An Introduction To Error Propagation: Derivation, Meaning and Examples C Y


 Wesley Emory Lynch
 2 years ago
 Views:
Transcription
1 SWISS FEDERAL INSTITUTE OF TECHNOLOGY LAUSANNE EIDGENÖSSISCHE TECHNISCHE HOCHSCHULE LAUSANNE POLITECNICO FEDERALE DI LOSANNA DÉPARTEMENT DE MICROTECHNIQUE INSTITUT DE SYSTÈMES ROBOTIQUE Autoomous Systems Lab Prof. Dr. Rolad Segwart A Itroducto To Error Propagato: Dervato, Meag ad Examples T of Equato F X C X F X C Y Ka Olver Arras Techcal Report Nº EPFLASLTR980 R3 of the Autoomous Systems Lab, Isttute of Robotc Systems, Swss Federal Isttute of Techology Lausae (EPFL) September 998
2 Cotets. Itroducto Error Propagato: From the Begg A Frst Expectato Whe s the Approxmato a Good Oe? The Almost Geeral Case: Approxmatg the Dstrbuto of Y f( X, X,, X ) Addedum to Chapter Gettg Really Geeral: Addg Z g( X, X,, X ) Dervatg the Fal Matrx Form Examples Probablstc Le Extracto From Nosy D Rage Data Kalma Flter Based Moble Robot Localzato: The Measuremet Model Exercses Lterature Appedx A Fdg the Le Parameters r ad α the Weghted Least Square Sese Appedx B Dervg the Covarace Matrx for r ad α B. Practcal Cosderatos
3 . Itroducto Ths report attempts to shed lght oto the equato C Y T F X C X F X, () where C X s a, C Y a p p covarace matrx ad F X some matrx of dmeso p. I estmato applcatos lke Kalma flterg or probablstc feature extracto we frequetly ecouter the patter F X C X F X. May texts lterature troduce ths equato wth T out further explaato. But relatoshp (), called the error propagato law, ca be explctely derved ad uderstood, beg mportat for a compreheso of ts uderlyg approxmatve character. Here, we wat to brdge the gap betwee these texts ad the ovce to the world of ucertaty modelg ad propagato. Applcatos of () are e.g. error propagato modelbased vso, Kalma flterg, relablty or probablstc systems aalyss geeral.. Error Propagato: From the Begg We wll frst forget the matrx form of () ad chage to a dfferet perspectve. Error propagato s the problem of fdg the dstrbuto of a fucto of radom varables. Ofte we have mathematcal models of the system of terest (the output as a fucto of the put ad the system compoets) ad we kow somethg about the dstrbuto of the put ad the compoets. X System Y Fgure : The smplest case: oe put radom varable ( N ), ad oe output radom varable ( P ). The, we desre to kow the dstrbuto of the output, that s the dstrbuto fucto of Y whe Y f( X) where f (). s some kow fucto ad the dstrbuto fucto of the radom varable X s kow. If f (). s a olear fucto, the probablty dstrbuto fucto of Y, p Y ( y), becomes quckly very complex, partcularly whe there s more tha oe put varable. Although a geeral method of soluto exsts (see [BREI70] Chapter 66), ts complexty ca be demostrated already for smple problems. A approxmato of p Y ( y) s therefore desrable. The approxmato cossts the propagato of oly the frst two statstcal momets, that s the mea µ Y ad the secod (cetral) momet σ Y, the varace. These momets do ot geeral descrbe the dstrbuto of Y. However f Y s assumed to be ormally dstrbuted they do. We do ot cosder pathologcal fuctos f (). where Y s ot a radom varable however, they exst That s smply oe of the favorable propertes of the ormal dstrbuto.
4 . A Frst Expectato Look at fgure where the smple case wth oe put ad oe output s llustrated. Suppose that X s ormally dstrbuted wth mea µ ad stadard devato X σ X. Now we would lke Y µ y + σ y f( X) µ y µ y σ y µ x σ x µ x µ x + σ x X Fgure : Oedmesoal case of a olear error propagato problem to kow how the 68% probablty terval [ µ X σ X, µ X + σ X ] s propagated through the system f ().. Frst of all, from fgure t ca be see that f the shaded terval would be mapped oto the y axs by the orgal fucto ts shape would be somewhat dstorted ad the resultg dstrbuto would be asymmetrc, certaly ot Gaussa aymore. Whe approxmatg f( X) by a frstorder Taylor seres expaso about the pot X µ X, Y f ( µ X ) + f X µ X ( X ), () we obta the lear relatoshp show fgure ad wth that a ormal dstrbuto for p Y ( y). Now we ca determe ts parameters µ Y ad σ Y. µ Y f ( µ X ), (3) σ Y f σ X X µ X. (4) Fally the expectato s rsed that the remader of ths text we wll aga bump to some geeralzed form of equato (3) ad (4). At ths pot, we should ot forget that the output dstrbuto, represeted by µ Y ad σ Y, s a approxmato of some ukow truth. Ths truth s mpertetly olear, oormal ad asymmetrc, thus hbtg ay exact closed form aalyss most cases. We are the supposed to ask the questo: µ X Remember that the stadard devato σ s by defto the dstace betwee the most probable value, µ, ad the curve s turg pots. Aother useful property of the ormal dstrbuto, worth to be remembered: Gaussa stays Gaussa uder lear trasformatos. 3
5 . Whe s the Approxmato a Good Oe? Some textbooks wrte equatos (3) ad (4) as equaltes. But f the left had sdes deote the parameters of the output dstrbuto whch s, by assumpto, ormal, we ca wrte them as equaltes. The frst two momets of the true (but ukow) output dstrbuto let s call them µ ad are deftely dfferet from these values 0 σ 0. Hece µ 0 µ Y σ 0 σ Y It s evdet that f f (). s lear we ca wrte (5) ad (6) as equaltes. However the followg factors affect approxmato qualty of µ Y ad σ Y by the actual values y ad s y. µ Y y σ Y s y Thus they apply for both cases; whe f (). s lear ad whe t s olear. The guess x : I geeral we do ot kow the expected value µ X. I ths case, ts best (ad oly avalable) guess s the actual value of X, for example the curret measuremet x. We the dfferetate f( X) at the pot X x hopg that x s close to E[ X] such that y s ot too far from E[ Y]. Extet of olearty of f (). : Equatos (3) ad (4) are good approxmatos as log as the frstorder Taylor seres s a good approxmato whch s the case f f (). s ot too far from lear wth the rego that s wth oe stadard devato of the mea [BREI70]. Some people eve ask f (). to be close to lear wth a ± σ terval. The olearty of f (). ad the devato of x from E[ X] have a combed effect o σ Y. If both factors are of hgh magtude, the slope at X x ca dffer strogly ad the resultg approxmato of σ Y ca be poor. At the other had, f already oe of the them s small, or eve both, we ca expect s y to be close to σ Y. The guess s x : There are several possbltes to model the put ucertaty. The model could corporate some ad hoc assumptos o σ X or t mght rely o a emprcally gaed relatoshp to µ X. Sometmes t s possble to have a physcally based model provdg true ucertaty formato for each realzato of X. By systematcally followg all sources of perturbato durg the emergece of x, such a model has the desrable property that t accouts for all mportat factors whch fluece the outcome x ad s x. I ay case, the actual value s x remas a guess of σ X whch s hopefully close to σ X. But there s also reaso for optmsm. The codtos suggested by Fgure are exaggerated. Mostly, σ X s very small wth respect to the rage of X. Ths makes () a approxmato of suffcet qualty may practcal problems. (5) (6) (7) (8) Remember that the expected value ad the varace (ad all other momets) have a geeral defto,.e. are depedet whether the dstrbuto fuctos exhbts some ce propertes lke symmetry. They are also vald for arbtrarly shaped dstrbutos ad always quatfy the most probable value ad ts spread. It mght be possble that some olear but cecodtoed cases the approxmato effects of oormalty ad the other factors compesate each other yeldg a very good approxmato. Nevertheless, the opposte mght also be true. 4
6 .3 The Almost Geeral Case: Approxmatg the Dstrbuto of Y f( X, X,, X ) The ext step towards equato () s aga a practcal approxmato based o a frstorder Taylor seres expaso, ths tme for a multpleput system. Cosder X X X 3 System Y X Fgure 3: Error propagato a multput sgleoutput system: N, P. Y f( X, X,, X ) where the X s are put radom varables ad Y s represeted by ts frstorder Taylor seres expaso about the pot µ, µ,, µ Equato (9) s of the form Y f ( µ, µ,, µ ) + f ( µ. (9), µ,, µ ) [ X µ ] Y a 0 + a ( X µ ) wth a 0 f ( µ, µ,, µ ), (0) a f ( µ, µ,, µ ). () As chapter., the approxmato s lear. The dstrbuto of Y s therefore Gaussa ad we have to determe µ Y ad σ Y. µ Y E[ Y] E[ a 0 + a ( X µ )] () E[ a 0 ] + E[ a X ] E[ a µ ] (3) a 0 + a E[ X ] a E [ µ ] (4) a 0 + a µ a µ (5) a 0 (6) µ Y f ( µ, µ,, µ ) (7) σ Y E[ ( Y µ Y ) ] E[ ( a ( X µ )) ] (8) E[ a ( X µ ) a j ( X j µ j )] (9) j E[ a ( X µ ) + a a j ( X µ )( X j µ j )] (0) j 5
