4 0.2 0 1
2 3 Exploration depth
4
5
F3" !+$zD " +," B 5-+,& $+ " * . + ' () 5+,& +
58 " (0#%" +,&1+,& - < +," !( +,&3
ID (BF=3, conf=0.1) 1.2
% of total num. rules
1 0.8 0.6 0.4
n=1000 n=5000 n=10000
0 2 3 Exploration depth
4
5
F3" 1j $zD " +," B 5-+,& $+ " * . + ' () 5+,& +
58 " (0#%" +,&1+,& - < +," !( +,&3
G @l-mm
0+,&" 2 +," 1#% 2 +2+,& N /K,c/dN" + +," ( 9 " .& " +," 0= ," ;1F8@-A%#%&" &1 " . +2" . +," 0 +D," " +-+,& +8 + (" G +," $^ \D+8" $#% ; +,& +- " -. . +, % +,& -/- . 5D + " ( P%#%" +,&0+,& mB& " +," 0 < +," *+, #%" +,&$+,& -+, " + (*" -" . + ;!" +56 T+,& .+ ++ + W3& #%9 " *+,&" .+,& +, " -& + " (*;$" +-," S 3. 9 , (1;1+,& ./- . 52 " . ," *+,& '+, " . (; < ('P2+ &!+
3' " 1" & *+$ < (*$+, " 56 H+,& $+, 0 ()+,& 0 1 5+,& $+ " ! . + $" + (D4%&/+,& 2," S 5+,& +, %" D
. + (' ;.; ," - " 0 +%+," . 53+,& - - 58+,& + " . . + . +8+,& D+ " " + " '+,& % < +," $+ +,& #%" $ #a " *" .& !+0 < (!0+, " %" )+,& '+, . (!+,& / +," / +," 8+,& -+ " $" + " 1" . + 4%& F8@:& " +," $ < +,& +, " +," ; 3" ! +, + #%" +,&0+,& mB& " +," #%& " 0 8 (" ," . < ," 58 (/" ( + 0+, " %" < () ++,& ' . '+," . 0C!& ! 9 +," ;' < " . $ (" ," < ," 0 53- (/" ( + -+, " +,& 0 +,& 0 < " +," ) (" ," ! < ," 1 5+,& 0 (/" ( + #%" D +," / < (P2 , ++,& $ J< 53+,& 2 , +," '+, " #%" +,&$ (" ," . < ," +," ( ( +. . < (4%&/" * ( -+.5 . " '+,& .( " +," 53+,& 2F8@)& " +," -#% % ('+" +,/(/ - #)+,; 2 53+, " +,&
c /d /c/bN-O+ L bK,c/d M P2! (/" ( + $,9,+, " 2" - J<1 5
+, " 2= " +,& -.. < " . % . (/" ( + '+, " 6A2#%&" & & 2 +% +2 - (" ," 6 . < ," *+," 8 < (2P2( +1 " +, + *+, " " '. < " . -" 5-" +.& .* (" ," < ," $ (!0+, " 2" .0 (/" ( + 0#%& !" +-& . +- + 0 (" ," * < ," B+" . +.0 J " 3%#% ' ( J" $+,& 2 +2 +% 58+, " +- -+,& - +%#%&" &0. < " " +,& ., 5%+,& " " +,"6 . 5D +,&*" +. < " . 2+, " - ( " +,9,+, " 1F 8" )F3" "' (15 % "$+,& ,k 5D+,& " "6+," 5D 8+,& -+, " " "
36
=+A & =+ A & =+% A & =+ A & =+
A ]+& "&2]+& h"& ]+& h" & ]+& " &]+& f \& ` F "$+,& '" .0 +,& ',&B + 5-+, " '#%" +,&!&" & " " +,;/
F '1 " D+,& ' +,9,+, " -" + " B 5 ,+ & $J 1 " (B& " +," * " $+,& #%& 1,H 5+, " " " +,"6 ." . < " . -=6" (/" *+,& . " " +,"6 ' 52+,& ,9,+, " A
F
U < +," -g% ; " &+ n h ] ] ]poq-rHstu.v.wwxy ;$h ] ] ]
4%&" % +,;$#%" 8 ($;." (/+," 3 \2+,& D +,;-& (%+," " ;',"
+ " %," 9
zD . .h/ , .\9D
-f!
