Modeling the Effect of Time in Bayesian Knowledge Tracing

3 downloads 0 Views 259KB Size Report
In the real world coming into class on a new day may result in a student forgetting ... model's prediction of new day events while the forget hypothesis could affect ...
'RHV7LPH0DWWHU"0RGHOLQJWKH(IIHFWRI7LPHLQ %D\HVLDQ.QRZOHGJH7UDFLQJ 

(EELQJKDXV @ UHVHDUFKHUV KDYH H[DPLQHG WKH PDQQHU LQ ZKLFK PHPRU\SHUIRUPDQFHGHFOLQHVZLWKWLPHRULQWHUYHQLQJHYHQWV>3DYOLN $QGHUVRQ@ ,QWKHUHDOZRUOGFRPLQJLQWRFODVVRQDQHZGD\PD\UHVXOWLQDVWXGHQWIRUJHWWLQJWKH PDWHULDO RU D KLJKHU SUREDELOLW\ RI WKHP VOLSSLQJ %\ WDNLQJ WKLV UHDO ZRUOG IDFW LQWR FRQVLGHUDWLRQLQWKLVSDSHUZHORRNLQWRKRZ.7SHUIRUPVRQHDFKQHZGD\¶VUHVSRQVHV :HGHILQHDQHZGD\¶VUHVSRQVHDVDUHVSRQVHWKDWRFFXUUHGRQDODWHUFDOHQGDUGDWHWKDQ WKHVWXGHQW¶VSUHYLRXVUHVSRQVHWRDTXHVWLRQRIWKHVDPHVNLOO:HIRXQGWKDW.7¶VQHZ GD\ HUURU LV IDU KLJKHU WKDQ VDPH GD\ HUURU $ UHVLGXDO DQDO\VLV VKRZHG WKDW .7 ZDV

BBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBBB  $XWKRUV¶DGGUHVVHV'HSDUWPHQWRI&RPSXWHU6FLHQFH:RUFHVWHU3RO\WHFKQLF,QVWLWXWH86$(PDLO \PTLX#ZSLHGX3DUGRV +HIIHUQDQ 5XL] DQG %HFN @ $OWHUQDWLYHO\ SRRU SHUIRUPDQFHRQDQHZGD\PD\DOVRVXJJHVWWKDWVWXGHQWVPD\QRWDFWXDOO\EH³IRUJHWWLQJ´ EXWLQVWHDGWKH\PLJKWMXVWEH³VOLSSLQJ´:HXVHG%D\HVLDQQHWZRUNV>5H\H@DQG ([SHFWDWLRQ0D[LPL]DWLRQ>'HPSVWHU/DLUGDQG5XELQ@WRGHWHFWZKHWKHUWLPHKDG DQ\LQIOXHQFHRQWKHIRUJHWSDUDPHWHUDQGWKHVOLSSDUDPHWHURIWKH.7PRGHO7KHPRGHO ZLWK WKH EHWWHU SUHGLFWLYH DFFXUDF\ ZLOO LQGLFDWH WKH EHWWHU FRJQLWLYH H[SODQDWLRQ RI WKH GDWD   

 



