Experimental Study on Classification
|
|
- Jacob Ramsey
- 5 years ago
- Views:
Transcription
1 Chapter 7. Expermental Study on Classfcaton 7.1 Characterzaton of Explosve Materals Atomc effect number and densty Theoretcally, most explosves fall wthn a relatvely narrow wndow n Z eff and n densty, and can be dstngushed from other organc and norganc materals (see Fgure 7.1-1). For a compound materal, the effectve atomc number Z eff s the characterstc value of a hypothetcal substance havng an equvalent x-ray attenuaton effect as the compound. We assume that a materal s a unform compound consstng of M elements, havng an atomc number Z and contrbutng mass m. Z eff can be estmated wth followng formula [JOH83] [EIL96], 1/ 3.5 M M 3.5 Z eff = a Z a (7.1) = 1 = 1 where a = m Z / A, and A s the atomc weght for element. From (7.1), t s known that to estmate Z eff of a compound, we need to estmate how many elements are n the compound, what they are, and to measure or estmate the contrbutng mass of each element. It s almost mpossble to get ths nformaton by usng dual-energy transmsson and scatter x-ray
2 magng technques so far. Nor can the densty of a compound be measured drectly wth sngle vew dual-energy system. Z eff Inorganc materals Other organc materals Explosves Densty Fgure Characterzaton of common materals found n luggage Classfcaton feature space n the prototype scannng system As stated n Chapter 1, the overall goal of ths research s to develop an x-ray scannng system that uses dual-energy transmsson and scatterng to detect explosves. To realze ths goal, sx types of mages are avalable n the prototype scannng system. Rght now, four of them are processed to classfy materal types. They are, Transmsson of hgh energy: T H Transmsson of low energy: T L Chapter 7: Expermental Study on Classfcaton Page 163
3 Back scatter of low energy: B L Forward scatter of low energy: F L By combnng these sensors together, a feature vector n 2-dmensonal space ( R, L ) s formed wth Equatons 7.2 and 7.3. Any object scanned n the luggage bags wll be mapped onto ths 2-dmensonal plane. ψ R = (7.2) ψ j 2 2 ( a F + a F + b B + b B ) log 0 L 1 L 0 L 1 L L = (7.3) log( T ) L In these equatons, ψ and ψ j are the area attenuaton coeffcents, estmated by Algorthm 5.1, at low and hgh x-ray energes respectvely; a 0, a 1, b 0, and b 1 are coeffcents found n Equaton 6.12 or Equaton Actually, R s close to Z eff, and s farly good at dstngushng organc materals from norganc materals. L s related to densty, and s very effectve for separatng thn materals from thck materals. Fgure gves an example scanned from real luggage bags. In the fgure, sx explosve smulants provded by FAA are represented as +. They are smulants for hgh-densty ammona ntrate, low densty ammona ntrate, smokeless powder, black powder, semtex, and dynamte. The explosve smulants wll be descrbed further n Secton 7.3. Three step wedges, represented as o, are shown n three dfferent boxes. Other materals, such as clothes, shoes, books, plastcs, woods, toletry formulatons, chocolates, and so on, are also represented as o. We see that (R, L) s a pretty good feature space. Based on (R, L), statstcal decson rules have been developed to dstngush explosves from other materals. Chapter 7: Expermental Study on Classfcaton Page 164
4 Fgure Measurements on real luggage materals, step wedges and explosve smulants: + represents explosve smulants, o represents the other materals. Chapter 7: Expermental Study on Classfcaton Page 165
5 7.2 Test Objects Descrpton of test objects To verfy the capablty and to study the lmtatons of materal characterzaton method, experments were performed usng partcular test objects and actual passenger luggage bags, wth and wthout explosve smulants. Twenty unclamed luggage bags were purchased from a major arlne. Also, we obtaned some explosve smulants n two orders from FAA; they are plastc smulants, and explosve smulants. By nsertng dfferent objects nto these luggage bags, hundreds of expermental scenaros were created. The objects that were nserted nto bags nclude: explosve smulants, whte and clear plastc step wedges, an alumnum step wedge, a walkman player, shampoo bottles, bottles of honey, sugar, har dryers, pllows, clothes, towels, books, a steel step wedge, etc. A typcal bag s shown on Fgure 7.2-1, where (a) gves the outward appearance, and (b) shows some contents nsde. Our research efforts focused on detectng explosves n luggage. But usng real explosves as the testng objects poses serous safety concerns at a unversty. The SDA lab does not have the necessary means to obtan, transport, and store these hazardous materals. However, to perform meanngful experments some types of llct materals have to be used. Explosve smulants are beleved to be resonable substtutes for the real explosves. Explosve smulants are nert materals that exhbt accurately controlled physcal propertes. They specfcally and relably duplcate selected characterstcs, such as densty and Z eff, of real explosve materals, and those characterstcs are recognzed by usng explosve detecton technologes such as x-ray detecton technologes [EIL96] [SPA96]. Chapter 7: Expermental Study on Classfcaton Page 166
6 Rgd plastcs were also used n ths work. They have lmted usefulness n accurately smulatng explosve devces, tranng operators, and testng automated detecton systems that use x-rays. The plastc smulants n our study are shown n Fgure They have a smlar Z eff to the actual explosves, but not the same densty. Fgure shows a pcture of sx explosve smulants. They are all members of a new class of explosve smulants. These smulants are mxtures of two or more components. The nert propertes of each component have been verfed by analyses, and none of these components s regulated by the government. They closely match both the Z eff and densty of real explosves. Also snce they are all powders, ths makes them lke real explosves as well. All these smulants have demonstrated stablty of more than sx months, as long as they are approprately wrapped and stored. Descrpton of both explosve and plastc smulants s lsted n Table Table Lst of standard smulants used n the expermental study Smulant name RXN-08-AJ RXN-11-GE-AB RXN-07-AE RXN-04-AF RXN-06-AF RXN-10-AF Plastc #1 Plastc #2 Plastc #3 Plastc #4 Descrpton Smulant for smokeless powder Smulant for hgh-densty ammona ntrate Smulant for black powder Smulant for semtex Smulant for dynamte Smulant for low-densty ammona ntrate Sold square plastc object Sold square plastc object Sold square plastc object Bundle of three sold cylnder plastc objects Chapter 7: Expermental Study on Classfcaton Page 167
