CS434a/541a: Pattern Recognition Prof. Olga Veksler. Lecture 9
|
|
- Dina Dorsey
- 6 years ago
- Views:
Transcription
1 CS434/541: Pttern Recognton Prof. Olg Veksler Lecture 9
2 Announcements Fnl project proposl due Nov prgrph descrpton Lte Penlt: s 1 pont off for ech d lte Assgnment 3 due November 10 Dt for fnl project due Nov. 15 Must be ported n Mtlb, send me.mt fle wth dt nd short descrpton fle of wht the dt s Lte penlt s 1 pont off for ech d lte Fnl project progress report Meet wth me the week of November ponts of f I wll see ou tht hve done NOTHNG et Assgnment 4 due December 1 Fnl project due December 8
3 Tod Lner Dscrmnnt Functons Introducton 2 clsses Multple clsses Optmzton wth grdent descent Perceptron Crteron Functon Btch perceptron rule Sngle smple perceptron rule
4 lghtness Lner dscrmnnt functons on Rod Mp No probblt dstrbuton (no shpe or prmeters re known) slmon bss slmon slmon Lbeled dt The shpe of dscrmnnt functons s known slmon bss lner dscrmnnt functon lot s known length Need to estmte prmeters of the dscrmnnt functon (prmeters of the lne n cse of lner dscrmnnt) lttle s known
5 lghtness lghtness Lner Dscrmnnt Functons: Bsc Ide bss slmon bss slmon bd boundr length good boundr length Hve smples from 2 clsses x 1, x 2,, x n Assume 2 clsses cn be seprted b lner boundr l(θ) wth some unknown prmeters θ Ft the best boundr to dt b optmzng over prmeters θ Wht s best? Mnmze clssfcton error on trnng dt? Does not gurntee smll testng error
6 Prmetrc Methods vs. Assume the shpe of denst for clsses s known p 1 (x θ ), 1 p 2 (x θ ), 2 Estmte θ 1, θ 2, from dt Use Besn clssfer to fnd decson regons c 3 c 2 Dscrmnnt Functons Assume dscrmnnt functons re or known shpe l(θ ), l(θ 1 2 ), wth prmeters θ, θ 1 2, Estmte θ 1, θ 2, from dt Use dscrmnnt functons for clssfcton c 3 c 2 c 1 c 1 In theor, Besn clssfer mnmzes the rsk In prctce, do not hve confdence n ssumed model shpes In prctce, do not rell need the ctul denst functons n the end Estmtng ccurte denst functons s much hrder thn estmtng ccurte dscrmnnt functons Some rgue tht estmtng denstes should be skpped Wh solve hrder problem thn needed?
7 LDF: Introducton Dscrmnnt functons cn be more generl thn lner For now, we wll stud lner dscrmnnt functons Smple model (should tr smpler models frst) Anltcll trctble Lner Dscrmnnt functons re optml for Gussn dstrbutons wth equl covrnce M not be optml for other dt dstrbutons, but the re ver smple to use Knowledge of clss denstes s not requred when usng lner dscrmnnt functons we cn s tht ths s non-prmetrc pproch
8 LDF: 2 Clsses A dscrmnnt functon s lner f t cn be wrtten s g(x) = w t x + w 0 w s clled the weght vector nd w 0 clled bs or threshold R 2 x (2) R 1 g(x) > 0 g g g ( x ) > 0 x clss 1 ( x ) < 0 x clss 2 ( x ) = 0 ether clss g(x) < 0 x (1) decson boundr g(x) = 0
9 LDF: 2 Clsses Decson boundr g(x) = w t x + w 0 =0 s hperplne set of vectors x whch for some sclrs α 0,, α d stsf α 0 +α 1 x (1) + + α d x (d) = 0 A hperplne s pont n 1D lne n 2D plne n 3D
10 LDF: 2 Clsses g(x) = w t x + w 0 w determnes orentton of the decson hperplne w 0 determnes locton of the decson surfce x (2) g(x) / w w x g(x) > 0 w 0 / w g(x) < 0 x (1) g(x) = 0
11 LDF: 2 Clsses
12 LDF: Mn Clsses Suppose we hve m clsses Defne m lner dscrmnnt functons g t ( x) w x + w 0 = = 1,...,m Gven x, ssgn clss c f g ( x) g ( x) j j Such clssfer s clled lner mchne A lner mchne dvdes the feture spce nto c decson regons, wth g (x) beng the lrgest dscrmnnt f x s n the regon R
13 LDF: Mn Clsses
14 LDF: Mn Clsses For two contguous regons R nd R j ; the boundr tht seprtes them s porton of hperplne H j defned b: g ( x) g ( x) w x + w = w x + w ( ) t w w x + ( w w ) 0 t t = j 0 j j0 j 0 j0 = Thus w w j s norml to H j And dstnce from x to H j s gven b d( x, H j ) = g ( x ) g ( x ) w w j j
15 LDF: Mn Clsses Decson regons for lner mchne re convex, z R α + 1 ( α ) z R z j g ( ) g j ( ) nd g ( z) g j ( z) ( α + ( 1 α ) z) g ( α + ( α ) z) j g 1 In prtculr, decson regons must be sptll contguous j R R R R R j s vld decson regon R j s not vld decson regon
16 LDF: Mn Clsses Thus pplcblt of lner mchne to mostl lmted to unmodl condtonl denstes p(x θ) even though we dd not ssume n prmetrc models Exmple: need non-contguous decson regons thus lner mchne wll fl
17 LDF: Augmented feture vector Lner dscrmnnt functon: t g( x) = w x + w 0 1 x Cn rewrte t: [ ] t t g x) = w w = = g( ) ( 0 new weght vector new feture vector s clled the ugmented feture vector Added dumm dmenson to get completel equvlent new homogeneous problem t g( x) = w x + w x old problem 1 x d 0 new problem g( ) = 1 x 1 x d t
18 LDF: Augmented feture vector Feture ugmentng s done for smpler notton From now on we lws ssume tht we hve ugmented feture vectors Gven smples x 1,, x n convert them to ugmented smples 1,, n b ddng new dmenson of vlue 1 = 1 x ( 2) R 2 g() < 0 R 1 g() > 0 g() / g() = 0 (1)
