Top Banner
Mining Frequent Patterns with Counting Inference Yves Bastide Rafik Taouil Nicolas Pasquier Gerd Stumme Lotfi Lakhal ABSTRACT 1. INTRODUCTION 1.1 Related work
10

Mining frequent patterns with counting inference

Jan 20, 2023

Download

Documents

Welcome message from author
This document is posted to help you gain knowledge. Please leave a comment to let me know what you think about it! Share it to your friends and learn new things together.
Transcript
Page 1: Mining frequent patterns with counting inference

Mining Frequent Patterns with Counting Inference

Yves Bastide�

Rafik Taouil�

Nicolas Pasquier�

Gerd Stumme�

Lotfi Lakhal�

���� ��� ������ ����� �� ������ �� ��� �� ! ��! ����� "� # �%$�� �&��! ���� '( ��)* �� +&���, �-"� �&+�� # �%��� ��.�/+��� �� # 01� "-�� 2 �� ��! ��� �3�+�� "+4! ��*3� �! �,! � ��� "-�� 5*2 �/����� # �

ABSTRACT6798&:1; <.=1>*=@?�ACB(D,?E=�A&F"=@F"<G?�8&:H?E>CI J"FCA�; 8�:HKMLONQP�R1NQSEDT:H; U�:V; 7�W8GA&F�XHYHU�?�</>Z71FC["?�I\F"=�8�; K]; ^�>C8&; F"7VF*_�8&:1?`D�?�I I Wab71F�Dc7d>*I J"F*A&; 8&:1Ke =�A&; F*A&;gfih�:1; <�F"=�8�; K]; ^�>C8&; F"7�; <�j1>*<G?�X�F"7E>.7H?�Dk<G8GA�>C8&?�J*l�U�>*I I ?�XmHn*o o�p�q�rks&t*u�rHogv r�wxv rCy�p�q�p�r@s�p 8�:1>C8TA�?�I ; ?�<.F*7z8&:1?/U�F"7HU�?�=H8{F*_c| p�}mHn*o o�p�q�rH~ f���?9<G:1F�D�8&:@>C8]8&:H?9<GY1=H=(FCA&8xFC_`_ A�?���Y1?�7�8x71F*7HW�a�?�l=1>C8G8�?�A&7H<�U�>C7�j@?x; 7�_ ?�AGA&?�X�_ A�F*KM_ A&?��bYH?�7b8�a�?�l�=1>C8G8&?�A&71<`DT; 8&:�WF"Y�8E>*U�U�?�<G<&; 71J98�:H?ZX1>�8�>Cj@>C<&?"f��\�H=@?�A&; KE?�7b8&<`U�F"KE=1>CA&; 71J�L�N�P��R1N�S/8&F�8&:1?T8�:�A&?�?`>*I J"F*A&; 8&:HK]< e =�A&; F*A&;gB@�,I F*<&?`>*71Xz�z>C�bWG�Z; 71?�A�B<G:1F�D�8&:@>C8xL�NQP�R1N�S�; <]>*KEF"7HJ�8&:1?ZKEF"<�8�?��EU�; ?�7b8]>CI J"F*A&; 8&:HK]<_ FCA{KE; 7H; 71J`_ A�?���Y1?�7�8T=1>C8G8�?�A&7H<�f

1. INTRODUCTION�.71F�DTI ?�XHJ*?]X�; <GU�FC[�?�AGl�; 7VX1>�8�>Cj@>*<G?�<E� �/�{�`��; </X�?��@7H?�X9>*<�8&:1?7HF"7HWg8GA�; [�; >CI{?��b8&A�>*U�8&; F*7�FC_�[">CI ; X(BT; K]=HI ; U�; 8�B�=@F*8&?�7b8&; >CI I l�Y1<G?�_ YHI>*71X�Y1I 8&; K]>C8&?�I l�YH7@X�?�A&<G8�>*71X1>Cj1I ?9; 7�_ F*A&K]>C8&; F"7�; 7�I >�A�J*?�X1>�8�>CWj1>*<G?�<{� �"����f,�HF*A�<G?�[�?�A�>*I1lb?�>�A�<�BH>.Dc; X�?�A�>*7HJ"?�F*_O>C=1=HI ; U�>C8&; F*71<�; 7[">CA&; F*Y1<`XHF*K]>*; 7H<`:1>�[�?�j@?�71?��H8&?�X�_ A&F"K��.�.��8&?�U�:H71; ��Y1?�</>*71XK]>*7�l�D�F*A&a�:@>*<Ej@?�?�7kU�F"7@X�Y1U�8&?�XdF"7d8�:H; <�8&F"=H; U"f�h�:H?z=HA&F"j�WI ?�K�FC_OKE; 71; 7HJ{_ A&?��bYH?�7b8,=1>C8G8�?�A�7H<,>CA&F"<G?��HA&<�8�>*<c>{<&YHj�W=�A�F*j1I ?�KF*_/KE; 71; 71JV>*<G<&F�U�; >C8&; F*7�A&YHI ?�<�� ���gB�jHYH8E; 8]8&:H?�7�8&Y�A�7H?�XdF"YH8]8&Fj@?�=�A&?�<G?�7b8/; 7V>x[">CA&; ?�8l�F*_,=HA&F"jHI ?�KE<�� �C�����{KE; 7H; 7HJx<&?���Y1?�7�8&; >*I=1>C8G8�?�A&7H</�  ���B1?�=1; <GFbXH?�</� ��¡C�gB(>C<&<GF�U�; >C8&; F"7�A�YHI ?�<.� ����B1U�FCA&A&?�I >�8�; F"71<� ¢�£{�* ���B�K�YHI 8&; W�X�; KE?�7H<&; F"7@>CI�=1>C8G8�?�A&7H<z� �� �£/��¤*�gB�K]>C��; K]>*I�=1>C8GW8&?�A&71<�� ¥b£O�*¡b£��C¦��gBQU�I F"<G?�X9=1>C8G8&?�A�7H<�� �C¤H£\�*§b£���¨�£\������f`©�; 71U�?/8&:1?U�F"KE=HI ?��H; 8�lZFC_i8�:H; <�=HA&F"jHI ?�Kª; <{?���=(F"7H?�7b8&; >CI�; 7z8&:1?/<G; ^�?`F*_i8&:1?jH; 7@>�A&lEXH>C8�>*j1>*<G?�; 7H=1Y�8�A&?�I >C8&; F*7Z>C7@X�<G; 7HU�?T8&:1; <�A&?�I >C8&; F"7]:1>*<�8&Fj@?�<GU�>*7H7H?�XV<G?�[�?�A�>*IO8&; KE?�</X�Y�A�; 71J]8&:1?�=�A&FbU�?�<G<�B�?��]U�; ?�7b8`>*I J"FCWA&; 8&:1KE<�_ F*A{KE; 71; 7HJ/_ A&?��bYH?�7b8T=1>C8G8&?�A&71<T>CA&?.A&?��bYH; A&?�X(f

1.1 Related workh�:�A&?�?x>*=1=�A&F�>*U�:H?�<�:1>�[�?Ej@?�?�7�=�A&F"=@F"<G?�X�_ FCAEKE; 71; 7HJZ_ A&?���Y1?�7�8=1>C8G8�?�A&7H<�f«h�:H?��1A&<�8Z; <]8GA�>�[�?�A&<G; 7HJd; 8&?�A�>C8&; [�?�I l�8&:H?�<G?�8ZF*_�>*I I�*¬ 6G��­.©(B"Y17H; [�?�A&<G; 8&®{¯�I >*; <G?.°\>*<GU�>*IgB�U�F"KE=1I ?��H?c<&U�; ?�7b8&; �@�bYH?{X�?�<�,®�^�?�>CYH�(BQ��¤]>�[(f1XH?�< ¬ >*7@XH>*; <�B@¡" b�C¥"¥ e YHj1; ±�A&?`�,?�X�?��QB1��A�>*71U�?� 6²{³�6 e ¬ FCA&A�>*; 71?"B�¦C¤�¦*§"¡µ´,>C7@X�¶/YH[bA&?�W�I ±�<�W²{>*7HU�l·�,?�X�?��QB��A�>C71U�?� 6� �©«���,²{³{©�¸.°i³{��© e ¡"§�¥C§��ZWEY17H; ["?�A�<G; 8&®�XH?�²�; U�?"B/§"¡"¨*§" ©bF"=1:H; > e 7�8&; =@F"I ; <�B1�HA�>*7HU�?� 671<�8&; 8&YH8�_ ¹HA e 7HJ"?�Dc>*71X�8&?T67�_ F*A&K]>C8&; a�Y171XE�1FCA&Kx>CI ?�¯,?�<GU�:�A�?�; WjHY17HJ"<G[�?�AG_g>*:�A&?�7�B�¸{7H; [�?�A�<G; 8�ºC8,�/>CA&I <�A&Y1:H?��gh�».��B���WG¥*¡b�C�*¢/�`>�A&I W<�A&Y1:H?"B(¼�?�A&K]>*7bl� ¬ 6&�½���,²�³.©��O³{�¾�"��¤�¡��/W.Y17H; [�?�A&<G; 8&®ZXH?xI >��Z®�X�; 8&?�A&A�>*7H®�?"B�� ��C¢"¢x�z>CA&<G?�; I I ?E��?�XH?��Z§"¨bB(��A�>*7HU�?

=1>C8G8&?�A�7H<�; 7x>{I ?�[�?�I DT; <G?{K]>*7H71?�ACfO�{Y�A�; 71J.?�>*U�:]; 8&?�A�>C8&; F"7]U�F*AGA&?�W<G=@F"7@X�; 7HJ.8&F/>/I ?�[�?�IgBb>.<G?�8,F*_QU�>*71X�; X1>�8�?c=@>�8&8&?�A&71<i; <�U�A&?�>C8&?�XEjbl¿F"; 7H; 71J/8&:1?�_ A&?��bYH?�7b8�=1>C8G8&?�A&71<,X�; <GU�FC[�?�A&?�X]X�YHA&; 7HJ.8&:H?T=HA&?�[b; F"YH<; 8&?�A�>C8&; F*7�B�8&:1?T<GYH=1=@F*AG8&<,F*_�>*I I1U�>*71XH; X1>C8&?T=@>C8G8&?�A&71<,>�A�?TU�F"YH7b8&?�X>*71XZ; 7�_ A&?��bYH?�7b8TF"7H?�<�>CA&?`XH; <GU�>CA�X�?�XQf�h�:H?`KEF"<�8T=�A&F"KE; 7H?�7b8�>CI WJ"FCA�; 8�:HK�j1>*<G?�XEF"7E8&:1; <�>*=1=�A&F�>*U�:]; <�8&:H? e =HA&; F*A&;@>*I J*F*A&; 8&:1K�� ����fe [">CA&; ?�8�lEF*_�KEF�X�; �1U�>C8&; F"7H<cF*_Q8&:1; <c>*I J*F*A&; 8&:1KÀ>�A&F"<G?E� ¨b£Q��¥b£@�"�b£�"¦���; 7�F*A�XH?�A�8&F�; K]=�A&FC[�?ZX�; Á(?�A&?�7b8�?��EU�; ?�71U�l�>*<G=@?�U�8�<�f�»�F�DcW?�[�?�A�BH>*I IQF*_�8&:H?�<&?{>*I J*F*A&; 8&:1KE<c:1>�[�?c8&FEX�?�8&?�A&K]; 71?c8�:H?{<GYH=1=@F*AG8&<F*_ n"  _ A�?���Y1?�7�8.=1>C8G8&?�A�7H<.>C7@X9FC_,<GF"KE?�; 7H_ A�?���Y1?�7�8.F*71?�<{; 798&:1?XH>C8�>*j1>*<G?"fh�:H?z<G?�U�F"71X�>C=1=�A�F">*U�:�; <Ej@>*<G?�XkF"7k8&:1?z?��b8&A�>CU�8�; F"7�FC_/K]>C��; WK]>*I Ã/_ A&?��bYH?�7b8/=1>C8G8&?�A&71<�B�_ A&F*KÄDT:H; U�:�>CI I�<GY1=@?�A&<G?�8�<`>CA&?�; 7�_ A&?�W��Y1?�7�8c>*71Xx>CI I(<&YHjH<&?�8&<c>�A�?T_ A&?��bYH?�7b8�fih�:1; <c>*=H=HA&F�>*U�:xU�F*K�jH; 7H?�<>�I ?�[�?�I Dc; <G?Tj@F*8G8&F"K�WYH=`8GA�>�["?�A�<&>*IbDT; 8&:E>�8&F"=�WXHF�Dc7�8GA�>�[�?�A&<&>*IH; 7F*A�X�?�AT8&FE�bYH; U�a�I l]�171X]8�:H?/K]>��H; K]>*I@_ A&?���Y1?�7�8T=1>C8G8�?�A&7H<�f�h�:H?�7�B>*I Ic_ A&?���Y1?�7�8�=1>C8G8&?�A�7H<E>CA&?zXH?�A&; ["?�X�_ A�F*KÅ8�:H?�<G?xF"71?�<E>C7@X�F"71?I >C<G8cX1>C8�>*j1>*<G?{<&U�>C7x; <�U�>CAGA&; ?�X]F"7]8&F/U�F"YH7b8,8&:H?�; Ac<GY1=H=@F*AG8Cf\h�:1?KEF"<�8\=�A�F*K]; 71?�7�8\>CI J"F*A&; 8&:HKÆY1<G; 7HJ�8&:H; <\>*=H=HA&F�>CU�:`; <i�z>���W��Z; 7H?�A� ¥���f`�\�H=@?�A&; KE?�7b8�>*IQA&?�<GY1I 8�<{:@>�[�?`<&:HF�DT7z8�:1>C8�8�:H; <.>*=1=�A&F�>*U�:9; <=1>CAG8&; U�Y1I >CA&I l�?��EU�; ?�7b8]_ FCAZ?��b8GA�>CU�8&; 71J�K]>C�H; Kx>CI�_ A&?��bYH?�7b8]=1>C8GW8&?�A&71<�BQjHYH8TDT:1?�7V>*=H=1I ; ?�X�8&F]?���8GA�>*U�8&; 71JZ>CI I\_ A&?���Y1?�7�8.=1>C8G8�?�A&7H<=@?�AG_ F*A&K]>*7HU�?�<TXbA�>C<G8&; U�>CI I lZX�?�U�A�?�>C<G?.j@?�U�>*YH<&?{F*_�8&:1?�U�F"<�8TF*_�8&:1?I >C<G8�<GU�>*7VDT:1; U�:VA&?��bYH; A&?�</A&F"YHJ":HI l�>C7�; 7HU�I YH<&; F"7�8&?�<�8�j@?�8D�?�?�7?�>*U�:�_ A&?��bYH?�7b8{=@>C8G8&?�A&7V>*71X�?�>*U�:9F"j�¿�?�U�8/F*_�8&:1?�X1>�8�>Cj@>C<&?"f e <_ F*A]8�:H?x�HA�<�8x>*=H=HA&F�>CU�:�Bc>*I J"FCA�; 8�:HKE<Ej@>C<&?�XkF"7k8&:1; <]>C=1=�A�F">*U�::1>�[�?E8&F�?���8GA�>*U�8`8&:1?]<GYH=1=@F*AG8&<�FC_�>*I I�_ A&?��bYH?�7b8�=@>�8&8&?�A&7H<`_ A�F*K8&:1?.XH>C8�>*j@>C<&?"fh�:H?�8&:1; A�X`>*=H=�A�F">*U�:�B�A�?�=�A&?�<G?�7b8&?�X/jbl{8�:H?��,I F*<&?c>*I J*F*A&; 8&:1KÇ� �C§C��B; <\j1>*<G?�X`F"7`8�:H?�8&:H?�F*A&?�8�; U�>*I�_ A�>*KE?�D�F*A&a/; 7b8GA&F�X�Y1U�?�X/; 7x� ��¢��H8&:@>�8YH<&?�</8&:H?xU�I F"<GYHA&?]F*_c8�:H?Z¼.>CI F"; <�U�F"7H7H?�U�8�; F"7�� ��§C��f96�7V8&:1; <�>*=�W=�A�F">*U�:�B�8&:1?c_ A&?��bYH?�7b8�U�I F"<G?�X]=@>�8&8&?�A&7H<���>C7@X�8&:1?�; A�<GY1=H=(FCAG8��i>�A�??��b8&A�>*U�8&?�X/_ A&F"K«8&:H?cXH>C8�>*j@>C<&?�; 7E>{I ?�[�?�I DT; <&?cK]>*717H?�A�f e U�I F"<G?�X=1>C8G8&?�A�79; <.8&:H?�JCA&?�>C8&?�<G8/=@>�8G8�?�A&7VU�F*KEK]F*7�8&FZ>x<&?�8.F*_�F*jH¿?�U�8&<F*_�8&:H?�X1>C8�>*j1>*<G?"£�>*71X�?�>*U�:V7HF"7�WU�I F"<G?�X�=@>�8G8�?�A&79:1>*<T8�:H?`<�>CK]?=�A�F*=(?�AG8&; ?�<9� ;gf ?"f�8&:H?z<&>*KE?Z<G?�8xFC_/F"j�¿?�U�8�<]U�F*7b8�>*; 7H; 7HJ�; 8x>*71X8&:�YH<`8�:H?Z<&>*KE?Z<GY1=H=@F*AG8��`>*<E; 8&<�U�I F*<&Y�A&?"B�8&:H?x<&K]>*I I ?�<�8]U�I F"<G?�X=1>C8G8&?�A�7xU�F*7b8�>*; 7H; 7HJ�; 8Cfih�:1?�7�B�>*I I@_ A&?���Y1?�7�8c=@>�8&8&?�A&7H<c>*<�D�?�I I�>C<8&:1?�; A{<GY1=H=@F*AG8{>CA&?`XH?�A&; ["?�Xx_ A&F"KÀ8�:H?{_ A&?��bYH?�7b8�U�I F"<G?�X�=1>C8G8�?�A&7H<>*71X�8�:H?�; AE<GY1=H=(FCAG8�Dc; 8&:1F"Y�8]>*U�U�?�<G<G; 71JV8&:1?zXH>C8�>*j1>*<G?"f�»�?�71U�?7HF*8E>*I Ic=1>C8G8&?�A&71<�>CA&?ZU�F"7H<G; XH?�A&?�XdXHY�A&; 71JZ8�:H?xKEF"<�8�?���=@?�71<G; [�?=1>CAG8zF*_`8�:H?V>CI J"F*A&; 8&:HK·� U�F"Y17�8&; 71J�8&:1?9<GYH=1=@F*AG8&<ZF*_`8&:1?9=1>C8GW8&?�A&71<��i>*71X�8&:1?c<G?�>CA&U�:]<G=@>CU�?T; <�XbA�>*<�8&; U�>*I I lEA&?�X�Y1U�?�X(B�?�<G=@?�U�; >*I I l_ F*A.<�8GA&F"71J*I l�U�F*AGA&?�I >C8&?�XVXH>C8�>�f��\�H=@?�A&; KE?�7b8&<c:@>�["?`<G:1F�DT7�8&:@>�88&:1; <.>*=H=�A�F">*U�:9; <�K�YHU�:ZKEF*A&?�?��EU�; ?�7b8�8&:1>*7z8&:1?/8D�F]=HA&?�[b; F"YH<F"7H?�<TF*7z<&YHU�:zX1>�8�>�f