7 (a) (b) Fgure A typcal luggage bag used n these experments. (a) Outward appearance. (b) Contents of the bag. Chapter 7: Expermental Study on Classfcaton Page 168
8 Plastc #3 Plastc #4 Plastc #1 Plastc #2 Fgure Plastc smulants used n ths research. RXN-06-AF RXN-11-GE-AB RXN-08-AJ RXN-04-AF RXN-07-AE RXN-10-GE-AF Fgure Explosve smulants used n ths research. Chapter 7: Expermental Study on Classfcaton Page 169
9 7.2.2 Image examples collected usng the prototype scanner For each scenaro, mages were collected usng the prototype x-ray scannng system. As stated n Secton 7.1.2, these mages nclude a hgh-energy transmsson mage, a low-energy transmsson mage, a low-energy backward scatterng mage and a low-energy forward scatterng mage. Fgure shows the mages scanned for sx explosve smulants. From left to rght, they represent smokeless powder, hgh-densty ammona ntrate, black powder, semtex, dynamte, and low-densty ammona ntrate. From the fgure, we can observe that explosve smulants exhbt only a small dfference between low and hgh transmssons, and have a hgh scatterng. Ths characterzaton s what we have seen n the prevous secton, where they fall n a block havng hgher value of L n (R, L) space. Fgure gves the mages scanned for four plastc smulants. They are packaged n a typcal test bag from FAA. They have smlar characterstcs as the explosve smulants. The scatterng mages for three step wedges are gven n Fgures to Ther transmsson mages have been shown n Secton through Fgures to Step wedges are very useful to our expermental study n ths dssertaton based on the followng consderatons: 1) alumnum step wedges stand for a typcal object of norganc materal, whle whte and clear plastc step wedges are typcal objects of organc materal; 2) by consderng step wedges, we can verfy the thckness effects on both transmsson and scatter magng; 3) by usng step wedges, t s very convenent to evaluate new methods for mprovng object classfcaton (such as wth or wthout copper flter, the numercal method, and so on). Several typcal magng examples of luggage bags are gven n Fgures to Chapter 7: Expermental Study on Classfcaton Page 170
10 Fgure Images scanned for sx explosve smulants: (a) low-energy transmsson, (b) hgh-energy transmsson. Chapter 7: Expermental Study on Classfcaton Page 171
11 Fgure 7.2-4, contnued. (c) Low-energy backward scatterng, and (d) low-energy forward scatterng. Chapter 7: Expermental Study on Classfcaton Page 172
12 Plastc #2 Plastc #1 Plastc #3 Plastc #4 Fgure Images scanned for plastc smulants: (a) low-energy transmsson, (b) hghenergy transmsson. Chapter 7: Expermental Study on Classfcaton Page 173
13 Fgure 7.2-5, contnued. (c) Low-energy backward scatterng, and (d) low-energy forward scatterng. Chapter 7: Expermental Study on Classfcaton Page 174
14 Fgure Images scanned for whte plastc step wedge: (a) low-energy backward scatterng, and (b) low-energy forward scatterng. Chapter 7: Expermental Study on Classfcaton Page 175
15 Fgure Images scanned for clear plastc step wedge: (a) low-energy backward scatterng, and (d) low-energy forward scatterng. Chapter 7: Expermental Study on Classfcaton Page 176
16 Fgure Images scanned for alumnum step wedge: (a) low-energy backward scatterng, and (d) low-energy forward scatterng. Chapter 7: Expermental Study on Classfcaton Page 177
17 RXN-04-AF Steel Book Fgure Images scanned for a common luggage bag wth one explosve smulant: (a) low-energy transmsson, and (b) hgh-energy transmsson. Chapter 7: Expermental Study on Classfcaton Page 178
18 Fgure 7.2-9, contnued. (c) Low-energy backward scatterng, and (d) low-energy forward scatterng. Chapter 7: Expermental Study on Classfcaton Page 179
19 Alumnum step wedge Clear plastc wedge Fgure Images scanned for a common luggage bag nserted wth two step wedges: (a) low-energy transmsson, and (b) hgh-energy transmsson. Chapter 7: Expermental Study on Classfcaton Page 180
20 Fgure , contnued. (c) Low-energy backward scatterng, and (d) low-energy forward scatterng. Chapter 7: Expermental Study on Classfcaton Page 181
21 Wood board RXN-06-AF Fgure Images scanned for a common luggage bag wth one explosve smulant: (a) low-energy transmsson, and (b) hgh-energy transmsson. Chapter 7: Expermental Study on Classfcaton Page 182
22 Fgure , contnued. (c) low-energy backward scatterng, and (d) low-energy forward scatterng. Chapter 7: Expermental Study on Classfcaton Page 183
23 Chocolate bar Alumnum wedge RXN-10-AF RXN-08-AJ Fgure Images scanned for a common luggage bag wth three explosve smulants: (a) low-energy transmsson, and (b) hgh-energy transmsson. Chapter 7: Expermental Study on Classfcaton Page 184
24 Fgure , contnued. (c) Low-energy backward scatterng, and (d) low-energy forward scatterng. Chapter 7: Expermental Study on Classfcaton Page 185
25 7.3 Expermental Study on Classfcaton Bayes decson theory Bayes decson theory serves as a fundamental statstcal approach to the problem of pattern recognton. Ths approach s based on the assumpton that the decson problem s posed n probablstc terms, and that all of the relevant probabltes are known [DUD73] [SCH96] [SCH92]. We assume ths s the case n our analyss on materal characterzaton. The feature vector x, whch wll be used n analyss, combnes dual-energy and scatterng measurements, (R, L). Let ( x ) p be the probablty densty functon for x gven that the state of nature s C j, C j where C Ω = { C, C, 2, } j 1 C s, and s s the number of states of nature. In ths dssertaton, C j represents the materal type j characterzed by R and L. Then the dfference between p ( x ) and ( x ) C k materal n p descrbes how lkely that materal k may be dstngushed from C R L feature space. Suppose that we know both the a pror probablty P(C j ) and the condtonal densty ( x C j ) posteror probablty, ( C x) condtonal densty by Bayes rule [DUD73]: p. Suppose further that we measure x = ( R, L), the a P j, then can be computed from a pror probablty and ( x) P C j ( x C ) P( C ) p j j = (7.4) p ( x) where Chapter 7: Expermental Study on Classfcaton Page 186
