A Theoretical Understanding of 2 Base Color Codes and Its Application to Annotation, Error Detection, and Error Correction

Size: px
Start display at page:

Download "A Theoretical Understanding of 2 Base Color Codes and Its Application to Annotation, Error Detection, and Error Correction"

Transcription

1 Whte Paper SOLD System heoretcal Understandng of 2 Base olor odes and Its pplcaton to nnotaton, Error Detecton, and Error orrecton Methods for nnotatng 2 Base olor Encoded Reads n the SOLD System Henz Breu Introducton he SOLD System enables massvely parallel sequencng of clonally amplfed DN fragments lned to beads. hs unque sequencng methodology s based on sequental lgaton of dye-labeled olgonucleotde probes whereby each probe assays two base postons at a tme. he system uses four fluorescent dyes to encode for the sxteen possble two-base combnatons. hs unque approach employs a scheme that represents a fragment of DN as an ntal base followed by a sequence of overlappng dmers (adacent pars of bases). he system encodes each dmer wth one of four colors usng a degenerate codng scheme that satsfes a number of rules. sngle color n the read can represent any of four dmers, but the overlappng propertes of the dmers and the nature of the color code allow for error-correctng propertes. In ths document, we dscuss the theory that explans these errorcorrectng propertes, show how to correct the msapplcatons of these propertes, and descrbe software algorthms to utlze and verfy the 2 base encodng scheme. For example, we can dentfy and annotate solated erroneous color calls, as well as color-reads that correspond to solated blocs of adacent nucleotde varants from a reference, most realstcally one, two, or three, but as many as the applcaton mght requre. onstructng the 2 Base olor ode he SOLD System s 2 base color codng scheme s shown n Fgure 1. code dye FM y3 XR y5 Use the followng steps to encode a DN sequence * : 1. start at the 5' end, Fgure 1: SOLD System s 2 base odng Scheme. he column under code lsts the correspondng dye and the d-bases (adacent nucleotdes) encoded by color. For example, s labeled wth y3 and coded as replace the d-base at ths poston wth ts correspondng code 3 from the table, 3. advance by one base, whch exposes the d-base, and 4. contnue, as shown below. Base Sequence: olor Strng: hs process encodes a -mer of bases as a (-1)-mer of colors. lthough ths color strng codes for four dfferent -mers, nowledge of the type and poston of any of ts bases helps to encode the sequence. For SOLD sequencng applcatons, prepend the leadng base to result n a -mer (from the example above) from whch the base sequence can be reconstructed. * he example depcted here uses an as the frst base. In practce, the current chemstry on the SOLD System uses a as the frst base.

2 SOLD Substrate D-base Probes 1 μm bead 5 P1 dapter emplate Sequence n n n z z z 5 n n n z z z 5 n n n z z z 5 EMPLE 2nd Base 1st Base n n n z z z 5 lass Slde leavage Ste 1. Prme and Lgate 5. Repeat steps 1-4 to Extend Sequence P OH PRIMER ROUND 1 + Lgase Lgaton cycle (n cycles) 1 μm bead Unversal seq prmer (n) P1 dapter emplate Sequence 2. Image Excte Fluorescence 6. Prmer Reset Unversal seq prmer (n-1) 1 μm bead 2. Prmer reset 1. Melt off extended sequence 3. ap Unextended Strands Phosphatase PO 4 7. Repeat steps 1-5 wth new prmer 1 base shft PRIMER ROUND leave off Fluor leavage gent HO Unversal seq prmer (n-1) 1 μm bead P 8. Repeat Reset wth, n-2, n-3, n-4 prmers Read Poston Unversal seq prmer (n) Prmer Round Unversal seq prmer (n-1) Unversal seq prmer (n-2) Unversal seq prmer (n-3) Brdge Probe Brdge Probe 5 Unversal seq prmer (n-4) Brdge Probe Indcates postons of nterogaton Lgaton ycle Fgure 2. Lgaton based sequencng wth d-base probes usng the SOLD System. hs schematc shows bases nterrogated by the d-base probes at postons 1 and 2.