È�É �z>C�H; K]>*IgÊ1KE?�>*7H< É K]>C��; K]>*I@DT; 8&:ZA&?�<G=(?�U�8c8&FE<G?�8T; 71U�I Y1<G; F"7QÊ f

©b6&¼��/�{�«�\�H=HI F*A�>C8&; F*71<�f���F"=blbA&; J":b8 UË �*§"§*§xÌ`Í.νÏ(Ð�Ñ]ÒEÓ�Ó]B1�.?�U�?�K�j@?�A��*§*§"§�f ´�F"I YHKE?`�bBH6<G<&YH?��.W,=1>*J"?/¡"¡

Page 2: Mining frequent patterns with counting inference

1.2 Contribution��?{=HA&?�<G?�7b8,8�:H?`L�NQP�R1NQS � >CI J"F*A&; 8&:HKzBH; 7b8GA&F�X�Y1U�; 7HJ�>�7HFC[�?�I BH?�_ W_ ?�U�8&; ["?�>*71X�<&; K]=HI ?9F*=H8&; KE; ^�>�8�; F"7�F*_/8�:H? e =HA&; F*A&;.>*I J*F*A&; 8&:1Kzfh�:H; <�F"=�8�; KE; ^�>C8&; F"7.; <�j1>*<G?�X.F"7 mHnCogo�p&q�r]s&t*ubr1o v rbw{v rCy�p�q�p�r@s�p 8&:@>�8A&?�I ; ?�<{F"7�8&:H?`7H?�DÆU�F"7HU�?�=H8�F*_T| p�}`mHn*o o�p�q�rH~ f e a�?�lZ=@>�8G8�?�A&7�; <>�KE; 71; K]>*I@=1>C8G8&?�A&7ZF*_\>*7Z?��bYH; [">CI ?�7HU�?.U�I >C<&<cJ�>C8&:H?�A�; 71JE>*I I(=1>C8GW8&?�A&71<�8&:@>�8]:1>�[�?x8&:H?Z<&>*KE?ZF"j�¿�?�U�8&<���f�h�:1?Z=1>C8G8&?�A&7�U�F"YH7b8&; 71J; 7�_ ?�A&?�7HU�?/>*I I F�DT<�8�F�X�?�8�?�A&KE; 7H?c8&:1?�<GY1=H=(FCA&8&<�FC_ ~�t��xp _ A&?���Y1?�7�8>*71X9; 7�_ A&?��bYH?�7b8{=@>�8G8�?�A&7H<�� 8&:1?`a�?�l�=1>C8G8&?�A�7H<��T; 798&:1?�X1>�8�>Cj@>*<G?F"7HI lbf�h�:H?z<GY1=H=(FCA&8&<�F*_/>*I IcF*8&:1?�AE_ A�?���Y1?�7�8E=@>C8G8&?�A&71<E>CA&?�X�?�WA&; [�?�X�_ A�F*K¾8&:1?c_ A&?��bYH?�7b8,a"?�lE=1>C8G8&?�A�7H<�fih�:H; <c>CI I F�DT<,8&F/A&?�XHYHU�?"B>C8`?�>*U�:dXH>C8�>*j1>*<G?]=@>C<&<�BO8&:H?E7bY1K�j@?�A/FC_c=@>�8&8&?�A&7H<`U�F*71<G; X�?�A�?�X(B>*71XQBQ?�["?�79KEF*A&?`; KE=(FCA&8�>*7�8�B@8&F]A&?�X�Y1U�?`8&:1?`7�YHK�j(?�ATF*_�=1>*<G<&?�<; 7�8&F*8�>*Igfih�:1; <�F"=�8�; K]; ^�>C8&; F"7�; <�[*>*I ; X�<G; 7HU�?Ta�?�l`=1>C8G8�?�A&7H<i:@>�["?c>=�A�F*=(?�AG8l�8&:1>C8�; <�U�F*K]=1>C8&; jHI ?�Dc; 8&:�8&:H?cF*A&; J*; 7@>CI e =HA&; FCA�;��gXH?�8&?�AGWKE; 7H; <�8�; UC�O:1?�Y�A&; <�8�; U"£bD,?T<G:1F�Dk8&:@>�8,>*I IH<&YHj1<G?�8&<�F*_Q>{a�?�l`=@>C8G8&?�A&7>CA&?`a�?�lz=1>C8G8�?�A&7H<{>*71X9>*I IO<&YH=@?�A�<G?�8&<{FC_,>E71F"7�W�a�?�lZ=@>�8&8&?�A&79>CA&?7HF"7HW�a�?�l.=@>�8&8&?�A&71<�f\h,:1?�7QB*8&:H?�U�F"YH7b8&; 7HJ�; 7�_ ?�A&?�7HU�?�; <\=@?�A&_ F*A&KE?�X; 7]>{I ?�[�?�I Dc; <G?�K]>*7H71?�A���6�_Q>.U�>*71X�; X1>�8�?�=@>C8G8&?�A&7�F*_Q<&; ^�?��`DT:H; U�:<GY1=H=@F*AG8T:@>C<�8&F�j@?{XH?�8&?�A&KE; 71?�X]; <T>`71F"7�W�a�?�lE=@>C8G8&?�A&7�Bb8&:1?�7x; 8�<<GY1=H=@F*AG8�; <�?��bY1>*I�8&F{8&:1?�KE; 71; Kx>CIH<GY1=H=@F*AG8,>*KEF"7HJ{8&:H?c=@>�8&8&?�A&7H<F*_�<G; ^�?��bW&�T8&:1>C8�>CA&?{; 8�<�<&YHj1<G?�8&<�fi67EU�F"KE=@>CA&; <&F*7]8&F/KEF"<�8�F*8&:H?�AKEF�X�; �1U�>C8&; F"7H<.F*_ e =�A�; F*A&;gB(8&:1; <�; 71XHYHU�?�<.>]KE; 7H; K]>*I�; KE=1>*U�8{F"78&:1?ZYH7@X�?�A&<�8�>C7@XH>*jH; I ; 8lk>C7@X�<&; KE=1I ; U�; 8ldF*_{; KE=1I ?�K]?�7�8�>C8&; F"7�F*_8&:1?V>*I J"FCA�; 8�:HKzfÀh,:1?V; KE=(FCA&8�>*7�8zXH; ÁQ?�A&?�7HU�?V; <Z8&FkXH?�8&?�A&KE; 71?>*<�K�Y1U�:�<&YH=H=(F*AG8�U�F"YH7b8&<E>*<�=@F"<G<&; j1I ?xDc; 8&:1F*YH8]>*U�U�?�<G<G; 7HJV8&:1?XH>C8�>*j@>C<&?�jbld; 7H_ F*A&K]>C8&; F"7�J�>C8&:H?�A�?�X�; 7�=�A�?�[b; F"YH<]=@>C<&<G?�<�f e <<G:1F�Dc7�jblz8�:H?�?��H=@?�A&; KE?�7b8&<�B(8&:1?�?��EU�; ?�7HU�lVJ�>*; 7�; <.Y1=98�FZF*71?F*A�X�?�A�F*_iK]>*J*71; 8&Y1XH?�F"7zU�F*AGA&?�I >C8&?�X�XH>C8�>�f

1.3 Organization of the paper67�8&:H?�71?��b8]<&?�U�8�; F"7�BcD�?zA&?�U�>*I I{8&:1?z=HA&F"jHI ?�K F*_/KE; 71; 71JV_ A&?�W�bYH?�7b8,=@>C8G8&?�A&71<�f�h,:1?{?�<&<G?�7�8�; >*IQ71FC8�; F"71<c>*7@XxX�?��171; 8&; F*71<�FC_\a�?�l=1>C8G8�?�A&7H<]>*7@Xk=1>C8G8�?�A&7�U�F"YH7b8&; 7HJ�; 7�_ ?�A&?�7HU�?9>CA&?zJ"; ["?�7�; 7�©b?�U�W8&; F"7Z �f�h�:H?`L�NQP�R1N�S/>*I J*F*A&; 8&:1K�; <cX�?�<&U�A�; j(?�X]; 7z©b?�U�8&; F"7x¤�>*71X?��H=@?�A&; KE?�7b8�>*I@A&?�<GY1I 8�<c_ F*A�U�F"KE=@>�A&; 71J�; 8&<�?��EU�; ?�71U�lx8&F�8�:HF"<G?.F*_e =�A&; F*A&;gB��z>C�bWG�Z; 7H?�AE>*71X��,I F"<G?z>CA&?Z=HA&?�<G?�7b8&?�X�; 7�©b?�U�8&; F"7�¦bf©b?�U�8�; F"7�¡�U�F"7HU�I Y@X�?�<c8&:H?.=1>*=@?�ACf

2. STATEMENT OF THE PROBLEM¬ ?�8��Vj@?`>`�@7H; 8&?/<G?�8{FC_ v op��~ B�«>��@7H; 8&?{<&?�8�F*_ t � ��pGs�og~ �g?"f�JHf B8GA�>*71<&>*U�8&; F*7�; XH<���>*71X��������� >Vj1; 7@>CAGl�A�?�I >C8&; F"7Æ� DT:H?�A�?� ��������� ��K]>�l]j@?.A&?�>"XZ>*<! �; 8&?�K � ; <T; 71U�I Y1X�?�Xx; 7Z8GA�>*71<&>*U�8&; F*7��" ��fih�:H?.8GA&; =HI ?$#&% � � � � � � ; <cU�>*I I ?�X�' n*o�n*~�p�o f��>*U�:�<GY1jH<G?�8)(�FC_��x; <\U�>CI I ?�X�> mHnCogop�q�r f\��?�<&>�l.8�:1>C8�>T=@>C8G8&?�A&7(¾; < v r@s� u ' p '/; 7V>*7zF"j�¿�?�U�8 �*� ¾; _ � �������+� � _ F*A.>*I I ��� (�f¬ ?�8-,zj(?.8&:1?._ YH71U�8&; F*7�DT:H; U�:9>*<G<G; J"7H<{8&FE?�>*U�:9=1>C8G8&?�A&7�(.�/�8&:1?�<G?�8cF*_O>*I I(F"j�¿�?�U�8&<,8�:1>C8c; 71U�I Y1X�?T8&:1; <,=@>C8G8&?�A&7��), � ( � %10 �2�.3 � ; 7HU�I Y1XH?�<�(-4�fh�:H? ~&uCm"mHt*q�o F*_�>ª=1>C8G8�?�A&75( ; <ÆJ"; ["?�7Åj�lQ�7698�: � ( � %U�>CA�X � , � ( �9�9; U�>�A�X � � f`�HF*A{>EJ*; [�?�7z8&:�A&?�<&:HF"I X=<2> ?�@�ACB �ED F��HGJI B>�=@>�8&8&?�A&7K(«; <cU�>*I I ?�X y�q&p9L�u@p�rHoOmHn*o o�p�q�r ; _M6�8N: � ( ��O <2> ?�@�ACB\fh�:H?,8�>*<Ga/FC_@KE; 7H; 7HJT_ A&?���Y1?�7�8\=1>C8G8&?�A&71<OU�F"7H<G; <�8&<i; 7�XH?�8&?�A&KE; 71; 71J>*I Ib_ A&?���Y1?�7�8i=@>C8G8&?�A&71<�8&F"J"?�8&:H?�AiDT; 8&:`8&:1?�; A�<GYH=1=@F*AG8&<O_ F*A,>TJ"; ["?�78&:HA&?�<G:1F*I XZKE; 7H<&YH=QfP h,:1?Ç��A&?�71U�:ÄK]>C8&:1?�K]>C8&; U�; >*7�¯,I >*; <&?�°\>*<GU�>*I �RQ\�,I ?�A&K]F*7b8GW�H?�A&A�>*71X«��¡��C �BTSO��¡"¡��k°\>CA&; <��Z; 7b[�?�7�8�?�X�>*7 ?�>CA&I l U�F*K]=HY�8�; 71JX�?�[�; U�?"fU e <G; KE; I >CAx7HF*8&; F"7�F*_.?��bYH; [">CI ?�71U�?ZU�I >*<G<G?�<]Dc>*<Z>*I <GFVA�?�U�?�7�8&I l=�A�F*=(F"<G?�X�jbl�³`f\¯�>�lb>CA�XHF9>*7@X�³`f e J*A�>�Dc>*I�� ¡C�c8&F9U�:1>CA�>*U�8&?�AGW; ^�?* >CW�Kx>��H; K]>*I " A&YHI ?�<x�g; f ?"f B�>*<G<&F�U�; >C8&; F*7�A&YHI ?�<`Dc; 8&:dK]>C��; K]>*I>*7�8�?�U�?�XH?�7�8���f

3. PATTERN COUNTING INFERENCE67�8�:H; <�<G?�U�8&; F"7QB�D�?TJ"; [�?c8&:H?�8&:1?�F*A&?�8&; U�>CI@j1>*<G; <,F*_(8&:1?�71?�D L�NQP��R1NQS9>*I J*F*A&; 8&:1Kzf�h�:H; <�j@>C<&; <]=�A&FC[�; XH?�<E>C8�8&:1?x<&>*KE?x8&; KE?]8&:1?=�A�F�F*_{F*_{U�F*AGA&?�U�8&71?�<G<EF*_T8�:H?z>*I J"F*A&; 8&:1Kzf�67�©b?�U�8&; F"7k¤HB,8&:H?�<&?8&:1?�FCA&?�K]<,DT; I I�j@?{8�Y�A&71?�X]; 7�8&FE=H<G?�Y@X�F*W�U�F�X�?"f¬ ; a"? e =HA&; FCA�; BiL�N�P�R1NQSEDT; I I\8&A�>�["?�A&<&?`8&:1?�=@F�D�?�A&<G?�8`F*_)�kI ?�["?�I WDT; <&?"� e 8T8&:1?T�WVYXz; 8&?�A�>C8&; F"7QB(8&:1?�>*I J*F*A&; 8&:1K�J"?�7H?�A�>�8&?�<.�HA�<�8.>*I Is&n*r ' v ' n*op �[Z mHn*ogop�q�rH~ f\ p ]irHv o v�t*r_^ ` e ��W=1>C8G8�?�A&7/( ; <x>�<GY1jH<G?�8]F*_a� <GYHU�:k8&:@>�8