26 p s ( x) = p( x C j ) P( C j ) j= 1 (7.5) Suppose that we observe a partcular measurement x and that we contemplate takng an acton α, where α = { α, α 2,, } A α a 1, and a s the number of possble actons. If the true state of nature s C j, we wll ncur the loss λ ( α ). Snce ( C x) C j p j s the probablty that the true state of nature s C j, the expected loss assocated wth takng acton α s s ( x) = λ( α C j ) P( C j x) Θ α (7.6) j= 1 In decson theoretc termnology, an expected loss s called a rsk, and ( α x) Θ s known as the condtonal rsk. Whenever we encounter a partcular observaton x, we can mnmze our expected loss by selectng the acton that mnmzes the condtonal rsk. Ths justfes the followng statement of the Bayes decson rule: To mnmze the overall rsk, compute the condtonal rsk ( α x) ( α x) Θ wth (7.6) for = 1, 2,, a and select the acton α for whch Θ s mnmum. Two possble actons, dentfcaton as explosves or other materals, could be taken n the context of luggage scannng Mnmum-error-rate classfcaton In classfcaton problems, each state of nature s usually assocated wth a certan class. If acton α s taken and the true state of nature s C j, then the decson s correct f n error f = j, and j. If errors are to be avoded, t s natural to seek a decson rule that mnmzes the average probablty of error,.e., the error rate. Chapter 7: Expermental Study on Classfcaton Page 187
27 A loss functon of partcular nterest for ths case s the so-called symmetrcal or zero-one loss functon [DUD73], 0 for = j λ ( α C j ) = (7.7) 1 for j Ths loss functon assgns no loss to a correct decson, and a unt loss for any error. From (7.4), the condtonal rsk s thus equal to, s ( x) = λ( α C j ) P( C j x) = 1 P( C x) Θ α (7.8) j= 1 Snce ( C x) P s the condtonal probablty that acton α s correct, the overall rsk s precsely the average probablty of error. Remember that Bayes rule mnmzes rsks by selectng the acton that mnmzes the condtonal rsk. Thus to mnmze the average probablty of error, we should select C that maxmzes the a posteror probablty ( C x) > for all j. In other words, for mnmum error rate, decde C f P( C x) P( C x) j P Classfers, dscrmnant functons and decson boundares There are many dfferent ways to represent pattern classfers. One way s n terms of a set of dscrmnant functons g (x), vector x to class C f = 1,, s, [DUD73]. The classfer s sad to assgn a feature g ( x) > g j ( x), for all j (7.9) Chapter 7: Expermental Study on Classfcaton Page 188
28 The classfer, therefore, s vewed as a machne that computes s dscrmnant functons and selects the category correspondng to the largest dscrmnant. Ths representaton of a classfer s llustrated n Fgure g 1 (x) x 1 g 1 g 2 (x) x 2 g 2 MAX α(x) x d g d (x) g s x Dscrmnant calculators Maxmum selector Decson Fgure A pattern classfer [DUD73]. A Bayes classfer can be easly represented n ths way. Generally, we can let g ( x) = Θ( α x), snce the maxmum dscrmnant functon wll then correspond to the mnmum condtonal rsk. For the mnmum error rate case, we can smplfy the expresson further by takng g x) P( C x) ( =, so that the maxmum dscrmnant functon corresponds to the maxmum a posteror probablty. It should be ponted out that the choce of the dscrmnant functons s not unque. We can always multply the dscrmnant functons by a postve constant or bas them by an addtve constant wthout nfluencng the decson. For a more general case, f we replace every g (x) by f(g (x)), the resultng classfcaton s unchanged as long as f() s a monotoncally Chapter 7: Expermental Study on Classfcaton Page 189
29 ncreasng functon. Ths observaton can lead to sgnfcant analytcal and computatonal smplfcatons. Even though the dscrmnant functons can be wrtten n a varety of forms as stated above, the decson rules are equvalent. The effect of any decson rule s to dvde the feature space nto s regons, whch can be represented as, R 1, R 2,, R s. If g ( x) > g j ( x) for all j, then we say that x s n R, and decson rule calls us to assgn x to C. The regons are separated by decson boundares, surfaces n the feature space where tes occur among the largest dscrmnant functons. If R and R j are contguous, the equaton for the decson boundary s g ( x) = g j ( x) (7.10) For ponts on the decson boundary the classfcaton s not unquely defned. Because the condtonal rsk assocated wth ether decson s the same for the Bayes classfer, t does not matter how tes are broken Classfcaton rules n the prototype x-ray magng system The two-category classfcaton problem s merely a partcular case of above dscusson, where Ω = { } and A = { } C 1,C 2 α 1,α 2. Let C 2 be explosves and C 1 stand for other materals, suppose we want to detect C 2 from C 1 ; then under the assumptons stated earler Θ( α x ) s the probablty of mssed detecton, and ( x ) rate. 1 C 2 Θ α 2 C 1 s the false alarm In ths dssertaton, a statstcally determned decson boundary s used to dstngush explosves from other materals. From theoretcal analyss and expermental measurements, t Chapter 7: Expermental Study on Classfcaton Page 190
30 s known that: 1) explosves fall n a regon of the ( R, L) space, not a pont; 2) we have only a lmted set of testng materals, both explosve smulants and nnocent materals; 3) decson surfaces of separatng explosves from other materals are not convex. So the proposed classfcaton procedure s as follows: Threshold on transmsson magng. A threshold s selected for removng the objects exhbtng hgh attenuaton. Ths may be caused ether by hgh Z materals or by very thck low Z materals. In ths case, x-rays are not strong enough to penetrate the materal, resultng n a bgger error. Dscrmnant functons. There are only two dscrmnant functons for our case. They are g ( x) = p( x C ) P( C ) and ( x) p( x C ) P( ) g = respectvely. The a 2 2 C2 pror probablty P(C j ) can be selected as follows: 1) for the system test at laboratores, we can set P ( C ) = P( C ) =, whch means that explosves and other materals are equally lkely to appear n luggage bags; 2) for the luggage scannng at arports, we may select ( C ) P( ) P >, dependng on the rsks of 1 C 2 mssed detecton and false alarm. The dscrmnant functon g (x), therefore, s only determned by the probablty densty functon p x C ). ( Decson boundary. Two methods have been used to desgn decson boundares n ths research. 1) Under the assumpton of normal dstrbutons, t s possble to estmate the parameters of p x C ) by usng the measurement data. The ( j procedure s as follows: (a) compute the mean values µ j and covarance matrx Ω j for both classes: explosves and other materals; (b) obtan the normal densty functons p(x C j ); (c) draw the decson boundary, whch satsfes p(x C 1 ) = p(x C 2 ). In Secton 7.3-5, a decson boundary based on the approach above wll be shown as an example. 2) A pecewse lnear boundary can also be used to dstngush explosves from other materals. The reason to use pecewse lnes s that t provdes us a way to treat concave decson regons. The pecewse lnes Chapter 7: Expermental Study on Classfcaton Page 191