7 a E[ ( X µ ) ] + a a j E[ ( X µ )( X µ j )] () a σ + a a j σ j () j σ f Y σ f f + σj j j (3) The vector µ, µ,, µ has bee omtted. If the X s are depedet the covarace σ j dsappears, ad the resultg approxmated varace s σ f Y σ. (4) Ths s the momet to valdate our expectato from chapter. for the oedmesoal case. Equato (7) correspods drectly wth (3) whereas (3) somewhat cotas equato (4)..3. Addedum to Chapter. I order to close the dscusso o factors affectg approxmato qualty, we have to cosder brefly two aspects whch play a role f there s more tha oe put. Idepedece of the puts: Equato (4) s a good approxmato f the stated assumpto of depedece of all s s vald. X It s fally to be metoed that, eve f the put dstrbutos are ot strctly Gaussa, the assumpto of the output beg ormal s ofte reasoable. Ths follows from the cetral lmt theorem whe the s somehow addtvely costtute the output Y. X j.4 Gettg Really Geeral: Addg Z g( X, X,, X ) Ofte there s ot just a sgle Y whch depeds o the X s but there are more system outputs, that s, more radom varables lke Y. Suppose our system has a addtoal output Z wth Z g( X, X,, X ). X X X 3 X System Y Z Fgure 4: Error propagato a multput multoutput system: N, P Obvously µ Z ad σ Z ca exactly be derved as show before. The addtoal aspect whch s troduced by Z s the questo of the statstcal depedece of Y ad Z whch s expressed by ther covarace σ YZ E[ ( Y µ Y )( Z µ Z )]. Let s see where we arrve whe substtutg Y ad Z by ther frstorder Taylor seres expaso (9). 6
8 σ YZ E[ ( Y µ Y )( Z µ Z )] (5) E[ Y Z] E[ Y]E[ Z] (6) E f µ Y + [ X (7) µ ] g µ Z + [ X µ ] µ Y µ Z f g f g E µ Y µ Z + µ Z [ X (8) µ ] + µ Y [ X µ ] + [ X µ ] [ X µ ] µ Y µ Z f f E [ µ Y µ Z ] + µ Z E µ + µ Y E + E X f g [ X j µ ][ X j µ j ] f µ Y µ Z µ Z E X f [ ] µ Z E µ g [ ] µ Y E X g + + [ ] µ Y + E µ Y µ Z g X g µ f g [ X µ ] f g + [ X j µ ][ X j µ j ] µ Y µ Z j E [ µ ] (9) (30) σ YZ X f g (3) E ( X µ ) f g [ ] + E [( X j µ )( X j µ j )] j f g f g σ (3) + σ j X j j j If ad are depedet, the secod term, holdg ther covarace, dsappears. Addg more output radom varables brgs o ew aspects. I the remader of ths text we shall cosder P wthout loss of geeralty. 3. Dervatg the Fal Matrx Form Now we are ready to retur to equato (). We wll ow see that we oly have to reformulate equatos (3) ad (3) order to obta the tal matrx form. We recall the gradet operator wth respect to the dmesoal vector X X. (33) f ( X ) s a p dmesoal vectorvalued fucto f ( X ) f ( X ) f ( X ) f p ( X ) T. The Jacoba F X s defed as the traspose of the gradet of f ( X ), whereas the gradet s the outer product of ad f ( X ) X T F X X f ( X ) T T f ( X ) ( X ) f f f f f (34) F X has dmeso ths case, p geeral. We troduce the symmetrc put covarace matrx C X whch cotas all varaces ad covaraces of the put radom varables X, X,, X. If the X s are depedet all σ j wth j dsappear ad C X s dagoal. 7
9 σ X σ X X σ X X C X σ X X σ X : : : σ X X σ X X σ X σ X X (35) We further troduce the symmetrc output covarace matrx C Y ( p p geeral wth outputs Y, Y,, Y p ) C Y σ Y σ Y Y σ Y σ Y Y (36) Now we ca statly form equato () C Y F X C X F X T (37) σ Y σ Y Y σ Y Y σ Y f f f f σ X σ X X σ X X σ X X σ X σ X X : : : σ X X σ X X σ X f f : : f f (38) f σx f σx f + + σx X X f + + σx X X f f σx X + + σx X f f σx X + + σx X f f : : f f (39) Looks ce. But what has t to do wth chapter? To aswer ths questo we evaluate the frst elemet, the varace of : σ Y σ Y Y f f f f f f f f f f σ X + σx X + + σx X X + σx X + σ X + + σx X X f f f f f + + σx X X + σx X + + σ X (40) f f f σ X + σx X j j j (4) If we ow retroduce the otato of chapter.3, that s, f ( X ) f( X), Y Y, ad σ X σ, we see that (4) equals exactly equato (3). Assumg the reader beg a otorous skeptc, we wll also look at the offdagoal elemet σ Y Y, the covarace of Y ad Z: f σ Y Y f f f f f f f f f f f σx + σx X + + σx X X + σx X + σx + + σx X X f f f f f + + σx X X + σx X + + X f σx (4) 8
10 f f f f σx + (43) σx X j j j Aga, by substtutg f ( X ) by f( X), f ( X ) by g( X) ad σ X by σ, equato (43) correspod exactly to the prevously derved equato (3) for σ YZ. We were obvously able, havg started from a smple oedmesoal error propagato problem, to derve the error propagato law C Y T F X C X F X. Puttg the results together yelded ts wdely used matrx form (). Now we ca also uderstad the formal terpretato of fgure 5. The put ucertaty... T C Y F X C X F X...ad approxmatvely mapped to the output....s popagated through the system f(.)... Fgure 5: Iterpretato of the error propagato law ts matrx form 4. Examples 4. Probablstc Le Extracto From Nosy D Rage Data Modelbased vso where geometrc prmtves are the sought mage features s a good example for ucertaty propagato. Suppose the segmetato problem has already bee solved, that s, the set of ler pots wth respect to the model s kow. Suppose further that the regresso equatos for the model ft to the pots have a closedform soluto whch s the case whe fttg straght les. Ad suppose fally, that the measuremet ucertates of the data pots are kow as well. The we ca drectly apply the error propagato law order to get the output ucertates of the le parameters. x (r, q ) r d α Fgure 6: Estmatg a le the least squares sese. The model parameters r (legth of the perpedcular) ad α (ts agle to the abscssa) descrbe uquely a le. 9
11 Suppose measuremet pots polar coordates x ( ρ, θ ) are gve ad modeled as radom varables X ( P, Q ) wth,,. Each pot s depedetly affected by Gaussa ose both coordates. P ~ N( ρ, σ ρ ) (44) Q ~ N( θ, σ θ ) (45) E[ P P j ] E[ P ]E[ P j ], j,, (46) E[ Q Q j ] E[ Q ]E[ Q j ], j,, (47) E[ P Q j ] E[ P ]E[ Q j ], j,, (48) Now we wat to fd the le x cosα + y sα r 0 where x ρcos( θ) ad y ρs( θ) yeldg ρcosθcosα + ρsθsα r 0 ad wth that, the le model ρcos( θ α) r 0. (49) Ths model mmzes the orthogoal dstaces from the pots to the le. It s mportat to ote that fttg models to data some least square sese yelds ot a satsfyg geometrc soluto geeral. It s crucal to kow whch error s mmzed by the ft equatos. A good llustrato s the paper of [GAND94] where several algorthms for fttg crcles ad ellpses are preseted whch mmze algebrac ad geometrc dstaces. The geometrc varety of solutos for the same set of pots demostrate the mportace of ths kowledge f geometrc meagful results are requred. The orthogoal dstace d of a pot ( ρ, θ ) to the le s just ρ cos( θ α) r d. (50) Let S be the (uweghted) sum of squared errors. The model parameters ( α, r) S d ( ρ cos( θ α) r) are ow foud by solvg the olear equato system (5) 0. (5) Suppose further that for each pot a varace modellg the ucertaty radal ad agular drecto s gve a pror or ca be measured. Ths varace wll be used to determe a weght for each sgle pot, e.g. w The, equato (5) becomes S 0 S w σ σ. (53) S w d w ( ρ cos( θ α) r). (54) The ssue of determg a adequate weght whe σ (ad perhaps some addtoal formato) s gve s complex geeral ad beyod the scope of ths text. See [CARR88] for a careful treatmet. 0
12 It ca be show (see Appedx A) that the soluto of (5) the weghted least square sese s α w ρ sθ w w j ρ ρ j cosθ sθ j w ata w ρ cosθ w w j ρ ρ j cos( θ + θ j ) w (55) r w ρ cos( θ α) (56) w Now we would lke to kow how the ucertates of the measuremets propagate through the system (55), (56). See fgure 7. X X X 3 X Model Ft A R Fgure 7: Whe extractg les from osy measuremet pots X, the model ft module produces le parameter estmates, modeled as radom varables A, R. It s the terestg to kow the varablty of these parameters as a fucto of the ose at the put sde. Ths s where we smply apply equato (). We are lookg for the matrx output covarace σ AR σ C A AR σ AR σ R, (57) gve the put covarace matrx C C P 0 dag( σ ρ ) 0 X 0 C Q 0 dag( σ θ ) (58) ad the system relatoshps (55) ad (56). The by calculatg the Jacoba P F P P PQ P P P Q Q Q Q Q Q (59) We follow here the otato of [DRAP88] ad dstgush a weghted least squares problem f C X s dagoal (put errors are mutually depedet) ad a geeralzed least squares problem f C X s odagoal.