3 he SOLD System generates ts reads n precsely ths encoded form. One way of accomplshng ths s shown n Fgure 2. Requrements/Propertes for a 2 Base olor ode Scheme he 2 base color codng scheme possesses certan propertes that wll be dscussed later n the document. Interestngly, ths scheme s essentally the only code that satsfes these propertes. hs can be observed by treatng the propertes as requrements and constructng the color code from them. hs document deals only wth bases, not other IUB (Internatonal Unon of Bochemstry) codes. So let B = {,,, }. he color code should satsfy the followng requrements: For all bases b, d, e n B: 1. he avalable colors are 0, 1, 2, and 3: color (bd) {0, 1, 2, 3}. 2 wo dfferent d-bases that have the same frst base get dfferent colors: color (bd) color (be) f d e. For example, color () color (). 3. d-base and ts reverse get the same color: color (bd) = color (db). For example, color () = color (). 4. Monodbases get the same color: color (bb) = color (dd). hat s, color () = color () = color () = color (). he followng are not requrements, but nterestng propertes that follow from these four. Property 5 follows from requrements 2 and 3, and wll mae our constructon easer. 5. wo dfferent d-bases that nevertheless have the same second base get dfferent colors: color (bd) color (ed), f b e. For example, color () color (). Property 6 also follows from requrements 1-4, but t s most easly verfed aganst the completed code (Fgure 3, Panel E). 6. d-base and ts complement get the same color: color (b c d c ) = color (d c b c ). For example, color () = color (). Satsfyng the Requrements for a 2 Base odng System he remander of ths document uses a notaton dfferent from Fgure 1. Fgure 3 lsts the colors for each d-base. For example, the value n row and column wll be the color (2) for d-base (Fgure 3, Panel ). Requrements 1 and 2 requre that all colors are present n the frst row. Because the system can use any one-to-one mappng between the actual dyes and the labels 0, 1, 2, and 3 (provded that requrements 1 and 2 reman satsfed) then the frst row (row ) can be Frst Base 2 Frst Base Frst Base Panel Panel B Panel Frst Base Frst Base Panel D Panel E Fgure 3. Requrements that ssgn the olor for a 2 Base ode.