.+, " D#%&" &*#% & 0 " +& +,& -&" & +- 9 " " +,;
7
1#% -& .+,& + + " +,& + + 5%+, " 5D5 ,+ & . +,;1#%" 2 +," / $+$& (*5 7\.," $+,& < (!+, " %#%" +,&*+,& &" & +- " " +,;*#% - (( (1+ " 1 ( -+. + "
71&
F3" `- " J +,& 2 + +," 0 53+,& 2 +D 5 (/" ( + +, " %+,& $ +. 5 (/" ( + 1,9,+, " ) %+,& $ + 5 ) =+,& . < " . D+, " A- (0 5%+,& + " $ . 9 + #%& !+,& ' (;1 ," ! 5+,& .F8@a& " +," $" " ( ++,& .. %" $F3" -i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
10
G " " ;/F3" -` " 2+,& +," * 53+,& -+, " 3 +%#%& +,& . +,9,+, " " + " )" ( !+,&" . $0 '" ," ( +,& - / +% +-+,& -+, " 3 " "6+,; 3," .#%" +,&0+,&" ," !+,& .+, " %#%" +,&1&" & +- " "6+,;1" -+,& . .& + - < ( (
% #! !!"" # ! ( $ . $ - #*,*/#! (! 1 #*#*,*,*//#!#! (!(! )! (!! " 3 3
$ &!&!'' (!(!!! ))'*'* (!(!!! #! ,!&!'' -(!"! )'* (!! #* ,!&! '!(!' ! ." ) '*/ #(!' ! (! #* ,*&!' (! !"/0 ')! -'! (!" ! 3
$$ $ $ $ #*#*,*,*//00 & 3
%% + + + -21!1% 3
F3" '`+4%& . + + 5%+,& . +- 5%+, " -" (/ (*;0+,& F @R& " +," $#%& *," 1+,& ' (;1" + " ) (!+,& .+,9 8 " + ]+& \'"/
0
% #! !! "" '#! ! ( $ . $ - )! (!! " 1 #*,*/# %1! 3 3
$ &!' (!! )' (!! #! &!,!'' -(!!" )' (!! #* ,!&! ' (!'*!"( -)!'*/ #*(!'*! %1! #*,!)! 4!'*'* (! ((!-! /#*'* %1! #*,!)! 4!'*'* (! ((!-! " " /#' %1! #*,* /0!'* (!(! 3
%% $ + $ + && 1!1!(! & 1!! 3 3 $$
F " '\]+%4%& - + +2 5D+,& - + 58+, " 3 " (/ (0;'F8@ & " +," !#%& B," B+,& 0 +,9,+, " " + " a (B+,& 1+,9 " + ]+& \'"/
0
C1 D,& ( + %+,& +3+,& DJ 9, " (& " +," 2#% 3( (
G @l-mm
#%" +,&1+,& +," ' 5D " " 0+,& ;/+ -&1 +, D ," ' +,& -/- - ($V " +,;1 5D+,& -" (0 2; ++," .+,& 5 %+,& + " '" + " 3%C!& $," .+,& F8@R& " +," $#% ' < ++,& ' ;/+.+0&/ '+0 < $ +," ;$. /- 2 5+, " +,& #%" 2& 2#% (. 2+,& E%F G
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
78
6
65
9
4%& (0./(/" J +," ) ," +" 1/ " $ + ;1+,#% . D + (1 + #%" +,&*+,& - + . +56 7 ( +,& % +,& %#%" +,&'+,& D +D . +856 ) W+,& 2," S 2 5+,& +8 (. % . + .+,&" % +,& % +3 +D+, " + < ( (0 (' 25 ($ . +,& +, " 2" $+,& -. + (0 + + +,& +2" 1+,& + ( (0& " +," ." 1 ( -+ J(1+,& + +-+, " D +,& ) a ( : +& .+' +," ;)+, () ." +'#% (! 0 ;!( . (/" !+1/ ,&* + + (-4%& .. + (1 +-#% (0& .+. J (7 (B ( (B ;)#%& 7 +,; B5 +%#%& . #:+, " D& (1+. -" + (1" +$ 5%+,& -