31 used n ths research were selected manually based on the actual measurement data. Ths method s not further dscussed n the dssertaton Classfcaton results As stated above, by nsertng the explosve smulants nto common bags, we can nvestgate object classfcaton under varous scenaros. More than one hundred scenaros were created and scanned to verfy the performance of the prototype x-ray scannng system. Some typcal mage examples scanned by usng the prototype scannng system have been shown n Secton The dual-energy and scatter-energy values for some materals are gven n the followng tables. Table shows observed transmsson and scatter values as well as the computed (R, L) values for sx explosve smulants (mages gven n Fgure 7.2-4). The transmssons and scatters as well as the computed (R, L) values for three step wedges are shown n Tables to (see Fgures to for scatter mages, and Fgures to for transmsson mages). Wth (R, L) values gven n Tables to 7.3-4, we wll show an example to derve the decson rules based on our dscusson n Secton Let s consder the 2-class case as follows, explosve materals (C 2 : data gven n Table 7.3-1) and other organc materals (C 1 : data gven n Tables and 7.3-4). Inorganc materals, such as alumnum, are not ncluded n our dscusson because t can be easly elmnated by pre-usng a pece-wse lne boundary due to ther bg dstance to explosve materals. Chapter 7: Expermental Study on Classfcaton Page 192
32 Table R and L values for sx explosve smulants. Smulants T L T H B L F L R L Smokeless powder Ammona ntrate (hgh densty) Black powder Semtex Dynamte Ammona ntrate (low densty) Table R and L values for alumnum step wedge. Thckness (cm) T L T H B L F L R L Chapter 7: Expermental Study on Classfcaton Page 193
33 Table R and L values for clear plastc step wedge. Thckness (cm) T L T H B L F L R L Chapter 7: Expermental Study on Classfcaton Page 194
34 Table R and L values for whte plastc step wedge. Thckness (cm) T L T H B L F L R L Chapter 7: Expermental Study on Classfcaton Page 195
35 (1) Normal densty functons For a two-dmensonal varable x [ x x ] T 1 2 normal densty functon p ( x) can be wrtten as follows: =, ts mean value µ, covarance matrx Ω, and µ = [ µ ] T 1 µ 2 (7.11) where N 1 µ =, and N s the number of samples; x j N j= 1 ω Ω = ω ω12 ω 22 (7.12) N 1 where ω ( x µ )( x µ ) kl = N 1 j= 1 kj k lj l ; p ( x) = 1 1 T 1 exp ( x µ ) Ω ( x µ ) (7.13) 1 / 2 2π Ω 2 (2) Dscrmnant functon The dscrmnant functon for class can be obtaned by takng logarthm on (7.13) and by removng the constant away from the equaton, yeldng, Chapter 7: Expermental Study on Classfcaton Page 196
36 T T g ( x) x W x + w x + w0 = (7.14) 1 where W = Ω, w 1 T 1 = 2 Ω µ, and w = µ Ω µ log( Ω ). 0 (3) Decson boundary For class C 1 : other organc materals, ts mean value µ 1 and covarance matrx Ω 1 can be obtaned drectly by usng (7.11) and (7.12): µ 1R µ 1 = = (7.15) µ 1L Ω1 = (7.16) For class C 2 : explosve materals, we have, µ 2R µ 2 = = (7.17) µ 2L Ω 2 = (7.18) By submttng Equatons (7.15) and (7.16) nto (7.14), we obtan the dscrmnant functon g ( R, ) for class 1, 1 L Chapter 7: Expermental Study on Classfcaton Page 197
37 g ( R, L) R L RL R L = (7.19) Smlarly we have g ( R, ) for class 2, 2 L 2 2 ( R, L) = R L RL R L g (7.20) 2 The decson boundary s therefore obtaned wth g R, L) g ( R, L) 0, yeldng, 2 ( 1 = 2 2 D ( R, L) = R L RL R L = 0 (7.21) (4) Decson rules Equatons (7.19) and (7.20) represent two ellpses n 2-dmensonal space. The te between them results n a hyperbola (7.21). The decson rules therefore can be stated as follows: for any object havng a (R, L) value, f ts D ( R, L) s greater than zero, we decde t as explosve materals (shown as dark area n Fgure 7.3-4); otherwse t s decded to be other organc materals. We started our test wth smple luggage bags, then wth overlappng materals havng regular shapes, fnally wth overlappng materals havng rregular shapes. The probablty of correct classfcaton was mproved to 82% based on lab verfcatons from system group. By correct classfcaton t means that no explosves mssed detecton, and no other types of materals were dentfed as threat. It should be noted that to acheve a good performance, every part n the prototype system should work properly. Durng our expermental study on object classfcaton for real luggage, t was found that the major problem les wth overlappng objects, where the true Chapter 7: Expermental Study on Classfcaton Page 198
38 sgnal ntensty of transmsson and scatter mages should be unquely provded for each object. More analyss on ths ssue s avalable n [LU99]. Also, we found that there s some overlap between nnocent artcles and the compostons of explosve smulants, for example, between whte plastc and smokeless powder. Therefore, we are not surprsed that plastc s used as an explosve smulant adopted by FAA. Fgure An example on decson boundary for dscrmnatng two-class materals. Chapter 7: Expermental Study on Classfcaton Page 199
P R. Lecture 4. Theory and Applications of Pattern Recognition. Dept. of Electrical and Computer Engineering /
Theory and Applcatons of Pattern Recognton 003, Rob Polkar, Rowan Unversty, Glassboro, NJ Lecture 4 Bayes Classfcaton Rule Dept. of Electrcal and Computer Engneerng 0909.40.0 / 0909.504.04 Theory & Applcatons
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 informationLecture 12: Classification
Lecture : Classfcaton g Dscrmnant functons g The optmal Bayes classfer g Quadratc classfers g Eucldean and Mahalanobs metrcs g K Nearest Neghbor Classfers Intellgent Sensor Systems Rcardo Guterrez-Osuna
More informationMIMA Group. Chapter 2 Bayesian Decision Theory. School of Computer Science and Technology, Shandong University. Xin-Shun SDU
Group M D L M Chapter Bayesan Decson heory Xn-Shun Xu @ SDU School of Computer Scence and echnology, Shandong Unversty Bayesan Decson heory Bayesan decson theory s a statstcal approach to data mnng/pattern
More informationModule 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:
More informationLinear Approximation with Regularization and Moving Least Squares
Lnear Approxmaton wth Regularzaton and Movng Least Squares Igor Grešovn May 007 Revson 4.6 (Revson : March 004). 5 4 3 0.5 3 3.5 4 Contents: Lnear Fttng...4. Weghted Least Squares n Functon Approxmaton...