4 labeled as shown (Fgure 3, Panel B.) Requrement 3, that color (bd) = color (db), gves a unque labelng for column (Fgure 3, Panel ). Requrement 4, that color (bb) = color (), gves a unque labelng for the dagonal (Fgure 3, Panel D). Fnally, requrements 1, 2, and 5, state that every color must appear n every row and every column exactly once (Fgure 3, Panel E). he table (Fgure 3, Panel E) s easy to memorze and wor wth because, by vrtue of Property 3, one can thn of d-bases as two-element sets for assgnng colors. he d-bases startng wth get colors 0, 1, 2, and 3 respectvely. herefore: 0.,,, all get color 0, 1. and get color 1, and so must and, 2. and get color 2, and so must and, 3. and get color 3, and so must and. Determnng the olor Strng of the Sequence Wthout the leadng base, t s not possble to determne f a partcular DN sequence s -rch from ts color strng alone. Here, we wll show that for any -rch sequence S, there exsts another sequence S', wth exactly the same color strng. Mae the new sequence S' from S by replacng wth, wth, wth, and wth (.e., replace the base wth the other purne or pyrmdne). he result s an -rch sequence that has the same color strng. Example: S = S' = hs s because, f a d-base bd has a color, then so does ts replacement b'd'. For example: 0:, 3:, 2:, 1:. If the d-base n S has color 0, then ts bases are equal and reman so n S'. If t has color 3, ts bases are complementary and reman so n S'. If t has color 2 and ts bases are both purnes, they wll both be pyrmdnes n S', and vce versa. Fnally, f t has color 1 then t s,,, or, and so replaced by,,, or respectvely, all of whch agan have color 1. olors as ransformatons of Bases Up untl ths pont, colors are assgned as a result of an encodng process, ether a specfc chemcal one, le the SOLD sequencng process, or a purely mathematcal one. Here, there s ust one functon, color: B B {0, 1, 2, 3}, that maps d-bases to colors, e.g., color () = 3. For transformaton of bases, color can be represented n a dfferent way. Each color d s a functon f d : B B that transforms base u to base v. here can be any number of colors, but n ths case there are four. he transformatons are also specfed n Fgure 3, Panel E. o transform base u wth color d, loo up color d n row u and report the unque column (by Requrement 2) n whch t resdes. For example, color 3 transforms base to base, whch can be wrtten n dfferent ways: f 3 () =, 3 =, 3, or even 3. Swappng Rules he followng rules can be appled to verfy the observatons from Fgure 3: 1. olor 0 s the dentty functon. hat s f 0 (b) = b for every base b. 2. olor 1 swaps wth, and t swaps wth. For example, f 1 () =. 3. olor 2 swaps wth, and wth. 4. olor 3 swaps wth, and wth. Functon omposton on olors Strngs of colors can also be treated as transformatons by smply applyng one color transformaton after another. hs s how to decode a color read. For example, to decode apply color 3 to to get f 3 () =. hen apply the next color, 2, to ths result, to get f 2 () =. ontnung n ths way, decode all bases, ncludng the last: hat s, decodes to. o compose all color functons n an entre strng, the next step s to gnore ntermedate bases. Just as a sngle color transforms one base nto another, so does a strng of colors. he example above transforms the frst base nto the last base of the sequence. hs s true of the whole strng and also of substrngs. In the example above, we can thn of the substrng 102 as transformng nto : v h r v h r v v r h h h r v r r h v Panel Panel B Fgure 4. ddton able to Obtan the ode for Strngs of olors as ransformatons of Bases. () he orgnal Klen four-group addton table. hs addton table has been obtaned by the Klen four-group en.wpeda.org/w/klen_four-group, whch s the symmetry group of a rectangle. (B) he correspondng addton table for strngs of colors.