6SOLW.70RGHO'HVLJQ ,QRUGHUWRGHWHUPLQHWKHYDOLGLW\RIWKLVPHWKRGZHUHSUHVHQWWKHDERYHWZRK\SRWKHVLV LQ WKH %D\HVLDQ .QRZOHGJH 7UDFLQJ PRGHO E\ LQWURGXFLQJ D QRYHO PRGLILFDWLRQ WR WKH PRGHO WKDW DOORZV XV WR ILW D VDPH GD\ DQG QHZ GD\ SDUDPHWHU IRU RQH SDUDPHWHU LQ D &RQGLWLRQDO 3UREDELOLW\ 7DEOH &37  ZKLOH NHHSLQJ WKH RWKHU SDUDPHWHU LQ WKH &37 FRQVWDQW,Q.QRZOHGJHWUDFLQJOHDUQDQGIRUJHWVKDUHD&37DQGJXHVVDQGVOLSVKDUHD &37$VVKRZQEHORZWKHGLIIHUHQFHEHWZHHQVSOLW.7DQGWKHRULJLQDO.7LVWKHDELOLW\ WR VHSDUDWH WKH IRUJHW OHDUQ JXHVV DQG VOLS SDUDPHWHUV LQGLYLGXDOO\ 7KH HTXLYDOHQFH EHWZHHQWKHVHWZR.7PRGHOVZDVFRQILUPHGHPSLULFDOO\E\OHDUQLQJSDUDPHWHUVIRUHDFK PRGHO IURP D VKDUHG GDWDVHW ZLWKRXW QHZ GD\ GDWD DQG FRQILUPLQJ WKDW WKH OHDUQHG SDUDPHWHUVDQGSUHGLFWLRQVZHUHWKHVDPH:HDOVRFRPSDUHGWKHFRPSXWDWLRQDOUXQWLPH RI WKH 6SOLW.7 DQG 5HJXODU.7 :H WHVWHG RQ RQH RI WKH GDWDVHW IURP $66,67PHQW ZKLFK FRQWDLQHG  GDWD SRLQWV DQG FDOFXODWHG WKH (0 SDUDPHWHU OHDUQLQJ WLPH RI WKH WZRPRGHOV,WWRRNDSSUR[LPDWHO\VHFRQGVIRUWKH5HJXODU.7WROHDUQWKHSDUDPHWHUV DQGVHFRQGVIRU6SOLW.7ZKLFKHTXDWHVWRDSHQDOW\RIDERXWWRUXQWKH6SOLW .7PRGHO%RWKPRGHOVUHVXOWHGLQWKHVDPHOHDUQHGSDUDPHWHUV 7KHLQGLYLGXDOL]DWLRQRIWKHIRXUSDUDPHWHUVZHUHDFKLHYHGE\DGGLQJDIRUJHWQRGH DQGDOHDUQQRGHWRWKHNQRZOHGJHQRGHDVZHOODVDGGLQJDJXHVVQRGHDQGVOLSQRGHWR WKH TXHVWLRQ QRGH 7KHUHIRUH WKH NQRZOHGJH QRGHV DQG TXHVWLRQ QRGHV DUH FRQGLWLRQHG XSRQWKHIRXUQHZQRGHV7KH&37IRUNQRZOHGJHQRGHLVJLYHQLQ7DEOH,,7KH&37IRU WKHTXHVWLRQQRGHLVDOVRRIWKLVIRUPWKHRQO\GLIIHUHQFHLVFKDQJLQJWKHOHDUQDQGIRUJHW SDUDPHWHUV WR JXHVV DQG VOLS SDUDPHWHUV DQG FKDQJLQJ WKH SUHYLRXV DQG FXUUHQW NQRZOHGJH WR SUHYLRXV DQG FXUUHQW VWXGHQW SHUIRUPDQFH 7KH TXHVWLRQ DQG NQRZOHGJH &37VDUHIL[HGDQGHVVHQWLDOO\VHUYHDVORJLFJDWHV7KHJXHVVVOLSOHDUQDQGIRUJHWQRGH &37V FRQWDLQ WKH FRQWLQXRXV SUREDELOLWLHV WKDW DUH IDPLOLDU WR WKH VWDQGDUG .7 PRGHO 7DNH WKH ILUVW URZ DV DQ H[DPSOH NQRZLQJ WKDW WKH VWXGHQWV GR QRW KDYH SUHYLRXV NQRZOHGJH RI WKH VNLOO .QRZOHGJHBSUHYLRXV )  DQG WKH\ QHLWKHU OHDUQ QRU IRUJHW OHDUQ ) IRUJHW )  WKHQ ZH FDQ LQIHU WKH SUREDELOLW\ WKDW VWXGHQWV KDYH WKH FXUUHQW NQRZOHGJHLV 3 .QRZOHGJHBFXUUHQW 7     7DEOH,,7KH&37IRU.QRZOHGJHQRGH /HDUQ )RUJHW .QRZOHGJHBSUHYLRXV 3 .QRZOHGJHBFXUUHQW 7  ) ) )  7 ) )  ) 7 )  7 7 )  ) ) 7  7 ) 7  ) 7 7  7 7 7 