U�>CA�X � ( � %b�(f e s�nCr ' v ' nCo�p �[Z mHn*o o�p�q�r ; <c>T��W�=@>C8G8&?�A&7EDT:1?�A&?{>*I I; 8&<T=�A&F"=@?�A�<GY1j�W�=@>C8G8&?�A&71<c>�A&?/_ A&?���Y1?�7�8�f¼�; [�?�7Z8&:H?`<G?�8�FC_�U�>*71XH; XH>C8&?a�bW�=@>C8G8&?�A&71<�B1F"7H?�XH>C8�>*j1>*<G?`=1>*<G<�; <YH<&?�Xd8&FdX�?�8&?�A�KE; 71?Z8&:H?�; A]<&YH=H=(F*AG8�f�67H_ A�?���Y1?�7�8]=@>�8&8&?�A&7H<x>�A�?8&:1?�7 =�A&Y17H?�XQfÄh�:H; <9>C=1=�A&F�>*U�: D�F*A&ab<9j(?�U�>CY1<G?�FC_E8&:H?�D�?�I I Wab71F�Dc7�_g>CU�8�8&:1>C8�>z=@>�8&8&?�A&7dU�>C717HF*8�j(?E_ A�?��bY1?�7b8�; _�; 8�:1>*<�>*7; 7�_ A&?���Y1?�7�8T<GY1j�W=1>C8G8&?�A�7Qf

3.1 Key Patterns­�YHA,>C=1=�A�F">*U�:�; <�j1>*<G?�XEF*7�8�:H?cF"jH<G?�A�[*>C8&; F"7�8&:@>�8i_ A&?��bYH?�7b8�=1>C8GW8&?�A&71<TU�>*7zj@?`U�F*71<G; X�?�A&?�X�>*<+ �?��bYH; [">CI ?�7b8 " ; _\8&:1?�lz>CA&?/; 7HU�I Y@X�?�X; 7�?��1>CU�8�I l�8&:1?]<&>*KE?]F"j�¿�?�U�8&<�f���?]XH?�<GU�A&; j@?]8&:1; </_g>CU�8Ej�lV8&:1?_ F"I I F�Dc; 71J]?���Y1; [">*I ?�7HU�?.A&?�I >C8&; F"7dc�F"7Z8&:H?{_ A�?���Y1?�7�8T=1>C8G8�?�A&7H<�f\ p ]irHv o v�t*rdeN` ¼{; ["?�7d8D�F9=1>C8G8&?�A�7H<2( ��f �g��BOI ?�8h(=c f ; _

>*71XdF"71I lk; _a, � ( � %5, ��fT� f�h�:H?z<G?�8EF*_.=1>C8G8&?�A�7H<�DT:H; U�:�>�A�?p9L�u�v i*n" p�r1o 8&F�>.=1>C8G8�?�A&7j(�; <�J*; [�?�7Ej�l D ( I %10 f �k��3l(=c f 4�f67z8&:1?`U�>*<G?�F*_�=1>C8G8�?�A&7H<$(Ç>C7@X f DT; 8&:_(=c f B@j@F*8&:9=1>C8G8�?�A&7H<F"jb[�; F"Y1<GI lZ:1>�[�?�8&:1?{<�>CKE?.<GY1=H=@F*AG8C�m�n�o+ocNÀ�"f�p p�o ( nCr ' f ��p.o q,t.mHn*ogop�q�r1~H`��rR� (=c f %[st6�8N: � ( � %/6�8N: �Yf-�

� rYrR� (u� f/v 698�: � ( � %/698�: �Yf-� %[sw(=c fL+xNyzyz{}| � r~� (=c f�� s�, � ( � %�, ��fT� %zs�698N: � ( � %

U�>CA�X � , � ( ���9; U�>�A�X � � %�U�>CA�X � , ��f-���9; U�>�A�X � � %/698�: ��fT� f� rYrR� ©b; 7HU�?�(�� f >C7@X�, ; <�KEF*71F*8&F"7HF"YH<�XH?�U�A�?�>C<&; 71JHB.D�?:1>�[�?k, � ( ��� , ��f-� f�6�8N: � ( � %7698�: �Yf-� ; <V?��bYH; [">CI ?�7b8�8&FU�>CA�X � , � ( ��� % U�>CA�X � , ��fT�9� Dc:1; U�: ; KE=HI ; ?�<ZDc; 8&: 8�:H?�_ F*A&KE?�A, � ( � %�, �Yf-� >C7@X]8&:�YH<+(�c f f»�?�71U�?.; _iD�?/a�7H?�D�8�:H?.A&?�I >C8&; F"7=c�; 7z>"XH[*>*71U�?"BHD�?.D�F"YHI Xz7H?�?�X8&F/U�F"Y17�8i8&:1?c<GYH=1=@F*AG8�F*_QF"7HI l�F"7H?T=@>�8&8&?�A&7�; 7�?�>*U�:E?��bYH; [">CI ?�71U�?U�I >C<&<�f,­�_�U�F"Y�A&<&?T8&:1; <,; <�71FC8�8&:1?TU�>*<G?"£HjHY�8,D�?{U�>C7]U�F"71<�8GA&Y1U�8�; 8<�8&?�=zjblx<�8&?�=}�*f�h�:�YH<�BHD,?/Dc; I I,� ; 7ZJ"?�7H?�A�>*I �c7H?�?�Xx8&FEXH?�8&?�A&KE; 7H?8&:1?�<GY1=H=(FCAG8cF*_\KEF*A&?�8&:1>*7xF"7H?{=1>C8G8�?�A&7Z; 7Z?�>*U�:ZU�I >*<G<�BHj1Y�8c71FC8F*_�>*I I(FC_Q8&:H?�KzfO6_(D�?�>*I A&?�>"Xbl]:1>�["?�XH?�8&?�A&KE; 7H?�X�8&:H?T<GY1=H=@F*AG8,F*_>x=1>C8G8&?�A�7�(Ç; 798&:H?EXH>C8�>*j1>*<G?E>*71X9?�71U�F*Y17�8�?�A.I >C8&?�A/>x=@>�8&8&?�A&7f���D ( I B*8&:1?�7`D�?c7H?�?�X�71FC8�>*U�U�?�<G<i8&:1?TXH>C8�>*j1>*<G?�_ FCA,; 8�j@?�U�>*YH<&?D�?.a�7HF�D�8&:@>C8!6�8N: �Yf-� %/6�8N: � ( � fh�:H?,�HA&<G8\=1>C8G8&?�A�7H<OF*_@>*7`?��bYH; [">*I ?�71U�?,U�I >*<G<O8�:1>C8OD�?,A&?�>*U�:`YH<&; 71J>�I ?�[�?�I DT; <&?Z>*=1=�A&F�>*U�:k>CA&?x?��1>*U�8&I l�8&:1?]KE; 7H; K]>*I ��=1>C8G8&?�A�7H<�; 78&:1?{U�I >C<&<��\ p ]irHv o v�t*r���` e =1>C8G8&?�A&7�(Æ; <T>E| p�}{mHn*o o�p�q�r ; _M( ���*� �zD ( I £

8&:@>�8.; <�B(; _�7HF]=HA&F*=(?�A{<&YHjH<&?�8�F*_�(¾; <�; 7�8&:H?`<&>*KE?`?��bYH; [">CI ?�71U�?U�I >C<&<�f e s&n*r ' v ' n*o�p | p�}/mHnCogop�q�r ; <{>�=@>C8G8&?�A&7�<GY1U�:Z8&:1>C8.>*I IO; 8&<=�A�F*=(?�AT<GY1j�W=1>C8G8&?�A�7H<c>CA&?._ A&?��bYH?�7b8�a�?�lx=@>�8&8&?�A&7H<�f� 67z8&:H?�>*I J"F*A&; 8&:HK�B@8&:H?`?���Y1; [">*I ?�7HU�?.A�?�I >C8&; F"79; <{7HF*8{?���=1I ; U�; 8&I lJ"?�7H?�A�>C8&?�XQB"j1Y�8O; <~��>C<\8&:1?�>CI J"F*A&; 8&:HK«; <\j@>C<&?�X�F*7�8&:H?�_ F"I I F�DT; 71J8&:1?�FCA&?�K]<~��; KE=1I ; U�; 8&I lxYH<&?�X(f��É �Z; 7H; K]>*IgÊ1KE?�>*7H< É KE; 7H; K]>*I@DT; 8�:ZA&?�<G=@?�U�8T8&F�<&?�8T; 7HU�I Y1<G; F*7�Ê f

©b6&¼��/�{�«�\�H=HI F*A�>C8&; F*71<�f���F"=blbA&; J":b8 UË �*§"§*§xÌ`Í.νÏ(Ð�Ñ]ÒEÓ�Ó]B1�.?�U�?�K�j@?�A��*§*§"§�f ´�F"I YHKE?`�bBH6<G<&YH?��.W,=1>*J"?/¡�¥

Page 3: Mining frequent patterns with counting inference

­�j1<G?�A&[�?`8&:@>C8.>*I IiU�>C7@X�; X1>�8&?�a"?�l�=@>�8&8&?�A&71<.>CA&?�>*I <GFzU�>*71XH; X1>�8�?=1>C8G8�?�A&7H<�f�O; J"Y�A&?Z��=HA&?�<G?�7b8&<{8&:H?�I >C8G8&; U�?EF*_,_ A�?���Y1?�7�8.=1>C8G8&?�A�7H<{_ F"YH7@X�F"7>*7k?��1>CK]=HI ?ZXH>C8�>*j1>*<G?ZDT; 8&:�>9KE; 71; K�YHK�<GY1=H=(FCA&8�<T> ?N@�A�B1%� ;�� BO:1; J":HI ; J":�8�; 71Jza"?�l�=@>�8&8&?�A&7H<`>*71X9?���Y1; [*>*I ?�7HU�?�U�I >*<G<G?�<x� 8&:1?XH>C8�>*j@>C<&?�; <{<G; KE; I >�A.8&F]8&:1?`F"7H?�X�?�<GU�A�; j(?�X�; 7�8&:1? q�ubr1rHv rbw9p�� Znl��m( p <GYHj1<G?�U�8&; F"7�FC_\8�:H?�L�NQP�R1NQSE>*I J"FCA�; 8�:HKzB1?��HU�?�=�8T8&:@>�8T8&:1?; 8&?�K��ÆDc>*<�A&?�KEFC["?�X@��f�67z8&:H; <{XH>C8�>*j1>*<G?"B1_ F*A{; 71<�8�>*7HU�?"B(?�[�?�AGlF"j�¿�?�U�8�U�F*7b8�>*; 7H; 7HJ��k>*I <GF{U�F"7b8�>*; 71<`£�jHYH8O7HF*8�>*I IbF*_H8�:H?�K U�F"7�W8�>*; 7��]f/»�?�71U�?h0 �a4E>C7@X_0 ��a4E>CA&?�; 798&:1?�<&>*KE?`?��bYH; [">*I ?�71U�?U�I >*<G<~��j1Y�8�7HF*8a0 ���24�f

Ø

{CE}{BC}{BE}{AE}{AB}{AC}

{A} {C} {B} {E}

{ABE} {ACE} {BCE}

{ABCE}

{ABC}

Key pattern

Equivalenceclass

�O; J"Y�A&?E�*�,�\�1>CKE=1I ?{I >C8G8&; U�?.F*_\_ A&?���Y1?�7�8T=@>�8&8&?�A&7H<

3.2 Counting Inference67`8&:H?�>*I J*F*A&; 8&:1KzBCD�?c>*=H=1I l{8&:1?,=�A&Y17H; 7HJ�<�8GA�>�8�?�J*l`8&FTj(F*8&:`U�>*7�WX�; X1>�8�?�=@>C8G8&?�A&71<�>C7@X`U�>*71XH; X1>C8&?ca"?�l`=1>C8G8&?�A�7H<�fih�:H; <�; <(¿Y1<�8&; �1?�Xjbl]8&:H?{_ F"I I F�Dc; 71JE8&:H?�F*A&?�Kz�

���[n�yzx�n�o �bf � rR��� y f v ~Tn,y&q�p9L�u1p�r1o | p�}cmHn*o o�p�q�r9nCr '�(u�f�� o �1p&r ( v ~/n" ~�txnTy�q&p9L�u1p�r1o | p�}.mHnCogop�q�r[`� rYrR��� y ( v ~{r@t*oTn�y&q�p9L�u1p�r1o | p�}/mHn*o o�p�q�rdnCr 'T(u� f�� o �1p�r fv ~{r(tCocnTy�q&p9L�u@p�rHo | p�}.mHnCogop�q�rdp�v o �1p�qC` �L+xNyzyz{}|c��?`DT; I I\�1A&<�8/=�A&FC[�?�>xKEF*A&?�J*?�71?�A�>*Ii<�8�>C8&?�KE?�7b8{F*_

8&:1; <c8&:1?�F*A&?�KÇDT; 8&:z7HF�:�lH=@F*8&:1?�<G; <TF"7x8&:1?.<GY1=H=@F*AG8Cf� rYrR� ¬ ?�8�(«>*71X f j@?.8D�F�=@>�8&8&?�A&71<,DT; 8&:d(u� f >*71X�(Æ71FC8j@?�; 71J9>�a�?�lV=@>C8G8&?�A&7�f9h�:H?�7d8&:1?�A&?]?���; <�8�<T(�� �b�h� �zD ( I Dc; 8&:(���� (�f9��?x71?�?�X�8&F9<G:1F�DÇ8&:@>C8h, �Yf-� % , ��f��h� ( � (�� ��� f��?�U�>*7�A&?�D�A�; 8�?_, ��f-� >*<K, ����f��*� ( � (�� �9� �&� ( � (�� �9� BTF*A, �Yf!� � ( � (�� ���#" , � ( � (�� � fO��?ca�7HF�Dk8&:1>C8�, � (�� � �/, � ( � (�� � Bj@?�U�>*Y1<G?�, � ( � � %�, � ( � >C7@X�,d; <�>�X�?�U�A&?�>*<G; 71Jz_ YH71U�8&; F"7%$*B\<GF, �Yf-�!� , ��f&��� ( � ( � �9�'" , � ( � � fch,:1; <c_ F*A&K�Y1I >�; <�?���Y1; [*>*I ?�7�88&Fk, ��fT��� , ����f(�h� ( � (�� �9�)� (�� � f ©b; 71U�?Z7@>C8&Y�A�>CI I l&(��T���f*�$� ( � (�� �9� B1; 8�_ F"I I F�DT<{8&:1>C8$, ��fT��� , ��f*�$� ( � (�� �9� fTh,:1?F"=H=(F*<&; 8&?E; 71?���Y@>*I ; 8l�; <.F*j�[b; F"YH<�f/»�?�71U�?"B f ; </7HF*8{K]; 71; K]>CIO; 7+ 67`Kx>�8&:1?�K]>C8&; U�>CI�8&?�A�KE<�B ��rR� >*71X � r�r~� <�8�>C8&?,8&:@>�8\8&:H?,<G?�8iF*_1_ A&?�W�bYH?�7b8ia"?�l�=1>C8G8&?�A&71<i; <,>C7�F*A�XH?�A,; X�?�>*I�� F*A�X�F�DT7�W<G?�8��OF*_ � �-, � �{��f. h,:@>C8T; <�B�( È �&( P s�, � ( P � �/, � ( È � f