More informationA Bayes Algorithm for the Multitask Pattern Recognition Problem Direct Approach
A Bayes Algorthm for the Multtask Pattern Recognton Problem Drect Approach Edward Puchala Wroclaw Unversty of Technology, Char of Systems and Computer etworks, Wybrzeze Wyspanskego 7, 50-370 Wroclaw, Poland
More informationComposite Hypotheses testing
Composte ypotheses testng In many hypothess testng problems there are many possble dstrbutons that can occur under each of the hypotheses. The output of the source s a set of parameters (ponts n a parameter
More informationThe Gaussian classifier. Nuno Vasconcelos ECE Department, UCSD
he Gaussan classfer Nuno Vasconcelos ECE Department, UCSD Bayesan decson theory recall that we have state of the world X observatons g decson functon L[g,y] loss of predctng y wth g Bayes decson rule s
More informationPsychology 282 Lecture #24 Outline Regression Diagnostics: Outliers
Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.
More informationChapter 5. Solution of System of Linear Equations. Module No. 6. Solution of Inconsistent and Ill Conditioned Systems
Numercal Analyss by Dr. Anta Pal Assstant Professor Department of Mathematcs Natonal Insttute of Technology Durgapur Durgapur-713209 emal: anta.bue@gmal.com 1 . Chapter 5 Soluton of System of Lnear Equatons
More informationHomework Assignment 3 Due in class, Thursday October 15
Homework Assgnment 3 Due n class, Thursday October 15 SDS 383C Statstcal Modelng I 1 Rdge regresson and Lasso 1. Get the Prostrate cancer data from http://statweb.stanford.edu/~tbs/elemstatlearn/ datasets/prostate.data.
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
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 informationSupport Vector Machines. Vibhav Gogate The University of Texas at dallas
Support Vector Machnes Vbhav Gogate he Unversty of exas at dallas What We have Learned So Far? 1. Decson rees. Naïve Bayes 3. Lnear Regresson 4. Logstc Regresson 5. Perceptron 6. Neural networks 7. K-Nearest
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 informationENG 8801/ Special Topics in Computer Engineering: Pattern Recognition. Memorial University of Newfoundland Pattern Recognition
EG 880/988 - Specal opcs n Computer Engneerng: Pattern Recognton Memoral Unversty of ewfoundland Pattern Recognton Lecture 7 May 3, 006 http://wwwengrmunca/~charlesr Offce Hours: uesdays hursdays 8:30-9:30
More information3.1 Expectation of Functions of Several Random Variables. )' be a k-dimensional discrete or continuous random vector, with joint PMF p (, E X E X1 E X
Statstcs 1: Probablty Theory II 37 3 EPECTATION OF SEVERAL RANDOM VARIABLES As n Probablty Theory I, the nterest n most stuatons les not on the actual dstrbuton of a random vector, but rather on a number
More informationSupporting Information
Supportng Informaton The neural network f n Eq. 1 s gven by: f x l = ReLU W atom x l + b atom, 2 where ReLU s the element-wse rectfed lnear unt, 21.e., ReLUx = max0, x, W atom R d d s the weght matrx to
More informationCSci 6974 and ECSE 6966 Math. Tech. for Vision, Graphics and Robotics Lecture 21, April 17, 2006 Estimating A Plane Homography
CSc 6974 and ECSE 6966 Math. Tech. for Vson, Graphcs and Robotcs Lecture 21, Aprl 17, 2006 Estmatng A Plane Homography Overvew We contnue wth a dscusson of the major ssues, usng estmaton of plane projectve
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 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 informationStructure and Drive Paul A. Jensen Copyright July 20, 2003
Structure and Drve Paul A. Jensen Copyrght July 20, 2003 A system s made up of several operatons wth flow passng between them. The structure of the system descrbes the flow paths from nputs to outputs.
