Evaluation of classifiers MLPs
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1 Lecture Evaluaton of classfers MLPs Mlos Hausrecht 539 Sennott Square Evaluaton For any data set e use to test the model e can buld a confuson matrx: Counts of examples th: class label ω that are classfed th a label predct α α ω 4 target ω 7 54 α
2 Evaluaton For any data set e use to test the model e can buld a confuson matrx: target ω ω α 4 7 predct α 54 agreement Error:? Evaluaton For any data set e use to test the model e can buld a confuson matrx: target predct α α Error: 37/3 Accuracy - Error 94/3 ω 4 ω 7 54 agreement
3 Evaluaton for bnary classfcaton Entres n the confuson matrx for bnary classfcaton have names: target ω ω α TP FP predct α FN TN TP: True postve ht FP: False postve false alarm TN: True negatve correct reecton FN: False negatve a mss Addtonal statstcs Senstvty recall Specfcty SENS SPEC TP TP + FN TN TN + FP Postve predctve value precson TP PPT TP + FP Negatve predctve value TN NPV TN + FN
4 Bnary classfcaton: addtonal statstcs Confuson matrx target predct 4 8 SENS 4/6 SPEC 8/9 PPV 4/5 NPV 8/ Ro and column quanttes: Senstvty SENS Specfcty SPEC Postve predctve value PPV Negatve predctve value NPV Bnary decsons: ROC..8.6 ω ω.4. x * Probabltes: SENS SPEC p p x > x* x ω x < x* x ω
5 Recever Operatng Characterstc ROC ROC curve plots : SN p x > x* x ω -SP p x > x* x ω for dfferent x* ω x * ω SN p x > x* x ω SPEC p x > x* x ω ROC curve Case Case Case p x > x* x ω p x > x* x ω
6 Recever operatng characterstc ROC shos the dscrmnablty beteen the to classes under dfferent decson bases Decson bas can be changed usng dfferent loss functon Zero-one loss functon Msclassfcaton error Based on the zero-one loss functon Any msclassfed example counts as Correctly classfed example counts as ω ω ω α 4 α α 4 76 agreement
7 General loss functon Error functon based on a more general loss functon Dfferent msclassfcatons have dfferent eght loss α our choce ω true label λ α ω loss for classfcaton Example: λ α ω α α α ω 3 3 ω ω 5 Bayesan decson theory More general loss functon Dfferent msclassfcatons have dfferent eght loss λ α ω Expected loss for the classfcaton choce R α x λ α ω P y ω x Also called condtonal rs Decson rule: α x Chooses label acton accordng to the nput The optmal decson rule α * x arg mn λ α ω P y ω x α α
8 Bayesan decson theory The optmal decson rule α * x arg mn λ α ω P y ω x α Ho to modfy classfers to handle dfferent loss? Dscrmnatve models: Drectly optmze the parameters accordng to the ne loss functon Generatve models: Learn probabltes as before Decsons about classes are based to mnmze the emprcal loss as seen above Calculatng the loss for data Confuson matrx: Counts of examples th: class label ω that are classfed th a label α α α α ω 4 7 ω 54 4 ω 8 76 agreement Loss N λ α ω N α ω
9 Multlayer neural netors Lnear unts x x Lnear regresson f x d + d x f x Logstc regresson f x p y x g x x d z d + x f x p y x x d On-lne gradent update: + α y f x x d The same On-lne gradent update: + α y f x + α y f x x + α y f x x
10 Lmtatons of basc lnear unts Lnear regresson f x + d x Logstc regresson f x p y x g d + x x x f x x x z p y x d d x d x d Functon lnear n nputs!! Lnear decson boundary!! Regresson th the quadratc model. Lmtaton: lnear hyper-plane only a non-lnear surface can be better
11 Classfcaton th the lnear model. Logstc regresson model defnes a lnear decson boundary Example: classes blue and red ponts Decson boundary Lnear decson boundary logstc regresson model s not optmal but not that bad
12 When logstc regresson fals? Example n hch the logstc regresson model fals Lmtatons of lnear unts. Logstc regresson does not or for party functons - no lnear decson boundary exsts Soluton: a model of a non-lnear decson boundary
13 f x x Extensons of smple lnear unts use feature bass functons to model nonlneartes Lnear regresson m + φ x φ x φ x φ x - an arbtrary functon of x Logstc regresson f x g + φ x m x d φ m x m f x Learnng th extended lnear unts Feature bass functons model nonlneartes Lnear regresson m + φ x x φ x φ x Logstc regresson f x m g + φ x x d x φ m m Important property: The same problem as learnng of the eghts for lnear unts the nput has changed but the eghts are lnear n the ne nput Problem: too many eghts to learn
14 Mult-layered neural netors Alternatve ay to ntroduce nonlneartes to regresson/classfcaton models Idea: Cascade several smple neural models th logstc unts. Much le neuron connectons. Multlayer neural netor Also called a multlayer perceptron MLP Cascades multple logstc regresson unts Example: layer classfer th non-lnear decson boundares x x x d z z z p y x Input layer Hdden layer Output layer
15 Multlayer neural netor Models non-lneartes through logstc regresson unts Can be appled to both regresson and bnary classfcaton problems Input layer x x x d Hdden layer z z Output layer regresson f x f x z classfcaton f x p y x opton Multlayer neural netor Non-lneartes are modeled usng multple hdden logstc regresson unts organzed n layers The output layer determnes hether t s a regresson or a bnary classfcaton problem Input layer x Hdden layers Output layer regresson f x f x x classfcaton x d opton f x p y x
16 Learnng th MLP Ho to learn the parameters of the neural netor? Gradent descent algorthm Weght updates based on the error: J D α J D We need to compute gradents for eghts n all unts Can be computed n one bacard seep through the net!!! The process s called bac-propagaton Bacpropagaton --th level -th level +-th level x x z + l + z l x l + x z - output of the unt on level - nput to the sgmod functon on level - eght beteen unts and on levels - and + x z g z x
17 Bacpropagaton δ x u u n u f y K δ Update eght usng a data pont D J α D J z δ Let Then: x z z D J D J δ S.t. s computed from and the next layer + δ l x x l l l + + δ δ Last unt s the same as for the regular lnear unts: It s the same for the classfcaton th the log-lelhood measure of ft and lnear regresson th least-squares error!!! } { > < y D x x Learnng th MLP Gradent descent algorthm Weght update: D J α x z z D J D J δ x αδ x δ - -th output of the - layer - dervatve computed va bacpropagaton α - a learnng rate
18 Learnng th MLP Onlne gradent descent algorthm Weght update: α J onlne D u J onlne Du z J onlne Du z δ x αδ x x - -th output of the - layer δ - dervatve computed va bacpropagaton α - a learnng rate Onlne gradent descent algorthm for MLP Onlne-gradent-descent D number of teratons Intalze all eghts for :: number of teratons do select a data pont D u <xy> from D set learnng rate α compute outputs x for each unt compute dervatves δ va bacpropagaton update all eghts n parallel αδ x end for return eghts
19 Xor Example. lnear decson boundary does not exst Xor example. Lnear unt
20 Xor example. Neural netor th hdden unts Xor example. Neural netor th hdden unts
21 MLP n practce Optcal character recognton dgts x Automatc sortng of mals 5 layer netor th multple output functons outputs 9 layer Neurons Weghts x 4 nputs
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