13 we ca statly form the error propagato equato (60) yeldg the sought. C AR C AR T F PQ C X F PQ (60) Appedx A s cocered about the stepbystep dervato of the ft equatos (55) ad (56), whereas Appedx B equato (60) s oce more derved. Uder the assumpto of eglgble agular ucertates, mplemetable expressos for the elemets of C AR are determed, also a stepbystep maer. 4. Kalma Flter Based Moble Robot Localzato: The Measuremet Model The measuremet model a Kalma flter estmato problem s aother place where we ecouter the error propagato law. The reader s assumed to be more or less famlar wth the cotext of moble robot localzato, Kalma flterg ad the otato used [BAR93] or [LEON9]. I order to reduce the ubouded growth of odometry errors the robot s supposed to update ts pose by some sort of exteral referecg. Ths s acheved by matchg predcted evromet features wth observed oes ad estmatg the vehcle pose some sese wth the set of matched features. The predcto s provded by a a pror map whch cotas the posto of all evromet features global map coordates. I order to establsh correct correspodece of observed ad stored features, the robot predcts the posto of all curretly vsble features the sesor frame. Ths s doe by the measuremet model ẑ( k + k) h( xˆ ( k + k) ) + wk ( + ). (6) The measuremet model gets the predcted robot pose xˆ ( k + k) as put ad asks the map whch features are vsble at the curret locato ad where they are supposed to appear whe startg the observato. The map evaluates all vsble features ad returs ther trasformed postos the vector ẑ( k + k). The measuremet model s therefore a worldtorobottosesor frame trasformato. However, due to osystematc odometry errors, the robot posto s ucerta ad due to mperfect sesors, the observato s ucerta. The former s represeted by the (predcted) state covarace matrx Pk ( + k) ad the latter by the sesor ose model wk ( + ) N ( 0, Rk ( + ) ). They are assumed to be depedet. Sesg ucertaty Rk ( + ) affects the observato drectly, whereas vehcle posto ucertaty Pk ( + k) whch s gve world map coordates wll propagate through the frame trasformatos worldtorobottosesor h ()., learzed about the predcto xˆ ( k + k). The the observato s made ad the matchg of predcted ad observed features ca be performed the sesor frame yeldg the set of matched features. The remag posto ucertaty of the matched features gve all observatos up to ad cludg tme k, Sk ( + ) cov[ z( k + ) Z k ], (6) Note that h (). s assumed to be tellget, that s, t cotas both, the mappg xˆ ( k + k) m j (wth m j as the posto vector of feature umber j ) ad the worldtosesor frame trasformato of all vsble m j for the curret predcto xˆ ( k + k). Ths s cotrast to e.g. [LEON9], where solely the frame trasformato s doe by h( xˆ ( k + k), m j ) ad the mappg xˆ ( k + k) m j s somewhere else.
14 s the supermposed ucertaty of observato ad the propagated oe from the robot pose, Sk ( + ) Sk ( + ) hpk ( + k) h + Rk ( + ). (63) s also called measuremet predcto covarace or ovato covarace. 5. Exercses As stated the troducto, the report has educatoal purposes ad accompaes a lecture o autoomous systems the mcroegeerg departemet at EPFL. Thus, the audece of ths report are people ot yet too famlarzed wth the feld of ucertaty treatmet. Some propostos for exercses are gve: Let them do the dervato of µ Y (equatos () to (7)) ad σ Y (equatos (8) to (3)) gve a few rules for the expected value. Let them do the dervato of σ YZ (equatos (5) to (3)) or, the cotext of example, equatos (93) to (0) of Appedx B for σ AR. Some rules for the expected value ad double sums mght be helpful. Let them make a llustrato of each factor affectg approxmato qualty dscussed chapter. wth drawgs lke fgure. If least squares estmato a more geeral sese s the ssue, dervatg the ft equatos for a regresso problem s qute structve. The stadard case, lear the model parameters, ad wth ucertates oly oe varable s much smpler tha the dervato of example Appedx A. Addtoally the output covarace matrx ca be determed wth (). 3
15 Lterature [BREI70] [GAND94] [DRAP88] [CARR88] [BAR93] [LEON9] [ARRAS97] A.M. Brepohl, Probablstc Systems Aalyss: A Itroducto to Probablstc Models, Decsos, ad Applcatos of Radom Processes, Joh Wley & Sos, 970. W. Gader, G. H. Golub, R. Strebel, LeastSquares Fttg of Crcles ad Ellpses, BIT, vol. 34, o. 4, p , Dec N.R. Draper, H. Smth, Appled Regresso Aalyss, 3rd edto, Joh Wley & Sos, 988. R. J. Carroll, D. Ruppert, Trasformato ad Weghtg Regresso, Chapma ad Hall, 988. Y. BarShalom, X.R. L, Estmato ad Trackg: Prcples, Techques, ad Software, Artech House, 993. J.J. Leoard, H.F. DurratWhyte, Drected Soar Sesg for Moble Robot Navgato, Kluwer Academc Publshers, 99. K.O. Arras, R.Y. Segwart, Feature Extracto ad Scee Iterpretato for Map Based Navgato ad Map Buldg, Proceedgs of SPIE, Moble Robotcs XII, Vol. 30, p. 453,
16 Appedx A: Fdg the Le Parameters r ad α the Weghted Least Square Sese Cosder the olear equato system S 0 (64) S 0 (65) where S s the weghted sum of squared errors S w ( ρ cosθ cosα + ρ sθ sα r). (66) We start solvg the system (64), (65) by workg out parameter r. S 0 w ( ρ cosθ cosα + ρ sθ sα r) ( ) (67) w ρ ( cosθ cosα + sθ sα) + w r (68) w ρ cos( θ α) + r w (69) rw w ρ cos( θ α) (70) w ρ r cos( θ α) (7) w Parameter α s slghtly more complcated. We troduce the followg otato cosθ c sθ s. (7) S w ( ρ c cosα + ρ s sα r) ( ρ c (73) cosα + ρ s sα r) w ρ c cosα + ρ s sα w j ρ j cos( θ j α) ρ (74) c sα + ρ s cosα r w j w ρ c cosα + ρ s sα w j ρ j cos( θ j α) ρ s (75) cosα ρ c sα w j ρ j s( θ j α) w j w j w ρ c s cos α ρ c cosα sα ρ c cosα w j ρ j s( θ j α) + ρ s cosα sα w j ρ s c s α ρ w s sαw j ρ j j s( θ j α) w w j ρ j j cos( θ j α) ρ s cosα w w j ρ j j cos( θ j α) ρ s cosα ( w j ) w j ρ j cos( θ j α) w j ρ j s( θ j α) w ρ c s cos α s ( α) ρ cosα sα( c s ) w j ρ ρ j c cosα s( θ j α) w w j ρ j ρ j s sα s( θ j α) w w j ρ j ρ j s cosα cos( θ j α) w w j ρ j ρ j c sα cos( θ j α) ( w j ) w j w k ρ j ρ k cos( θ j α) s( θ k α) j k w j (76) (77) 5