5 For nput color strng 102 acts ust le color 3, and ths also transforms to. lso, 102 acts ust le color 3, for all nputs,,, and. o understand ths concept, begn by composng ust two adacent colors as follows: 10 = f 0 o f 1 () = f 0 (f 1 ()) = f 0 () = = f 1 () In ths example, color strng 10 behaves ust le the sngle color 1, whch also maps to. Usng the swappng rules, color strng 10 behaves le color 1 for all nput bases. olor 1 swaps wth, and wth. olor 0 does not swap any of the bases. he rules for all parwse combnatons of colors, showng that each two-color strng behaves le a partcular sngle color, have already been determned n the Klen four-group. he Klen four-group (http://en.wpeda.org/ w/klen_four-group) s the symmetry group of a rectangle, whch has four elements, the dentty, the vertcal reflecton, the horzontal reflecton, and a 180 degree rotaton, as shown n Fgure 4. he rectangle symmetry group has the addton table shown n Fgure 4, Panel. he symbol means followed by. For example, v h = r means that a vertcal reflecton followed by a horzontal reflecton s the same, as f t had rotated the rectangle by 180 degrees. he set of color operatons wth functonal composton that were dscussed above s somorphc to the Klen fourgroup. hs can be observed by labelng the corners of a rectangle wth the bases and wtnessng ther rearrangements (transformatons), as shown n Fgure 5. he dentty leaves the bases unchanged, exactly le color 0. he vertcal reflecton swaps wth, and wth, exactly le color 1. he horzontal reflecton swaps wth, and wth, exactly le color 2; and the 180 degree rotaton swaps wth, and wth, exactly le color 3. n addton table can be created for colors from the addton table for rectangle symmetres smply by substtutng, v, h, and r wth 0, 1, 2, and 3 respectvely, as n Fgure 4, Panel B. he symbol means followed by n ths context too, but t s usually more convenent to leave t out n the wrtten expressons. For example, nstead of , ust wrte , and = means the same as f2(f2(f0(f3(f2(f0(f1(f2(f3 ())))))))) =, the former beng somewhat easer to read, wrte, and parse. Propertes of roups group (http://en.wpeda.org/w/roup_theory) s a set of elements (colors, n our case) wth an operator that satsfes the followng four propertes: 1) losure: If a and b are elements, then a b s also an element. 2) ssocatve: (a b) c = a (b c) 3) Identty: here s an element such that a = a = a. 4) Inverse: For every element a, there s an element a-1, such that a a-1 = a-1 a =. v In addton, the Klen four-group s also belan, whch s to say that t s 5) ommutatve: a b = b a. h r In ths group of color calls, 0 s the dentty element and every element s ts own nverse (the dagonal of Fgure 4, Panel B are all 0 s). It should be noted that except for the dentty color 0, any color composed wth any other color s the thrd color. hese propertes can be used when desgnng protocols, wrtng programs, and provng correctness. Fgure 5: Usng the Klen four-group to Obtan the ode for Strngs of olors as ransformatons of Bases. Rectangle symmetres, v, h, and r transform bases exactly le color operators 0, 1, 2, and 3 respectvely. pplcatons to SOLD Sequencng Events Usng the theoretcal methods dscussed above, here are a few examples on how to analyze typcal SOLD resequencng events. Each of the next few sectons analyzes common resequencng events by usng an example of a nucleotde sequence read, the correspondng reference, and ther colorstrng representatons. In general, a color strng that does NO satsfy these requrements s sad to be nvald.

6 Msmatched Leadng Base In ths applcaton, a gven color read and ts algnment wth a reference s shown below. If all colors match, but the leadng base does not, then LL decoded bases wll be msmatches. hs s observed n the example below he color code group propertes can be used to prove ths wll always be the case. Begn wth ust the algnment of color reads: For all postons, the reference base at s not equal to the read base at. In consderng an arbtrary poston, t has already been shown that the base n reference poston s b, where b s the sum of the reference colors n postons 1 through, usng the addton table n Fgure 4, Panel B. Smlarly the base n read poston s d, where d s the sum of read colors n postons 1 through. However, b must be equal to d because the read colors are the same as the reference colors. herefore, by consderng color code requrement 2, b b, whch shows all bases must be msmatches. n Isolated Sngle-Base Varant sngle-base varant s a read base that dffers from ts algned reference base. he varant s sad to be solated f t s not adacent to a varant on ether sde. For example, the dot mars the solated sngle base varant n the algnment below Reference: Read: o analyze ths case, let s focus on the varant and ntroduce varables, as shown n the algnment below. u v Reference: B D F Read: B E F w x Here D E are msmatched bases, and they are solated because they are flaned by matched bases B and F. In order for colors u, v, w, and x to be consstent wth ths varant, they must satsfy the followng propertes: 1. u w because color (BD) color (BE) by ode Requrement v x follows from the frst two requrements. hs does not need to be tested snce the frst two are true. wo dacent Varants In the prevous applcaton, both relevant color postons were msmatches. he two examples n ths secton show that some colors n other nds of resequencng events, although otherwse constraned, may result n matches or msmatches. herefore, when annotatng color reads, t s not suffcent to annotate only colors that are msmatches. Here s an example where all relevant postons are color msmatches: Reference: Read: nalyss: u v w Reference: B D E F Read: B H F x y z Propertes: 1. u x by ode Requrement uv xy because they must transform B to dfferent bases E and H. 3. uvw = xyz because both uvw and xyz must transform B to F. 4. v and y could be equal or they could be dfferent. here are two addtonal nterestng propertes that follow from the propertes above, therefore, there s no reason to test for them when dentfyng two-base varants above. 5. vw yz because they must transform dfferent bases D and to F. 6. w z because they must transform dfferent bases E and H to F. Here s another example of two adacent varants resultng n three color postons n whch only the outer two are msmatches Reference: Read: Even though the center two color postons, both wth color 1, are not msmatches, the analyss above stll apples, and ths new example satsfes all of the above propertes. 2. uv = wx because both colors uv and wx transform base B to base F, so that uv = color (BF) = wx.