+" + #% ( D " J (" 5+,& D+, " #% 3 ++ 8+,& . +3 +8 %+, " " *+,& , (/" 1 + (1 + 5D+,& . #% -#% 2 ," +," +,& B+,& $ #k+, " - (! $" + (B . 1+,& $#% + +, " 3" 1+,& - + ($ +-=( (/" $ 0#%& +,& 2 +2+,& - + #% %56 6A
:
6
3.4
;
<
Experimental Evaluation
48 2+,& - +," - 58+,& -F3" 9@2 " (0& " +," < 9 " . +D#% 8 (/+ (.#%" +,& +,&.; +,& +," % (- ( + 4%& D; +,& +," %( +2
+," '. +,&/(- ," + (" - ( ; +," .>?%@D " 0-/- 2 5
$
2/- 5D +,9 "
(0 " " +,;*(/" +," +," *5 +,& " Q/#% " &+D4%& D/- D 5+ + 3 " ( +,#% 0\] ] ] (1h ] ] ] ]'+ + +,& - J( +,& ,& (0 +,#% 1] \ ( ] ` ($+,& , +2+,& ,& (0#% DJ ($#%" +,&$+,& '\ 5 &0 .. ," S %F 2 &1 J +," *^ ]2#% 5 . (D4%& D / J %#% D + " (.56 :+,& % +,& 5[ ' \ "_% (* , ()+1h$. +,& 52 0( +1#%&" &!#% (/" " ( (*" +$i$#% 5D
.( +4%& + " 0" + " #% 2 +2+. ;." 1+,&
58+ ] & \-+.+ ] & ` i
7
$.7
P$+ + (B" G +," a^ ^+,& 1. " a +," * 5.+,& 1J 9
" ('& " +," " %+ " ( -+," &+% +, 2+,& %/- 5 %" (';. ++," +,& 2 - 53+,& 2+ " '" + " 3 + %+,& +8" 5+,& 2+ " " + " $+ / D+,& % ]+,& D 9 +3" (/ (.;-+,& D& " +," 2" D +, ;+,& % . 2 8+,& +8 " ;'+,& - (+,&9J+ &0
" +,&$
6
U < +," -g% ; " &+ n h ] ] ]poq-rHstu.v.wwxy ;$h ] ] ]
zD . .h/ , .\9D
-f/j
#%" +,&1+,& - ]+& j ] FG (n=10000, BF=5) 35 Conf=0.1, Grd Conf=0.1, Bt Conf=0.3, Grd Conf=0.3, Bt Conf=0.5, Grd Conf=0.5, Bt
25
FG (Conf=0.3, BF=5) 160 SC=0.5, Bt SC=0.5, Grd SC=0.7, Bt SC=0.7, Grd BFS
140
20
120 Iterations (x1E6)
Num. rules (x1000)
30
15 10 5
100 80 60 40
0 0
0.2
0.4 0.6 Stopping criterion
0.8
1
F " *\ \$4%& $ " +," 7 5-+,& '/- ' 5 '#%" +,&)+,& 3 J 9, " ('+ " " + " $#%& . " ('++,& %; +,& +," ( +
F3" !\ \0,& #2'+,& 0 " +," : 5-+,& $/- 0 5 $ 9 + " (.#%" +,&'(/" +D D 5+,& %+ " -" + " '5 8 + 5D ( ;0
+ (1 .. #%" +,&!\] ] ] ]$+ + (0
% &" -5 + D 5 "2 +,9 " D D
84%& D , + ,& #a+,& +-+,& -/- . 5D -" (/ (*( ./ +,& ; #%" +,&B+,& 0 1 5-+,& 0+ " )" + " 3+,& 0+,#%1 " & 0$ +," ,&" :#%" +,&)0
+," 0, 0#%& $ 0( 9 #%" +,&*+,& ' J( '4%& 5 8+,& .+ " 1" + 9 " 0 " 2+,& ;/+
//(' +, 8 2+,& 2/- 5 + " (%Z0 /F3" -\ \%,& #28+,& +8 ;-5 8. 9 D 53+,& 2 J( 2" +D" 8 ," -+-(/" +," " ,&1+,& %/- 58 2 +, ($;'+,& - (;$ ($ +,9,+, " 8 ,"
FG (n=10000, BF=5, Conf=0.1)
20 0 0
5000
10000
15000
20000
Num. States
F3" )\^+!4%& 0 5 . * 5.+,& 0J 9, " (7& " +," #%" +,&1+,& -; +,& +," ( + F3" -\^2 " D+,& D " +," $ 5+,& 8/- D 5" + +," %#%" +,& +,& ," S 5D+,& .. 5 (/" +- - 5D+,& + "
" + " *= G g2AW +,& % 5 . 2 53+,& 2E%F G
" +,&R" D