D f�I >*71Xx8&:�YH<cjblzX�?��@7H; 8&; F"7Z71F*8�>�a�?�lx=@>�8&8&?�A&7Qf � r~� ; <{>�X�; A&?�U�8I F*J"; U�>*IOU�F*71<G?��bYH?�7HU�?.F*_ � r�r~� f��:1?�7�8�>*ab; 71J�8&:1?9<GYH=1=@F*AG8Z; 7�8&Fk>*U�U�F*Y17�8CB�; 8z; <Z<&Y��EU�; ?�7b8x8&F7HF*8&; U�?�8&:1>C8Z>*I I{<GY1jH<&?�8&<xF*_`>�_ A&?��bYH?�7b8x=1>C8G8�?�A&7�>CA&?�_ A&?���Y1?�7�8=1>C8G8&?�A�7H<�B1<GF�8&:1?{=HA&F�F*_\A&?�Kx>C; 7H<c[">*I ; XQfh�:H?z>*I J"F*A&; 8&:1K X�?�8&?�A�KE; 71?�<�B,>C8E?�>CU�:d; 8&?�A�>C8&; F"7QB,8&:H?Za�?�l�=1>C8GW8&?�A&71<c>*KEF"71J/8&:1?�U�>*71XH; X1>C8&?{a�?�lE=@>�8&8&?�A&7H<cjblEYH<&; 71J � rYrR� F*_�8&:1?_ F"I I F�Dc; 71JE8&:H?�F*A&?�Kz����[n�yzx�n�oÅ �f+p p�o ( ��p`n�y&q�p9L�u1p�r1oOmHnCogop�q�r[`��rR� p p�o �_� ( `0/'�1p�r ( �/D ( � 0 � 4 I v 1 698N: � ( � %u698�: � ( �0 � 4 � `� rYrR� ( v ~{n | p�}�mHn*o o�p�q�r�v 1 698�: � ( ��2% �h� �43#546�� 698�: � ( � 0 � 4 �9� `L+xNyzyz{}| � r~� h�:H? É ; _�Êc=@>CAG8Z_ F"I I F�DT<�_ A&F"K ¬ ?�KEK]>�� ��r�r~� f

h�:H? É F*71I l�; _�Ê,=1>CAG8x; <xF"jb[�; F"Y1<�f � r�r~� �HA&F"K � rR� D,?9XH?�X�Y1U�?8&:@>�8h(À; <�>�a�?�lV=@>C8G8&?�A&7d; Á/698�: � ( �72%�698N: � ( � 0 � 4 � Bi_ F*AE>*I I�d� (�fT©b; 7HU�?a698�:Z; <{>�KEF"7HF*8&F"7HF"Y1<�XH?�U�A&?�>*<G; 7HJ�_ YH71U�8&; F*7�B18&:1; <; <c?��bYH; [">*I ?�7b8c8&F � rYrR� fe <�>CI I�U�>*71XH; X1>C8&?Ea�?�lV=@>C8G8&?�A&71<�>CA&?x>*I <&F�U�>*7@X�; XH>C8&?]=@>C8G8&?�A&71<�BDT:H?�7EJ"?�7H?�A�>�8&; 71J�>*I I1U�>*71XH; X1>�8�?T=@>�8&8&?�A&71<\_ F*A�8&:H?T7H?���8�I ?�["?�I1D�?U�>*7z>C8T8&:1?{<&>*KE?.8&; KE?.XH?�8&?�A&KE; 7H?{8&:1?{U�>*71XH; X1>C8&?.a�?�lx=1>C8G8�?�A&7H<>*KEF*71J`8&:1?�Kzf6�_iD�?`A&?�>*U�:V>EU�>*71XH; XH>C8&?T�bW�=@>�8&8&?�A&7�DT:H; U�:9; <{7HF*8/>EU�>*71X�; X1>�8�?a�?�l/=1>C8G8�?�A&7QB"8&:H?�7`D,?T>*I A&?�>"Xbl�=@>*<G<G?�X�>*I F*71J.>C8�I ?�>*<�8�F"7H?cFC_@8&:1?a�?�lz=@>C8G8&?�A&71<{; 7V; 8�<{?��bYH; [">CI ?�71U�?�U�I >C<&<.; 7V>*7V?�>CA&I ; ?�A`; 8&?�A�>C8&; F*7�f»�?�71U�?�D,?V>*I A&?�>"Xbl�a�71F�D ; 8&<x<GY1=H=@F*AG8CfƸ{<G; 7HJ�8&:H?z_ F"I I F�DT; 71J8&:1?�FCA&?�K�B�D�?�X�?�8�?�A&KE; 7H?x8&:1; <]<GYH=1=@F*AG8 qiv o �Ht*uboEn"s&s�p&~�~&v r�w�o �1p' nCon ��nC~�p ����[n�yzx�n�o�¤Hf � y ( v ~/nEr@t*r Zg| p&}`m�n*ogop�q�r � o �1p�r

6�8N: � ( � % �*� �3#546 � 698�: � ( � 0 � 4 �9�98

L+xNyzyz{}|! #: " _ F"I I F�DT<{_ A&F"Kª8&:1?._g>*U�8{8&:@>C8�698�:�; <.>EKEF"7HF*8&F*W7HF"YH<]X�?�U�A&?�>*<G; 7HJ�_ YH71U�8&; F*7�fE O�" �Z6�_$(�; <E7HF*8]>Va�?�l�=@>�8&8&?�A&78&:1?�7x?���; <�8&< �j� ( DT; 8&:d(�cM( � 0 � 4�fi»{?�7HU�?�698�: � ( � %b698�: � ( �0 � 4 ��Ok�h� �<;-5=6�� 698N: � ( � 0?>W4 � fh�:bY1<T8&:1?`X1>�8�>Cj@>*<G?�=1>*<G<.7H?�?�XH<c8�F]U�F*Y17�8{8&:1?/<GY1=H=@F*AG8�<�F*_�8&:1?U�>*71X�; X1>�8�?za�?�lk=1>C8G8&?�A�7H<EF"7HI lbf��.71F�Dc; 71JV8&:1; <�B,D�?�U�>*7�<GY1K�WK]>CA&; ^�?EL�NQP�R@NQSE>*<T_ F"I I F�DT<���68TD�F*A&a�<T?��1>CU�8�I l9>*< e =HA&; FCA�; B@j1Y�8U�F"YH7b8&<cF*71I lx8&:HF"<G?/<GYH=1=@F*AG8&<c; 7Z8&:H?/XH>C8�>*j1>*<G?/=1>*<G<�DT:H; U�:zU�>*7�W7HF*8Ej(?zX�?�A&; [�?�X�_ A&F"K <GYH=1=@F*AG8&<]>CI A&?�>"X�lkU�F"KE=HYH8&?�X(f���?xU�>*78&:�YH<�BbF*7x?�>CU�:xI ?�[�?�IgB�A&?�<�8GA�; U�8,8�:H?�?���=@?�71<G; [�?TU�F"YH7b8,; 7E8&:H?{XH>C8�>CWj1>*<G?�8&F�<&F*KE?dF*_�8&:1?�U�>*71XH; XH>C8&?�<�fM¯,?�8G8&?�A9lb?�8�B._ A&F"Kµ<&F*K]?I ?�["?�IQF*7�B�>*I I(U�>*71X�; X1>�8�?{=@>�8G8�?�A&7]K]>�lEj(?�a�7HF�DT7]8&F`j(?�71F*7HW�a�?�l=1>C8G8&?�A�7H<�f�h�:1?�7�>*I IcA&?�K]>*; 7H; 71J9_ A&?��bYH?�7b8E=1>C8G8&?�A�7H<E>*7@Xd8&:1?�; A<GY1=H=@F*AG8�U�>*7�j@?/XH?�A&; ["?�XxDT; 8&:HF"YH8{>*U�U�?�<G<G; 71JE8&:H?`XH>C8�>*j1>*<G?�>C7blKEF*A&?"f\67�8&:1?cD�F*A&<G8�U�>*<G?�� ;gf&?"f Bb; 7ED�?�>*a�I lEU�FCA&A&?�I >�8�?�XxXH>C8�>���B�>*I IU�>*71X�; X1>�8�?{=@>�8&8&?�A&7H<c>CA&?/>CI <GFEU�>C7@X�; XH>C8&?.a�?�l]=1>C8G8&?�A&71<�f�h�:H?.>CI WJ"FCA�; 8�:HK�j@?�:@>�[�?�<i8&:1?�7�?��H>*U�8&I lE>*< e =�A&; F*A&;gB"DT; 8&:]7HF/<G; J*71; �1U�>*7�8FC["?�A�:H?�>"X(f

4. THE PASCAL ALGORITHM67�8�:H; <x<G?�U�8&; F"7QB�D�?�8GA�>C71<�_ FCA�K 8�:H?�8&:1?�FCA�?�K]<]_ A&F"K 8&:1?9I >*<�8<G?�U�8&; F"79; 7b8&FZ>*7V>CI J"F*A&; 8&:HKzf`h�:1?`=H<&?�Y1XHFCWU�FbXH?`; <{J"; [�?�7V; 7 e I WJ"FCA�; 8�:HK �"f e I ; <�8�FC_c71FC8�>�8�; F"7H<�; </=�A�F�[�; X�?�XV; 7�hO>*j1I ?��"f]��?>*<G<GYHK]?�8�:1>C8)�z; <,I ; 7H?�>CA&I l]F*A�X�?�A�?�X(B�?"fGJHf B �d%10 G �?8-8?8H��@ 4�fih�:1; <DT; I I�j(?.YH<&?�Xx; 7VL�N�P�R@NQS*��Ñhn<A�fh�:H?T>*I J"F*A&; 8&:HK«<G8�>CAG8&<\DT; 8�:`8&:1?,?�KE=H8l/<G?�8�B"Dc:1; U�:�>*I Dc>�lH<�:1>*<i><GY1=H=@F*AG8,F*_i��>C7@X�DT:H; U�:E; <�� jbl�X�?��@7H; 8&; F"7@�i>{a�?�l�=@>C8G8&?�A&7��g<�8&?�=

©b6&¼��/�{�«�\�H=HI F*A�>C8&; F*71<�f���F"=blbA&; J":b8 UË �*§"§*§xÌ`Í.νÏ(Ð�Ñ]ÒEÓ�Ó]B1�.?�U�?�K�j@?�A��*§*§"§�f ´�F"I YHKE?`�bBH6<G<&YH?��.W,=1>*J"?/¡"¢

Page 4: Mining frequent patterns with counting inference

hO>*jHI ?��"��²�F*8�>C8&; F"7H<TYH<&?�XZ; 79L�N�P�R@NQS� ; <.8&:1?�U�F"YH7b8&?�A.DT:1; U�:�; 71XH; U�>C8&?�</8&:H?EU�Y�AGA&?�7b8/; 8GW

?�A�>C8&; F"7Qf�6�7�8�:H?��b8&:d; 8&?�A�>C8&; F"7QB�>*I I�_ A&?��bYH?�7b82��W=1>C8G8�?�A&7H<�>*71Xx>*I I(a�?�lE=1>C8G8�?�A&7H<�>*KEF"7HJ`8&:H?�KÇ>CA&?X�?�8�?�A&KE; 7H?�XQf

���%U�F"7�8�>C; 7H<T>C_ 8&?�Ac8&:H?a��8&:x; 8&?�A�>�8�; F"7�>*I I(_ A�?���Y1?�7�8+��W=1>C8G8�?�A&7H<)(�8�F*J"?�8&:1?�A,Dc; 8&:�8�:H?�; A,<GYH=1=@F*AG8�( 8 698�:QB>*71X�>/j@FbF*I ?�>*7][">CA&; >Cj1I ?+( 8 a�?�l�; 71X�; U�>C8&; 7HJ`; _ (�; <>Z� U�>*7@X�; XH>C8&?C�,a�?�lx=1>C8G8&?�A�7Qf

� � <�8&F*A&?�<�8&:1?`U�>*71XH; X1>C8&?T�bW�=1>C8G8�?�A&7H<T8&F"J"?�8&:H?�A.Dc; 8&:8&:1?�; Ac<GY1=H=(FCA&8.�g; _\ab71F�DT7(��B�8&:1?�j(F�F"I ?�>*7x[">CA&; >*jHI ?( 8 a"?�l(Bb>*7@XE>{U�F"YH7b8&?�A�( 8 =�A&?�X��c<&YH=�DT:H; U�:E<G8&F*A&?�<8&:1?xKE; 7H; K�Y1K�F*_�8&:1?Z<GYH=1=@F*AG8&<EFC_/>*I I � ��� G�� W<GY1j�W=1>C8G8&?�A&71<�FC_)(�f

� � � ��� ������� L�NQP�R1NQS����� 8 698�:�� G £ � 8 a�?�l��Ä8&A&YH?*£��� �! "� 0 � 4�£#�� � È � 0�_ A&?��bYH?�7b8/��W=1>C8G8�?�A&7H<J4�£$��&% � ��' ��� � È�( �*),+-�.�0/1�� � 8 =HA&?�X��c<GY1=2� G £ � 8 a"?�l2� � � 8 698N: 2% GC� £3�� + / ( £45�&% � � ���h% � £6����7 È 2% � £}�98:8.� ( �*),+;�<� /=�� � �>� L�NQP�R@NQS*��Ñhn A � ����7 È � £?�� � %&@ �0A�� � �.DT:H?�A&? A#8 a"?�l*%�8GA&Y1? � ����+ /�CB�� % �.��' �T� # ( �*),+-�.� /�5��� �-D � <GY1jH<G?�8 � � � �9� � £�C��� % �.�5'. A$� � D DT:H?�A&? A#8 a"?�l*%�8GA&Y1? ( ��C#�� A#8 698N:>898`£�E$�� + / ( £�C1��F% �.��' A�� � � ( ��C3�� � % A#8 698�: O <T> ?N@�A�B ����+6/G),+-�.� /��45� � % A#8 a"?�lz>*71X A#8 6�8N:*% A#8 =HA&?�X<�T<GYH= ���H+6/�C=�� A#8 a"?�l��Ä_ >*I <G?b£�C?�� ���>�I��� � 0 A 4"£�5B�� + / ( £�J��� + / ( £�5��� ��+-��K��J/ML � � � f

�V>C7@X��"��f 67�<�8&?�=� �Bc_ A&?��bYH?�7b8 G W�=@>�8&8&?�A&7H<Z>�A&?VXH?�8&?�A&KE; 7H?�XQfh�:H?�l�>CA&?EK]>CA&a�?�XV>C<`a"?�l9=@>�8&8&?�A&7H</YH71I ?�<G<.8&:1?�; A/<GY1=H=@F*AG8`; <��� <�8�?�=H<{¤�Nb¡���f{h�:1?/K]>*; 7zI FbF"=9; <{<G; KE; I >CA{8&F�8&:1?/F"7H?`; 7 e =�A&; F*A&;� <�8�?�=H<�¥T8�F`�b�C��f��O; A�<�8�B1L�N�P�R1NQS*��Ñhn<A]; <�U�>*I I ?�X�8&F.U�F"KE=1Y�8&?,8&:1?U�>*71XH; X1>C8&?`=@>�8&8&?�A&7H<�fTh�:H?�<GYH=1=@F*AG8{FC_�a�?�lZF"7H?�<{; <.X�?�8�?�A&KE; 7H?�X[�; >d>�X1>C8�>*j1>*<G?9=1>*<G<9�g<�8&?�=H<���§�N1��¤b��f¾h�:H? É <GY1jH<G?�8�Ê,_ YH71U�8&; F"7� <�8�?�=��*�C��; <c8&:1?{<�>CKE?`>*<c; 7 e =�A&; F*A&;g�i; 8TA&?�8&YHA&7H<T>*I IQ8&:H?.U�>*71X�; WXH>C8&?`=1>C8G8&?�A�7H<{; 71U�I Y1X�?�XZ; 7z8&:1?/F"j�¿�?�U�8 � f���>*71XH; X1>C8&?`=@>�8&8&?�A&7H<>CA&?T<�8&F*A&?�X�; 7E>/XH>C8�>�<G8GA&Y1U�8&Y�A&?T>*I I F�DT; 7HJ`_ >*<�8�A&?�8GA�; ?�[">*IgB�I ; a"?�8&:1?:1>*<G:HWg8&A&?�?�; 7 e =�A�; F*A&;�� ����f�679F"Y�A/; KE=1I ?�KE?�7�8�>�8&; F"7QB@D,?EYH<G?�X9>8GA&; ?.Dc; 8&:z:@>C<&:z7HF�X�?�</� ��¢C��fh�:H?�79� <G8&?�=H<T�C¦5Nb�*§��i8&:H? É 8GA�>"X�; 8&; F"71>*IgÊ�=HA&YH71; 71J{; <�X�F"71?"f e 8i8&:1?<&>*KE?x8&; KE?"B�_ F*Ax>*I I�A�?�K]>*; 71; 7HJVU�>C7@X�; X1>�8&?Za�?�ld=1>C8G8&?�A�7H<�B,; 8E; <X�?�8�?�A&KE; 7H?�X]DT:H?�8�:H?�Ac8&:H?�lZ>CA&?/a�?�lZFCA�7HF*8`� <�8�?�=V�C¥�>*7@X9��¢���fh�:H?,Dc>�l.8&:@>C8�L�NQP�R1N�SC��Ñ*n A�F"=@?�A�>C8&?�<\; <Oj@>C<&; U�>*I I l`a�7HF�DT7/_ A&F"K8&:1?cJ"?�71?�A�>�8�FCA,_ YH71U�8&; F"7 e =HA&; FCA�; WG¼�?�7�DT:1; U�:ED�>*<,; 7�8&A&FbXHYHU�?�X�; 7� ���gfZ��:1?�7VU�>*I I ?�X�>C8/8&:1?h�b8&:�; 8&?�A�>�8�; F"7QB\; 8`Y1<G?�<�>*</; 71=HYH8.8&:1?<G?�8.F*_i_ A&?��bYH?�7b8 � ��� G�� W=1>C8G8&?�A�7H<"���57 È f�6�8&<{F"Y�8&=1Y�8{; <T8&:1?`<G?�8F*_�U�>C7@X�; X1>�8�?-��W=1>C8G8&?�A�7H<�f e XHX�; 8&; F"71>*I I lx8&F e =�A&; F*A&; W�¼{?�7QÊ < ��t*v r

� �.� ��� �����PO L�N�P�R1NQS*��Ñhn<A671=HYH8���� ��7 È B*8&:1?�<G?�8iF*_@_ A�?���Y1?�7�8 � �Q� G�� W�=@>�8&8&?�A&7H< � DT; 8�:�8&:1?�; A<GY1=H=@F*AG8 � 8 698N:z>*71Xx8�:H? �'8 a�?�l2R@>*JHf­�YH8&=HYH8�� � �bB�8&:H?{<G?�8TFC_\U�>*7@X�; XH>C8&?$��W=1>C8G8&?�A�7H< A ?�>*U�:xDT; 8&:x8&:1?R@>*J A#8 a�?�l(B�8&:H?.[">CI YH? A#8 =�A&?�X��c<GY1=(B@>C7@X]8&:H?.<GY1=H=@F*AG8 A#8 698�:Z; _ A; <c71FC8{>�a�?�l]=@>�8&8&?�A&7Qf