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

 PRGHOGLVFXVVHGLQWKHSUHYLRXVVHFWLRQ%\DGGLQJDWLPHQRGHWRWKH6SOLW.7PRGHOZH DUHDEOHWRHDVLO\VSHFLI\ZKLFKSDUDPHWHUVRIWKHPRGHOVKRXOGEHDIIHFWHGE\DQHZGD\ 7KHQHZGD\QRGHLVIL[HGZLWKDSULRUSUREDELOLW\RIZKLFKLVWKHRYHUDOOSURSRUWLRQ RIWKHQHZGD\LQVWDQFHVLQWKHGDWDVHW7KHWRSRORJ\RIWKH.7)RUJHWPRGHOLVVKRZQLQ )LJ7KHIRUJHWQRGHLVRQO\FRQGLWLRQHGRQWKHDGGHGQHZWLPHQRGHVRWKHUHLVRQO\ RQH QHZ SDUDPHWHU ³IRUJHWBQ´ LQWURGXFHG LQ WKLV .7)RUJHW PRGHO DQG UHSUHVHQWV WKH IRUJHW UDWH RQ D QHZ GD\ :H XVH ³IRUJHWBV´ WR GHQRWH WKH IRUJHW UDWH RQ D VDPH GD\ ZKLFK ZH VHW WR EH  MXVW DV WKH IRUJHW SDUDPHWHU LQ WKH RULJLQDO .QRZOHGJH 7UDFLQJ PRGHOLPSO\LQJWKDWWKHUHLVQRIRUJHWWLQJEHWZHHQRSSRUWXQLWLHVLQWKHVDPHGD\  7DEOH,,,&37RIWKHIRUJHWQRGH 1HZ'D\ 3 )RUJHW 7  ) 7

 IRUJHWBQ

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

7DEOH,9&37RIWKHVOLSQRGH 1HZ'D\ 3 VOLS 7  ) 7

VOLSBV VOLSBQ

 6LQFHWKHVOLSQRGHLVRQO\FRQGLWLRQHGRQWKHDGGHGQHZWLPHQRGHWKHUHLVDOVRRQHQHZ SDUDPHWHU VOLSBQ LQWURGXFHG LQ WKLV .7VOLS PRGHO ZKLFK UHSUHVHQWV WKH VOLS UDWH RQ D QHZ GD\ DQG WKH RULJLQDO VOLS SDUDPHWHU LV GHQRWHG DV VOLSBV KHUH ZKLFK LV VKRZQ LQ 7DEOH ,9 7KLV WDEOH VD\V WKDW ZKHQ D QHZ GD\ UHVSRQVH RFFXUV 1HZ 'D\ 7 WKH SUREDELOLW\ RI VOLSSLQJ LV VOLSBQ 3 VOLS 7_1HZ 'D\ 7  DQG LV VOLSBV RWKHUZLVH   

 



7RSRORJ\RIWKHPRGHOV 7KH6SOLW.7PRGHO¶VWRSRORJ\LVVKRZQWRJHWKHUZLWK.76OLSDQG.7)RUJHWLQ)LJ %R[HVLQWKHILJXUHGHQRWHWKHSRUWLRQVRIWKHILJXUHWKDWDUHXVHGLQHDFKPRGHO:KLOH DOOPRGHOVDUHVKRZQLQWKLVILJXUHVRWKHUHODWLRQVKLSEHWZHHQWKHPFDQEHVHHQ:KHQ WKHPRGHOVDUHUXQWKH\DUHUXQVHSDUDWHO\DVDVHSDUDWHWRSRORJ\DQGQRWRQHELJPRGHO dŚĞDŽĚĞůƐ