" 3-P < + (3+,& - , +-,& #:+,& ++,& F8@:& " +," " D-&'. 2+," . 2 ," .+,& $+,& %E%F G /C!&" -+,& 2E%F G
" +,&a < &" " +%% " D & " %+,& 8 , +8 " .& " ,
+D+,& +8+,& %F8@*& " +," 2" 3 ; " , -5 + #% D#% 2+%J+82V (/ +," %56+," ." ' +,& % " ' #% ;/ #%" +,&*$ +," B/ Y'" + + +,& !+ ] & ` ` `.> #% .#% .+ + (*" G +," B^ ^D+,& ' 5 . 0#% +. " , 1#%& 7( " ," )+,& 0& " +," * (B . 0(/" +," $5 +,& D" . +D 5" +3 5 . 2#% D - " 3DF3" ; +,& +,#%- ," 53+,& 2& " +," 2" (/" +," "6,& 6 -" 0+,& /- . 5D" + +," - 5 . (* +,& &!#%& * ," ( "
+,& D" -+," . 2+,& D (; ," . 5 .8, " &+, ; ++
4 3.5
FG aggregated weeks (Conf.=0.1) 5 Grd Bt
2.5 4
2 1.5
Num. rules (x1000)
Length
3
MRL, Grd MRL, Bt ARL, Grd ARL, Bt
1 0.5 0 0
0.2
0.4 0.6 Stopping criterion
0.8
2
1 1
F3" '\
h D4%& - " +," * 5D+,&
+,&1 5D+,& #%" +,&0+,& J 9, " ('+ " " + " $#%& . " ('++,& %; +,& +," ( + F3" B\ h*,& #20+,& 1 " +," R 5.+,& *
1 1
+,& =P(8A2 (*+,& . < " . %
+,&B=Z 8A2#%" +,&*+,& .+ 9 " '" + " 1#%& 0+,& - J( +,& ,& (0#% 2 +% +2]+& \ +2 $ - 0+,& +2 05 %&" &1 - 58+,& -+ " $" 9 + " 3#%&" &$ , (+-. 9, +" +D" D ," + + " $ - 2 , " ;$#%" +,&'+,& (;. ," 3 .5 + #%" +,&!+,& ' (;1 ," !" +-" ," 0+$ + " ) . . .+'+,& -. < " . D
+,&*#%" +,&1+,& + " 0" + " * +
G @l-mm
3
0 0
0.2
0.4 0.6 Stopping criterion
0.8
1
F3" \f+34%& D " +," ' 5+,& 8/- D 5/ 8#%" +,&.+,& 8J 9
" (1+ " $" + " 1#%& 1 " (1+.+,& 8( + F3" !\f*,& #2'+,& 0 " +," : 5-+,& $/- 0 5 $ 9 + " (!#%" +,&!+,& '+ " *" + " )5 +,& . 2( + U & " +" !+,& . + , ('+$+,& '
. , +. 52+,& 1( +. +%5 " * J +," 3W3 &0( +. +2 9 , (.+$0#% 1 5% 0( +$4%& , +. JT+,&
U < +," -g% ; " &+ n h ] ] ]poq-rHstu.v.wwxy ;$h ] ] ]
zD . .h/ , .\9D
-f i
+ " ($#%" +,&'; +,& +," -( +;,& #%" +,& +%+,& %" D 9 +," " +,;! +,#% !+,& ./- ' 5% . (!+,& . $ 5 +,& 2+ " ." + " 3 .5 + -. " +," $" $+,& 2 52+,& '+ " 1" + " B#%" % (*+0$," " $ " +," !" +,& /- 5D - + " (* (1+,& +- " +," *" - .+ " 5 +,& & +-+,&
58+,& -+ " $" + " 3
FG greedy; aggregated weeks 0.7
RS, Conf=0.1 CT, Conf=0.1 RS, Conf=0.3 CT, Conf=0.3
0.6
Probability
0.5
FG aggregated weeks (Conf.=0.1) 10
MRL, Grd MRL, Bt ARL, Grd ARL, Bt
Length
8
0.4 0.3 0.2 0.1 0 0
6
0.2
0.4 0.6 Stopping criterion
0.8
1
F3" -\ j 84%& % +," $ 5 +,&'+,& % (/" ( + (+,& D 9 +#%" +,&)+,& '+ " *" + " B#%& !+,& ' (;* ," )" " (1 1+,& 8( +
4
2
0 0
0.2
0.4 0.6 Stopping criterion