19 LDF: Trnng Error For the rest of the lecture, ssume we hve 2 clsses Smples 1,, n some n clss 1, some n clss 2 Use these smples to determne weghts n the t dscrmnnt functon g( ) = Wht should be our crteron for determnng? For now, suppose we wnt to mnmze the trnng error (tht s the number of msclssfed smples 1,, n ) g( ) > 0 clssfed c 1 Recll tht g( ) < 0 clssfed c 2 Thus trnng error s 0 f g( g( ) > 0 ) < 0 c c 1 2
20 LDF: Problem Normlzton Thus trnng error s 0 f Ths suggest problem normlzton : 1. Replce ll exmples from clss c 2 b ther negtve c 2 2. Seek weght vector s.t. t > 0 < > c c t t Equvlentl, trnng error s 0 f ( ) > > 2 t 1 t c 0 c 0 If such exsts, t s clled seprtng or soluton vector Orgnl smples x 1,, x n cn ndeed be seprted b lne then
21 LDF: Problem Normlzton before normlzton ( 2) fter normlzton ( 2) (1) (1) Seek hperplne tht seprtes ptterns from dfferent ctegores Seek hperplne tht puts normlzed ptterns on the sme (postve) sde
22 LDF: Soluton Regon Fnd weght vector s.t. for ll smples 1,, n t ( 2) d = k = 0 k ( k ) > 0 best (1) In generl, there re mn such solutons
23 LDF: Soluton Regon Soluton regon for : set of ll possble solutons defned n terms of norml to the seprtng hperplne ( 2) (1) soluton regon
24 Optmzton Need to mnmze functon of mn vrbles ( x ) = J( x ) J,..., 1 x d We know how to mnmze J(x) Tke prtl dervtves nd set them to zero x x 1 d J J ( x ) ( x ) = J ( x ) = 0 grdent However solvng nltcll s not lws es Would ou lke to solve ths sstem of nonlner equtons? sn cos x 4 ( x1 + x2 ) + e = ( x + x ) + log( x ) Sometmes t s not even possble to wrte down n nltcl expresson for the dervtve, we wll see n exmple lter tod x 2 4 = 0
25 Optmzton: Grdent Descent ( ) ( ) Grdent J x ponts n drecton of steepest ncrese of J(x), nd J x n drecton of steepest decrese one dmenson two dmensons J(x) dj dx ( ) J( ) dj dx ( ) x dj dx ( )
26 Optmzton: Grdent Descent (1 ) J( x ) J(x) J ( 2 ) ( x ) s (1) s (2) J ( k ) ( x ) = 0 x x (1) x (2) x (3) x (k) Grdent Descent for mnmzng n functon J(x) set k = 1 nd x (1) to some ntl guess for the weght vector ( k ) ( ) whle ( k η J x ) > ε choose lernng rte η (k) x (k+1) = x (k) η (k) k = k + 1 J( x ) (updte rule)
27 Optmzton: Grdent Descent Grdent descent s gurnteed to fnd onl locl mnmum J(x) x x (1) x (2) x (3) x (k) globl mnmum Nevertheless grdent descent s ver populr becuse t s smple nd pplcble to n functon
28 Optmzton: Grdent Descent Mn ssue: how to set prmeter η (lernng rte ) If η s too smll, need too mn tertons J(x) x If η s too lrge m overshoot the mnmum nd possbl never fnd t (f we keep overshootng) J(x) x (1) x (2) x
29 Tod Contnue Lner Dscrmnnt Functons Perceptron Crteron Functon Btch perceptron rule Sngle smple perceptron rule
30 lghtness LDF: Augmented feture vector Lner dscrmnnt functon: t g( x) = w x + w 0 bss slmon need to estmte prmeters w nd w 0 from dt length Augment smples x to get equvlent homogeneous problem n terms of smples : g x) = ( 0 1 x [ ] t w w t = = g( ) normlze b replcng ll exmples from clss c 2 b ther negtve c 2
31 LDF Augmented nd normlzed smples 1,, n t Seek weght vector s.t. > 0 ( 2) ( 2) (1) before normlzton (1) fter normlzton If such exsts, t s clled seprtng or soluton vector orgnl smples x 1,, x n cn ndeed be seprted b lne then
32 Optmzton: Grdent Descent J(x) (1 ) J( x ) J ( 2 ) ( x ) s x (1) s (1) x (2) s (2) x (3) J ( k ) ( x ) = 0 x (k) Grdent Descent for mnmzng n functon J(x) set k = 1 nd x (1) to some ntl guess for the weght vector ( k ) ( ) whle ( k η J x ) > ε choose lernng rte η (k) x (k+1) = x (k) η (k) k = k + 1 ( k + 1) ( k + 1) ( k ) ( k ) ( k ) = x J( x ) x = η ( J( x )) x (updte rule)
33 LDF: Crteron Functon Fnd weght vector s.t. for ll smples 1,, n t d = k = 0 k Need crteron functon J() whch s mnmzed when s soluton vector Let Y M be the set of exmples msclssfed b Y Frst nturl choce: number of msclssfed exmples J ( k ) ( ) { t = smple s.t. 0 } M < > ( ) = Y ( ) pecewse constnt, grdent descent s useless M 0 J()
34 LDF: Perceptron Crteron Functon Better choce: Perceptron crteron functon J p ( ) ( t = ) If s msclssfed, t 0 Thus ( ) 0 J p J p () s tmes sum of dstnces of msclssfed exmples to decson boundr Y M t / J p () s pecewse lner nd thus sutble for grdent descent J()
35 LDF: Perceptron Btch Rule Grdent of J p () s J ( ) p = ( ) Y M re smples msclssfed b (k) It s not possble to solve J p = nltcll Thus grdent decent btch updte rule for J p () s: J p ( ) ( t = ) becuse of Y M ( ) 0 Y M ( k + 1) ( k ) ( k = + η ) It s clled btch rule becuse t s bsed on ll msclssfed exmples Updte rule for grdent descent: x (k+1) = x (k) η (k) Y M Y M J( x )
36 LDF: Perceptron Sngle Smple Rule Thus grdent decent sngle smple rule for J p () s: ( k +1) ( k ) ( k ) = + note tht M s one smple msclssfed b (k) must hve consstent w of vstng smples η M Geometrc Interpretton: M msclssfed b (k) ( ( k )) t 0 M M s on the wrong sde of decson hperplne ddng η M to moves new decson hperplne n the rght drecton wth respect to M (k) (k+1) η M M