17 cosα w ρ c s cosα sα w ρ c w w w j ρ ρ j ( c cosα + s sα) ( s j cosα c j sα) cosα w ρ s sα w ρ c w w w j ρ ρ j [ c s j cos α c c j sα cosα + s s j sα cosα s c j s α c s j cos α + c c j sα cosα s s j sα cosα + s c j s α s c j cos α s s j sα cosα + c c j sα cosα + c s j s α ] cosαw ρ s sαw ρ c w w w j ρ ρ j ( s c j cos α s s j sα cosα + c c j sα cosα + c s j s α) cosαw ρ s sαw ρ c w w w j ρ ρ j [ sα cosα( c c j s s j ) + c s j s α s c j cos α] cosαw ρ s sαw ρ c sα cosα w w w j ρ ρ j c + j s + αw w w j ρ ρ j c s j cos α w w w j ρ ρ j s c j cosα w ρ s sα w ρ c sα w w j ρ ρ j c + j (78) (79) (80) (8) (8) (83) cosα w ρ s w w j ρ ρ j c s j sα w (84) ρ c w w j ρ ρ j c + j From (84) we ca obta the result for α or taα respectvely w w w j ρ ρ j [ c cosα( s j cosα c j sα) s sα( s j cosα c j sα) + s cosα( c j cosα + s j sα) + c sα( c j cosα + s j sα) ] w w w w j ρ ρ j c s j ( cos α s α) w w sα cosα w w w j ρ ρ j c s j w ρ s w w w j ρ ρ j c + j w ρ c (85) taα w w w j ρ ρ j cosθ s θ j w ρ sθ w w w j ρ ρ cos( j θ + θ ) j w ρ cosθ Equato (86) cotas double sums whch may ot be fully evaluated. Due to the symmetry of trgoometrc fuctos the correspodg offdagoal elemets ca be added ad thus smplfes calculato. For the fal result (87), the fourquadrat arc taget has bee take. Ths soluto, (7) ad (87), geerates sometmes ( α, r) pars wth egatve r values. They must be detected order to chage the sg of r ad to add π to the correspodg α. All αvalues le the the terval π < α 3π. (86) w α ata w w j ρ ρ s( j θ + θ ) j w < j w w + ( j)w ρ sθ w w w j ρ ρ cos( j θ + θ ) j ( w < j w w + j)w ρ cosθ (87) 6
18 Appedx B: Dervg the Covarace Matrx for r ad α The model parameter α ad r are just half the battle. Besdes these estmates of the mea posto of the le we would also lke to have a measure for ts ucertaty. Accordg to the error propagato law (), a approxmato of the output ucertaty, represeted by C AR, s subsequetly determed. At the put sde, mutually depedet ucertates radal drecto oly are assumed. The put radom vector X T [ P T Q T ] cossts of the radom vector P [ P, P,, P ] T [ Q, Q,, Q ] T deotg the varables of the measured rad, ad the radom vector Q holdg the correspodg agular varates. The put covarace matrx s therefore of the form C X C C P 0 dag( σ ρ ) 0 X 0 C Q 0 0. (88) We represet both output radom varables A, R by ther frstorder Taylor seres expaso about the mea µ T T T [ µ ρ µ θ ]. The vector µ has dmeso ad s composed of the two mea vectors µ ρ [ µ ρ, µ ρ,, µ ρ ] T ad µ θ [ µ θ, µ θ,, µ θ ] T. A αµ ( ) + [ P P µ ρ ] + X µ R r ( µ ) + [ P P µ ρ ] + X µ Q Q Q X µ [ µ θ ] Q X µ [ µ θ ] (89) (90) The relatoshps r (). ad α. () correspod to the results of Appedx A, equatos (7) ad (87). Referrg to the dscusso of chapter., we do ot kow µ advace ad ts best avalable guess s the actual value of the measuremets vector µ. It has bee show that uder the assumpto of depedece of P ad Q, the followg holds σ A σ R σ P ρ + σ P ρ + σ Q θ σ Q θ (9) (9) uder the abovemetoed assumpto of egl We further wat to kow the covarace gble agular ucertates: σ AR COV[ A, R] E[ A R] E[ A]E[ R] (93) E α + [ P µ (94) P ρ ] r + P [ P µ ρ ] αr E αr + r [ P µ (95) P ρ ] + α P [ P µ ρ ] + [ P µ P ρ ] P [ P µ ρ ] αr P E[ αr] + re P µ ρ + αe P P P P µ ρ 7
19 + E [ P P P µ ρ ][ P j µ ρ ] j αr (96) αr r P ( α)e P + [ ] r P E [ µ ρ ] + α P E [ P ] α P ()E r [ µ ρ ] + E j [ P P P µ ρ ][ P j µ ρ ] + j [ P P P µ ρ ] αr (97) E ( P (98) P P P j P µ P P j j µ P + µ P µ P ) + E j [ P P P µ ρ ] j E[ P (99) P P P j P µ P P j j µ P + µ P µ P ] + j E[ ( P P P µ ρ ) ] j E[ P (00) P P P j P µ P P j j µ P + µ P µ P ] + j σ P P ρ P j P j Sce ad are depedet, the expected value of the bracketed expresso dsappears. Hece σ AR σ P P ρ (0) If, however, the put ucertaty model provdes oeglgble agular varaces, t s easy to show that uder the depedecy assumpto of P ad Q the expresso keepg track of Q ca be smply added to yeld σ AR σ P P ρ + σ Q Q θ. (0) As demostrated chapter 3, the results (9), (9) ad (0) ca also be obtaed the more compact but less tutve form of equato (). Let F PQ P Q F P F Q P Q P P P P P P Q Q Q Q Q Q (03) the composed p Jacoba matrx cotag all partal dervates of the model parameters wth respect to the put radom varables about the guess µ. The, the sought covarace matrx ca be rewrtte as C AR C AR T F PQ C X F PQ. (04) Uder the codtos whch lead to equato (0), the rght had sde ca be decomposed yeldg T T C AR F P C P F P + F Q C Q F Q (05) 8
20 B. Practcal Cosderatos We determe mplemetable expressos for the elemets of covarace matrx C AR. Uder the assumpto of eglgble agular ucertates, cocrete expressos of σ A, σ R ad σ AR wll be derved. We must furthermore keep md that our problem mght be a real tme problem, requrg a effcet mplemetato. Expresso are therefore sought whch mmze the umer of floatg pot operatos, e.g. by reuse of already computed subexpressos. Although calculatg wth weghts does ot add much dffculty, we wll omt them ths chapter. For the sake of brevty we troduce the followg otato wth N α ata D  N (06)  P, (07) P j cosq s Q j P sq D  P. (08) P cos( j Q + Q ) j P cosq We wll use the result that the parameters α (equato (87)) ad r (equato (7)) ca also be wrtte Cartesa form, where x ρ cosθ ad y ρ sθ : α, (09)  ( y y )( x x ) ata ( y y ) ( x x ) r xcosα + y sα. (0) They use the meas x x y y. () From equatos (9), (9) ad (0) we see that the covarace matrx s defed whe both partal dervates of the parameters wth respect to P are gve. Let us start wth the dervate of α. D N D N N D N D P P P P + N D D  P D + N The partal dervates of the umerator ad the deomator wth respect to obtaed as follows: P () ca be N P  P (3) P j P k cosq j sq k { P P j sq j } P sq +  P { P c s + P c P s + P c P 3 s 3 + P c P 4 s P c P s + P c s + P c P 3 s 3 + P c P 4 s P 3 c 3 P s + P 3 c 3 P s + P 3c3 s 3 + P 3 c 3 P 4 s 4 + } (4) See [ARRAS97] for the results wth weghts. Performace comparso results of three dfferet ways to determe α ad r are also brefly gve. 9
21  P { P j cosq j P sq } + P P Q s { Q j j P s Q (5)  ( sq P j cosq j + cosq P j sq j ) P sq (6)  ( sq x + cosq y) P sq (7) ( xsq + ycosq P sq ) (8) The mea values x ad y are those of equato (). D  P (9) P P j P k cos( Q + Q j ) { P P j cosq j } P cosq +  { P P c + + P P c + + P P 3 c P P 4 c P P c + + P c + + P P 3 c P P 4 c P 3 P c P 3 P c P 3c P 3 P 4 c } (0)  { P P j P cos( Q + Q j )} + { P P P j cos( Q j + Q )} P cosq ()  { P P j P cos( Q + Q j )} P cosq ()  P j cos( Q + Q j ) P cosq (3)  P j cosq cosq j  P j sq sq j P cosq (4)  cosq P j cosq j  sq P j sq j P cosq (5)  cosq x  sq y P cosq (6) ( xcosq ysq P cosq ) (7) Substtutg to equato () gves D N N D P  P (8) P D + N ( xcosq  ysq P cosq )N ( xsq + ycosq P sq )D (9) D + N N( xcosq ysq P cosq ) D( xsq + ycosq P sq ) (30) D + N It remas the dervate of r : P  P (3) P j cos( Q j α)  (3) { P P j cos( Q j α) } 0