��� �0/HST+;�����0/.�T� � �ST+6 +6U�� �'8 � VEWH� È B � 8 � VEWH� P B 8-8-8 B � 8 � VCWH� ��7 È B<> 8 � VCWH� ��7 È% ����� � ��7 È � B-� ��7 È >X ��+-��+ � 8 � VEWH� È % > 8 � VEWH� È B 8-8-8 B� 8 � VEWH� ��7 P %*> 8 � VEWH� ��7 P B �'8 � VEWH� ��7 È�Y > 8 � VCWH� ��7 È £���&% �.��' A�� � � ( �*),+;�<� /#�� A#8 a"?�l��Ä8GA&Y1?"£ A#8 =�A&?�X��c<&YH=��Z89[�£

$��\% �.��' � �]� GC� W�<GY1jH<G?�8�<�^/FC_ A ( �*),+;�<� /1�� � % ^ ;� � ��7 È ���H+6/3�� ( +- +;�T+ A _ A&F"K � � £45� +- ST+�),+;�<� /=�� A#8 =HA&?�X��c<GY1=2� �h� �z� A#8 =�A&?�X��c<GY1= � ^ 8 698�: � £?�� � % 7HF*8>^ 8 a�?�l ����+ / A#8 a�?�l2�Ä_g>CI <G?b£�CB�� +6/ ( £�5��� +6/ ( £�C��� � % 7HF*8 A#8 a"?�l ����+ / A�8 6�8N:�� A�8 =HA&?�X<�T<GYH=Q£�C#�� + / ( £�E$�� ��+;��KH��/ � �bf

>*71X m@q�u�r(p <�8&?�=H<�BbL�NQP�R@NQS*��Ñhn<A�K]>*a"?�<(8�:H?,7H?�DdU�>*71XH; X1>C8&?�<O; 7�W:H?�A�; 8O8�:H?i_g>CU�8\F*_1j@?�; 7HJ�FCAi7HF*8\>TU�>C7@X�; XH>C8&?,a�?�l{=@>C8G8&?�A&7x� <�8&?�=�¨"�j�l�YH<G; 71J/h�:H?�F*A&?�K��b£�>*71X�; 8�XH?�8&?�A&KE; 7H?�<�>C8,8&:1?�<&>*KE?c8&; KE?�8&:1?<GY1=H=@F*AG8,F*_Q>*I I171F*7HW�a�?�l`U�>*71XH; XH>C8&?�=1>C8G8&?�A&71<T� <�8�?�=z���"�\jbl�YH<&; 71Jh�:H?�F*A&?�KÀ¤�f

Running example.��?�; I I Y1<�8GA�>C8&?{8&:H?/L�NQP�R1N�S/>*I J*F*A&; 8&:1KÇF"7

8&:1?�_ F*I I F�DT; 71JZXH>C8�>*<G?�8T_ F*A-<T> ?N@�A�B�% � ;�� �

ID Items

1� `_ �

2� ba �

3� � ca �

4�ba �

5� � ca �

h�:H?�>CI J"F*A&; 8&:HK�=@?�A&_ F*A&KE<i�1A&<�8,F"7H?�XH>C8�>*j@>C<&?c=@>C<&<i8�F.U�F"YH7b8i8&:1?<GY1=H=@F*AG8EF*_.8&:H?V��W=1>C8G8&?�A&71<�f�h�:H?zU�>*71XH; XH>C8&?z=1>C8G8&?�A�7b0�_j49; <=�A�YH7H?�XZj(?�U�>CY1<G?`; 8.; <{; 7�_ A&?��bYH?�7b8�f e <$0 �T4�:1>*<c8�:H?/<�>CK]?/<GY1=�W=@F*AG8�>*<�8�:H?{?�K]=�8l]<G?�8�B}0 �-4`; <TK]>CA&a�?�X]>*<T>�7HF"7�Wa"?�l]=@>C8G8&?�A&7��

� È 6�8N: a�?�l0 �$4ed ;�� 80 �24gf ;�� 80#a4hf ;�� 80�aT4gf ;�� 80 �-4 G _

e 8/8&:1?E7H?���8`; 8&?�A�>C8&; F"7QB�>*I I,U�>*71XH; XH>C8&? � W�=@>�8&8&?�A&71<`>CA&?]U�A&?�>C8&?�X>*71XZ<�8&F*A&?�Xz; 7 � P f e 8c8&:1?.<�>CKE?.8&; KE?"BH8&:1?.<GY1=H=(FCA&8TF*_,>*I IO=1>C8GW8&?�A&71<�U�F"7b8�>*; 71; 7HJ*0 �T4/>*<c>�<GYHjHW�=@>�8&8&?�A&7x; <�U�F"KE=1Y�8&?�XQfih,:1?�7Z>XH>C8�>*j1>*<G?�=1>*<G<�; <,=@?�AG_ F*A&KE?�X�8�F/X�?�8�?�A&KE; 7H?�8&:1?�<&YH=H=(F*AG8&<�F*_(8&:1?A&?�K]>*; 7H; 7HJE<G; �xU�>C7@X�; XH>C8&?/=1>C8G8&?�A�7H<��

©b6&¼��/�{�«�\�H=HI F*A�>C8&; F*71<�f���F"=blbA&; J":b8 UË �*§"§*§xÌ`Í.νÏ(Ð�Ñ]ÒEÓ�Ó]B1�.?�U�?�K�j@?�A��*§*§"§�f ´�F"I YHKE?`�bBH6<G<&YH?��.W,=1>*J"?/¡"¨

Page 5: Mining frequent patterns with counting inference

� P =HA&?�X��c<GY1= a"?�lt698N:0?���24 d ;�� 8 �0?��a4 d ;�� 8 �0?�>aT4 f ;�� 8 �0?���-4 d ;�� _ d ;��0 � a4 f ;�� 8 �0 �]aT4 f ;�� 8 �0 � �-4 f ;�� _ f ;��0 aT4 f ;�� 8 �0 ��-4 f ;�� _ f ;��05a �-4 f ;�� _ f ;��

� P 698�: a"?�l0 ���h4 � ;#� 80 ��-4ed ;#� _0 �"a24 � ;#� 80 ���T4ed ;#� _0 � -4hd ;#� 80 �]a24gf ;#� _0 � �T4gf ;#� _0#>a24 d ;#� 80#��T4hf ;#� _0�a��T4hf ;#� _

e 8O8&:H?�8�:H; A�X.; 8&?�A�>C8&; F"7QB�; 8O8&Y�A�7H<OF"Y�8\; 7EL�N�P�R1NQS*��Ñhn<A�8&:@>�8\?�>*U�:7H?�DTI l/J"?�71?�A�>�8�?�X/U�>*71XH; X1>C8&?,=1>C8G8&?�A&7`U�F"7�8�>*; 7H<\>C8OI ?�>*<�8iF"7H?,<GY1j�W=1>C8G8�?�A&7�DT:H; U�:�; <,7HF*8,>{a�?�l�=@>C8G8&?�A&7�fO»{?�7HU�?T>*I I171?�D�U�>*71XH; X1>�8�?=1>C8G8�?�A&7H<�>�A�?x7HF*8�U�>*71XH; X1>C8&?]a�?�l�=@>�8&8&?�A&7�B\>*7@X�>*I I�8�:H?�; A�<GY1=�W=@F*AG8�</>CA&?�XH?�8&?�A&KE; 7H?�X9XH; A�?�U�8&I l9; 7dL�NQP�R1N�SC��Ñhn A�f�� qGtl� o �1p�q&pt*r � o �Hp ' n*o�n �&n*~�p-qivg Â(r@t*o!��p`n�s&s�p�~&~�p ' n*rH}*�Et*q&pC`� U =�A�?�X<�c<&YH= a�?�l 698�:

0?��� �T4 � ;�� _ � ;��0?��� -4 � ;�� _ � ;��0?���]a24 � ;�� _ � ;��0?��>a24 � ;�� _ d ;��0?����T4 d ;�� _ d ;��0?�>a��T4 � ;�� _ � ;��0 � >a24 d ;�� _ d ;��0 � ��T4 d ;�� _ d ;��0 �]a��T4 f ;�� _ f ;��0 a��T4 d ;�� _ d ;��

� U 6�8N: a�?�l0 ��� �-4 � ;�� _0 ����a4 � ;�� _0 ���&aT4 � ;�� _0 �� aT4 � ;�� _0 ����-4ed ;�� _0 �"a �-4 � ;�� _0 � aT4gd ;�� _0 � ��-4hd ;�� _0 �]a �-4 f ;�� _0#>a �-4hd ;�� _

67�8&:H?�_ F*YHAG8&:E>*7@X/�H_ 8&:�; 8&?�A�>�8�; F"7QB�>*I I@<GYH=1=@F*AG8&<�>CA&?TX�?�8�?�A&KE; 7H?�XX�; A&?�U�8�I l�; 7�L�NQP�R@NQS*��Ñhn AOf�67d8&:H?z<G; ��8&:k; 8&?�A�>C8&; F"7QB{L�NQP�R1N�SC�Ñhn<AxJ"?�71?�A�>C8&?�<c7HF`71?�D�U�>*71XH; X1>C8&?T=@>C8G8&?�A&71<�Bb8&:�YH<,71F/_ A&?���Y1?�7�8� W�=@>�8&8&?�A&7H<�>CA&?/U�F*KE=1Y�8�?�Xx>C7@X]8&:H?`>*I J"F*A&; 8&:HKÀ<�8�F*=1<��

� � =�A�?�X<�c<&YH=da"?�l�698�:0?��� aT4 � ;�� _ � ;#�0?��� ��-4 � ;�� _ � ;#�0?���]a �-4 � ;�� _ � ;#�0?��>a �-4 � ;�� _ d ;#�0 � >a �-4 d ;�� _ d ;#�

� � 698�:da�?�l0 ����>a24 � ;�� _0 ������T4 � ;�� _0 ���&a��-4 � ;�� _0 �� a��T4 � ;�� _0 � a��-4 d ;�� _

� � =HA&?�X��c<GY1=�a�?�l�698N:0 ��� a��T4 � ;�� _ � ;��

� � 698N:da�?�l0 ����>a �-4 � ;�� _

»{?�7HU�?ZL�NQP�R@NQSx7H?�?�XH<{8D�F�X1>�8�>Cj@>*<G?]=1>*<G<&?�<`; 7VDT:H; U�:V8&:1?]>CI WJ"F*A&; 8�:HKÄU�F*Y17�8�<T8&:1?`<GY1=H=(FCAG8�<{F*_ � 8 � % G � =1>C8G8&?�A&71<�f e =�A&; WF*A&;(D�F"YHI X]:1>�[�?{7H?�?�X�?�XE�@[�?{X1>�8�>Cj@>C<&?{=@>C<&<G?�<�_ F*A�U�F*Y17�8�; 71J/8&:1?<GY1=H=@F*AG8�<TF*_ � 8 GCF 8 GHF 8 � 8 G % d � =1>C8G8&?�A�7H<T_ F*A{8&:H?`<&>*KE?XH>C8�>*<G?�8Cf e I IQFC8�:H?�A�U�Y�AGA�?�7�8{>*I J"F*A&; 8&:HK]</� Dc; 8&:Z8&:1?.F"7HI lZ?��HU�?�=�W8&; F"7�F*_{�,I F"<G?C�/K]>�l�71?�?�XVI ?�<G<`8&:1>*7V�1[�?�=@>C<&<G?�<�B\jHYH8.8&:1?�lV>*I I:1>�[�?{8&F�=@?�AG_ F*A&KÀ8&:H?/ ��`U�F*Y17�8�<�f

5. EXPERIMENTAL EVALUATION��?,?�[">*I Y1>C8&?�XEL�N�P�R1NQS{>CJ�>*; 7H<�8�8�:H?c>CI J"F*A&; 8&:HK]< e =�A�; F*A&;gB���I F"<G?"B>*71X��z>���W��Z; 7H?�ACf��z>C�bWG�x; 71?�A�Dc>C<�?��b8&?�7@X�?�X�8&FzA&?�8GA�; ?�[�?]8&:1?_ A&?���Y1?�7�89=@>�8&8&?�A&71<zDT; 8&: 8&:1?�; AV<GY1=H=(FCA&8�jbl >�=1>*<G<VFC["?�A�8&:1?

XH>C8�>*j1>*<G?�<�£O8�:H?�8�D,Fz=1:1>*<G?�<�>�A&?]<G:HF�DT7�; 7V8&:1?�8�>*jHI ?�</j(?�I F�D��"fL�NQP�R1N�SbB e =HA&; FCA�; B@��I F"<G?`>C7@X�8&:H; <,�17@>*I@<�8&?�=]8�F��z>C�bWG�x; 71?�AT>*I I<G:@>�A&?�X�8&:1?E<&>*KE?]X1>�8�>z<G8GA&YHU�8�Y�A&?�<�>C7@XVJ"?�71?�A�>*I�F*A&J�>*7H; ^�>C8&; F*7�f­�=H8&; KE; ^�>�8�; F"7H<�<&YHU�:Z>*<c<G=@?�U�; >*I(:1>*71XHI ; 71J`F*_O=@>C<G<�8�D,F�FCAT; 8&?�KE<A&?�F*A�X�?�A�; 71J/D�?�A&?{X�; <&>*jHI ?�XQfi�\�H=@?�A&; KE?�7b8&<iD�?�A&?�U�F*7@X�Y1U�8&?�X�F"7x>°,��°O?�7b8&; YHKª �¡"§*§��Z»�^.DT; 8�:�¦b�C�"�Z; ¯�F*_i³ e �Vf�,:1>CA�>*U�8&?�A&; <�8&; U�<xF*_{8�:H?zXH>C8�>*<G?�8�<]YH<G?�X�>�A�?ZJ"; [�?�7�; 7�hO>*jHI ?9�bfh�:H?�<G?�X1>�8�>C<G?�8�<\>�A�?�8&:H?��c�*§C����§C��>C7@X`�c¥* C����§C��U�?�7H<GY1<\XH>C8�>CW<G?�8&<9_ A&F"K 8�:H?k°i¸/��© <&>*KE=HI ?��1I ?��"B`8&:H?khT�*¦C6���§*����§*� >*71XhT�*¦C6G�*§*����§"§*� Ã� <Gl�7b8&:1?�8�; U�XH>C8�>*<G?�8{8&:1>C8{KE; KE; U�<cK]>CA&a�?�8�j1>*<�Wa�?�8/XH>C8�>�BO>*71X�8&:1?EÎ�QP?��xNyzy ocP*ÃGÃ�XH>C8�>*<G?�8`XH?�<GU�A&; jH; 71JZK�YH<G:HWA&FbF*K]<TU�:@>�A�>CU�8�?�A&; <�8&; U�<�� ¦���fT67�>*I IO?���=(?�A&; KE?�7�8&<�BHD�?`>C8G8&?�KE=H8&?�X8&F�U�:1F�F"<G?`<G; J"7H; �1U�>*7�8�KE; 7H; K�Y1K�<GYH=1=@F*AG8c8&:HA&?�<G:HF"I XZ[">CI Y1?�<�f

$, ��/+ ����#(��0� g+&�&����@5"�"�Q��� �&+�����#Q� ��+&�.�� �5B�� 3��"�CB5B�� �CB5B�� B5B5B �5B ��� B5B5B� �51�� �EB��"�CB�� �CB�� B5B5B �51 ��� B5B5B� �51�� �5B��"�CB5B�� �CB5B�� B5B5B �51 �CB�� B5B5B