37 LDF: Perceptron Sngle Smple Rule ( k +1) ( k ) ( k ) = + η M (k) (k+1) M (k) (k+1) M k k η s too lrge, prevousl correctl clssfed smple k s now msclssfed η s too smll, M msclssfed s stll
38 LDF: Perceptron Exmple fetures grde nme good ttendnce? tll? sleeps n clss? chews gum? Jne es (1) es (1) no (-1) no (-1) A Steve es (1) es (1) es (1) es (1) F Mr no (-1) no (-1) no (-1) es (1) F Peter es (1) no (-1) no (-1) es (1) A clss 1: students who get grde A clss 2: students who get grde F
39 LDF Exmple: Augment feture vector fetures grde nme extr good ttendnce? tll? sleeps n clss? chews gum? Jne 1 es (1) es (1) no (-1) no (-1) A Steve 1 es (1) es (1) es (1) es (1) F Mr 1 no (-1) no (-1) no (-1) es (1) F Peter 1 es (1) no (-1) no (-1) es (1) A convert smples x 1,, x n to ugmented smples 1,, n b ddng new dmenson of vlue 1
40 LDF: Perform Normlzton fetures grde nme extr good ttendnce? tll? sleeps n clss? chews gum? Jne 1 es (1) es (1) no (-1) no (-1) A Steve -1 es (-1) es (-1) es (-1) es (-1) F Mr -1 no (1) no (1) no (1) es (-1) F Peter 1 es (1) no (-1) no (-1) es (1) A Replce ll exmples from clss c 2 b ther negtve c 2 Seek weght vector s.t. t > 0
41 LDF: Use Sngle Smple Rule fetures grde nme extr good ttendnce? tll? sleeps n clss? chews gum? Jne 1 es (1) es (1) no (-1) no (-1) A Steve -1 es (-1) es (-1) es (-1) es (-1) F Mr -1 no (1) no (1) no (1) es (-1) F Peter 1 es (1) no (-1) no (-1) es (1) A 4 = k = 0 t ( k ) Smple s msclssfed f k < 0 grdent descent sngle smple rule: Set fxed lernng rte to η (k) = 1: ( k + 1) ( k ) ( k = + η ) ( k +1) ( k ) = + M Y M
42 LDF: Grdent decent Exmple set equl ntl weghts (1) =[0.25, 0.25, 0.25, 0.25] vst ll smples sequentll, modfng the weghts for fter fndng msclssfed exmple nme Jne Steve t 0.25*1+0.25*1+0.25*1+0.25*(-1)+0.25*(-1) >0 0.25*(-1)+0.25*(-1)+0.25*(-1)+0.25*(-1)+0.25*(-1)<0 msclssfed? no es new weghts ( 2) ( 1) = + = [ ]+ M [ ] = [ ] + 1 =
43 LDF: Grdent decent Exmple ( 2) = [ ] nme Mr t -0.75*(-1)-0.75* * *1-0.75*(-1) <0 msclssfed? es new weghts ( 3) ( 2) = + = [ ]+ M = [ ] = + 1 [ ]
44 LDF: Grdent decent Exmple ( 3) = [ ] nme Peter t * * * (-1) *(-1)-1.75*1 <0 msclssfed? es new weghts ( 4) ( 3) = + = [ ]+ M [ ] = + 1 = [ ]
45 LDF: Grdent decent Exmple ( 4) = [ ] nme Jne Steve Mr Peter t * *1-0.75* *(-1) *(-1) *(-1)+1.25*(-1) -0.75*(-1) -0.75*(-1)-0.75*(-1)> *(-1)+1.25*1-0.75* *1 0.75*(-1) > * *1-0.75* (-1)-0.75* (-1) *1 >0 msclssfed? no no no no Thus the dscrmnnt functon s ( 0) ( 1) ( 2) ( 3) ( 4) g( ) = 0.75 * * 0.75 * 0.75 * 0.75 * Convertng bck to the orgnl fetures x: ( 1) ( 2) ( 3) ( 4) g( x ) = 1.25 * x 0.75 * x 0.75 * x 0.75 * x 0.75
46 LDF: Grdent decent Exmple Convertng bck to the orgnl fetures x: 1.25 * x 1.25 * x ( 1) ( 2) ( 3) ( 4) 0.75 * x 0.75 * x 0.75 * x > ( 1) ( 2) ( 3) ( 4) 0.75 * x 0.75 * x 0.75 * x < grde grde A F good ttendnce tll sleeps n clss chews gum Ths s just one possble soluton vector If we strted wth weghts (1) =[0,0.5, 0.5, 0, 0], soluton would be [-1,1.5, -0.5, -1, -1] 1.5 * 1.5 * x x ( 1) ( 2 ) ( 3 ) ( 4 ) 0.5 * x x x ( 1) ( 2 ) ( 3 ) ( 4 ) 0.5 * x x x > 1 grde A < 1 grde F In ths soluton, beng tll s the lest mportnt feture
47 LDF: Nonseprble Exmple Suppose we hve 2 fetures nd smples re: Clss 1: [2,1], [4,3], [3,5] Clss 2: [1,3] nd [5,6] These smples re not seprble b lne Stll would lke to get pproxmte seprton b lne, good choce s shown n green some smples m be nos, nd t s ok f the re on the wrong sde of the lne Get 1, 2, 3, 4 b ddng extr feture nd normlzng 1 = = = = = 5 1 6
48 LDF: Nonseprble Exmple Let s ppl Perceptron sngle smple lgorthm ntl equl weghts ( 1 ) = [ 1 1 1] ths s lne x (1) +x (2) +1=0 fxed lernng rte η = 1 ( k +1) ( k ) = + M (1) 1 = = = = = t 1 (1) = [1 1 1]*[1 2 1] t > 0 t 2 (1) = [1 1 1]*[1 4 3] t > 0 t 3 (1) = [1 1 1]*[1 3 5] t > 0
49 LDF: Nonseprble Exmple ( ) [ 1 1 1] 1 = ( k +1) ( k ) = + M 1 = = = = = t 4 (1) =[1 1 1]*[ ] t = -5< 0 (1) (2) ( 2 ) ( 1) = + = [ 1 1 1] + [ 1 1 3] = [ 0 0 2] M t 5 (2) =[0 0-2]*[ ] t = 12 > 0 t 1 (2) =[0 0-2]*[1 2 1] t < 0 ( 3 ) ( 2 = ) + = [ ] + [ ] = [ ] M
50 LDF: Nonseprble Exmple ( ) = [ 1 2 1] 3 ( k +1) ( k ) = + M 1 = = = = = (3) t 2 (3) =[1 4 3]*[1 2-1] t =6 > 0 t 3 (3) =[1 3 5]*[1 2-1] t > 0 t 4 (3) =[ ]*[1 2-1] t = 0 ( 4 ) ( 3 ) = + = [ 1 2 1] + [ 1 1 3] = [ 0 1 4] M (2)
51 LDF: Nonseprble Exmple ( ) = [ 0 1 4] 4 ( k +1) ( k ) = + M 1 = = = = = (3) t 2 (3) =[1 4 3]*[1 2-1] t =6 > 0 t 3 (3) =[1 3 5]*[1 2-1] t > 0 t 4 (3) =[ ]*[1 2-1] t = 0 ( 4 ) ( 3 ) = + = [ 1 2 1] + [ 1 1 3] = [ 0 1 4] M (4)
52 LDF: Nonseprble Exmple we cn contnue ths forever there s no soluton vector stsfng for ll t 5 = k = 0 k ( k ) > 0 need to stop but t good pont: solutons t tertons 900 through 915. Some re good some re not. How do we stop t good soluton?