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS. x, where. = y  ˆ " 1
STATISTICAL PROPERTIES OF LEAST SQUARES ESTIMATORS Recall Assumpto E(Y x) η 0 + η x (lear codtoal mea fucto) Data (x, y ), (x 2, y 2 ),, (x, y ) Least squares estmator ˆ E (Y x) ˆ " 0 + ˆ " x, where ˆ
More informationChapter Eight. f : R R
Chapter Eght f : R R 8. Itroducto We shall ow tur our atteto to the very mportat specal case of fuctos that are real, or scalar, valued. These are sometmes called scalar felds. I the very, but mportat,
More informationSimple Linear Regression
Smple Lear Regresso Regresso equato a equato that descrbes the average relatoshp betwee a respose (depedet) ad a eplaator (depedet) varable. 6 8 Slopetercept equato for a le m b (,6) slope. (,) 6 6 8
More informationAPPENDIX III THE ENVELOPE PROPERTY
Apped III APPENDIX III THE ENVELOPE PROPERTY Optmzato mposes a very strog structure o the problem cosdered Ths s the reaso why eoclasscal ecoomcs whch assumes optmzg behavour has bee the most successful
More informationMeasures of Dispersion, Skew, & Kurtosis (based on Kirk, Ch. 4) {to be used in conjunction with Measures of Dispersion Chart }
Percetles Psych 54, 9/8/05 p. /6 Measures of Dsperso, kew, & Kurtoss (based o Krk, Ch. 4) {to be used cojucto wth Measures of Dsperso Chart } percetle (P % ): a score below whch a specfed percetage of
More informationThe simple linear Regression Model
The smple lear Regresso Model Correlato coeffcet s oparametrc ad just dcates that two varables are assocated wth oe aother, but t does ot gve a deas of the kd of relatoshp. Regresso models help vestgatg
More informationIDENTIFICATION OF THE DYNAMICS OF THE GOOGLE S RANKING ALGORITHM. A. Khaki Sedigh, Mehdi Roudaki
IDENIFICAION OF HE DYNAMICS OF HE GOOGLE S RANKING ALGORIHM A. Khak Sedgh, Mehd Roudak Cotrol Dvso, Departmet of Electrcal Egeerg, K.N.oos Uversty of echology P. O. Box: 163151355, ehra, Ira sedgh@eetd.ktu.ac.r,
More informationRecurrence Relations
CMPS Aalyss of Algorthms Summer 5 Recurrece Relatos Whe aalyzg the ru tme of recursve algorthms we are ofte led to cosder fuctos T ( whch are defed by recurrece relatos of a certa form A typcal example
More information6.7 Network analysis. 6.7.1 Introduction. References  Network analysis. Topological analysis
6.7 Network aalyss Le data that explctly store topologcal formato are called etwork data. Besdes spatal operatos, several methods of spatal aalyss are applcable to etwork data. Fgure: Network data Refereces
More informationANOVA Notes Page 1. Analysis of Variance for a OneWay Classification of Data
ANOVA Notes Page Aalss of Varace for a OeWa Classfcato of Data Cosder a sgle factor or treatmet doe at levels (e, there are,, 3, dfferet varatos o the prescrbed treatmet) Wth a gve treatmet level there
More informationStatistical Pattern Recognition (CE725) Department of Computer Engineering Sharif University of Technology
I The Name of God, The Compassoate, The ercful Name: Problems' eys Studet ID#:. Statstcal Patter Recogto (CE725) Departmet of Computer Egeerg Sharf Uversty of Techology Fal Exam Soluto  Sprg 202 (50
More informationSHAPIROWILK TEST FOR NORMALITY WITH KNOWN MEAN
SHAPIROWILK TEST FOR NORMALITY WITH KNOWN MEAN Wojcech Zelńsk Departmet of Ecoometrcs ad Statstcs Warsaw Uversty of Lfe Sceces Nowoursyowska 66, 787 Warszawa emal: wojtekzelsk@statystykafo Zofa Hausz,
More informationChapter 3 31. Chapter Goals. Summary Measures. Chapter Topics. Measures of Center and Location. Notation Conventions
Chapter 3 3 Chapter Goals Chapter 3 umercal Descrptve Measures After completg ths chapter, you should be able to: Compute ad terpret the mea, meda, ad mode for a set of data Fd the rage, varace, ad stadard
More informationThe Digital Signature Scheme MQQSIG
The Dgtal Sgature Scheme MQQSIG Itellectual Property Statemet ad Techcal Descrpto Frst publshed: 10 October 2010, Last update: 20 December 2010 Dalo Glgorosk 1 ad Rue Stesmo Ødegård 2 ad Rue Erled Jese
More informationLecture 4. Materials Covered: Chapter 7 Suggested Exercises: 7.1, 7.5, 7.7, 7.10, 7.11, 7.19, 7.20, 7.23, 7.44, 7.45, 7.47.
TT 430, ummer 006 Lecture 4 Materals Covered: Chapter 7 uggested Exercses: 7., 7.5, 7.7, 7.0, 7., 7.9, 7.0, 7.3, 7.44, 7.45, 7.47.. Deftos. () Parameter: A umercal summary about the populato. For example:
More informationChapter 11 Systematic Sampling
Chapter Sstematc Samplg The sstematc samplg techue s operatoall more coveet tha the smple radom samplg. It also esures at the same tme that each ut has eual probablt of cluso the sample. I ths method of
More informationSettlement Prediction by Spatialtemporal Random Process
Safety, Relablty ad Rs of Structures, Ifrastructures ad Egeerg Systems Furuta, Fragopol & Shozua (eds Taylor & Fracs Group, Lodo, ISBN 97877 Settlemet Predcto by Spataltemporal Radom Process P. Rugbaapha
More informationAverage Price Ratios
Average Prce Ratos Morgstar Methodology Paper August 3, 2005 2005 Morgstar, Ic. All rghts reserved. The formato ths documet s the property of Morgstar, Ic. Reproducto or trascrpto by ay meas, whole or
More informationThe GompertzMakeham distribution. Fredrik Norström. Supervisor: Yuri Belyaev
The GompertzMakeham dstrbuto by Fredrk Norström Master s thess Mathematcal Statstcs, Umeå Uversty, 997 Supervsor: Yur Belyaev Abstract Ths work s about the GompertzMakeham dstrbuto. The dstrbuto has
More informationOn Error Detection with Block Codes
BULGARIAN ACADEMY OF SCIENCES CYBERNETICS AND INFORMATION TECHNOLOGIES Volume 9, No 3 Sofa 2009 O Error Detecto wth Block Codes Rostza Doduekova Chalmers Uversty of Techology ad the Uversty of Gotheburg,
More informationOptimal multidegree reduction of Bézier curves with constraints of endpoints continuity
Computer Aded Geometrc Desg 19 (2002 365 377 wwwelsevercom/locate/comad Optmal multdegree reducto of Bézer curves wth costrats of edpots cotuty GuoDog Che, GuoJ Wag State Key Laboratory of CAD&CG, Isttute
More informationof the relationship between time and the value of money.
