Outline. Learning with Missing Data. Examples. Modeling Missing Values ( ) Missing Data Algorithms Experiments Conclusions Future Directions
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- Cornelius Tucker
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1 Outlne earnng wth Mssng Data Onur Ozan Koluoglu Naveen Ramakrshnan Mssng Data lgorthms Eerments Conlusons Future Dretons Mssng Data Eamles What s mssng data? Consder n observatons eatures resonse n. s mssng Fae reognton wth olusons Surve datasets Sensors (unrelable/nos measurements DN datasets Mssng values n eatures and/or observatons Modelng Mssng Values Mssng Data et ( M M be the mssng data ndator matr D be the omlete dataset φ be the unknown arameter (aordng to some model haraterzng mssngness o the data MCR: Mssng omletel at random P( M D φ P( M φ MR: Mssng at random P( M D φ P( M D φ [ D D] φ ask: Fnd a model ( Fnd a lnear model?
2 What we oused on: Mssng Data Regresson roblems wth mssng values n tranng hase Mssng values an be n eatures and/or n resonse and Sequental roah Frst mute the mssng values usng an mutaton method then al a learnng algorthm. Imutaton muted earnng 7 Imutaton Deleton o Observatons Delete row t has mssng values o eamle: n Imutaton muted.. Deleton o Observatons Deleton o Features Delete olumn t has mssng values o eamle: n.. Deleton o Features veragng Fous on olumn Denote mssng row ndes b m Fll the mssng values b averagng Data( m Data( n m m lso alled mean mutaton [Haste bshran Fredman]
3 o eamle:.. veragng n veragng K NN veragng Fous on olumn Denote row ndes o K nearest neghbors b Fll the mssng values b averagng Data( NN Data( NN NN Choosng nearest neghbors based on a dstane metr NN K NN veragng earnng o eamle:.. n NN veragng Standardzed values are used muted earnng Eerments Full data (Boston Housng Data wo tes o deletons: observaton o [ o o o] ol where ol s the requen o observatons wth l mssng values and eatures [ ] where o l denotes the mssng req. or eature 7 Full Data veragng K-NN veragng Deleton o Samle o. Full data s obvousl the best For B.H. averagng s losest to ull data (or ratal deleton vetors For a gven mutaton are smlar Full Data veragng K-NN veragng Deleton o Samle o.7 Full Data veragng K-NN veragng Deleton o Samle o. Sequental observaton K
4 o Full Data veragng K-NN veragng Deleton o Samle. [ ] Full Data veragng K-NN veragng Deleton o Samle o. [ ] Full Data veragng K-NN veragng Deleton o Feature. Full Data veragng K-NN veragng Deleton o Feature.7 Deleton o samles deends onl on the sum o requenes Other methods also erorm well K NN s worse than averagng Full Data veragng K-NN veragng Deleton o Samle o.7 [ ] Sequental observaton For a gven mutaton are smlar Sequental eature Full Data veragng K-NN veragng Deleton o Feature. K NN an be better than averagng s hghl orrelated wth deletng t worsens the erormane o Deleton o Feature Full Data veragng K-NN veragng Deleton o Feature.7 Sequental eature Full Data veragng -NN -NN -NN -NN o. K NN meets averagng as K nreases Full Data veragng -NN -NN -NN -NN o.7 Full Data veragng -NN -NN -NN -NN o. Sequental observatons Snthet Data n Entres are hosen..d. ~ ntodalt: sgn( Ν ( sgn( - Full Data veragng -NN -NN -NN -NN. K NN s otmal or some ntermedate K value veragng s the worst - Full Data veragng -NN -NN -NN -NN.7 Sequental Snthet Data eatures
5 What else an we do? Proosed roah # Imutaton Sde normaton muted earnng earnng algorthms to mute Order the observatons Ft a lnear model (wth rdge or eah mssng olumn o eah new observaton Use the tted models to ll the mssng values n Regress. Regress Proosed roah # Proosed roah # Sde normaton or learnng ssoate a ost to eah observaton b ountng ts number o mssng values sa Sale down eah eature o ths observaton wth g( earn the oeents b rdge regresson wth the moded dataset n RSS λ ( g( λ g( 7 earnng algorthm or mutaton Sde normaton or learnng ssoate a ost to eah eature b ountng the number o mssng values sa Mod rdge regresson usng these osts RSS n λ λ g( ( g( Proosed roah # Proosed roah # Onlne (ase b ase learnng Order the data and learn the oeents rom the observatons wthout mssng values sa For observaton untl n Imute the mssng values wth rdge regresson Comute the assoated ost g ( Udate the oeents where ( g ( g ( arg mn k k k λ Eent omutaton o nverson lemma E E E E I ( Onlne (ase b ase udaton g( : (usng matr
6 Full Data v K-NN DOF N N N Full Data v K-NN DOF N N N Full Data v K-NN DOF N N N o. o.7 Full Data v K-NN DOF N N N o. o.7 Full data s obvousl the best New algorthms ( and erorm as good as averagng and K NN New lgorthms observaton K Full data s obvousl the best New algorthms ( and are robust to antodalt unlke averagng New lgorthms Snthet Data observaton K Full Data v K-NN DOF N N N Full Data v K-NN DOF N N N Full Data v K-NN DOF N N N Full Data v K-NN DOF N N N New lgs. and are smlar to averagng and K NN DOF s worst due to deleton o mortant eature New lgorthms eature New lgorthms eature Conlusons Full Data v K-NN DOF N N N. New lgs and are muh better than averagng New lg s smlar to K NN Full Data v K-NN DOF N N N.7 New lgorthms Snthet Data eatures Mssng data roblem Pratal & Cool Hard to model & data deendent New algorthms based on the rnles: earnng the mssng values (not a naïve mutaton Consderng the relablt o muted values Snthet Data & Boston Housng Data
7 Future Dretons Other modatons Mult resonse Onlne udate rule wth eatures New datasets Mssng values n test observatons Mssng data n lassaton earnng wth osts Cost o eatures/observatons aordng to ther relablt 7
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