53 LDF: Convergence of Perceptron rules If clsses re lnerl seprble, nd use fxed lernng rte, tht s for some constnt c, η (k) =c both sngle smple nd btch perceptron rules converge to correct soluton (could be n n the soluton spce) If clsses re not lnerl seprble: lgorthm does not stop, t keeps lookng for soluton whch does not exst b choosng pproprte lernng rte, cn lws ensure ( k ) convergence: η 0 s k ( 1) ( k ) η for exmple nverse lner lernng rte: η = k for nverse lner lernng rte convergence n the lnerl seprble cse cn lso be proven no gurntee tht we stopped t good pont, but there re good resons to choose nverse lner lernng rte
54 LDF: Perceptron Rule nd Grdent decent Lnerl seprble dt perceptron rule wth grdent decent works well Lnerl non-seprble dt need to stop perceptron rule lgorthm t good pont, ths mbe trck Btch Rule Smoother grdent becuse ll smples re used Sngle Smple Rule eser to nlze Concentrtes more thn necessr on n solted nos trnng exmples
Machine Learning Support Vector Machines SVM
Mchne Lernng Support Vector Mchnes SVM Lesson 6 Dt Clssfcton problem rnng set:, D,,, : nput dt smple {,, K}: clss or lbel of nput rget: Construct functon f : X Y f, D Predcton of clss for n unknon nput
More informationAn Introduction to Support Vector Machines
An Introducton to Support Vector Mchnes Wht s good Decson Boundry? Consder two-clss, lnerly seprble clssfcton problem Clss How to fnd the lne (or hyperplne n n-dmensons, n>)? Any de? Clss Per Lug Mrtell
More information18.7 Artificial Neural Networks
310 18.7 Artfcl Neurl Networks Neuroscence hs hypotheszed tht mentl ctvty conssts prmrly of electrochemcl ctvty n networks of brn cells clled neurons Ths led McCulloch nd Ptts to devse ther mthemtcl model
More information4. Eccentric axial loading, cross-section core
. Eccentrc xl lodng, cross-secton core Introducton We re strtng to consder more generl cse when the xl force nd bxl bendng ct smultneousl n the cross-secton of the br. B vrtue of Snt-Vennt s prncple we
More informationChapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter.4 Newton-Rphson Method of Solvng Nonlner Equton After redng ths chpter, you should be ble to:. derve the Newton-Rphson method formul,. develop the lgorthm of the Newton-Rphson method,. use the Newton-Rphson
More informationReview of linear algebra. Nuno Vasconcelos UCSD
Revew of lner lgebr Nuno Vsconcelos UCSD Vector spces Defnton: vector spce s set H where ddton nd sclr multplcton re defned nd stsf: ) +( + ) (+ )+ 5) λ H 2) + + H 6) 3) H, + 7) λ(λ ) (λλ ) 4) H, - + 8)
More informationRemember: Project Proposals are due April 11.
Bonformtcs ecture Notes Announcements Remember: Project Proposls re due Aprl. Clss 22 Aprl 4, 2002 A. Hdden Mrov Models. Defntons Emple - Consder the emple we tled bout n clss lst tme wth the cons. However,
More informationESCI 342 Atmospheric Dynamics I Lesson 1 Vectors and Vector Calculus
ESI 34 tmospherc Dnmcs I Lesson 1 Vectors nd Vector lculus Reference: Schum s Outlne Seres: Mthemtcl Hndbook of Formuls nd Tbles Suggested Redng: Mrtn Secton 1 OORDINTE SYSTEMS n orthonorml coordnte sstem
More informationDennis Bricker, 2001 Dept of Industrial Engineering The University of Iowa. MDP: Taxi page 1
Denns Brcker, 2001 Dept of Industrl Engneerng The Unversty of Iow MDP: Tx pge 1 A tx serves three djcent towns: A, B, nd C. Ech tme the tx dschrges pssenger, the drver must choose from three possble ctons:
More informationChapter Newton-Raphson Method of Solving a Nonlinear Equation
Chpter 0.04 Newton-Rphson Method o Solvng Nonlner Equton Ater redng ths chpter, you should be ble to:. derve the Newton-Rphson method ormul,. develop the lgorthm o the Newton-Rphson method,. use the Newton-Rphson
More informationUNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS. M.Sc. in Economics MICROECONOMIC THEORY I. Problem Set II
Mcroeconomc Theory I UNIVERSITY OF IOANNINA DEPARTMENT OF ECONOMICS MSc n Economcs MICROECONOMIC THEORY I Techng: A Lptns (Note: The number of ndctes exercse s dffculty level) ()True or flse? If V( y )
More informationMachine Learning. Support Vector Machines. Le Song. CSE6740/CS7641/ISYE6740, Fall Lecture 8, Sept. 13, 2012 Based on slides from Eric Xing, CMU
Mchne Lernng CSE6740/CS764/ISYE6740 Fll 0 Support Vector Mchnes Le Song Lecture 8 Sept. 3 0 Bsed on sldes fro Erc Xng CMU Redng: Chp. 6&7 C.B ook Outlne Mu rgn clssfcton Constrned optzton Lgrngn dult Kernel
More informationRank One Update And the Google Matrix by Al Bernstein Signal Science, LLC
Introducton Rnk One Updte And the Google Mtrx y Al Bernsten Sgnl Scence, LLC www.sgnlscence.net here re two dfferent wys to perform mtrx multplctons. he frst uses dot product formulton nd the second uses
More informationFall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. with respect to λ. 1. χ λ χ λ ( ) λ, and thus:
More on χ nd errors : uppose tht we re fttng for sngle -prmeter, mnmzng: If we epnd The vlue χ ( ( ( ; ( wth respect to. χ n Tlor seres n the vcnt of ts mnmum vlue χ ( mn χ χ χ χ + + + mn mnmzes χ, nd
More informationPrinciple Component Analysis
Prncple Component Anlyss Jng Go SUNY Bufflo Why Dmensonlty Reducton? We hve too mny dmensons o reson bout or obtn nsghts from o vsulze oo much nose n the dt Need to reduce them to smller set of fctors
More informationSVMs for regression Multilayer neural networks
Lecture SVMs for regresson Muter neur netors Mos Husrecht mos@cs.ptt.edu 539 Sennott Squre Support vector mchne SVM SVM mmze the mrgn round the seprtng hperpne. he decson functon s fu specfed suset of