0.8
1
F3" -\'" 34%& D " +," $ 5+,& D," S 2 5+,& 8 D#%" +,&+,& DJ 9
" (1+ " $" + " 1#%& 1 " (1+.+,& 8( + F3" !\'"1,& #2.& #k +,&BP(: (BZ : ;!#%" +,&B+,& + " $ . + 5 -+,& - 8( +' +4%& , +- 9 J +,& 1 + " (:#%" +,&7; +,& +," *( +RZ0 2" +$" " + +," $+- + +,& +2#%" +,&0+,& (;' ," 1+,& -
'
+,&!& -" +. < " - $#%& !+,& .+ " *" 9 + " !" . + ++ ] & j/D#%&" &!" +,& ." (/" +," :+,& + . $+, " + (0+. < (1J+P (/" $+'+,& ( J9 " +," 2" G +," 0^ ^+,& 2 9, +D" 5 ($ + " 2 +,&$+,& (/" ( + 1 (!+,& $. < " . -+, " .5 '0 " 7 0 5+,& + " ! . + BP2 + +," ; -+,& '& " +," * () +,1+. " +,& - ;/++,& -. < " . D+, " - ;
" " +,;:+,& +$ , (0+* (/" ( + !+, " B5 + +,& 2J %,& #B+,& +," 1 53+,& 2,R 53+,& 2 " " 9 +," D 5+,& %. < " . +, " 2= G A3 (.+,& % (/" ( + +, " 2=,g%4A #%" +,&)+,& ' 0 5+,& $+ " *" + " 3!C!&" 0F3" * \ ! , (+.+,& - +,9,+, " % ," *F3" $\ j. , ( +'+,& . (;0 ," 3.4%& , +-,& #a+,& +-+,& - +,9,+, " ," )+ (-+'/ !$-&*
. +- 52 (/" ( + $+, " (! V +, ;!0. $ + 52 14%&" " < " ( ;.+,& " " +,;$ 58+,& (;. ," 0+ + " 0 . . < " . +, " +- ;0+
" !+,& < +," 3$4%&" 5 +2& + < " 1#%&;+,& %" .+," . 53+,& 2 (;. ," $" D ++ ," -#%& $& " '+- / + -+,& -+, " 3+ < ($ +% .
%+,& (/" ( + $ (1 (" ," $+, " 3" +& -+'+, . . ' +- 52 (/" ( + .+,& *+,& ' , (/" 0 +,9,+, " ," 3
4.
CONCLUSIONS
FG best-trail; aggregated weeks 0.7
)0 " ' $#% ' ()+1./( -1 +," 7 5 "
+," * ++ &; + + " " +,"6' 9 . 2#%& 2&" & 2 " " +,;$+,"
2 , ($++,& 2 5 (1+, " .[ h_,4%& " 38 1
" +,&T#% 2 (*+ J(! +,& '+,"
'#%" +,&! " " +,;) 00, " J ()+,9 " +
RS, Conf=0.1 CT, Conf=0.1 RS, Conf=0.3 CT, Conf=0.3
0.6
Probability
0.5 0.4 0.3 0.2 0.1 0 0
0.2
0.4 0.6 Stopping criterion
0.8
1
F3" -\!+84%& % +," $ 5 +,&.+,& % (/" ( + (+,& % 9 +D#%" +,&$+,& + " ." + " 0#%& '+,& 2 +,9,+, " 3 ," $" " (* 0+,& 8( + F " 0\!0 (*F3" 0\ j. " ( $$( + " () ;/," . 52+,& 3 9, + ," +," 1;$ " " $+,& - . +2 58 " "6+,;$" " +$+,& +$ , (0+!. < " . +, " 0 (B+,& 1 . +$ 5
G @l-mm
1+,&" #% #% . #7& " +," .#%&" &1 " .- +2 9 & " B+,& 1>?%@T " " +,"6 1;) " 7+,& 1 ;/+0+ & 1. 0 +, - $+,& $/- ," S 2 () " " +,; 5D+,& - " (/ (4%& -& " +," " . + 0" + +," ( " 1 &!+,& +" . + ;*" (.+,& 9, +'P + " - . + %" 8 " ( ('#%&" &$. , D+,& 2(/" + +,#% 1+,& -" (/ (0 9, + (1+,& - + 5D D+,"