TIME AND THE VALUE OF MONEY Most agrbusess maagers are famlar wth the terms compoudg, dscoutg, auty, ad captalzato. That s, most agrbusess maagers have a tutve uderstadg that each term mples some relatoshp
More informationInduction Proofs. ) ( for all n greater than or equal to n. is a fixed integer. A proof by Mathematical Induction contains two steps:
CMPS Algorthms ad Abstract Data Types Iducto Proofs Let P ( be a propostoal fucto, e P s a fucto whose doma s (some subset of) the set of tegers ad whose codoma s the set {True, False} Iformally, ths meas
More informationAbraham Zaks. Technion I.I.T. Haifa ISRAEL. and. University of Haifa, Haifa ISRAEL. Abstract
Preset Value of Autes Uder Radom Rates of Iterest By Abraham Zas Techo I.I.T. Hafa ISRAEL ad Uversty of Hafa, Hafa ISRAEL Abstract Some attempts were made to evaluate the future value (FV) of the expected
More informationCurve Fitting and Solution of Equation
UNIT V Curve Fttg ad Soluto of Equato 5. CURVE FITTING I ma braches of appled mathematcs ad egeerg sceces we come across epermets ad problems, whch volve two varables. For eample, t s kow that the speed
More informationOnline Appendix: Measured Aggregate Gains from International Trade
Ole Appedx: Measured Aggregate Gas from Iteratoal Trade Arel Burste UCLA ad NBER Javer Cravo Uversty of Mchga March 3, 2014 I ths ole appedx we derve addtoal results dscussed the paper. I the frst secto,
More informationAn Approach to Evaluating the Computer Network Security with Hesitant Fuzzy Information
A Approach to Evaluatg the Computer Network Securty wth Hestat Fuzzy Iformato Jafeg Dog A Approach to Evaluatg the Computer Network Securty wth Hestat Fuzzy Iformato Jafeg Dog, Frst ad Correspodg Author
More informationThe analysis of annuities relies on the formula for geometric sums: r k = rn+1 1 r 1. (2.1) k=0
Chapter 2 Autes ad loas A auty s a sequece of paymets wth fxed frequecy. The term auty orgally referred to aual paymets (hece the ame), but t s ow also used for paymets wth ay frequecy. Autes appear may
More informationCredibility Premium Calculation in Motor ThirdParty Liability Insurance
Advaces Mathematcal ad Computatoal Methods Credblty remum Calculato Motor Thrdarty Lablty Isurace BOHA LIA, JAA KUBAOVÁ epartmet of Mathematcs ad Quattatve Methods Uversty of ardubce Studetská 95, 53
More informationT = 1/freq, T = 2/freq, T = i/freq, T = n (number of cash flows = freq n) are :
Bullets bods Let s descrbe frst a fxed rate bod wthout amortzg a more geeral way : Let s ote : C the aual fxed rate t s a percetage N the otoal freq ( 2 4 ) the umber of coupo per year R the redempto of
More informationECONOMIC CHOICE OF OPTIMUM FEEDER CABLE CONSIDERING RISK ANALYSIS. University of Brasilia (UnB) and The Brazilian Regulatory Agency (ANEEL), Brazil
ECONOMIC CHOICE OF OPTIMUM FEEDER CABE CONSIDERING RISK ANAYSIS I Camargo, F Fgueredo, M De Olvera Uversty of Brasla (UB) ad The Brazla Regulatory Agecy (ANEE), Brazl The choce of the approprate cable
More informationn. We know that the sum of squares of p independent standard normal variables has a chi square distribution with p degrees of freedom.
UMEÅ UNIVERSITET Matematskstatstska sttutoe Multvarat dataaalys för tekologer MSTB0 PA TENTAMEN 00409 LÖSNINGSFÖRSLAG TILL TENTAMEN I MATEMATISK STATISTIK Multvarat dataaalys för tekologer B, 5 poäg.
More informationApplications of Support Vector Machine Based on Boolean Kernel to Spam Filtering
Moder Appled Scece October, 2009 Applcatos of Support Vector Mache Based o Boolea Kerel to Spam Flterg Shugag Lu & Keb Cu School of Computer scece ad techology, North Cha Electrc Power Uversty Hebe 071003,
More informationNumerical Methods with MS Excel
TMME, vol4, o.1, p.84 Numercal Methods wth MS Excel M. ElGebely & B. Yushau 1 Departmet of Mathematcal Sceces Kg Fahd Uversty of Petroleum & Merals. Dhahra, Saud Araba. Abstract: I ths ote we show how
More information22. The accompanying data describe flexural strength (Mpa) for concrete beams of a certain type was introduced in Example 1.2.
. The accompayg data descrbe flexural stregth (Mpa) for cocrete beams of a certa type was troduced Example.. 9. 9.7 8.8 0.7 8.4 8.7 0.7 6.9 8. 8.3 7.3 9. 7.8 8.0 8.6 7.8 7.5 8.0 7.3 8.9 0.0 8.8 8.7.6.3.8.7
More informationModels for Selecting an ERP System with Intuitionistic Trapezoidal Fuzzy Information
JOURNAL OF SOFWARE, VOL 5, NO 3, MARCH 00 75 Models for Selectg a ERP System wth Itutostc rapezodal Fuzzy Iformato Guwu We, Ru L Departmet of Ecoomcs ad Maagemet, Chogqg Uversty of Arts ad Sceces, Yogchua,
More informationFractalStructured Karatsuba`s Algorithm for Binary Field Multiplication: FK
FractalStructured Karatsuba`s Algorthm for Bary Feld Multplcato: FK *The authors are worg at the Isttute of Mathematcs The Academy of Sceces of DPR Korea. **Address : U Jog dstrct Kwahadog Number Pyogyag
More informationREGRESSION II: Hypothesis Testing in Regression. Excel Regression Output. Excel Regression Output. A look at the sources of Variation in the Model
REGRESSION II: Hypothess Testg Regresso Tom Ilveto FREC 408 Model Regressg SAT (Y o Percet Takg (X Y s the Depedet Varable State average SAT Score 999  SATOTAL X s the Idepedet Varable Percet of hgh school
More informationChapter 12 Polynomial Regression Models
Chapter Polyomal Regresso Models A model s sad to be lear whe t s lear parameters. So the model ad y = + x+ x + β β β ε y= β + β x + β x + β x + β x + β xx + ε are also the lear model. I fact, they are
More informationChapter 6: Testing Linear Hypotheses
Chapter 6: Testg Lear ypotheses Prerequstes: Chapter 5 6 The Dstrbuto of the Regresso Model Estmator Accordg to Theorem (49), f we have a radom vector a such that the varace of a s kow, V(a) C, lets say,
More informationPreprocess a planar map S. Given a query point p, report the face of S containing p. Goal: O(n)size data structure that enables O(log n) query time.
Computatoal Geometry Chapter 6 Pot Locato 1 Problem Defto Preprocess a plaar map S. Gve a query pot p, report the face of S cotag p. S Goal: O()sze data structure that eables O(log ) query tme. C p E
More informationForecasting Trend and Stock Price with Adaptive Extended Kalman Filter Data Fusion
2011 Iteratoal Coferece o Ecoomcs ad Face Research IPEDR vol.4 (2011 (2011 IACSIT Press, Sgapore Forecastg Tred ad Stoc Prce wth Adaptve Exteded alma Flter Data Fuso Betollah Abar Moghaddam Faculty of
More informationReinsurance and the distribution of term insurance claims
Resurace ad the dstrbuto of term surace clams By Rchard Bruyel FIAA, FNZSA Preseted to the NZ Socety of Actuares Coferece Queestow  November 006 1 1 Itroducto Ths paper vestgates the effect of resurace
More informationCHAPTER 2. Time Value of Money 61
CHAPTER 2 Tme Value of Moey 6 Tme Value of Moey (TVM) Tme Les Future value & Preset value Rates of retur Autes & Perpetutes Ueve cash Flow Streams Amortzato 62 Tme les 0 2 3 % CF 0 CF CF 2 CF 3 Show
More informationRegression Analysis. 1. Introduction
. Itroducto Regresso aalyss s a statstcal methodology that utlzes the relato betwee two or more quattatve varables so that oe varable ca be predcted from the other, or others. Ths methodology s wdely used
More informationLoad and Resistance Factor Design (LRFD)
53:134 Structural Desg II Load ad Resstace Factor Desg (LRFD) Specfcatos ad Buldg Codes: Structural steel desg of buldgs the US s prcpally based o the specfcatos of the Amerca Isttute of Steel Costructo
More informationSecurity Analysis of RAPP: An RFID Authentication Protocol based on Permutation
Securty Aalyss of RAPP: A RFID Authetcato Protocol based o Permutato Wag Shaohu,,, Ha Zhje,, Lu Sujua,, Che Dawe, {College of Computer, Najg Uversty of Posts ad Telecommucatos, Najg 004, Cha Jagsu Hgh
More informationChapter 3. AMORTIZATION OF LOAN. SINKING FUNDS R =
Chapter 3. AMORTIZATION OF LOAN. SINKING FUNDS Objectves of the Topc: Beg able to formalse ad solve practcal ad mathematcal problems, whch the subjects of loa amortsato ad maagemet of cumulatve fuds are
More information1. The Time Value of Money
Corporate Face [000345]. The Tme Value of Moey. Compoudg ad Dscoutg Captalzato (compoudg, fdg future values) s a process of movg a value forward tme. It yelds the future value gve the relevat compoudg
More informationMEASURES OF CENTRAL TENDENCY
MODULE  6 Statstcs Measures of Cetral Tedecy 25 MEASURES OF CENTRAL TENDENCY I the prevous lesso, we have leart that the data could be summarsed to some extet by presetg t the form of a frequecy table.