� �5B��"�CB�� �CB�� B5B5B �5B #5=53� 4T#��"�CB�� �CB�� B5B5B 4T# ��� ��4T=

����� �"!�#"#%$&� =�� $ �C3 �5# �C�5=

hO>*j1I ?��b�i�/>�8�>C<&?�8&<

³T?�I >C8&?�X.D�F*A&a/:1>�[�?,<G:HF�DT7/8&:@>C8O8&:H?�j(?�:1>�[b; F*A\F*_@>*I J*F*A&; 8&:1KE<O_ FCA?��b8&A�>*U�8&; 7HJ`_ A&?��bYH?�7b8,=@>C8G8&?�A&71<cX�?�=@?�7@X�<�K]>*; 7HI lEF"7]8&:H?.XH>C8�>*<G?�8U�:1>CA�>*U�8&?�A�; <�8�; U�<�f���?�>Ca�I l�U�F*AGA&?�I >C8&?�X�XH>C8�>�B�<&YHU�:�>C<�<�l�7b8&:1?�8&; UXH>C8�>�BHU�F"7H<G8&; 8&Y�8&?/?�>*<�lxU�>*<G?�<c_ F*AT8&:H?.?��b8GA�>CU�8�; F"7z<G; 7HU�?._ ?�D =1>C8GW8&?�A&71<E>CA&?x_ A�?���Y1?�7�8�fk�HF*AE<GY1U�:kX1>C8�>�B�>*I Ic>*I J"FCA&; 8&:1KE<EJ"; ["?z>*U�WU�?�=�8�>*j1I ?/A&?�<G=(F*71<G?.8�; KE?�<T>*<TD,?`U�>C7zF"j1<G?�A&[�?/; 79©b?�U�8&; F"79¦�f �`; 7DT:H; U�:k?��H=@?�A&; KE?�7b8�>*I�A&?�<GYHI 8&<EF*jH8�>*; 71?�Xd_ F*AE8&:H?ZhT�*§*6�¡*����§"§C�EBhT�*¦C6���§*����§*�¾>*71X]hT�"¦�6&�C§*����§"§*�¾XH>C8�>*<G?�8&<c>CA&?�=�A&?�<&?�7�8&?�XQf�­�78&:1?zU�F"7b8GA�>CAGlbBTU�F*AGA�?�I >C8&?�X�X1>C8�>�U�F"71<�8&; 8&Y�8�?z_g>CAzKEF*A&?9X�; �EU�Y1I 8U�>*<G?�<T_ F*AT8�:H?.?���8GA�>*U�8&; F"79X�Y1?{8&F�8�:H?/; KE=(F*AG8�>*7�8�=�A�F*=(FCA&8&; F*7�F*_=1>C8G8&?�A�7H<`8�:1>C8�>CA&?x_ A&?���Y1?�7�8E>CK]F*71J9>CI Ic=1>C8G8�?�A&7H<�fd©bY1U�:�X1>C8�>A&?�=HA&?�<G?�7�8x>�:�YHJ"?Z=@>�A&8�F*_�A&?�>*I W�I ; _ ?9X1>�8�>C<&?�8&<�Bc>C7@XdXH; ÁQ?�A&?�7HU�?�<j@?�8�D,?�?�7�?��b8GA�>CU�8�; F"7k8�; K]?�<�F"j�8�>*; 7H?�XdDT; X�?�I l�[*>CAGl�X�?�=(?�71XH; 71JF"7d8&:1?Z>CI J"F*A&; 8&:HK YH<G?�XQfd�\��=(?�A&; KE?�7�8�>CI�A&?�<&YHI 8&<�F"j�8�>C; 71?�X�_ FCA8&:1?9�c�*§*����§*��BT��¥* *����§*�M>*71X�Î�(P?��x�yzyzoTPzXH>C8�>*<G?�8&<�Bc8&:@>�8>CA&?.K]>"XH?{Y1=xF*_\U�F*AGA&?�I >C8&?�X�XH>C8�>�B1>CA&?/J"; [�?�7z; 7�©�?�U�8&; F"7�¦�f �bf

5.1 Weakly correlated datah�:H?�hT�C§*6G¡C����§*§*��B"hT�"¦�6���§*����§*��>*7@X{hT�"¦�6&�C§*����§"§*��<�lH7�8�:H?�8&; UXH>C8�>*<G?�8&<�>CA&?�U�F"7H<�8GA�YHU�8&?�XE>*U�U�F*A�XH; 71J.8&F.8&:H?T=HA&F"=@?�AG8&; ?�<,F*_QK]>CAGWa�?�8�j@>C<&a"?�8{XH>C8�>�8�:1>C8{>CA&?/8lH=H; U�>*IQD�?�>Ca�I l�U�F*AGA�?�I >C8&?�X9X1>�8�>bfT678&:1?�<G?,XH>C8�>*<G?�8&<�BC8&:1?�7bY1K�j@?�AQF*_�_ A�?���Y1?�7�8�=@>C8G8&?�A&71<�; <O<GK]>*I I�U�F"K�W=1>CA&?�X�8&Fz8&:1?�8&F*8�>*I�7�YHK�j@?�A/F*_�=1>C8G8�?�A�7H<�>C7@X(B\; 7�KEF"<�8�U�>C<&?�<�B7H?�>CA&I lz>*I I(8&:H?._ A�?���Y1?�7�8T=1>C8G8�?�A&7H<T>CA&?/>*I <GFEa�?�lx=1>C8G8�?�A&7H<�f³T?�<&=@F"7H<G?�8&; KE?�<�_ F*A.8&:1?EhT�C§*6�¡*����§"§*��XH>C8�>*<G?�8`>CA&?�=HA&?�<G?�7b8&?�X7bY1KE?�A&; U�>*I I lz; 7VhO>*jHI ?� ]>*71XzJ*A�>*=H:1; U�>*I I l9; 79�O; J"Y�A&?]��f{67z8&:1; <XH>C8�>*<G?�8�B(>CI I(_ A&?��bYH?�7b8&<c=1>C8G8&?�A&71<�>�A�?.a�?�lx=1>C8G8&?�A�7H<T>C7@X e =HA&; F*A&;>*71XxL�NQP�R@NQS{j(?�:1>�[�?�; XH?�7�8&; U�>*I I lx>C7@X`A&?�<G=(F*71<G?T8&; KE?�<�F*jH8�>*; 7H?�X_ A�F*K«8�:H?�K¾>*71XE�z>C�bWG�Z; 7H?�A�>CA&?T<G; K]; I >CA�f�h�:H?���I F"<G?{>*I J"F*A&; 8&:HKJ"; [�?�<.:1; J*:1?�A.A&?�<&=@F"7H<G?�8&; KE?�<.XHYH?�8�F]8&:H?�7�YHK�j@?�A{FC_c; 7�8�?�A&<G?�U�W8&; F"7�F"=@?�A�>C8&; F*71<Z7H?�?�XH?�X�_ FCA�U�F"KE=1Y�8&; 7HJ�8&:1?�U�I F"<GYHA&?�<zFC_`8&:1?' h,:1; <x<G?�U�F*7@X�=1:1>*<G?�; <]j�lk_ >CAx8&:1?�KEF"<�8xU�F"<�8&I l�=@>�A&8xFC_/8&:1?É �Z>C��W��Z; 7H?�A)(�Ê�A&YH7HWg8&; KE?�<�f*)+�,�-/.10�02+�,�-"3&465�57498�:�;=<?>&4@�A�8�0=-�8�B"0�C�-�-�B�D"0E5=@2<%>�8?>�0�-E8�F?>�0-�82F%>2G�H2:?>74I"C�-È )J ,�,�-K.L0�0�M�M�MK41;EN�F";�AE@=</46C�B�FO4L5�P�F"0E5E>202Q�8E@">=,E0E>2R2<EAE;2,�;S4 J ,�F"NÈ&È +�,�-/.10�02+�,�-/4LC�5�>7498?5�C741@�A28"0=-�8�B"0F";E5 J C�<�@"TEN�@�;�D�<?C�<EU?T�AE;2,E;2BE;�>2@E>�0�FE8?> J DEPEP�F"0;�U�;�D"C�5�8?>�T�N2@2-?C�P=,E;S49AE;�,�;

©b6&¼��/�{�«�\�H=HI F*A�>C8&; F*71<�f���F"=blbA&; J":b8 UË �*§"§*§xÌ`Í.νÏ(Ð�Ñ]ÒEÓ�Ó]B1�.?�U�?�K�j@?�A��*§*§"§�f ´�F"I YHKE?`�bBH6<G<&YH?��.W,=1>*J"?`¥*§

Page 6: Mining frequent patterns with counting inference

U�>*71XH; X1>C8&?.=@>�8&8&?�A&71<�f

� ��1� ��# �+&)�� '( ����� �! ������� ����� � ! ����+��E ��*2��E� -�+&� (� � B5B ��� 15#�$ �C# � �E$ �C# � 1J� �T1 � ?J� � � 35B 1 � B5#B � 4T1 $�� 4��EB �5B � $ � �5B � 3�4 #T1 � �5? $ � $5$ �5� � B53B � 15B �53�� ?515B $5$ � B5B $5$ � #5= 354 � =5� 3 � =�4 #51 � #�4B � �51 �C151�� 3�4T# �5��4 � ?�4 �T��4 � 4T? �C=T� � ?51 �C1 � 3T$ �CB5? � �E$

hO>*j1I ?� b�i³�?�<G=@F"71<G?{8&; KE?�<�_ FCA.hT�*§C6G¡C����§*§*�

��

���

��

�� � �� �� � �

�� ��� ��

��� ��� �! ��#"$ �%�%'& (*),+ -/.

021 354�1 67 %�(5� &8(5�9 6 & 35:�;1�<$= ��� ��:>(>?

�O; J"YHA&?��b���\��=@?�A�; K]?�7b8�>*I@A�?�<&YHI 8&<T_ F*A{hT�C§*6�¡*����§"§*�

� ��H� ��# ��+&)�� '( ���� �! �@���� ���� � ! ���+��E ��*25��� -"+�� (� � B5B 15=5# 1 � �C1 1 � 4T3 �5� � �C1 � � ��$ � � #B � 4T1 ��� �C151 ? � 4T# �5� � �C# #51 � 3�4 � � ?T? � � 4T4B � 15B ��� ��4�?�� �51T$ ?T35= � 3T$b?5#51 � �E$ ��� �C1J� � #T$ �T$ � ?�$ =�4T? � =51

h\>Cj1I ?/¤H�i³�?�<G=@F"71<G?{8�; KE?�<�_ F*A{hT�"¦�6&�C§*����§"§*�

³�?�<GYHI 8&<�_ FCA,8�:H?�hc�"¦C6G�*§*����§"§C�«X1>C8�>*<G?�8c>�A�?T=HA&?�<G?�7b8&?�X�7�YHKE?�A&; WU�>*I I l`; 7�hO>*jHI ?�¤�>*[email protected]*A�>*=H:H; U�>*I I l`; 7��O; J"Y�A&?� �f\�HF*AO8&:1; <OX1>C8�>*<G?�8�B7H?�>CA&I lE>*I I�_ A&?��bYH?�7b8i=1>C8G8&?�A�7H<�>CA&?Ta"?�l�=1>C8G8&?�A�7H<�B�>C7@X`A&?�<GY1I 8&<�>CA&?<G; KE; I >�A�8�F�8&:1F*<&?�F"jH8�>*; 71?�X/_ FCA�8&:1?chc�*§*6�¡*����§"§*�ÆXH>C8�>*<G?�8���L�N�P��R1N�SZ>C7@X e =�A�; F*A&;�J"; ["?]; XH?�7�8�; U�>*IiA&?�<G=(F*71<G?�8&; KE?�<`>C7@XV<GY�ÁQ?�A`><GI ; J*:b8�=(?�AG_ FCA�K]>C71U�?]I F*<&<�FC["?�AE�z>���W��Z; 7H?�A�Dc:1; I ?Z�,I F"<G?x; </8&:1?D�F*A&<�8{=@?�AG_ F*A&KE?�A�f67`hO>*jHI ?,¦�>*7@X.�O; J"Y�A&?i¤HB*?���?�U�YH8&; F*7.8&; KE?�<@_ FCAO8&:H?�hT�"¦�6���§C����§*�XH>C8�>*<G?�8c>CA&?T=�A�?�<G?�7�8&?�XQf\67�8&:1; <,XH>C8�>*<G?�8�B�8�:H?c=�A�F*=(FCA&8&; F*7]F*_(_ A&?�W�bYH?�7b8T=@>�8&8&?�A&71<T8&:@>�8.>CA&?`71F*8�a�?�lZ=1>C8G8�?�A�7H<{; <{K�Y1U�:ZKEF*A&?/; K�W=@F*AG8�>C7b8�8&:@>C7k_ FCAE8�:H?zhT�"¦�6&�C§*����§"§*�ÅX1>�8�>C<G?�8Cf��HF*AE8&:1?9�"f §"§>*71X«§�f ¥*¦�KE; 71<GY1= 8&:�A�?�<G:HF"I X�<�BE�z>C��W��Z; 7H?�A�=@?�AG_ F*A&KE<Vj@?�8G8&?�A8&:@>C7 e =HA&; F*A&;T>*7@X�L�NQP�R@NQS�8�:1>C8E8&:1?�KE<G?�I ["?�<E=@?�AG_ FCA�K j@?�8G8&?�A8&:@>C79�,I F"<G?"f,�1F*Ac8&:1?{I F�D�?�A{§�f ¦*§E>*71Xz§�f �"¦�KE; 71<GYH=]8�:�A&?�<G:1F"I XH<�BL�NQP�R@NQSzj@?�U�F"KE?�</8&:1?Ej@?�<�8�=@?�AG_ F*A&KE?�A�>*7@X�; <�<GI ; J":�8&I l�j@?�8G8&?�A8&:@>C7 ��I F"<G?9DT:H?�A&?�>*<x8&:1?�lkj@F*8&:�U�I ?�>CA&I l�F*YH8&=@?�AG_ F*A&K e =�A&; F*A&;>*71X9�z>C�bWG�Z; 71?�A�����:H?�7Z8&:1?/=HA&F"=@F*AG8&; F"7zF*_i_ A&?��bYH?�7b8T=@>�8&8&?�A&7H<8&:@>�8T>CA&?.7HF*8Ta"?�lE; <c<G; J"7H; �1U�>*7b8�B�8&:1?�KE?�U�:@>*7H; <GKÇYH<&?�X]j�lzL�N�P��R1N�SV� A&?�<G=�fZ�,I F"<G?C�{8&FZU�F"7H<&; XH?�A�F"7HI lVa�?�lk� A&?�<G=Qf]U�I F"<G?�X1�.=1>C8GW8&?�A&71<c?�71>*jHI ?�<�8&F�A&?�XHYHU�?{U�F*71<G; X�?�A�>Cj1I lx8&:H?.7bY1K�j@?�A�F*_O<GY1=H=@F*AG8U�F"YH7b8&<T=@?�AG_ F*A&KE?�X(f

A

B�A8A

C�A A8A

C�B�A8A

D A A8A

D8B�A8A

A�E BA�E F8BC

GH IJK LM

N�O P�O Q!R�QTS$R�U�U'V8W5X!Y Z\[

]2^ _5`�^�ab U�W5O V W*Oc a V _*dNe^�f�g N;O P�d>W>h

�O; J"Y�A�?/ ����\�H=@?�A&; KE?�7b8�>*I@A&?�<&YHI 8&<c_ F*A.hc�"¦C6G�*§*����§*�

� ��1� ��# �+&)��½'Q ����� �! ������� ����� � ! ���+i�E ��*25��� -"+&� (� � B5B #�� #5B5B # � �T$ # � 35� 3 � 3�4 B � 35# � � BT1B � 4T1 ��42� 15=5# 1 � ��4 3 � ?51 ? � #5= � � B5? # � =T#B � 15B #5#J��� �5=5B ��4 � =5� $ � � B53 �53 � $�# � � 4T3 #51 � 1T#B � �51 ��� ��4TB�� 1�4T# 4TB � #�4 �C=�4 � ?5� =53 � B5= 3 � ?5? �C1T$ � =T?

hO>*jHI ?�¦���³�?�<G=@F"7H<&?{8&; KE?�<�_ FCA.hT�*¦C6���§*����§*�

j

k�j

l�j j

l�k�j

m�j j

j�n m kj�n kj�n o kl

pq rst uv

w�x y�x z!{�z#|${�}�}'~ �*�,� �/�

�2� �5��� �� }��5x ~8�5x� � ~ �5�w;���$� w�x y��>�>�

�O; J"Y�A�?.¤H���\�H=@?�A&; KE?�7b8�>*I@A&?�<&YHI 8&<c_ F*A.hc�"¦C6���§*����§*�

©b6&¼��/�{�«�\�H=HI F*A�>C8&; F*71<�f���F"=blbA&; J":b8 UË �*§"§*§xÌ`Í.νÏ(Ð�Ñ]ÒEÓ�Ó]B1�.?�U�?�K�j@?�A��*§*§"§�f ´�F"I YHKE?`�bBH6<G<&YH?��.W,=1>*J"?`¥b�

Page 7: Mining frequent patterns with counting inference

5.2 Correlated data³�?�<G=@F"7H<&?�8&; KE?�<�F"j�8�>*; 7H?�XE_ A&F"K�8&:H?/�c�C§*����§*�¾>*71XZ�c¥* C����§C�U�?�7H<&YH<,XH>C8�>*<G?�8�<�>CA&?TJ"; ["?�7�7�Y1KE?�A&; U�>CI I l`; 7]hO>*jHI ?�<,¡.>C7@XE¥bBb>*71XJ*A�>*=H:1; U�>*I I l]; 7x�O; J"YHA&?�<c¦.>*71XE¡�fi³T?�<&YHI 8&<�_ FCA,8&:1?{Î (P?��x�yzyzoTPXH>C8�>*<G?�8�>CA&?c=HA&?�<G?�7b8&?�X/; 7�hO>*jHI ?c¢�>*7@X`�O; J"Y�A&?�¥bfO67/8&:1?�<G?,8&:�A�?�?XH>C8�>*<G?�8�<�B�U�F"71<�8&; 8&Y�8�?�X�F*_�U�FCA&A&?�I >�8&?�X�XH>C8�>�Bi8&:H?]=HA&F"=@F*AG8&; F"7dF*_=1>C8G8�?�A&7H<�8&:@>�8T>CA&?{_ A�?���Y1?�7�8�; <�; KE=(FCA&8�>*7�8cjHYH8�_ ?�D�F*_�8�:H?�K�>CA&?>*I <GFxa"?�lZ=1>C8G8&?�A&71<�f�»{?�7HU�?"B(YH<G; 7HJ]=1>C8G8&?�A&7zU�F"YH7b8&; 7HJ]; 7H_ ?�A�?�7HU�?"BL�NQP�R@NQS{:@>C<�8&F/=@?�AG_ F*A&K�K�Y1U�:`_ ?�D,?�A�<GY1=H=(FCA&8,U�F*Y17�8�<\8�:1>*7�8&:1?e =�A&; F*A&;O>*71XZ8&:1?`�z>C�bWG�Z; 71?�A{>*I J"FCA�; 8�:HKE<�fTh�:H?`<&>CK]?.F"jH<G?�A�[*>CW8&; F"7k<�8�>C7@X�<�_ F*AE8&:1?���I F"<G?�>*I J"F*A&; 8&:1KzB�8&:@>C8�YH<&?�<�8&:H?ZU�I F"<GYHA&?KE?�U�:@>C71; <GK 8&F�A&?�X�Y1U�?z8&:H?97�YHK�j(?�AEF*_`<GY1=H=@F*AG8xU�F"Y17�8&<�BT>*71Xj@F*8&: L�NQP�R1N�S9>*71X��,I F"<G?V>CA&?�>*7�F*A�X�?�A]FC_/K]>*J*71; 8&Y1XH?x_ >*<�8�?�A8&:@>C7 e =�A&; F*A&;\>*71X9�z>C�bWG�Z; 7H?�A�f��{; Á(?�A&?�71U�?�<Tj(?�8D�?�?�7z8&:H?`?���?�WU�Y�8�; F"7x8&; KE?�<�F*_{L�NQP�R1NQS�>C7@XZ�,I F"<G?`>C7@X]8&:HF"<G?.F*_ e =HA&; F*A&;Q>*71X�z>C�bWG�x; 71?�AcU�>*7xj@?{U�F"YH7b8&?�X]; 7]8&?�7H<�F*_OKE; 7�Y�8&?�<�_ FCA{�c�*§C����§C�>*71X�Î�(P?��x�yzyzoTP,>C7@X/; 7`:1F*YHA&<�_ F*A,��¥* *����§*��fb�ZFCA�?�F�[�?�A�B"=1>C8GW8&?�A&7�U�F"YH7b8&; 7HJ]; 7H_ ?�A�?�7HU�?�>*7@XZU�I F"<GYHA&?`KE?�U�:@>C71; <GKª>CI I F�DÆ8&F�A&?�WX�Y1U�?�8&:H?,7�YHK�j@?�A�F*_1=@>C<&<G?�<\F"7/8&:1?,X1>C8�>*<G?�8&<\<G; 7HU�?�8�:H?,<GY1=H=@F*AG8&<F*_�>CI I,U�>*71X�; X1>�8�?�=1>C8G8&?�A�7H</F*_T<GF"KE?�; 8&?�A�>C8&; F"7d>CA&?x>CI I�X�?�XHYHU�?�X_ A&F"KÀ8&:1?.<GY1=H=(FCA&8&<TFC_ia�?�lbBHF*A�U�I F"<G?�XQB@=1>C8G8&?�A�7H<TFC_�=HA&?�[�; F"Y1<c; 8GW?�A�>C8&; F"7H<�fV­�7k�c¥C *����§*��Dc; 8&:b<T> ?N@�A�B&% � F�� BO_ F*A�; 7H<G8�>*7HU�?"BL�NQP�R@NQSV>*7@X��,I F*<&?zj(FC8�:kK]>*a�?V�� V=1>*<G<&?�<EDc:1; I ?Z8�:H?ZI >�A�J*?�<G8_ A&?���Y1?�7�8�=1>C8G8&?�A�7H<`>CA&?]FC_T<G; ^�?���¨�fZ�1FCA`8&:H; <�XH>C8�>*<G?�8�>*71XV8&:1; <8&:HA&?�<G:1F*I X�[">*I YH?"BC_ A&?��bYH?�7b8i=1>C8G8�?�A&7H<iU�F"YHI X`71F*8ij(?cX�?�A&; [�?�X/_ A&F"K8&:1?zK]>��H; K]>*I�_ A&?��bYH?�7b8E=@>�8G8�?�A&7H<E?���8GA�>*U�8&?�XdDT; 8�:��z>���W��Z; 71?�A<G; 7HU�?xD�?ZXH; X�7HF*8�; KE=1I ?�KE?�7�8`KE?�KEF*AGl�K]>C7@>*J*?�K]?�7�8/_ F*A�8&:1; <=H:@>*<G?/>*7@Xx; 8TA&?��bYH; A&?�Xx; 7Z8&:H; <TU�>C<&?/KEF*A&?.KE?�KEF*AGl]<G=@>CU�?.8&:1>*7>�[">C; I >*jHI ?"f

� ��H� ��# ��+&)��½'Q ���� �! ������� ���� � ! ���+ �E ��*25��� -"+�� (�TB � B �5B�� �5#5? ? � $5$ 1�4 � �E1F�E$ � #53 B � ��4 454 � $�B�E1 � B #53�� #515? �C� � #J� =51 � #T1F�C= � ?5? B � �53 �5�C# � �5��EB � B =5?�� =5=5# �C? � �5? �C3T$ � =J� �5? � 15= B � #T$ �5BJ� � #5#4 � 1 �C15#�� �C35# �5# � 15#`�5#5� � $�B #53 � B5� B � #51 �535= � =5B1 � B #515��� 3J�5� #5# � B53`#5?51 � #5� 15B � $�3 B � $�= $��5= � 351� � 1 ��� �C35B�� #535# 151 � #5#c4T1T$ � 3T$\4T= � 35# B � =J� 454T1 � 153

hO>*jHI ?`¡b��³�?�<G=@F"7H<&?�8&; KE?�<c_ FCA.�c�C§*����§*�

� ��H� ��# ��+&)�� '( ���� �! �@����� ���� � ! ���+ �E ��*25��� -�+&� (=5B �CB5?�� �C15? ��454 � $�? #�� 353J� � ��4 �T$ � � ?J� B � =�4 #�� 4���4 � ?5?4T1 �5#51�� ��4�� #5?5� � =5B 42� 3515# � 15= 1T$�? � ��4 � � B53 42� 4T#5B � #534TB 1�4T��� B5=�4 4T=53 � $�? ��42� $�351 � �CB ��� �5�C� � $�� � � �5= ��42� 3J�C= � $�B35B $�� #5151�� 1T$�# #�� ?�4T� � �CB �CB5?�� �TBT$ � B5B 1�� 35BT$ � ?J� 4 � 4T� ���5�

���5� $i����+&-��� ���3��.+&�/�����C�hO>*jHI ?�¥���³�?�<G=@F"7H<&?�8&; KE?�<c_ FCA.�c¥C *����§*�

� ��H� ��# ��+&)�� '( ���� �! ���"�� ����� � ! ���+ �E ��*25��� -"+&� (�5B � B 1T#�� #5#�4 3 � $�= �5�C1 � =5� ? � 35# B � #J� �C#T$ � #J��C1 � B ?T?�� B�4T? ? � =J� �C?5B � ?T$ �E$ � 1�4 B � 15B �J�C= � ?5#�CB � B 35BTB�� =J��4 �T# � �C� 4T�T$ � #51 �5? � =5# B � =5? 4�$�1 � 4T�4 � 1 ?5#T3�� �T$�4 #T� � B5= ��� B5�5# � �T$ $ � � B51 � � �51 ��� B5#51 � $�=1 � B $�� �E$5B�� $�15# ?54 � �C� ��� 4T35# � $�� ?5= � =J� � � ?5? �2� 4T15� � B51

hO>*j1I ?`¢��i³�?�<G=@F"71<G?{8&; KE?�<�_ FCA.Î (P?��xNy yzocP

����������� ���������� ��������

���������

�� ��� ��

��� ��� �! ��#"$ &%�%(')+*-, .0/

132�4+5�2�67 %&)+� ')+�8 6 '4+9��2�:<; �=� �&9>)@?

AB�AC�ADAE AF�AGAH AIAJA

FB�AB�FC�A

KL MNO PQ

R=S T&S U!V&UXW$V&Y�Y(Z[+\-] ^0_

`3a�b+c�a�de d Zb+f

�O; J"Y�A�?�¦b�i�\�H=@?�A&; KE?�7b8�>*I@A&?�<GY1I 8&<c_ F*A.���*§*����§*�

©b6&¼��/�{�«�\�H=HI F*A�>C8&; F*71<�f���F"=blbA&; J":b8 UË �*§"§*§xÌ`Í.νÏ(Ð�Ñ]ÒEÓ�Ó]B1�.?�U�?�K�j@?�A��*§*§"§�f ´�F"I YHKE?`�bBH6<G<&YH?��.W,=1>*J"?`¥"�

Page 8: Mining frequent patterns with counting inference

���������

� �������

���������

���������

����������

�����������

������ �� ���

� ��� ��

��� ��� �������������! �"$#�% &('

)+*�,.-*�/0 ��"$� �".�1 / �,.2�3*�4�5 ��� ��26"67

8

98�8�8

:�8�8�8

;�8�8�8

< 8�8�8

=�8�8�8

>�8�8�8

>�8>�=? 8? =@�8

AB CDE FG

H�I J�I K�L�K�M�L�N�N!O�P$Q�R S(T

U+V�W.XV�YZ Y O�W.[

�O; J"Y�A�?/¡����\�H=@?�A&; KE?�7b8�>*I@A&?�<&YHI 8&<c_ F*A.��¥* *����§*�

\

]�\�\

^\�\�\

^�]�\�\

_�\�\�\

_�]�\�\

`�\�\�\

]^\^�]_�\

ab cde fg

hji kli mon�m�p�nlq�q!r�s.t�u vxw

y+z�{$|z�}~ q�s.i r�s.i� } r�{.�h3z���� hji k���s6�

���

���

���

���

����

���������

�� ��� ��

�j� ��� �o�������l���!���.��� �(�

 +¡�¢$£¡�¤¥ ¤ ��¢.¦

�O; J"Y�A�?�¥b���\�H=@?�A&; KE?�7b8�>*I1A&?�<&YHI 8&<T_ F*A.Î�(P?�[xNyzyzoTP

©b6&¼��/�{�«�\�H=HI F*A�>C8&; F*71<�f���F"=blbA&; J":b8 UË �*§"§*§xÌ`Í.νÏ(Ð�Ñ]ÒEÓ�Ó]B1�.?�U�?�K�j@?�A��*§*§"§�f ´�F"I YHKE?`�bBH6<G<&YH?��.W,=1>*J"?`¥* 

Page 9: Mining frequent patterns with counting inference

6. CONCLUSION��?�=�A�?�<G?�7�8�?�X�L�NQP�R@NQS�BO>x71FC["?�I\F"=H8&; KE; ^�>�8&; F"7�FC_�8&:1? e =�A&; F*A&;>*I J*F*A&; 8&:1K�_ F*A�_ >*<�8�X�; <GU�FC[�?�AGlEF*_Q_ A&?��bYH?�7b8,=@>C8G8&?�A&71<�f�L�NQP�R1NQS/; <j@F*8&:9?�Á(?�U�8�; [�?�>*7@Xz?�>*<�lz8&Fx; KE=1I ?�K]?�7�8�F*A�8&Fx; 7b8&?�J*A�>C8&?`; 7V?��bW; <�8&; 7HJ`; KE=HI ?�KE?�7b8�>C8&; F"71<\j1>*<G?�X�F"7�8&:1? e =HA&; F*A&;@>*=H=HA&F">*U�:�fih,:1; <F"=�8&; KE; ^�>C8&; F"7zY1<G?�<�=1>C8G8&?�A&7ZU�F"Y17�8&; 71JE; 7H_ ?�A&?�7HU�?"B1Y1<G; 7HJ�8�:H?.a�?�l=1>C8G8�?�A&7H<,; 7]?��bYH; [">CI ?�7HU�?�U�I >*<G<G?�<�8&F.A&?�XHYHU�?T8&:1?c7�YHK�j@?�A�F*_�=1>C8GW8&?�A&71<TU�F*Y17�8�?�Xx>*7@XZXH>C8�>*j1>*<G?/=1>*<G<G?�<�f��?EU�F"71X�Y1U�8&?�XV=@?�AG_ F*A&K]>*7HU�?E?�[">*I Y1>C8&; F*71</8&FzU�F"KE=@>�A�?�8&:H?]?�_ W�1U�; ?�71U�l�F*_xL�N�P�R@NQSzDc; 8&:k8&:1F*<&?xFC_ e =HA&; F*A&;gB,�z>C�bWG�Z; 71?�A]>*71X�,I F*<G?"f,h�:H?,A&?�<&YHI 8&<i<G:1F�D�?�X`8&:1>C8�L�NQP�R@NQSTJ"; ["?�<OA�?�<G=@F"7H<&?,8�; KE?�<?��bYH; [">CI ?�7b8i8&F{8&:1F"<G?TFC_ e =HA&; FCA�;1>*71XE�z>C��W��Z; 7H?�A�DT:H?�7�?���8GA�>*U�8GW; 7HJ�>CI I1_ A&?��bYH?�7b8,=1>C8G8&?�A�7H<,>*71X�8&:1?�; A�<&YH=1=@F*AG8�_ A�F*K�D�?�>*abI lEU�F*AGWA&?�I >C8&?�XEX1>�8�>�B�>C7@X`8&:@>�8�; 8�; <i8&:H?cKEF"<�8i?��EU�; ?�7b8�>CK]F*71JT8�:H?�_ F"YHA>*I J*F*A&; 8&:1KE<cDT:1?�7zXH>C8�>�>CA&?`X�?�7H<&?.F*ATU�F*AGA�?�I >C8&?�XQf��A�?���Y1?�7�8�a�?�l�=@>�8&8&?�A&7H<,>CA&?�>*I <&F/YH<&?�X�_ F*A�<G; KE=1I ; _ l�; 71J{A&Y1I ?TJ"?�7�W?�A�>C8&; F"7QB�>C<O8�:H?�l/U�>*7�j(?,<&?�?�7�>*<O8&:H?�I ?�_ 8�:@>*71X/<&; XH?�<iFC_(KE; 7H; K]>*I7HF"7HWgA&?�XHYH7@XH>*7�8T>*<G<GFbU�; >�8�; F"7�A&Y1I ?�</� ¤*�gf

7. ACKNOWLEDGEMENTS��?{D�F"YHI XzI ; a�?.8�F�8&:1>*71a]³TF"j@?�A&8&F]¯�>�lb>CA�XHF�B1Dc:1FE=�A&FC[�; X�?�XZYH<DT; 8&:�8&:1?�; KE=1I ?�KE?�7�8�>C8&; F"7`F*_Q�z>C�bWG�Z; 7H?�AiYH<&?�X`; 7`8&:1?�8�?�<�8&<i>*71XU�F"KEKE?�7b8&?�X]F"7z>C7z?�>CA&I ; ?�A�[�?�A&<G; F"7zFC_\8&:H; <cD,F*A&a(f

8. REFERENCES� ���/³/f e J*A�>�Dc>*I B(h�fH6KE; ?�I ; 71<Gab;gBH>C7@X e f1©�Dc>*KE;gf1�Z; 71; 7HJ�>*<G<&FCWU�; >C8&; F"7�A&YHI ?�</j(?�8D�?�?�7V<&?�8&</F*_c; 8�?�KE<.; 7�I >CA&J"?]X1>�8�>Cj@>*<G?�<�f67�� qGt�s�`������� �� � � \ � r1o�� Â���tCrCyH`�tCr��Vn*r@n�wbp�]p�rHo,tGy\ n*o�n B1=@>CJ"?�<{�*§"¥5N��b��¡�B��z>�lV��¨*¨" �f

� ���/³/f e J*A�>�Dc>*IO>C7@Xx³/f@©�A&; a">*7�8�fH�1>*<�8{>*I J"F*A&; 8&:HK]<,_ F*A�KE; 71; 71J>C<&<GFbU�; >C8&; F"7EA&YHI ?�<\; 7�I >CA&J"?cX1>C8�>*j1>*<G?�<�f�67�� qGt�s�`Qt�y,o �1p!e��Co �� rHo�� Â���t*r�yC`�tCr��1p�q�} p n*qgwbp \ nCon���n*~�p�~���� p \ ��� B�=1>*J"?�<¤�¥*¢�N�¤"¨"¨�B��"YH71?���¨*¨*¤Hf