EŽĚĞZĞƉƌĞƐĞŶƚĂƚŝŽŶ ĞĂƌŶŶŽĚĞ Y͗YƵĞƐƚŝŽŶŶŽĚĞ'͗'ƵĞƐƐŶŽĚĞ &͗ĨŽƌŐĞƚŶŽĚĞ ^͗^ůŝƉŶŽĚĞ d͗dŝŵĞŶŽĚĞ

d

d

ZĞŐƵůĂƌͲ

&

>


3DUGRV +HIIHUQDQ,Q3UHVV@,QWKLVWXWRUVWXGHQWVDQVZHUDOJHEUDSUREOHPVIURPWKHLU



 PDWKFXUULFXOXPZKLFKLVVSOLWLQWRVHFWLRQV7KHSUREOHPVFRQVLVWRIPDQ\VWHSVWKDWWKH VWXGHQWV PXVW DQVZHU WR JR WR WKH QH[W SUREOHP $ VWXGHQW QR ORQJHU QHHGV WR DQVZHU VWHSVRIDJLYHQVNLOOZKHQWKH&RJQLWLYH7XWRU¶V.QRZOHGJH7UDFLQJPRGHOEHOLHYHVWKH VWXGHQWNQRZVWKHVNLOOZLWKSUREDELOLW\RUJUHDWHU:KHQDVWXGHQWKDVPDVWHUHGDOO WKHVNLOOVLQWKHLUFXUUHQWVHFWLRQWKH\DUHDOORZHGWR PRYHRQWRWKHQH[W7KHWLPHIRU VWXGHQWV XVLQJ WKLV V\VWHP LV GHWHUPLQHG E\ WHDFKHUV 7ZHOYH VNLOOV ZHUH FKRVHQ DW UDQGRPIURPWKLVGDWDVHWIRUDQDO\VLV H[FOXGLQJVNLOOVVXFKDV³SUHVVHQWHU´ZKLFKGRQRW UHSUHVHQWPDWKVNLOOV 7KHUHZDVDQDYHUDJHRIVWXGHQWSHUVNLOOLQWKLVGDWDVHW $QRWKHUGDWDVHWLVFROOHFWHGIURP$66,67PHQWV3ODWIRUP¶V6NLOO%XLOGHUSUREOHPVHWV 7KH $66,67PHQWV 3ODWIRUP LV DQ HGXFDWLRQDO UHVHDUFK SODWIRUP EHWWHU NQRZQ IRU LWV H OHDUQLQJ >)HQJ +HIIHUQDQ 0DQL DQG +HIIHUQDQ @ WKDW SURYLGHV ZHE EDVHG PDWK WXWRULQJ WR WKWK JUDGH VWXGHQWV 8QOLNH WKH &RJQLWLYH 7XWRU 6\VWHP VWXGHQWV DUH IRUFHGWROHDYHWKHWXWRUDIWHUTXHVWLRQVKDYHEHHQILQLVKHGLQRQHGD\DQGZLOOFRPH EDFNWRWKHWXWRULQDQHZGD\,IDVWXGHQWDQVZHUVWKUHHTXHVWLRQVFRUUHFWLQDURZWKH\ DUH³JUDGXDWHG´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

 

 



d

'

d

d

&

WƌĞĚŝĐƚϯƌĚ ƋƵĞƐƚŝŽŶ

WƌĞĚŝĐƚϮŶĚ ƋƵĞƐƚŝŽŶ

WƌĞĚŝĐƚϭƐƚ ƋƵĞƐƚŝŽŶ

>

&

&

>

d

d >

&

d

&

>

>

&

>

Suggest Documents