Estimasi Prob. Density Function dengan EM

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1 Estimasi rob. Desity Fuctio dega EM Sumber: -Forsyth & oce Chap. 7 -Stadford Visio & Modelig robability Desity Estimatio arametric Represetatios o-arametric Represetatios Miture Models age

2 Metode estimasi o-parametric Tapa asumsi apapu tetag distribusi Estimasi sepeuhya bergatug ada DATA cara mudah megguaa: Histogram Histograms Disritisasi, latas ubah dalam betu batag: age

3 Histograms Butuh omputasi baya, amu sagat umum diguaa Dapat diterapa pada sembarag betu desitas arbitrary desity Histograms ermasalaha: Higher dimesioal Spaces: - umlah batag bis yg. Epoetial - umlah traiig data yg epoetial - Curse of Dimesioality size batag? Terlalu sediit: >> asar Terlalu baya: >> terlalu halus age 3 3

4 edeata secara prisip: diambil dari uow p probabiliti bahwa ada dalam regio R adalah: p ' d' p V R edeata secara prisip: diambil dari uow p probabiliti bahwa ada dalam regio R adalah: R K p ' d' p V age 4 4

5 edeata secara prisip: diambil dari uow p probabiliti bahwa ada dalam regio R adalah: R K p ' d' p V p K V edeata secara prisip: p K V Dega Fi V Tetua K Metoda Kerel-Based K-earest eighbor Dega Fi K Tetua V age 5 5

6 Metoda Kerel-Based: p K V arze Widow: u < / H u 0 otherwise Metoda Kerel-Based: p K V arze Widow: u < / H u 0 otherwise K H age 6 6

7 Metoda Kerel-Based: p K V arze Widow: u < / H u 0 otherwise K H p h H d Metoda Kerel-Based: p K V Gaussia Widow: p πh d / ep h age 7 7

8 Metoda Kerel-Based: K-earest-eighbor: eighbor: p K V Kembaa V sampai dia mecapai K poits. age 8 8

9 K-earest-eighbor: eighbor: K-earest-eighbor: eighbor: Klasifiasi secara Bayesia : p C p p C K V K V age 9 9

10 K-earest-eighbor: eighbor: Klasifiasi secara Bayesia : p C p p C K V K V p C K K atura lasifiasi -earest-eighbour robability Desity Estimatio arametric Represetatios o-arametric Represetatios Miture Models Model Gabuga age 0 0

11 Miture-Models Models Model Gabuga: Gaussias: - Mudah -Low Memory - Cepat - Good roperties o-arametric: -Umum - Memory Itesive -Slow Miture Models Campura fugsi Gaussia miture of Gaussias: p Jumlah dari Gaussias tuggal age

12 Campura fugsi Gaussia: p Jumlah dari Gaussias tuggal Keuggula: Dapat medeati betu desitas sembarag Arbitrary Shape Campura fugsi Gaussia: p Geerative Model: z 3 p age

13 Campura fugsi Gaussia: p p M p p ep σ d / πσ Campura fugsi Gaussia: Maimum Lielihood: E l L l p age 3 3

14 Campura fugsi Gaussia: Maimum Lielihood: E l L l p E 0 E Campura fugsi Gaussia: Maimum Lielihood: E l L l p E 0 age 4 4

15 5 age 5 Campura Campura fugsi fugsi Gaussia: Gaussia: Campura Campura fugsi fugsi Gaussia: Gaussia: M p p

16 6 age 6 Campura Campura fugsi fugsi Gaussia: Gaussia: / ep d p σ πσ M p p Campura Campura fugsi fugsi Gaussia: Gaussia: / ep d p σ πσ M p p

17 Campura fugsi Gaussia: Maimum Lielihood: E l L l p Tida ada solusi pede! E 0 E Campura fugsi Gaussia: Maimum Lielihood: E l L l p E Gradiet Descet age 7 7

18 Campura fugsi Gaussia: Maimum Lielihood: E l L l p E f,..., M, σ,..., σ M, α,..., α M Campura fugsi Gaussia: Optimasi secara Gradiet Descet: Comple Gradiet Fuctio highly oliear coupled equatios Optimasi sebuah Gaussia tergatug dari seluruh campura laiya. age 8 8

19 Campura fugsi Gaussia: -> Dega strategi berbeda: p Observed Data: Campura fugsi Gaussia: p Desitas yg dihasila Observed Data: age 9 9

20 Campura fugsi Gaussia: p Variabel Hidde y Observed Data: Campura fugsi Gaussia: p Variabel Hidde y Observed Data: Uobserved: y age 0 0

21 Cotoh populer ttg.. Chice ad Egg roblem: p Ma.Lielihood Ut. Gaussia # Ma.Lielihood Ut. Gaussia # Aggap ita tahu y Chice+Egg roblem: Aggap ita tahu p y y y age

22 Chice+Egg roblem: p Tapi yg ii ita tida tau sama seali?? y Chice+Egg roblem: p Coba pura tahu y age

23 Clusterig: Tebaa bear? y K-mea clusterig / Basic Isodata egelompoa Clusterig: rocedure: Basic Isodata,...,. Choose some iitial values for the meas M Loop:. Classify the samples by assigig them to the class of the closest mea. 3. Recompute the meas as the average of the samples i their class. 4. If ay mea chaged value, go to Loop; otherwise, stop. age 3 3

24 Isodata: Iisialisasi Isodata: Meyatu Covergece age 4 4

25 Isodata: Beberapa permasalaha Diteba Eggs / Terhitug Chice p Ma.Lielihood Ut. Gaussia # Ma.Lielihood Ut. Gaussia # Disii ita berada y age 5 5

26 GaussiaAproimasi yg. bai p amu tida optimal! ermasalaha: Highly overlappig Gaussias Epectatio Maimizatio EM EM adalah formula umum dari problem seperti Chice+Egg Mi.Gaussias, Mi.Eperts, eural ets, HMMs, Bayes-ets, Isodata: adalah cotoh spesifi dari EM Geeral EM for mi.gaussia: disebut Soft-Clusterig Dapat overge meadi Maimum Lielihood age 6 6

27 7 age 7 Igat Igat rumusa rumusa ii ii?:?: / ep d p σ πσ M p p Soft Chice ad Egg roblem: Soft Chice ad Egg roblem: p

28 Soft Chice ad Egg roblem: p Aggap ita tahu: Weighted Mea of Data Soft Chice ad Egg roblem: p Step-: Hitug ulag posteriors age 8 8

29 Lagah prosedur EM: rocedure: EM. Choose some iitial values for the meas,..., M E-Step:. Compute the posteriors for each class ad each sample: M-Step: 3. Re-compute the meas as the weighted average of their class: 4. If ay mea chaged value, go to Loop; otherwise, stop. EM da Gaussia miture θ arg ma Q θ, θ i i θ i p, θ p, θ i i age 9 9

30 30 age 30 EM EM da da Gaussia miture Gaussia miture, arg ma i i Q θ θ θ θ i T i i i i p p,, θ θ EM EM da da Gaussia miture Gaussia miture, arg ma i i Q θ θ θ θ i i p, θ α

31 Cotoh-cotoh EM: Traiig Samples Cotoh-cotoh EM: Traiig Samples Iitializatio age 3 3

32 Cotoh-cotoh EM: Traiig Samples Ed Result of EM Cotoh-cotoh EM: Traiig Samples Desity Isocotours age 3 3

33 Cotoh-cotoh EM: Color Segmetatio Cotoh-cotoh EM: Yair Weiss Layered Motio age 33 33

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