More informationSVMs for regression Non-parametric/instance based classification method
S 75 Mchne ernng ecture Mos Huskrecht mos@cs.ptt.edu 539 Sennott Squre SVMs for regresson Non-prmetrc/nstnce sed cssfcton method S 75 Mchne ernng Soft-mrgn SVM Aos some fet on crossng the seprtng hperpne
More informationCISE 301: Numerical Methods Lecture 5, Topic 4 Least Squares, Curve Fitting
CISE 3: umercl Methods Lecture 5 Topc 4 Lest Squres Curve Fttng Dr. Amr Khouh Term Red Chpter 7 of the tetoo c Khouh CISE3_Topc4_Lest Squre Motvton Gven set of epermentl dt 3 5. 5.9 6.3 The reltonshp etween
More informationDCDM BUSINESS SCHOOL NUMERICAL METHODS (COS 233-8) Solutions to Assignment 3. x f(x)
DCDM BUSINESS SCHOOL NUMEICAL METHODS (COS -8) Solutons to Assgnment Queston Consder the followng dt: 5 f() 8 7 5 () Set up dfference tble through fourth dfferences. (b) Wht s the mnmum degree tht n nterpoltng
More informationQuiz: Experimental Physics Lab-I
Mxmum Mrks: 18 Totl tme llowed: 35 mn Quz: Expermentl Physcs Lb-I Nme: Roll no: Attempt ll questons. 1. In n experment, bll of mss 100 g s dropped from heght of 65 cm nto the snd contner, the mpct s clled
More information6 Roots of Equations: Open Methods
HK Km Slghtly modfed 3//9, /8/6 Frstly wrtten t Mrch 5 6 Roots of Equtons: Open Methods Smple Fed-Pont Iterton Newton-Rphson Secnt Methods MATLAB Functon: fzero Polynomls Cse Study: Ppe Frcton Brcketng
More informationPartially Observable Systems. 1 Partially Observable Markov Decision Process (POMDP) Formalism
CS294-40 Lernng for Rootcs nd Control Lecture 10-9/30/2008 Lecturer: Peter Aeel Prtlly Oservle Systems Scre: Dvd Nchum Lecture outlne POMDP formlsm Pont-sed vlue terton Glol methods: polytree, enumerton,
More informationLecture 4: Piecewise Cubic Interpolation
Lecture notes on Vrtonl nd Approxmte Methods n Appled Mthemtcs - A Perce UBC Lecture 4: Pecewse Cubc Interpolton Compled 6 August 7 In ths lecture we consder pecewse cubc nterpolton n whch cubc polynoml
More informationAbhilasha Classes Class- XII Date: SOLUTION (Chap - 9,10,12) MM 50 Mob no
hlsh Clsses Clss- XII Dte: 0- - SOLUTION Chp - 9,0, MM 50 Mo no-996 If nd re poston vets of nd B respetvel, fnd the poston vet of pont C n B produed suh tht C B vet r C B = where = hs length nd dreton
More informationDefinition of Tracking
Trckng Defnton of Trckng Trckng: Generte some conclusons bout the moton of the scene, objects, or the cmer, gven sequence of mges. Knowng ths moton, predct where thngs re gong to project n the net mge,
More informationPhysics 121 Sample Common Exam 2 Rev2 NOTE: ANSWERS ARE ON PAGE 7. Instructions:
Physcs 121 Smple Common Exm 2 Rev2 NOTE: ANSWERS ARE ON PAGE 7 Nme (Prnt): 4 Dgt ID: Secton: Instructons: Answer ll 27 multple choce questons. You my need to do some clculton. Answer ech queston on the
More informationLinear and Nonlinear Optimization
Lner nd Nonlner Optmzton Ynyu Ye Deprtment of Mngement Scence nd Engneerng Stnford Unversty Stnford, CA 9430, U.S.A. http://www.stnford.edu/~yyye http://www.stnford.edu/clss/msnde/ Ynyu Ye, Stnford, MS&E
More informationGAUSS ELIMINATION. Consider the following system of algebraic linear equations
Numercl Anlyss for Engneers Germn Jordnn Unversty GAUSS ELIMINATION Consder the followng system of lgebrc lner equtons To solve the bove system usng clsscl methods, equton () s subtrcted from equton ()
More informationModel Fitting and Robust Regression Methods
Dertment o Comuter Engneerng Unverst o Clorn t Snt Cruz Model Fttng nd Robust Regresson Methods CMPE 64: Imge Anlss nd Comuter Vson H o Fttng lnes nd ellses to mge dt Dertment o Comuter Engneerng Unverst
More informationVariable time amplitude amplification and quantum algorithms for linear algebra. Andris Ambainis University of Latvia
Vrble tme mpltude mplfcton nd quntum lgorthms for lner lgebr Andrs Ambns Unversty of Ltv Tlk outlne. ew verson of mpltude mplfcton;. Quntum lgorthm for testng f A s sngulr; 3. Quntum lgorthm for solvng
More informationJens Siebel (University of Applied Sciences Kaiserslautern) An Interactive Introduction to Complex Numbers
Jens Sebel (Unversty of Appled Scences Kserslutern) An Interctve Introducton to Complex Numbers 1. Introducton We know tht some polynoml equtons do not hve ny solutons on R/. Exmple 1.1: Solve x + 1= for
More informationINTERPOLATION(1) ELM1222 Numerical Analysis. ELM1222 Numerical Analysis Dr Muharrem Mercimek
ELM Numercl Anlss Dr Muhrrem Mercmek INTEPOLATION ELM Numercl Anlss Some of the contents re dopted from Lurene V. Fusett, Appled Numercl Anlss usng MATLAB. Prentce Hll Inc., 999 ELM Numercl Anlss Dr Muhrrem
More informationInternational Journal of Pure and Applied Sciences and Technology
Int. J. Pure Appl. Sc. Technol., () (), pp. 44-49 Interntonl Journl of Pure nd Appled Scences nd Technolog ISSN 9-67 Avlle onlne t www.jopst.n Reserch Pper Numercl Soluton for Non-Lner Fredholm Integrl
More informationLeast squares. Václav Hlaváč. Czech Technical University in Prague
Lest squres Václv Hlváč Czech echncl Unversty n Prgue hlvc@fel.cvut.cz http://cmp.felk.cvut.cz/~hlvc Courtesy: Fred Pghn nd J.P. Lews, SIGGRAPH 2007 Course; Outlne 2 Lner regresson Geometry of lest-squres
More informationApplied Statistics Qualifier Examination
Appled Sttstcs Qulfer Exmnton Qul_june_8 Fll 8 Instructons: () The exmnton contns 4 Questons. You re to nswer 3 out of 4 of them. () You my use ny books nd clss notes tht you mght fnd helpful n solvng