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 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 informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for P Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Control Lmts for P Charts Copyrght 2017 by Taylor Enterprses, Inc., All Rghts Reserved. Control Lmts for P Charts Dr. Wayne A. Taylor Abstract: P charts are used for count data
More informationErrors for Linear Systems
Errors for Lnear Systems When we solve a lnear system Ax b we often do not know A and b exactly, but have only approxmatons  and ˆb avalable. Then the best thng we can do s to solve ˆx ˆb exactly whch
More informationA LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS. Dr. Derald E. Wentzien, Wesley College, (302) ,
A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS Dr. Derald E. Wentzen, Wesley College, (302) 736-2574, wentzde@wesley.edu ABSTRACT A lnear programmng model s developed and used to compare
More information2016 Wiley. Study Session 2: Ethical and Professional Standards Application
6 Wley Study Sesson : Ethcal and Professonal Standards Applcaton LESSON : CORRECTION ANALYSIS Readng 9: Correlaton and Regresson LOS 9a: Calculate and nterpret a sample covarance and a sample correlaton
More informationNumerical Heat and Mass Transfer
Master degree n Mechancal Engneerng Numercal Heat and Mass Transfer 06-Fnte-Dfference Method (One-dmensonal, steady state heat conducton) Fausto Arpno f.arpno@uncas.t Introducton Why we use models and
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationUnified Subspace Analysis for Face Recognition
Unfed Subspace Analyss for Face Recognton Xaogang Wang and Xaoou Tang Department of Informaton Engneerng The Chnese Unversty of Hong Kong Shatn, Hong Kong {xgwang, xtang}@e.cuhk.edu.hk Abstract PCA, LDA
More informationEcon107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)
I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes
More informationThe Multiple Classical Linear Regression Model (CLRM): Specification and Assumptions. 1. Introduction
ECONOMICS 5* -- NOTE (Summary) ECON 5* -- NOTE The Multple Classcal Lnear Regresson Model (CLRM): Specfcaton and Assumptons. Introducton CLRM stands for the Classcal Lnear Regresson Model. The CLRM s also
More informationTemperature. Chapter Heat Engine
Chapter 3 Temperature In prevous chapters of these notes we ntroduced the Prncple of Maxmum ntropy as a technque for estmatng probablty dstrbutons consstent wth constrants. In Chapter 9 we dscussed the
More informationUNIVERSITY OF TORONTO Faculty of Arts and Science. December 2005 Examinations STA437H1F/STA1005HF. Duration - 3 hours
UNIVERSITY OF TORONTO Faculty of Arts and Scence December 005 Examnatons STA47HF/STA005HF Duraton - hours AIDS ALLOWED: (to be suppled by the student) Non-programmable calculator One handwrtten 8.5'' x
More informationSimulated Power of the Discrete Cramér-von Mises Goodness-of-Fit Tests
Smulated of the Cramér-von Mses Goodness-of-Ft Tests Steele, M., Chaselng, J. and 3 Hurst, C. School of Mathematcal and Physcal Scences, James Cook Unversty, Australan School of Envronmental Studes, Grffth
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Experment-I MODULE VII LECTURE - 3 ANALYSIS OF COVARIANCE Dr Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur Any scentfc experment s performed
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationLinear Regression Analysis: Terminology and Notation
ECON 35* -- Secton : Basc Concepts of Regresson Analyss (Page ) Lnear Regresson Analyss: Termnology and Notaton Consder the generc verson of the smple (two-varable) lnear regresson model. It s represented
More informationLab 2e Thermal System Response and Effective Heat Transfer Coefficient
58:080 Expermental Engneerng 1 OBJECTIVE Lab 2e Thermal System Response and Effectve Heat Transfer Coeffcent Warnng: though the experment has educatonal objectves (to learn about bolng heat transfer, etc.),
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 informationEEE 241: Linear Systems
EEE : Lnear Systems Summary #: Backpropagaton BACKPROPAGATION The perceptron rule as well as the Wdrow Hoff learnng were desgned to tran sngle layer networks. They suffer from the same dsadvantage: they
More informationLecture 10 Support Vector Machines II
Lecture 10 Support Vector Machnes II 22 February 2016 Taylor B. Arnold Yale Statstcs STAT 365/665 1/28 Notes: Problem 3 s posted and due ths upcomng Frday There was an early bug n the fake-test data; fxed
More informationChapter 12 Analysis of Covariance
Chapter Analyss of Covarance Any scentfc experment s performed to know somethng that s unknown about a group of treatments and to test certan hypothess about the correspondng treatment effect When varablty
More information4 Analysis of Variance (ANOVA) 5 ANOVA. 5.1 Introduction. 5.2 Fixed Effects ANOVA
4 Analyss of Varance (ANOVA) 5 ANOVA 51 Introducton ANOVA ANOVA s a way to estmate and test the means of multple populatons We wll start wth one-way ANOVA If the populatons ncluded n the study are selected
More informationDUE: WEDS FEB 21ST 2018
HOMEWORK # 1: FINITE DIFFERENCES IN ONE DIMENSION DUE: WEDS FEB 21ST 2018 1. Theory Beam bendng s a classcal engneerng analyss. The tradtonal soluton technque makes smplfyng assumptons such as a constant
More informationUncertainty as the Overlap of Alternate Conditional Distributions
Uncertanty as the Overlap of Alternate Condtonal Dstrbutons Olena Babak and Clayton V. Deutsch Centre for Computatonal Geostatstcs Department of Cvl & Envronmental Engneerng Unversty of Alberta An mportant
More informationCHALMERS, GÖTEBORGS UNIVERSITET. SOLUTIONS to RE-EXAM for ARTIFICIAL NEURAL NETWORKS. COURSE CODES: FFR 135, FIM 720 GU, PhD
CHALMERS, GÖTEBORGS UNIVERSITET SOLUTIONS to RE-EXAM for ARTIFICIAL NEURAL NETWORKS COURSE CODES: FFR 35, FIM 72 GU, PhD Tme: Place: Teachers: Allowed materal: Not allowed: January 2, 28, at 8 3 2 3 SB
More informationStatistical pattern recognition
Statstcal pattern recognton Bayes theorem Problem: decdng f a patent has a partcular condton based on a partcular test However, the test s mperfect Someone wth the condton may go undetected (false negatve
More informationOnline Appendix to: Axiomatization and measurement of Quasi-hyperbolic Discounting
Onlne Appendx to: Axomatzaton and measurement of Quas-hyperbolc Dscountng José Lus Montel Olea Tomasz Strzaleck 1 Sample Selecton As dscussed before our ntal sample conssts of two groups of subjects. Group
More informationWhy Bayesian? 3. Bayes and Normal Models. State of nature: class. Decision rule. Rev. Thomas Bayes ( ) Bayes Theorem (yes, the famous one)
Why Bayesan? 3. Bayes and Normal Models Alex M. Martnez alex@ece.osu.edu Handouts Handoutsfor forece ECE874 874Sp Sp007 If all our research (n PR was to dsappear and you could only save one theory, whch
More informationInductance Calculation for Conductors of Arbitrary Shape
CRYO/02/028 Aprl 5, 2002 Inductance Calculaton for Conductors of Arbtrary Shape L. Bottura Dstrbuton: Internal Summary In ths note we descrbe a method for the numercal calculaton of nductances among conductors
More informationChapter 13: Multiple Regression
Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to
More informationCHAPTER 4 SPEECH ENHANCEMENT USING MULTI-BAND WIENER FILTER. In real environmental conditions the speech signal may be
55 CHAPTER 4 SPEECH ENHANCEMENT USING MULTI-BAND WIENER FILTER 4.1 Introducton In real envronmental condtons the speech sgnal may be supermposed by the envronmental nterference. In general, the spectrum
More informationOnline Classification: Perceptron and Winnow
E0 370 Statstcal Learnng Theory Lecture 18 Nov 8, 011 Onlne Classfcaton: Perceptron and Wnnow Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton In ths lecture we wll start to study the onlne learnng
More informationPredictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore
Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.