�  C�/³/f e J*A�>�Dc>*I@>*71X�³/fb©�A&; a">*7�8�fb�Z; 71; 7HJ`<G?���Y1?�7�8&; >*IH=@>C8G8&?�A&71<�f67�� q&t�s�`{tGy]o �Hp�^ ^Co � � rHo�� Â���tCrCyC`�t*r \ nCon! ,r�w*v rQp&p�q�v r�w� � � \ �� B1=1>*J"?�<� �N1��¤HB(�z>CA�f���¨"¨"¦bf

� ¤*�#"�f�¯�>*<�8&; X�?"Bc²`f�°\>*<&��Y1; ?�A�B,³`f�hO>*F*Y1; I B�¼`f�©�8�YHKEKE?"B�>*71X¬ f ¬ >*ab:@>CIgf\�Z; 7H; 7HJZKE; 71; Kx>CI\71F"7�WgA�?�X�Y171XH>*7b8�A�YHI ?�</Y1<G; 71J_ A�?���Y1?�7�8TU�I F*<&?�Xx; 8&?�K]<G?�8&<�f�6�7$� qGt�s�`it�y�o �1pT^C~&o � rHo�� Â%�it*r�yC`tCr&�itl��m@u�o�n*o v�t*r@n" p t&w*vgs'�)(Co � � r1o�� Â*��t*r�yC`Tt*r \ n*o�n �&n*~�p�H}*~&o�p��~,+ \ �#� \ � B*=1>*J"?�<,¨"¥"�5N�¨"¢"¡�f�©�=�A&; 7HJ"?�A�B-�*Y1I l��*§"§*§�f

� ¦���©@f\��fi¯�>�lbfih,:1?x¸.�,6/�.�{� e A&U�:H; [�?�� J ,�,�-K.L0�0�:EA�AS46C�5�>748"5�C&4@�A�8 ��f16A&[�; 71?"B1� e �H¸�71; [�?�A&<G; 8lxFC_��,>*I ; _ F*A&71; >�B(�.?�=1>CAG8GWKE?�7�8cF*_i67H_ F*A&K]>C8&; F"7z>*71Xz�,F"KE=HYH8&?�A�©�U�; ?�7HU�?"f

� ¡C�/³/fc¯�>�lb>CA�XHFd>*71X�³/f e J*A�>�Dc>*IgfT�Z; 7H; 7HJd8&:H?�KEF"<�8x; 7b8&?�AGW?�<�8&; 7HJ]A�YHI ?�<�fQ67.� qGt�s�`,tGy/o �1p�/Co � � rHo�� Â��it*r�yC`�tCr'0/r@tlqc Zp ' w�p \ v ~�s�t�i"p�q�}�nCr '=\ n*o�n'��v r1v r�w1� 0 \�\ � B�=1>*J"?�<���¤�¦5N��¦C¤HB e YHJHfQ��¨"¨*¨�f

� ¥��/³/f2�Hf�¯�>�l�>�A�XHFHfc�\�EU�; ?�7b8&I l�KE; 7H; 7HJ�I F"7HJd=1>C8G8&?�A&71<E_ A&F"KXH>C8�>*j1>*<G?�<�f*67�� qGt�s�`-�����3� �� � � \ � r1o�� Â4�it*r�yC`Ht*r5�Vn*r Zn�wbp�]p�r1oTt�y \ n*o�n B1=1>*J"?�<T¢�¦5Nb¨* �B��"YH71?���¨"¨"¢bf

� ¢C��©@f*¯�A&; 7QB"³`f"�ZFC8�D�>*71; Bb>*71X���f�©b; I ["?�A&<G8&?�; 7Qf"¯�?�lbF"71X`K]>CA&a"?�8j1>*<Ga�?�8&<��Z¼�?�71?�A�>*I ; ^�; 71JV>*<G<GFbU�; >C8&; F*7�A&YHI ?�<`8&F9U�F*AGA&?�I >C8&; F"7Qf67�� qGt�s�`������� �� � � \ � r1o�� Â���tCrCyH`�tCr��Vn*r@n�wbp�]p�rHo,tGy\ n*o�n B1=@>CJ"?�<{�*¡"¦5N��"¥C¡�B��z>�lV��¨*¨�¥bf

� ¨C��©@fO¯�A&; 7QBi³/f\�ZF*8Dc>*7H;gB%�Hf���fO¸�I I K]>*7QB\>*71Xd©@fOh�<GYHA�fO��l�W71>*KE; U/; 8&?�KE<G?�8.U�F"YH7b8&; 7HJx>*71X�; KE=HI ; U�>C8&; F*7�A&Y1I ?�<�_ FCA/K]>CAGWa"?�8ij@>C<&a"?�8,X1>�8�>bf�676� q&t�s�`7�8���� �� � � \ � rHo�� Â9�it*rCyH`@t*r�9n*r(n�wbp�]p�r1oTt�y \ nCon B1=1>*J"?�<��"¦"¦5Nb�*¡*¤HB��Z>�l9��¨"¨�¥�f

� ��§C��¯TfE¼.>*7�8&?�A�>*71X�³/f`��; I I ?"f � t*q�En"Â$��t*r@s�p�m@o!�cr(n* }*~&v ~;:�9n*o �Hp�]nCogvgs&n" � tCu�r ' n*o v�tCr1~ fQ©�=�A&; 71J*?�A�B���¨"¨"¨�f

� �*���5��fi».>C7�>C7@X��Vfi�/>*K�j(?�A�f)\ n*on!��v rHv rbw-: �it*r@s�p�m1og~znCr '/(pGs9��r1v�L�u@p�~ f(�ZF*A&J�>C7��`>CYH_ K]>C717QB@©b?�=�8Cf1�*§*§"§�f

� �����5��f\».>C7�B���fi°O?�;gB�>*7@X<"�f9"{; 7Qfi�Z; 7H; 7HJz_ A&?��bYH?�7b8�=1>C8G8�?�A&7H<Dc; 8&:1F*YH8�U�>C7@X�; X1>�8�?cJ"?�7H?�A�>�8�; F"7Qf�67�� q&t�s�`4������ �� �.� \� rHo�� Â���t*r�yC`�t*r.�VnCr(n�wbp�xp&r1o.tGy \ n*on BO=@>CJ"?�<E��N1�C��Bi�z>�l�C§"§"§�f

� �� C���9fO�`>CK�j@?�A�B=�HfO»{>*7QBi>*71X<"�f\�,:1; >*71J�fi�Z?�8�>CA&Y1I ?�WJ*Y1; X�?�XKE; 71; 7HJ¾F*_�K�YHI 8&; WXH; K]?�7H<G; F"71>*I�>*<G<GFbU�; >C8&; F"7�A&Y1I ?�<�YH<&; 71JXH>C8�>�U�Y1j@?�<�f@67'� q&t�s�`�tGy/o �1pT�"q ' 0 \�\ � r1o�� Â%��tCrCyH` B e YHJHf��¨"¨�¥bf

� ��¤*��¯Tf ¬ ?�7b8�B e fO©bDc>*KE; B\>*71X'�Hf���; X�F"KzfO�,I YH<�8&?�A�; 71J9>C<&<GF�U�; W>�8&; F"7�A�YHI ?�<�f,6�7>� qGt�s�`/t�yzo �1p_^��"o � � r1o�� Â#��t*r�yC`{t*r \ nCon �r�w*v rQp&p�q�v rbw$� � � \ �� BH=@>CJ"?�<.�"�C§�N��C ��"BQ�z>CA�f���¨"¨"¥bf

� ��¦��`��f ¬ ; 7�>*71X�?\fc�Vf,�/?�X�?�Kzf,°i; 7HU�?�AGWG©b?�>CA&U�:�� e 7H?�D >CI WJ*F*A&; 8&:1K�_ FCA�X�; <GU�FC[�?�A&; 7HJZ8&:1?�K]>C��; K�Y1KÀ_ A&?��bYH?�7b8.<&?�8�fQ67� q&t�s�`@tGy�o �1p�(�o � � r1o�� Â��it*r�yC` tCr5 �*o�p�r ' v r�w \ n*o�n �&n*~�p�/(p&s9� Zr@t" t&w*}$�@ \ � /4� BH=1>*J"?�<`��§�¦5NH�"��¨�B��z>�ACfQ��¨*¨"¢�f

� ��¡C�`»`f��z>C717H; I >bBb»`f�h�F"; [�F"7H?�7�B�>*71X e f*6�f"´i?�A�a*>*KEFHfC�{; <GU�FC[�?�AGlFC_{_ A�?���Y1?�7�8�?�=1; <GF�X�?�<E; 7d?�[�?�7b8�<G?��bYH?�7HU�?�<�f�\ n*o�n!��v r1v r�wnCr ' 0.r(t�qT p ' wbp \ v ~�s&tli"p�q�} B����g ���� �*¦*¨�N��C¢"¨�BO©b?�=H8�f(��¨"¨"¥bf

� ��¥��5��f1©(fH°\>CA&a(B1�Vf@©(fH�,:H?�7QB1>C7@XZ°Of1©(fA",Y�f e 7x?��]U�; ?�7b8T:1>*<G:j1>*<G?�X�>CI J"F*A&; 8&:HK�_ F*A`KE; 71; 71JZ>*<G<GFbU�; >C8&; F"7�A�YHI ?�<�f�67 � qGt�s�`�8���B� �� � � \ � rHo�� Â8��t*r�yC`{t*r��VnCr(n�wbp��xp�rHoEtGy \ n*o�n B=1>*J*?�</�C¥"¦TN1��¢"¡bB��z>�lV��¨"¨"¦bf

� ��¢C�`²`f{°\>*<&�bYH; ?�A�B�"�f.¯�>*<�8&; X�?"B�³`f{hO>*F"YH; IgBE>*71X ¬ f ¬ >Ca�:1>*Igf°OA�YH71; 71J�U�I F*<&?�XZ; 8�?�KE<G?�8cI >C8G8�; U�?�<c_ FCA{>*<G<GFbU�; >�8�; F"7�A&Y1I ?�<�fH67�cs�o�p�~ ' p�~a^DC7EM��tCu�q�r�F&p�~�G#��nC~�p�~ ' p ' tCr1r�F&p�~`nli*nCr(sHF&p�~�I B=1>*J*?�</�C¥"¥TN1��¨"¡bB�­{U�8�fQ��¨"¨*¢�f

� ��¨C�`²`fb°i>C<&�bYH; ?�A�B-"�fb¯�>C<G8&; XH?"B�³`fbhO>*F"YH; IgBH>*71X ¬ f ¬ >*ab:@>*I f��{; <�WU�F�[�?�A&; 7HJ]_ A&?��bYH?�7b8{U�I F"<G?�XV; 8&?�KE<G?�8&<{_ F*A`>C<&<GF�U�; >C8&; F"7�A&Y1I ?�<�f67J� qGt�s�`xtGy�o �1p�K�o � � rHo�� Â��it*r�yC`Et*r \ n*o�n �&n*~�p / �1pGt*q�}� � � \ /4� BH=@>CJ"?�<� "¨*¢�N�¤H��¡�B���>*7QfQ��¨"¨*¨�f

� �C§C�`²`f@°\>*<&�bYH; ?�A�B�"�f@¯�>*<�8&; X�?"B@³`f@hO>*F"YH; IgBQ>C7@X ¬ f ¬ >*ab:@>*I f(�O_ W�1U�; ?�7�8.KE; 7H; 7HJEF*_�>*<G<GFbU�; >�8�; F"7�A&YHI ?�<{YH<G; 71JxU�I F"<G?�X9; 8&?�KE<G?�8I >C8G8&; U�?�<�f2L t*ubq�r(n*Â�tGy � r�y�tCqJ�En*o v�tCr �H}*~&o�p��~ B,�C¤H�G����� �"¦5N�¤"¡�B�Z>CA�f���¨"¨"¨bf

� �����5��fc°O?�;gB,�Hf,».>*7QBc>*7@X�³`f,�z>*FHfT�,I F*<&?�8�� e 7�?��EU�; ?�7b8Z>CI WJ*F*A&; 8&:1KÅ_ F*AEK]; 71; 7HJz_ A�?���Y1?�7�8�U�I F"<G?�Xk; 8�?�KE<G?�8&<�fi67M� qGt�s�`N t*q | ~��Ht�mÄt*r3OTp&~�pGnCq&s9� � ~&~&u@p&~ tCr \ nConP��v r1v r�w¾nCr '0.r@tlqT p ' w�p \ v ~�s�t�i"p�q�}Q� \ ��0 \ � B�=1>*J"?�<Ç����Nb "§bBÆ�z>�l�C§"§"§�f

� �*��� e f,©�>�[">*<G?�A&?"BT��f�­{KE; ?�U�; 71<Gab;gB�>*71X�©(fi²{>�[">C8&:H?"f e 7�?���WU�; ?�7b8V>CI J"F*A&; 8&:HK _ FCAVKE; 71; 71J�>*<G<GFbU�; >�8�; F"7ÆA&Y1I ?�<�; 7 I >CA&J"?XH>C8�>*j1>*<G?�<�fC675� qGt�s�`HtGyio �1p)eN^Co � � rHo�� Â7��tCrCyH`Ht*r.�1p�q�} p nCq�wbp\ n*o�nR��n*~�p�~���� p \ ��� BH=1>*J"?�<T¤� ��5N"¤"¤"¤HBQ©b?�=H8�fQ��¨"¨"¦bf

� �C C���Tf�©b; I ["?�A�<�8&?�; 7QBb©@f*¯�A&; 7QBb>C7@X/³`f*�ZF*8Dc>*7H;gf"¯,?�lbF"71X/Kx>�A�a"?�8j1>*<Ga�?�8&<��]¼�?�71?�A�>*I ; ^�; 71J9>*<G<GFbU�; >C8&; F*7kA&YHI ?�<`8�FzXH?�=@?�7@X�?�71U�?A&YHI ?�<�f \ n*on���v rHv rbw.nCr ' 0/r@tlqT p ' wbp \ v ~�s�t�i"p�q�} B������C��BS��>C7�f��¨"¨"¢�f

©b6&¼��/�{�«�\�H=HI F*A�>C8&; F*71<�f���F"=blbA&; J":b8 UË �*§"§*§xÌ`Í.νÏ(Ð�Ñ]ÒEÓ�Ó]B1�.?�U�?�K�j@?�A��*§*§"§�f ´�F"I YHKE?`�bBH6<G<&YH?��.W,=1>*J"?`¥C¤

Page 10: Mining frequent patterns with counting inference

� �C¤C�`³/f"hO>*F*Y1; I B�²`f"°\>*<&�bYH; ?�A�B "�f"¯,>*<�8&; XH?"B�>*71X ¬ f ¬ >*a�:1>*Igf��Z; 7�W; 71J{j@>*<G; <\_ F*A�>*<G<GFbU�; >C8&; F"7EA&Y1I ?�<iY1<G; 7HJ/U�I F"<G?�X�<&?�8&<�f"6�76� qGt�s�`t�yxo �1p=^S(Co � � r1o�� Â#�it*rCyH`{tCr \ n*o�n< ,rbw*v r(p&p�q�v rbwM� � � \ �� B=1>*J"?/ "§�¥�B@�1?�jQf N��z>�ACf(�C§"§"§�f

� �"¦��`»`fQh�F"; [�F*71?�7Qf�©H>CK]=HI ; 71JxI >�A�J*?]XH>C8�>*j1>*<G?�<._ F*A`>*<G<&F�U�; >C8&; F*7A&YHI ?�<�fi67�� qGt�s�`�tGy]o �1pje e�r ' � r1o�� Â��it*rCyH`TtCr �1p�q�} p n*qgwbp\ n*o�nR��n*~�p�~�� � p \ ��� BH=@>CJ"?�<`�� *¤�N1��¤b¦bB�©b?�=H8�f(��¨"¨*¡�f

� �*¡����Vf9��f9?(>*a�;gB�©(fQ°\>CAG8&:@>C<�>�A�>�8&:blbBO�VfO­�J"; :1>CA�>�BO>*71Xz�Æf ¬ ;gf²�?�D�>*I J*F*A&; 8&:1KE<{_ F*A._g>*<�8`XH; <&U�F�[�?�AGl9F*_c>*<G<GFbU�; >C8&; F"7�A�YHI ?�<�f67$� qGt�s�`OtGy�o �1p$�"q ' � rHo�� Â��it*r�yC`�t*r�0.r(t�qT p ' wbp \ v ~�s&tli"p�q�}v r \ nCon �&nC~�p&~�� 0 \�\ � BH=1>*J"?�<.�C¢" �Nb�*¢"¡�B e Y1JHfQ��¨*¨�¥bf

©b6&¼��/�{�«�\�H=HI F*A�>C8&; F*71<�f���F"=blbA&; J":b8 UË �*§"§*§xÌ`Í.νÏ(Ð�Ñ]ÒEÓ�Ó]B1�.?�U�?�K�j@?�A��*§*§"§�f ´�F"I YHKE?`�bBH6<G<&YH?��.W,=1>*J"?`¥"¦