7 hree dacent Varants Here s an example of three adacent varants Reference: Read: hs example has two sngle-color msmatches solated by two color matches. In the analyss below, the B are bases and the c are colors: c 1 c 2 c 3 c 4 Reference: B 1 B 2 B 3 B 4 B 5 Read: B 1 B 6 B 7 B 8 B 5 c 5 c 6 c 7 c 8 Propertes: 1. c 1 c 5 2. c 1 c 2 c 5 c 6 3. c 1 c 2 c 3 c 5 c 6 c 7 4. c 1 c 2 c 3 c 4 = c 5 c 6 c 7 c 8 Inserton and Deletons (Indels) In these examples, a base deleted from the read s depcted as a n the read. Smlarly, a base nserted nto the read s depcted as a n the reference. It should be noted that the reference would never be stored wth a n those places, nor does the SOLD System ever call a for any of ts colors. he would be generated only by algnng the read wth the reference. For example, here s a sngle-base deleton: Reference: Read: he analyss must explan two stuatons, whch t does: u v u v Reference: B D E B D E and Read: B - E B - - w w - Propertes: 1. uv = w because both colors uv and w must transform base B to base E. sngle base deleton causes two elementary colors, u and v, to be replaced by one, w. In the above deleton, for example, 31 = 2, as s easly confrmed by the addton table. 2. Property 1 s a suffcent test, but t may be of nterest that the addton table mples that ether u, v, and w are all dfferent, or one s 0 and the other two are dentcal. Property 1 apples also for multple bases. For example, here s a three base nserton: Reference: Read: hs algnment shows the 3 n the nserton algnng wth a 3 n the reference, even though these two have nothng to do wth one another; the 3 n the reference encodes, whose bases bracet the nserton, and the nserton has a, whch s concdently also encoded wth color 3. nalyss: u Reference: B H Read: B D E F H v w x y he algnment above shows the reference color u algned wth read color v, but n fact mght algn wth color w, x, or y nstead. he followng property holds n all cases: u = vwxy. One elementary color, u, n the reference s replaced by four, c 1, c 2, c 3, and c 4, n the read. orrectng wo ommon Msunderstandngs cursory overvew of the SOLD 2 base color codng system may lead to some common msunderstandngs. he frst s that an solated color msmatch must always correspond to a sequencng error. he second s that certan (3/4 of the total) solated two-color changes always correspond to errors. hese msunderstandngs do not actually appear n prnt, but there s publc nformaton that can be easly msread that way f taen out of context. In ths secton, two examples are presented that dscuss these common msunderstandngs. he frst nvolves an solated par of adacent base varants and addresses only the solated color change problem. he second example nvolves an solated cluster of three adacent base changes. he second example has both an solated color change and an mpossble two color change. ounter Example to an solated color change always corresponds to a sequencng error Reference: Read: hs counter example shows two sngle-color msmatches, both 1 2, solated from one another by a color match. It s easy to verfy that ths counter example does ndeed satsfy the requrements for a two-base varant: 1 2, 13 = 2 1 = 23, and 131 = 3 = 232.