More informationSalmon: Lectures on partial differential equations. Consider the general linear, second-order PDE in the form. ,x 2
Salmon: Lectures on partal dfferental equatons 5. Classfcaton of second-order equatons There are general methods for classfyng hgher-order partal dfferental equatons. One s very general (applyng even to
More informationMASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.265/15.070J Fall 2013 Lecture 12 10/21/2013. Martingale Concentration Inequalities and Applications
MASSACHUSETTS INSTITUTE OF TECHNOLOGY 6.65/15.070J Fall 013 Lecture 1 10/1/013 Martngale Concentraton Inequaltes and Applcatons Content. 1. Exponental concentraton for martngales wth bounded ncrements.
More informationInner Product. Euclidean Space. Orthonormal Basis. Orthogonal
Inner Product Defnton 1 () A Eucldean space s a fnte-dmensonal vector space over the reals R, wth an nner product,. Defnton 2 (Inner Product) An nner product, on a real vector space X s a symmetrc, blnear,
More informationINF 5860 Machine learning for image classification. Lecture 3 : Image classification and regression part II Anne Solberg January 31, 2018
INF 5860 Machne learnng for mage classfcaton Lecture 3 : Image classfcaton and regresson part II Anne Solberg January 3, 08 Today s topcs Multclass logstc regresson and softma Regularzaton Image classfcaton
More informationChapter 6. Supplemental Text Material
Chapter 6. Supplemental Text Materal S6-. actor Effect Estmates are Least Squares Estmates We have gven heurstc or ntutve explanatons of how the estmates of the factor effects are obtaned n the textboo.
More informationConstitutive Modelling of Superplastic AA-5083
TECHNISCHE MECHANIK, 3, -5, (01, 1-6 submtted: September 19, 011 Consttutve Modellng of Superplastc AA-5083 G. Gulano In ths study a fast procedure for determnng the constants of superplastc 5083 Al alloy
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationPop-Click Noise Detection Using Inter-Frame Correlation for Improved Portable Auditory Sensing
Advanced Scence and Technology Letters, pp.164-168 http://dx.do.org/10.14257/astl.2013 Pop-Clc Nose Detecton Usng Inter-Frame Correlaton for Improved Portable Audtory Sensng Dong Yun Lee, Kwang Myung Jeon,
More informationOutline. Communication. Bellman Ford Algorithm. Bellman Ford Example. Bellman Ford Shortest Path [1]
DYNAMIC SHORTEST PATH SEARCH AND SYNCHRONIZED TASK SWITCHING Jay Wagenpfel, Adran Trachte 2 Outlne Shortest Communcaton Path Searchng Bellmann Ford algorthm Algorthm for dynamc case Modfcatons to our algorthm
More informationGlobal Sensitivity. Tuesday 20 th February, 2018
Global Senstvty Tuesday 2 th February, 28 ) Local Senstvty Most senstvty analyses [] are based on local estmates of senstvty, typcally by expandng the response n a Taylor seres about some specfc values
More informationLimited Dependent Variables
Lmted Dependent Varables. What f the left-hand sde varable s not a contnuous thng spread from mnus nfnty to plus nfnty? That s, gven a model = f (, β, ε, where a. s bounded below at zero, such as wages
More informationExperiment 1 Mass, volume and density
Experment 1 Mass, volume and densty Purpose 1. Famlarze wth basc measurement tools such as verner calper, mcrometer, and laboratory balance. 2. Learn how to use the concepts of sgnfcant fgures, expermental
More informationChapter 3 Describing Data Using Numerical Measures
Chapter 3 Student Lecture Notes 3-1 Chapter 3 Descrbng Data Usng Numercal Measures Fall 2006 Fundamentals of Busness Statstcs 1 Chapter Goals To establsh the usefulness of summary measures of data. The
More informationThe big picture. Outline
The bg pcture Vncent Claveau IRISA - CNRS, sldes from E. Kjak INSA Rennes Notatons classes: C = {ω = 1,.., C} tranng set S of sze m, composed of m ponts (x, ω ) per class ω representaton space: R d (=
More informationThe Geometry of Logit and Probit
The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.
More informationEVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES
EVALUATION OF THE VISCO-ELASTIC PROPERTIES IN ASPHALT RUBBER AND CONVENTIONAL MIXES Manuel J. C. Mnhoto Polytechnc Insttute of Bragança, Bragança, Portugal E-mal: mnhoto@pb.pt Paulo A. A. Perera and Jorge
More information5. POLARIMETRIC SAR DATA CLASSIFICATION
Polarmetrc SAR data Classfcaton 5. POLARIMETRIC SAR DATA CLASSIFICATION 5.1 Classfcaton of polarmetrc scatterng mechansms - Drect nterpretaton of decomposton results - Cameron classfcaton - Lee classfcaton
More information2010 Black Engineering Building, Department of Mechanical Engineering. Iowa State University, Ames, IA, 50011
Interface Energy Couplng between -tungsten Nanoflm and Few-layered Graphene Meng Han a, Pengyu Yuan a, Jng Lu a, Shuyao S b, Xaolong Zhao b, Yanan Yue c, Xnwe Wang a,*, Xangheng Xao b,* a 2010 Black Engneerng
More informationDr. Shalabh Department of Mathematics and Statistics Indian Institute of Technology Kanpur
Analyss of Varance and Desgn of Exerments-I MODULE III LECTURE - 2 EXPERIMENTAL DESIGN MODELS Dr. Shalabh Deartment of Mathematcs and Statstcs Indan Insttute of Technology Kanur 2 We consder the models
More informationError Probability for M Signals
Chapter 3 rror Probablty for M Sgnals In ths chapter we dscuss the error probablty n decdng whch of M sgnals was transmtted over an arbtrary channel. We assume the sgnals are represented by a set of orthonormal
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationCHAPTER IV RESEARCH FINDING AND ANALYSIS
CHAPTER IV REEARCH FINDING AND ANALYI A. Descrpton of Research Fndngs To fnd out the dfference between the students who were taught by usng Mme Game and the students who were not taught by usng Mme Game
More informationBasically, if you have a dummy dependent variable you will be estimating a probability.