More informationEcon107 Applied Econometrics Topic 4: Hypothesis Testing (Studenmund, Chapter 5)
Page Eco07 Appled Ecoometrcs Topc 4: Hypothess Testg (Studemud, Chapter 5). Statstcal ferece: Revew Statstcal ferece... draws coclusos from (or makes fereces about) a populato from a radom sample take
More informationBayesian Network Representation
Readgs: K&F 3., 3.2, 3.3, 3.4. Bayesa Network Represetato Lecture 2 Mar 30, 20 CSE 55, Statstcal Methods, Sprg 20 Istructor: SuI Lee Uversty of Washgto, Seattle Last tme & today Last tme Probablty theory
More informationCH. V ME256 STATICS Center of Gravity, Centroid, and Moment of Inertia CENTER OF GRAVITY AND CENTROID
CH. ME56 STTICS Ceter of Gravt, Cetrod, ad Momet of Ierta CENTE OF GITY ND CENTOID 5. CENTE OF GITY ND CENTE OF MSS FO SYSTEM OF PTICES Ceter of Gravt. The ceter of gravt G s a pot whch locates the resultat
More informationADAPTATION OF SHAPIROWILK TEST TO THE CASE OF KNOWN MEAN
Colloquum Bometrcum 4 ADAPTATION OF SHAPIROWILK TEST TO THE CASE OF KNOWN MEAN Zofa Hausz, Joaa Tarasńska Departmet of Appled Mathematcs ad Computer Scece Uversty of Lfe Sceces Lubl Akademcka 3, 95 Lubl
More informationA New Bayesian Network Method for Computing Bottom Event's Structural Importance Degree using Jointree
, pp.277288 http://dx.do.org/10.14257/juesst.2015.8.1.25 A New Bayesa Network Method for Computg Bottom Evet's Structural Importace Degree usg Jotree Wag Yao ad Su Q School of Aeroautcs, Northwester Polytechcal
More informationSpeeding up kmeans Clustering by Bootstrap Averaging
Speedg up meas Clusterg by Bootstrap Averagg Ia Davdso ad Ashw Satyaarayaa Computer Scece Dept, SUNY Albay, NY, USA,. {davdso, ashw}@cs.albay.edu Abstract Kmeas clusterg s oe of the most popular clusterg
More informationLecture Sheet 02: Measures of Central Tendency and Dispersion
EE 3 Numercal Methods & Statstcs Lecture Sheet 0: Measures o Cetral Tedecy ad Dsperso Lecture o Descrptve Statstcs Measures o Cetral Tedecy Measures o Locato Measures o Dsperso Measures o Symmetry Measures
More informationPerformance Attribution. Methodology Overview
erformace Attrbuto Methodology Overvew Faba SUAREZ March 2004 erformace Attrbuto Methodology 1.1 Itroducto erformace Attrbuto s a set of techques that performace aalysts use to expla why a portfolo's performace
More informationwhere p is the centroid of the neighbors of p. Consider the eigenvector problem
Vrtual avgato of teror structures by ldar Yogja X a, Xaolg L a, Ye Dua a, Norbert Maerz b a Uversty of Mssour at Columba b Mssour Uversty of Scece ad Techology ABSTRACT I ths project, we propose to develop
More informationRobust Realtime Face Recognition And Tracking System
JCS& Vol. 9 No. October 9 Robust Realtme Face Recogto Ad rackg System Ka Che,Le Ju Zhao East Cha Uversty of Scece ad echology Emal:asa85@hotmal.com Abstract here s some very mportat meag the study of realtme
More informationDECISION MAKING WITH THE OWA OPERATOR IN SPORT MANAGEMENT
ESTYLF08, Cuecas Meras (Meres  Lagreo), 79 de Septembre de 2008 DECISION MAKING WITH THE OWA OPERATOR IN SPORT MANAGEMENT José M. Mergó Aa M. GlLafuete Departmet of Busess Admstrato, Uversty of Barceloa
More informationBeta. A Statistical Analysis of a Stock s Volatility. Courtney Wahlstrom. Iowa State University, Master of School Mathematics. Creative Component
Beta A Statstcal Aalyss of a Stock s Volatlty Courtey Wahlstrom Iowa State Uversty, Master of School Mathematcs Creatve Compoet Fall 008 Amy Froelch, Major Professor Heather Bolles, Commttee Member Travs
More informationTaylor & Francis, Ltd. is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Experimental Education.
The Statstcal Iterpretato of Degrees of Freedom Author(s): Wllam J. Mooa Source: The Joural of Expermetal Educato, Vol. 21, No. 3 (Mar., 1953), pp. 259264 Publshed by: Taylor & Fracs, Ltd. Stable URL:
More informationBanking (Early Repayment of Housing Loans) Order, 5762 2002 1
akg (Early Repaymet of Housg Loas) Order, 5762 2002 y vrtue of the power vested me uder Secto 3 of the akg Ordace 94 (hereafter, the Ordace ), followg cosultato wth the Commttee, ad wth the approval of
More informationAn Effectiveness of Integrated Portfolio in Bancassurance
A Effectveess of Itegrated Portfolo Bacassurace Taea Karya Research Ceter for Facal Egeerg Isttute of Ecoomc Research Kyoto versty Sayouu Kyoto 606850 Japa arya@eryotouacp Itroducto As s well ow the
More informationMaintenance Scheduling of Distribution System with Optimal Economy and Reliability
Egeerg, 203, 5, 48 http://dx.do.org/0.4236/eg.203.59b003 Publshed Ole September 203 (http://www.scrp.org/joural/eg) Mateace Schedulg of Dstrbuto System wth Optmal Ecoomy ad Relablty Syua Hog, Hafeg L,
More informationProjection model for Computer Network Security Evaluation with intervalvalued intuitionistic fuzzy information. Qingxiang Li
Iteratoal Joural of Scece Vol No7 05 ISSN: 834890 Proecto model for Computer Network Securty Evaluato wth tervalvalued tutostc fuzzy formato Qgxag L School of Software Egeerg Chogqg Uversty of rts ad
More informationSTOCHASTIC approximation algorithms have several
IEEE TRANSACTIONS ON INFORMATION THEORY, VOL 60, NO 10, OCTOBER 2014 6609 Trackg a MarkovModulated Statoary Degree Dstrbuto of a Dyamc Radom Graph Mazyar Hamd, Vkram Krshamurthy, Fellow, IEEE, ad George
More informationNear Neighbor Distribution in Sets of Fractal Nature
Iteratoal Joural of Computer Iformato Systems ad Idustral Maagemet Applcatos. ISS 2507988 Volume 5 (202) 3 pp. 5966 MIR Labs, www.mrlabs.et/jcsm/dex.html ear eghbor Dstrbuto Sets of Fractal ature Marcel
More informationThe Analysis of Development of Insurance Contract Premiums of General Liability Insurance in the Business Insurance Risk
The Aalyss of Developmet of Isurace Cotract Premums of Geeral Lablty Isurace the Busess Isurace Rsk the Frame of the Czech Isurace Market 1998 011 Scetfc Coferece Jue, 10.  14. 013 Pavla Kubová Departmet
More informationClassic Problems at a Glance using the TVM Solver
C H A P T E R 2 Classc Problems at a Glace usg the TVM Solver The table below llustrates the most commo types of classc face problems. The formulas are gve for each calculato. A bref troducto to usg the
More information10.5 Future Value and Present Value of a General Annuity Due
Chapter 10 Autes 371 5. Thomas leases a car worth $4,000 at.99% compouded mothly. He agrees to make 36 lease paymets of $330 each at the begg of every moth. What s the buyout prce (resdual value of the
More informationU t + u U x µ 2 U = 0. (101)
Chapter 3 Fte Dfferece Methods I the prevous chapter we developed fte dfferece appromatos for partal dervatves. I ths chapter we wll use these fte dfferece appromatos to solve partal dfferetal equatos
More informationA Bayesian Networks in Intrusion Detection Systems
Joural of Computer Scece 3 (5: 5965, 007 ISSN 5493636 007 Scece Publcatos A Bayesa Networs Itruso Detecto Systems M. Mehd, S. Zar, A. Aou ad M. Besebt Electrocs Departmet, Uversty of Blda, Algera Abstract:
More informationOptimal replacement and overhaul decisions with imperfect maintenance and warranty contracts
Optmal replacemet ad overhaul decsos wth mperfect mateace ad warraty cotracts R. Pascual Departmet of Mechacal Egeerg, Uversdad de Chle, Caslla 2777, Satago, Chle Phoe: +5626784591 Fax:+562689657 rpascual@g.uchle.cl