-#%" +,& " " +,;a !+,& *+,9 " + U < " . +1#%" +,&7 +,& ; +,& +," ( ( +-#% 2 (/+ (' (.+,& D , +%,& # +,& +-+,& .+ " 0 . + " +,& . ;/+. +, % +,& /- - 58 " (0 (0+,& " 2 " "6+," P-56+, '#% *#% )+$" + $" !+,& '>?%@k./( .. , 5%+,& -#% 0
+ (056 k. V ;." 0 ( 2+ ," ++,& ;/+%" $J(/" '+, " 2 + +.. " 1+ " - + ($;$. +2 53/ ;/#% (
U < +," -g% ; " &+ n h ] ] ]poq-rHstu.v.wwxy ;$h ] ] ]
zD . .h/ , '\98
f `
5.
REFERENCES
! " #
! $% &' ( ) #*# # + ,-./0123054765894:0;/658<=6-9/=?>:65@-0=?>:A B0=C 49/9=?19D0=FEG=0H3A 9IJ 9KG@ L10M:9/NF>:=?IK>:6->O!@ =@ =J:P ) Q R:S?:T:UP:V3WDX
Y?PZ7[\GP\3$ $#7R R ] ^
_7 #!` :a7$ : #
) ## ,-./01205476589b!9cedL>J 93f3=?>:A N:L@ L7>:=?I!dL9/./0C g A @ =J b0/h:L80iP ) USU jP [ _7 P:k % a5
P:\3$ $# R R R
^ ^s7?~l?X P?: "
+P}G p " :(-
% P ZG_7:' %5
`$ 3 "" ) ## # ' ` "3&' ) #*# % Y ,- ./012054 6589 g 46583<=6-9/=?>:65@-0=?>:Ab0/A Ieb @5I 9Gb!9cGB0=49/9=?19P ) 3r rS?rQPv P " P :' R R j ^:U:VG? ! ! k
# $"# W +(- W3a5
` $# :( ` #: #
,-!./01270546589GfBOKG@ J:@ 6->:A?@ C c/>:/@59Lb0/h:L80iw0=l/5J >:=@ @ =Jlb!9cey i> 19P ) GT:Q:ST:]P Y % ' P k % a5
P \3$ $#3R R R
U:
:\l&$ # "# " ) #$ %
$ W +!: #
) ## ,-./012G0546589 g /L6<=6-9/=?>:65@-0=?>:A B0=49/9=?190=mA 9165/0=@51eB0nn9/19>:=?I b!9co?918 =0A 0-J:@59LP p W "&P:ZG q P?[ ) #`+? ^:r r r s ))
Q:,?k t P?[? p :u' P v[`'#& \w % )
+ + % # " a5 `W
Ya5
"% $# $ % ,-D./0199I:@ =J:L054e6589 L@ x:658!fBOzy< { E7K7K|<=6-9/=?>:65@-0=?>:AB0=49/9=?190=EG=0H3A C 9IJ 9!KG@ L10M:9/NF>:=?IK>:6->O!@ =@ =J:P3
#
P "&$##P \3$ $#^:r r rs )) T,k t P_}3"Y ` P k3? Y?Pv?[`'#&P: [?~D& # $ % t #
D :ae: #
) ## e
wFW +w #$ ` %3+ D"% $# ,-l./0199I:@ =J:Ll0546589!L@ x:658fBO y< { E7K7K<=6-9/=?>:65@-0=?>:AB0=49/9=?19D0=FEG=0H3A 9IJ 9!KG@ L10MC 9/Ne>:=?IK>:6->O!@ =@ =J:P
#
P "&$##P\3$ $#3^:r r r s ))
j: _ k #% 3v #Y :W7k & "# t + :W # # 3 #&W
% W W +GB0n7i;6-9/G965H30/h:L >:=?I <yK7yN:L6-9nLP^
5j r jTSrUP:\ ) %R RT 3 k & Y? y6->:65@ L65@51>:A>:=J:;>J 9G9>:/=@ =J:s&3 R R j
,-sv P
]:k3 k & #-?% [# # # " % a5 "3 ` Y : ` % fi iA @59I y6->:65@ L65@51LP^ ^ S^:rPR:U
"&
R: ([?k &P?[?v Y?P ev[?X $" # #` a5
?# : % ) ## <m m mlo/>:=L> 165@-0=L30=EG=0H3A 9IJ 9G>:=?I K>:6->m=J:@ =?99/@ =J:Pr
-^ ^:r R:S^ ^ P "&?:\ ) %R R ] r: k
Y% ?}3' ) #*#G\3 # $"#
e $ ' <m m m B0n7i;6-9/P?^:r
5R :SQ:P [ ) #`+? R ] G k
% ' P7:
+ & P [ : #::_7 # ) ) #
a5
` !W
% DW !W +F+ :W ) ## EG=0H3A 9IJ 9 >:=?I<=4:0/n>:65@-0=!yN:L6-9nLP?