More informationMultiple view geometry
EECS 442 Computer vson Multple vew geometry Perspectve Structure from Moton - Perspectve structure from moton prolem - mgutes - lgerc methods - Fctorzton methods - Bundle djustment - Self-clrton Redng:
More informationKatholieke Universiteit Leuven Department of Computer Science
Updte Rules for Weghted Non-negtve FH*G Fctorzton Peter Peers Phlp Dutré Report CW 440, Aprl 006 Ktholeke Unverstet Leuven Deprtment of Computer Scence Celestjnenln 00A B-3001 Heverlee (Belgum) Updte Rules
More informationIn this Chapter. Chap. 3 Markov chains and hidden Markov models. Probabilistic Models. Example: CpG Islands
In ths Chpter Chp. 3 Mrov chns nd hdden Mrov models Bontellgence bortory School of Computer Sc. & Eng. Seoul Ntonl Unversty Seoul 5-74, Kore The probblstc model for sequence nlyss HMM (hdden Mrov model)
More informationPattern Classification
Pattern Classfcaton All materals n these sldes ere taken from Pattern Classfcaton (nd ed) by R. O. Duda, P. E. Hart and D. G. Stork, John Wley & Sons, 000 th the permsson of the authors and the publsher
More informationLecture Notes on Linear Regression
Lecture Notes on Lnear Regresson Feng L fl@sdueducn Shandong Unversty, Chna Lnear Regresson Problem In regresson problem, we am at predct a contnuous target value gven an nput feature vector We assume
More informationLOCAL FRACTIONAL LAPLACE SERIES EXPANSION METHOD FOR DIFFUSION EQUATION ARISING IN FRACTAL HEAT TRANSFER
Yn, S.-P.: Locl Frctonl Lplce Seres Expnson Method for Dffuson THERMAL SCIENCE, Yer 25, Vol. 9, Suppl., pp. S3-S35 S3 LOCAL FRACTIONAL LAPLACE SERIES EXPANSION METHOD FOR DIFFUSION EQUATION ARISING IN
More informationLecture 20: Hypothesis testing
Lecture : Hpothess testng Much of statstcs nvolves hpothess testng compare a new nterestng hpothess, H (the Alternatve hpothess to the borng, old, well-known case, H (the Null Hpothess or, decde whether
More informationLecture 36. Finite Element Methods
CE 60: Numercl Methods Lecture 36 Fnte Element Methods Course Coordntor: Dr. Suresh A. Krth, Assocte Professor, Deprtment of Cvl Engneerng, IIT Guwht. In the lst clss, we dscussed on the ppromte methods
More information2E Pattern Recognition Solutions to Introduction to Pattern Recognition, Chapter 2: Bayesian pattern classification
E395 - Pattern Recognton Solutons to Introducton to Pattern Recognton, Chapter : Bayesan pattern classfcaton Preface Ths document s a soluton manual for selected exercses from Introducton to Pattern Recognton
More informationLinear Regression & Least Squares!
Lner Regresson & Lest Squres Al Borj UWM CS 790 Slde credt: Aykut Erdem Ths&week Lner®resson&prolem&& ' con0nuous&outputs& ' smple&model Introduce&key&concepts:&& ' loss&func0ons& ' generlz0on& ' op0mz0on&
More informationSupport vector machines for regression
S 75 Mchne ernng ecture 5 Support vector mchnes for regresson Mos Huskrecht mos@cs.ptt.edu 539 Sennott Squre S 75 Mchne ernng he decson oundr: ˆ he decson: Support vector mchnes ˆ α SV ˆ sgn αˆ SV!!: Decson
More informationChemical Reaction Engineering
Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F E F E + Q! 0
More informationFeature Selection: Part 1
CSE 546: Machne Learnng Lecture 5 Feature Selecton: Part 1 Instructor: Sham Kakade 1 Regresson n the hgh dmensonal settng How do we learn when the number of features d s greater than the sample sze n?
More informationChemical Reaction Engineering
Lecture 20 hemcl Recton Engneerng (RE) s the feld tht studes the rtes nd mechnsms of chemcl rectons nd the desgn of the rectors n whch they tke plce. Lst Lecture Energy Blnce Fundmentls F 0 E 0 F E Q W
More informationLogarithms. Logarithm is another word for an index or power. POWER. 2 is the power to which the base 10 must be raised to give 100.
Logrithms. Logrithm is nother word for n inde or power. THIS IS A POWER STATEMENT BASE POWER FOR EXAMPLE : We lred know tht; = NUMBER 10² = 100 This is the POWER Sttement OR 2 is the power to which the
More informationThe Number of Rows which Equal Certain Row
Interntonl Journl of Algebr, Vol 5, 011, no 30, 1481-1488 he Number of Rows whch Equl Certn Row Ahmd Hbl Deprtment of mthemtcs Fcult of Scences Dmscus unverst Dmscus, Sr hblhmd1@gmlcom Abstrct Let be X
More informationTopic 1 Notes Jeremy Orloff
Topic 1 Notes Jerem Orloff 1 Introduction to differentil equtions 1.1 Gols 1. Know the definition of differentil eqution. 2. Know our first nd second most importnt equtions nd their solutions. 3. Be ble
More informationLectures - Week 4 Matrix norms, Conditioning, Vector Spaces, Linear Independence, Spanning sets and Basis, Null space and Range of a Matrix
Lectures - Week 4 Matrx norms, Condtonng, Vector Spaces, Lnear Independence, Spannng sets and Bass, Null space and Range of a Matrx Matrx Norms Now we turn to assocatng a number to each matrx. We could
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationINTRODUCTION TO COMPLEX NUMBERS
INTRODUCTION TO COMPLEX NUMBERS The numers -4, -3, -, -1, 0, 1,, 3, 4 represent the negtve nd postve rel numers termed ntegers. As one frst lerns n mddle school they cn e thought of s unt dstnce spced
More informationCENTROID (AĞIRLIK MERKEZİ )
CENTOD (ĞLK MEKEZİ ) centrod s geometrcl concept rsng from prllel forces. Tus, onl prllel forces possess centrod. Centrod s tougt of s te pont were te wole wegt of pscl od or sstem of prtcles s lumped.