More informationA Study of Unrelated ParallelMachine Scheduling with Deteriorating Maintenance Activities to Minimize the Total Completion Time
Joural of Na Ka, Vol. 0, No., pp.59 (20) 5 A Study of Urelated ParallelMache Schedulg wth Deteroratg Mateace Actvtes to Mze the Total Copleto Te SuhJeq Yag, JaYuar Guo, HsTao Lee Departet of Idustral
More informationand postulate that dependencies among the regression residuals (errors), u
Part III. Areal Data Aalyss 6. Spatal Regresso Models for Areal Data Aalyss The prmary models of terest for areal data aalyss are regresso models. I the same way that georegresso models were used to study
More information20092015 Michael J. Rosenfeld, draft version 1.7 (under construction). draft November 5, 2015
009015 Mchael J. Rosefeld, draft verso 1.7 (uder costructo). draft November 5, 015 Notes o the Mea, the Stadard Devato, ad the Stadard Error. Practcal Appled Statstcs for Socologsts. A troductory word
More informationCIS603  Artificial Intelligence. Logistic regression. (some material adopted from notes by M. Hauskrecht) CIS603  AI. Supervised learning
CIS63  Artfcal Itellgece Logstc regresso Vasleos Megalookoomou some materal adopted from otes b M. Hauskrecht Supervsed learg Data: D { d d.. d} a set of eamples d < > s put vector ad s desred output
More informationConversion of NonLinear Strength Envelopes into Generalized HoekBrown Envelopes
Covero of NoLear Stregth Evelope to Geeralzed HoekBrow Evelope Itroducto The power curve crtero commoly ued lmtequlbrum lope tablty aaly to defe a olear tregth evelope (relatohp betwee hear tre, τ,
More informationPreparation of Calibration Curves
Preparato of Calbrato Curves A Gude to Best Practce September 3 Cotact Pot: Lz Prchard Tel: 8943 7553 Prepared by: Vck Barwck Approved by: Date: The work descrbed ths report was supported uder cotract
More informationCSSE463: Image Recognition Day 27
CSSE463: Image Recogto Da 27 Ths week Toda: Alcatos of PCA Suda ght: roject las ad relm work due Questos? Prcal Comoets Aalss weght grth c ( )( ) ( )( ( )( ) ) heght sze Gve a set of samles, fd the drecto(s)
More informationStudy on prediction of network security situation based on fuzzy neutral network
Avalable ole www.ocpr.com Joural of Chemcal ad Pharmaceutcal Research, 04, 6(6):0006 Research Artcle ISS : 09757384 CODE(USA) : JCPRC5 Study o predcto of etwork securty stuato based o fuzzy eutral etwork
More informationM. Salahi, F. Mehrdoust, F. Piri. CVaR Robust MeanCVaR Portfolio Optimization
M. Salah, F. Mehrdoust, F. Pr Uversty of Gula, Rasht, Ira CVaR Robust MeaCVaR Portfolo Optmzato Abstract: Oe of the most mportat problems faced by every vestor s asset allocato. A vestor durg makg vestmet
More informationDynamic Twophase Truncated Rayleigh Model for Release Date Prediction of Software
J. Software Egeerg & Applcatos 3 6369 do:.436/jsea..367 Publshed Ole Jue (http://www.scrp.org/joural/jsea) Dyamc Twophase Trucated Raylegh Model for Release Date Predcto of Software Lafe Qa Qgchua Yao
More informationProbability, Statistics, and Reliability for Engineers and Scientists MULTIPLE RANDOM VARIABLES
CHAPTR Probablt, Statstcs, ad Relablt or geers ad Scetsts MULTIPL RANDOM VARIABLS Secod dto A. J. Clark School o geerg Departmet o Cvl ad vrometal geerg 6b Probablt ad Statstcs or Cvl geers Departmet o
More informationUSEFULNESS OF BOOTSTRAPPING IN PORTFOLIO MANAGEMENT
USEFULNESS OF BOOTSTRAPPING IN PORTFOLIO MANAGEMENT Radovaov Bors Faculty of Ecoomcs Subotca Segedsk put 911 Subotca 24000 Emal: radovaovb@ef.us.ac.rs Marckć Aleksadra Faculty of Ecoomcs Subotca Segedsk
More informationFuzzy Reliability of a Marine Power Plant Using Interval Valued Vague Sets
Iteratoal Joural of Appled Scece ad Egeerg 006. 4 : 78 uzzy Relablty of a Mare Power Plat Usg Iterval alued ague Sets Amt Kumar a Shv Prasad Yadav a * ad Suredra Kumar b a Departmet of Mathematcs Ida
More informationα 2 α 1 β 1 ANTISYMMETRIC WAVEFUNCTIONS: SLATER DETERMINANTS (08/24/14)
ANTISYMMETRI WAVEFUNTIONS: SLATER DETERMINANTS (08/4/4) Wavefuctos that descrbe more tha oe electro must have two characterstc propertes. Frst, scll electros are detcal partcles, the electros coordates
More informationA Realtime Visual Tracking System in the Robot Soccer Domain
Proceedgs of EUEL obotcs, Salford, Eglad, th  th Aprl A ealtme Vsual Trackg System the obot Soccer Doma Bo L, Edward Smth, Huosheg Hu, Lbor Spacek Departmet of Computer Scece, Uversty of Essex, Wvehoe
More informationConstrained Cubic Spline Interpolation for Chemical Engineering Applications
Costraed Cubc Sple Iterpolato or Chemcal Egeerg Applcatos b CJC Kruger Summar Cubc sple terpolato s a useul techque to terpolate betwee kow data pots due to ts stable ad smooth characterstcs. Uortuatel
More informationStatistical Intrusion Detector with InstanceBased Learning
Iformatca 5 (00) xxx yyy Statstcal Itruso Detector wth IstaceBased Learg Iva Verdo, Boja Nova Faulteta za eletroteho raualštvo Uverza v Marboru Smetaova 7, 000 Marbor, Sloveja va.verdo@sol.et eywords:
More informationCommon pbelief: The General Case
GAMES AND ECONOMIC BEHAVIOR 8, 738 997 ARTICLE NO. GA97053 Commo pbelef: The Geeral Case Atsush Kaj* ad Stephe Morrs Departmet of Ecoomcs, Uersty of Pesylaa Receved February, 995 We develop belef operators
More informationDETERMINISTIC AND STOCHASTIC MODELLING OF TECHNICAL RESERVES IN SHORTTERM INSURANCE CONTRACTS
DETERMINISTI AND STOHASTI MODELLING OF TEHNIAL RESERVES IN SHORTTERM INSURANE ONTRATS Patrck G O Weke School of Mathematcs, Uversty of Narob, Keya Emal: pweke@uobacke ABSTART lams reservg for geeral surace
More informationA particle swarm optimization to vehicle routing problem with fuzzy demands
A partcle swarm optmzato to vehcle routg problem wth fuzzy demads Yag Peg, Yeme Qa A partcle swarm optmzato to vehcle routg problem wth fuzzy demads Yag Peg 1,Yeme Qa 1 School of computer ad formato
More informationAnalysis of onedimensional consolidation of soft soils with nondarcian flow caused by nonnewtonian liquid
Joural of Rock Mechacs ad Geotechcal Egeerg., 4 (3): 5 57 Aalyss of oedmesoal cosoldato of soft sols wth odarca flow caused by onewtoa lqud Kaghe Xe, Chuaxu L, *, Xgwag Lu 3, Yul Wag Isttute of Geotechcal
More informationCompressive Sensing over Strongly Connected Digraph and Its Application in Traffic Monitoring
Compressve Sesg over Strogly Coected Dgraph ad Its Applcato Traffc Motorg Xao Q, Yogca Wag, Yuexua Wag, Lwe Xu Isttute for Iterdscplary Iformato Sceces, Tsghua Uversty, Bejg, Cha {qxao3, kyo.c}@gmal.com,
More informationThe impact of serviceoriented architecture on the scheduling algorithm in cloud computing
Iteratoal Research Joural of Appled ad Basc Sceces 2015 Avalable ole at www.rjabs.com ISSN 2251838X / Vol, 9 (3): 387392 Scece Explorer Publcatos The mpact of servceoreted archtecture o the schedulg
More informationGreen Master based on MapReduce Cluster
Gree Master based o MapReduce Cluster MgZh Wu, YuChag L, WeTsog Lee, YuSu L, FogHao Lu Dept of Electrcal Egeerg Tamkag Uversty, Tawa, ROC Dept of Electrcal Egeerg Tamkag Uversty, Tawa, ROC Dept of
More informationANNEX 77 FINANCE MANAGEMENT. (Working material) Chief Actuary Prof. Gaida Pettere BTA INSURANCE COMPANY SE
ANNEX 77 FINANCE MANAGEMENT (Workg materal) Chef Actuary Prof. Gada Pettere BTA INSURANCE COMPANY SE 1 FUNDAMENTALS of INVESTMENT I THEORY OF INTEREST RATES 1.1 ACCUMULATION Iterest may be regarded as
More information