: TSU^P:+ $ 'eR R R :^ 7 p
t
$ \ )
+ + % # " ))
"&# : ( #
l &' ) #*#3<=4:0/n>:65@-0=|y?1@59=?19LP Q jTS?] jP R R R U:V3 % ?s& #7 :a: # #&
$ & & ' ) #*# B0nC n;=@51>:65@-0=L054G65897fBO!PU U
5U ^:R j:SUrP: "&!R R r Q:v Y :W #t GG #t
5\3 ) # W +G #? G\3,?"& % ( % ,-D./012054 g 46-99=658<=6-9/=?>:65@-0=?>:A3?0@ =6eB0=49/9=?19 0= f3/65@ g 1@5>:A<=6-9A A @ J 9=?19 5<Bf3<C:P ) 3j:S^ PV3 :' P : ) P \3$ $#7R R :T v Y :W #t eG #t
5\3 ) # #\3$#
` # " % % ' '#& t eW + ) ,-F./012 054 6589 g 46-99=658G>:65@-0=?>:A B0=49/9=?190=f3/65@ g 1@5>:A<=6-9A A @ J 9=?195f3f3f3<C::P ) ^ S :U^P
P ~D "
P:$% ' R R ] j:G7GG[ $YY: 5 Y ) "#
D ) #&l % ' e$ Y :D"& ,-w./0199I:@ =J:Ll0546589=@ =658D<=6-9/=?>:65@-0=?>:A b0/A Ib @5I 9eb!9c B0=49/9=?19P?\3`# ` P:}3
% % P^:r r r :[? ["&"&# P q7 & P _[` # &Z3 ) #& 965H30/h:L>:=?I ) :?% # ) "#&## ) $#GB0n7i;6-9/ G <yK7yN:L6-9nLPU r QT SQ jP R R ] ]:[ ) % )
$%
$ ! k3? $% # "&~DZ3e#
%?a5
7W + $# % t #
% ' :,-!./0127054G65897<=6-9/=?>:65@-0=?>:A7b0/h:L80i 0=e6589b!9c7>:=?I7K>:6-> c>:L9LG:b!9cK7 ::P ) ] Q:S^:r UP :( % " P[ ) P: "&!R R ] R: [ : #::PG k
% ' P::_7& ) P v (-VG:s :~!+ $ 7` _7 " : ' e )) % " #
:a $ ) ## a5
`W + #y< { E7K7Kwm xiA 0/>:65@-0=LP
-^ :^S^:UP ^:r r r ^:r:G [#
$#?b!9c3y@ 6-93y6->:65L65/> 1h:@ =J8 @ 65L3>:=?I>:=?>:A N@ =J65/>-D1 3+?
3 " p :W (-}3 % %5P?R R ^k3[?:~!#& % %5:v
+ + % # "% $ \F W
`
) $#
B0n7i;65@ =Jy;/M:9N:LP:^
5Q U jSU :RP_7"`( +? 7R ] r
G @l-mm
U < +," -g% ; " &+ n h ] ] ]poq-rHstu.v.wwxy ;$h ] ] ]
zD . .h/ , '\98
%" ]