More informationCOMPLEX NUMBERS AND QUADRATIC EQUATIONS
COMPLEX NUMBERS AND QUADRATIC EQUATIONS INTRODUCTION We know that x 0 for all x R e the square of a real number (whether postve, negatve or ero) s non-negatve Hence the equatons x, x, x + 7 0 etc are not
More information6.6 The Marquardt Algorithm
6.6 The Mqudt Algothm lmttons of the gdent nd Tylo expnson methods ecstng the Tylo expnson n tems of ch-sque devtves ecstng the gdent sech nto n tetve mtx fomlsm Mqudt's lgothm utomtclly combnes the gdent
More informationGeneralized Linear Methods
Generalzed Lnear Methods 1 Introducton In the Ensemble Methods the general dea s that usng a combnaton of several weak learner one could make a better learner. More formally, assume that we have a set
More informationMultilayer Perceptron (MLP)
Multlayer Perceptron (MLP) Seungjn Cho Department of Computer Scence and Engneerng Pohang Unversty of Scence and Technology 77 Cheongam-ro, Nam-gu, Pohang 37673, Korea seungjn@postech.ac.kr 1 / 20 Outlne
More informationKernel Methods and SVMs Extension
Kernel Methods and SVMs Extenson The purpose of ths document s to revew materal covered n Machne Learnng 1 Supervsed Learnng regardng support vector machnes (SVMs). Ths document also provdes a general
More informationThe Schur-Cohn Algorithm
Modelng, Estmton nd Otml Flterng n Sgnl Processng Mohmed Njm Coyrght 8, ISTE Ltd. Aendx F The Schur-Cohn Algorthm In ths endx, our m s to resent the Schur-Cohn lgorthm [] whch s often used s crteron for
More informationORDINARY DIFFERENTIAL EQUATIONS
6 ORDINARY DIFFERENTIAL EQUATIONS Introducton Runge-Kutt Metods Mult-step Metods Sstem o Equtons Boundr Vlue Problems Crcterstc Vlue Problems Cpter 6 Ordnr Derentl Equtons / 6. Introducton In mn engneerng
More informationTorsion, Thermal Effects and Indeterminacy
ENDS Note Set 7 F007bn orson, herml Effects nd Indetermncy Deformton n orsonlly Loded Members Ax-symmetrc cross sectons subjected to xl moment or torque wll remn plne nd undstorted. At secton, nternl torque
More information7.2 Volume. A cross section is the shape we get when cutting straight through an object.
7. Volume Let s revew the volume of smple sold, cylnder frst. Cylnder s volume=se re heght. As llustrted n Fgure (). Fgure ( nd (c) re specl cylnders. Fgure () s rght crculr cylnder. Fgure (c) s ox. A
More informationLecture 5 Single factor design and analysis
Lectue 5 Sngle fcto desgn nd nlss Completel ndomzed desgn (CRD Completel ndomzed desgn In the desgn of expements, completel ndomzed desgns e fo studng the effects of one pm fcto wthout the need to tke
More informationMath 426: Probability Final Exam Practice
Mth 46: Probbility Finl Exm Prctice. Computtionl problems 4. Let T k (n) denote the number of prtitions of the set {,..., n} into k nonempty subsets, where k n. Argue tht T k (n) kt k (n ) + T k (n ) by
More informationAnnouncements. Image Formation: Outline. The course. Image Formation and Cameras (cont.)
nnouncements Imge Formton nd Cmers (cont.) ssgnment : Cmer & Lenses, gd Trnsformtons, nd Homogrph wll be posted lter tod. CSE 5 Lecture 5 CS5, Fll CS5, Fll CS5, Fll The course rt : The phscs of mgng rt
More information8. INVERSE Z-TRANSFORM
8. INVERSE Z-TRANSFORM The proce by whch Z-trnform of tme ere, nmely X(), returned to the tme domn clled the nvere Z-trnform. The nvere Z-trnform defned by: Computer tudy Z X M-fle trn.m ued to fnd nvere
More informationModule 2. Random Processes. Version 2 ECE IIT, Kharagpur
Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be
More informationIntroduction to Numerical Integration Part II
Introducton to umercl Integrton Prt II CS 75/Mth 75 Brn T. Smth, UM, CS Dept. Sprng, 998 4/9/998 qud_ Intro to Gussn Qudrture s eore, the generl tretment chnges the ntegrton prolem to ndng the ntegrl w
More informationCS 3710: Visual Recognition Classification and Detection. Adriana Kovashka Department of Computer Science January 13, 2015
CS 3710: Vsual Recognton Classfcaton and Detecton Adrana Kovashka Department of Computer Scence January 13, 2015 Plan for Today Vsual recognton bascs part 2: Classfcaton and detecton Adrana s research
More informationCOMPLEX NUMBER & QUADRATIC EQUATION
MCQ COMPLEX NUMBER & QUADRATIC EQUATION Syllus : Comple numers s ordered prs of rels, Representton of comple numers n the form + nd ther representton n plne, Argnd dgrm, lger of comple numers, modulus
More informationPolynomial Approximations for the Natural Logarithm and Arctangent Functions. Math 230
Polynomil Approimtions for the Nturl Logrithm nd Arctngent Functions Mth 23 You recll from first semester clculus how one cn use the derivtive to find n eqution for the tngent line to function t given
More informationPyramid Algorithms for Barycentric Rational Interpolation
Pyrmd Algorthms for Brycentrc Rtonl Interpolton K Hormnn Scott Schefer Astrct We present new perspectve on the Floter Hormnn nterpolnt. Ths nterpolnt s rtonl of degree (n, d), reproduces polynomls of degree
More informationDemand. Demand and Comparative Statics. Graphically. Marshallian Demand. ECON 370: Microeconomic Theory Summer 2004 Rice University Stanley Gilbert
Demnd Demnd nd Comrtve Sttcs ECON 370: Mcroeconomc Theory Summer 004 Rce Unversty Stnley Glbert Usng the tools we hve develoed u to ths ont, we cn now determne demnd for n ndvdul consumer We seek demnd
More informationElectrochemical Thermodynamics. Interfaces and Energy Conversion
CHE465/865, 2006-3, Lecture 6, 18 th Sep., 2006 Electrochemcl Thermodynmcs Interfces nd Energy Converson Where does the energy contrbuton F zϕ dn come from? Frst lw of thermodynmcs (conservton of energy):
More informationUnit 5: Quadratic Equations & Functions
Date Perod Unt 5: Quadratc Equatons & Functons DAY TOPIC 1 Modelng Data wth Quadratc Functons Factorng Quadratc Epressons 3 Solvng Quadratc Equatons 4 Comple Numbers Smplfcaton, Addton/Subtracton & Multplcaton
More informationHaddow s Experiment:
schemtc drwng of Hddow's expermentl set-up movng pston non-contctng moton sensor bems of sprng steel poston vres to djust frequences blocks of sold steel shker Hddow s Experment: terr frm Theoretcl nd
More informationU.C. Berkeley CS294: Beyond Worst-Case Analysis Luca Trevisan September 5, 2017
U.C. Berkeley CS94: Beyond Worst-Case Analyss Handout 4s Luca Trevsan September 5, 07 Summary of Lecture 4 In whch we ntroduce semdefnte programmng and apply t to Max Cut. Semdefnte Programmng Recall that
More informationME 501A Seminar in Engineering Analysis Page 1
More oundr-vlue Prolems nd genvlue Prolems n Os ovemer 9, 7 More oundr-vlue Prolems nd genvlue Prolems n Os Lrr retto Menl ngneerng 5 Semnr n ngneerng nlss ovemer 9, 7 Outlne Revew oundr-vlue prolems Soot
More informationW. We shall do so one by one, starting with I 1, and we shall do it greedily, trying
Vitli covers 1 Definition. A Vitli cover of set E R is set V of closed intervls with positive length so tht, for every δ > 0 nd every x E, there is some I V with λ(i ) < δ nd x I. 2 Lemm (Vitli covering)
More informationVectors and Tensors. R. Shankar Subramanian. R. Aris, Vectors, Tensors, and the Equations of Fluid Mechanics, Prentice Hall (1962).