ECON 497: Lecture Notes 13 Page 1 of 1 Metropoltan State Unversty ECON 497: Research and Forecastng Lecture Notes 13 Dummy Dependent Varable Technques Studenmund Chapter 13 Bascally, f you have a dummy
More informationLINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity
LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 31 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 6. Rdge regresson The OLSE s the best lnear unbased
More informationFeb 14: Spatial analysis of data fields
Feb 4: Spatal analyss of data felds Mappng rregularly sampled data onto a regular grd Many analyss technques for geophyscal data requre the data be located at regular ntervals n space and/or tme. hs s
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 informationSupplementary Notes for Chapter 9 Mixture Thermodynamics
Supplementary Notes for Chapter 9 Mxture Thermodynamcs Key ponts Nne major topcs of Chapter 9 are revewed below: 1. Notaton and operatonal equatons for mxtures 2. PVTN EOSs for mxtures 3. General effects
More informationChapter 8 Indicator Variables
Chapter 8 Indcator Varables In general, e explanatory varables n any regresson analyss are assumed to be quanttatve n nature. For example, e varables lke temperature, dstance, age etc. are quanttatve n
More informationFor now, let us focus on a specific model of neurons. These are simplified from reality but can achieve remarkable results.
Neural Networks : Dervaton compled by Alvn Wan from Professor Jtendra Malk s lecture Ths type of computaton s called deep learnng and s the most popular method for many problems, such as computer vson
More informationHidden Markov Models
Hdden Markov Models Namrata Vaswan, Iowa State Unversty Aprl 24, 204 Hdden Markov Model Defntons and Examples Defntons:. A hdden Markov model (HMM) refers to a set of hdden states X 0, X,..., X t,...,
More informationCopyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for U Charts. Dr. Wayne A. Taylor
Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts Copyrght 207 by Taylor Enterprses, Inc., All Rghts Reserved. Adjusted Control Lmts for U Charts Dr. Wayne A. Taylor Abstract: U charts are used
More informationCredit Card Pricing and Impact of Adverse Selection
Credt Card Prcng and Impact of Adverse Selecton Bo Huang and Lyn C. Thomas Unversty of Southampton Contents Background Aucton model of credt card solctaton - Errors n probablty of beng Good - Errors n
More informationStatistics Chapter 4
Statstcs Chapter 4 "There are three knds of les: les, damned les, and statstcs." Benjamn Dsrael, 1895 (Brtsh statesman) Gaussan Dstrbuton, 4-1 If a measurement s repeated many tmes a statstcal treatment
More informationLearning from Data 1 Naive Bayes
Learnng from Data 1 Nave Bayes Davd Barber dbarber@anc.ed.ac.uk course page : http://anc.ed.ac.uk/ dbarber/lfd1/lfd1.html c Davd Barber 2001, 2002 1 Learnng from Data 1 : c Davd Barber 2001,2002 2 1 Why
More informationCS 468 Lecture 16: Isometry Invariance and Spectral Techniques
CS 468 Lecture 16: Isometry Invarance and Spectral Technques Justn Solomon Scrbe: Evan Gawlk Introducton. In geometry processng, t s often desrable to characterze the shape of an object n a manner that
More informationPHYS 450 Spring semester Lecture 02: Dealing with Experimental Uncertainties. Ron Reifenberger Birck Nanotechnology Center Purdue University
PHYS 45 Sprng semester 7 Lecture : Dealng wth Expermental Uncertantes Ron Refenberger Brck anotechnology Center Purdue Unversty Lecture Introductory Comments Expermental errors (really expermental uncertantes)
More informationStatistical Foundations of Pattern Recognition
Statstcal Foundatons of Pattern Recognton Learnng Objectves Bayes Theorem Decson-mang Confdence factors Dscrmnants The connecton to neural nets Statstcal Foundatons of Pattern Recognton NDE measurement
More informationLecture 4: November 17, Part 1 Single Buffer Management
Lecturer: Ad Rosén Algorthms for the anagement of Networs Fall 2003-2004 Lecture 4: November 7, 2003 Scrbe: Guy Grebla Part Sngle Buffer anagement In the prevous lecture we taled about the Combned Input
More informationAffine transformations and convexity
Affne transformatons and convexty The purpose of ths document s to prove some basc propertes of affne transformatons nvolvng convex sets. Here are a few onlne references for background nformaton: http://math.ucr.edu/
More informationMore metrics on cartesian products
More metrcs on cartesan products If (X, d ) are metrc spaces for 1 n, then n Secton II4 of the lecture notes we defned three metrcs on X whose underlyng topologes are the product topology The purpose of
More informationChapter - 2. Distribution System Power Flow Analysis
Chapter - 2 Dstrbuton System Power Flow Analyss CHAPTER - 2 Radal Dstrbuton System Load Flow 2.1 Introducton Load flow s an mportant tool [66] for analyzng electrcal power system network performance. Load
More informationChapter 2 - The Simple Linear Regression Model S =0. e i is a random error. S β2 β. This is a minimization problem. Solution is a calculus exercise.
Chapter - The Smple Lnear Regresson Model The lnear regresson equaton s: where y + = β + β e for =,..., y and are observable varables e s a random error How can an estmaton rule be constructed for the
More information