005 Vectors nd Tensors R. Shnkr Subrmnn Good Sources R. rs, Vectors, Tensors, nd the Equtons of Flud Mechncs, Prentce Hll (96). nd ppendces n () R. B. Brd, W. E. Stewrt, nd E. N. Lghtfoot, Trnsport Phenomen,
More informationLecture 1. Functional series. Pointwise and uniform convergence.
1 Introduction. Lecture 1. Functionl series. Pointwise nd uniform convergence. In this course we study mongst other things Fourier series. The Fourier series for periodic function f(x) with period 2π is
More informationThe Regulated and Riemann Integrals
Chpter 1 The Regulted nd Riemnn Integrls 1.1 Introduction We will consider severl different pproches to defining the definite integrl f(x) dx of function f(x). These definitions will ll ssign the sme vlue
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationSummary: Method of Separation of Variables
Physics 246 Electricity nd Mgnetism I, Fll 26, Lecture 22 1 Summry: Method of Seprtion of Vribles 1. Seprtion of Vribles in Crtesin Coordintes 2. Fourier Series Suggested Reding: Griffiths: Chpter 3, Section
More informationTwo Coefficients of the Dyson Product
Two Coeffcents of the Dyson Product rxv:07.460v mth.co 7 Nov 007 Lun Lv, Guoce Xn, nd Yue Zhou 3,,3 Center for Combntorcs, LPMC TJKLC Nnk Unversty, Tnjn 30007, P.R. Chn lvlun@cfc.nnk.edu.cn gn@nnk.edu.cn
More informationCourse Review Introduction to Computer Methods
Course Revew Wht you hopefully hve lerned:. How to nvgte nsde MIT computer system: Athen, UNIX, emcs etc. (GCR). Generl des bout progrmmng (GCR): formultng the problem, codng n Englsh trnslton nto computer
More informationSubstitution Matrices and Alignment Statistics. Substitution Matrices
Susttuton Mtrces nd Algnment Sttstcs BMI/CS 776 www.ostt.wsc.edu/~crven/776.html Mrk Crven crven@ostt.wsc.edu Ferur 2002 Susttuton Mtrces two oulr sets of mtrces for roten seuences PAM mtrces [Dhoff et
More informationAPPENDIX A Some Linear Algebra
APPENDIX A Some Lnear Algebra The collecton of m, n matrces A.1 Matrces a 1,1,..., a 1,n A = a m,1,..., a m,n wth real elements a,j s denoted by R m,n. If n = 1 then A s called a column vector. Smlarly,
More informationMath 497C Sep 17, Curves and Surfaces Fall 2004, PSU
Mth 497C Sep 17, 004 1 Curves nd Surfces Fll 004, PSU Lecture Notes 3 1.8 The generl defnton of curvture; Fox-Mlnor s Theorem Let α: [, b] R n be curve nd P = {t 0,...,t n } be prtton of [, b], then the
More informationEngineering Tensors. Friday November 16, h30 -Muddy Charles. A BEH430 review session by Thomas Gervais.
ngneerng Tensors References: BH4 reew sesson b Thoms Gers tgers@mt.ed Long, RR, Mechncs of Solds nd lds, Prentce-Hll, 96, pp - Deen, WD, nlss of trnsport phenomen, Oford, 998, p. 55-56 Goodbod, M, Crtesn
More informationSingular Value Decomposition: Theory and Applications
Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real
More information5.1 Estimating with Finite Sums Calculus
5.1 ESTIMATING WITH FINITE SUMS Emple: Suppose from the nd to 4 th hour of our rod trip, ou trvel with the cruise control set to ectl 70 miles per hour for tht two hour stretch. How fr hve ou trveled during
More informationMaximal Margin Classifier
CS81B/Stat41B: Advanced Topcs n Learnng & Decson Makng Mamal Margn Classfer Lecturer: Mchael Jordan Scrbes: Jana van Greunen Corrected verson - /1/004 1 References/Recommended Readng 1.1 Webstes www.kernel-machnes.org
More informationMath1110 (Spring 2009) Prelim 3 - Solutions
Math 1110 (Sprng 2009) Solutons to Prelm 3 (04/21/2009) 1 Queston 1. (16 ponts) Short answer. Math1110 (Sprng 2009) Prelm 3 - Solutons x a 1 (a) (4 ponts) Please evaluate lm, where a and b are postve numbers.
More informationProblem Set 9 Solutions
Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem
More informationLECTURE NOTE #12 PROF. ALAN YUILLE
LECTURE NOTE #12 PROF. ALAN YUILLE 1. Clustering, K-mens, nd EM Tsk: set of unlbeled dt D = {x 1,..., x n } Decompose into clsses w 1,..., w M where M is unknown. Lern clss models p(x w)) Discovery of
More information8.1 Arc Length. What is the length of a curve? How can we approximate it? We could do it following the pattern we ve used before
.1 Arc Length hat s the length of a curve? How can we approxmate t? e could do t followng the pattern we ve used before Use a sequence of ncreasngly short segments to approxmate the curve: As the segments
More information