8 ounter Example for mpossble two poston changes always correspond to measurement errors Reference: Read: he example shows a forbdden (.e., mpossble adacent msmatches when only a sngle base change s present) twocolor change, 12 31, solated by a color match (0 0) from a sngle-color change (3 2). Both mpossble color changes are nevertheless perfectly consstent wth a three-base varant, n partcular, the one shown above. gan, t s easy to verfy that the colors satsfy the requrements for a three-base varant: 1 3, 12 31, , and 1203 = clarfcaton of the msunderstandng follows easly by understandng that they are equvalent to assumng that the only base varants permtted n sequences are solated snglebase substtutons. nnotatng lgnment Msmatches n the SOLD System From the prevous dscusson, t s clear that when a SOLD read s algned to a reference sequence, only a subset of possble msmatches represent underlyng sequence varatons. It s therefore mportant to dentfy msmatches that are canddate sequence varatons and to dstngush them from other patterns that are the result of sequencng error. hs nformaton can then be used to correct errors from algned reads, resultng n a sgnfcant mprovement n the accuracy of the system, and to pass putatve sequence varaton to downstream algorthms for the assessment of genotype calls and dentfcaton of new alleles n the sample. hese annotatons are typcally conveyed n fles that descrbe the attrbutes of sequence reads, such as the SOLD FF fle. he SOLD FF s an mportant fle format for the SOLD System. In essence, a fle of ths type s a lst of reads, each wth several attrbutes. One of those attrbutes s the s, or annotatons attrbute. Here s the defnton of the annotatons attrbute of the SOLD FF v 0.2 fle: hs s a comma-separated strng representng annotatons on the sequence. he format s {char}{poston} where {char} s a character representng the type of annotaton (typcally a formattng request to vsualzaton software) and {poston} s the poston of ths annotaton n the read. he poston s 1-based on the strng recorded n the g attrbute. hat s, the prepended base has poston 1, and the frst color has poston 2. For example, a5, g7, g8 means format the color call at poston 5 gray, format call 7 green, and format call 8 green. he SOLD System follows ths conventon: a (ray) s an solated msmatch; t s a msmatch and nether the color-call on ts left nor the color-call on ts rght s a msmatch, g (reen) s a vald adacent msmatch; t s a msmatch and ths msmatch, together wth the adacent msmatch on ts left or rght, could correspond to an solated SNP, y (Yellow) s a color call that s consstent wth an solated two-base change. In general, these wll be msmatches, but a conserved color between two msmatches s also a possblty. r (Red) s a color call that s consstent wth an solated three-base change. b (Blue) s an nvald adacent msmatch; t s any other msmatch. n effcent algorthm for calculatng these annotatons, gven an algnment of the color reads wth a correspondng color reference, can be derved from the theory descrbed prevously. o begn, the followng theorem correlates the applcatons n ths secton to SOLD sequencng events. heorem 1: Let c = c 1 c 2 c 3 c be a -color substrng of a read algned wth the correspondng color reference r = r 1 r 2 r 3 r. hen c encodes an solated (-1)-base change f and only f the base poston precedng c s not a varant, and the followng two equatons hold under the olor ddton able (Fgure 4, Panel B): Equaton 1: c = r Equaton 2: c = r For all from 1 to -1 c r Proof: For the forward drecton, suppose that c encodes an solated -1 base change. r 1 r 2 r -1 r Reference: B 0 B 1 B 2 B -1 B Read: B 0 D 1 D 2 D -1 B c 1 c 2 c -1 c hen the base B 0 precedng c s not a varant because t must serve to solate the -1 base change. Smlarly, the base B followng c s also not a varant because t too must serve to solate the base change. Equaton c =color 1 and ( B02 Bare ) derved by usng the color addton propertes. c =color ( B0B ) and r =color ( B0B ) Equaton 2 stays the same, because all bases between B 0 to B r are =color varants. ( B0B )

9 o prove the reverse drecton, suppose that the base B 0 precedng color c s not a varant, as shown below, and that Equatons 1 and 2 hold. r 1 r 2 r -1 r Reference: B 0 B 1 B 2 B -1 B Read: B 0 D 1 D 2 D -1 D c 1 c 2 c -1 c hen Equaton 1 mples that D = B, as follows: D = B0 c = B0 r = B Bases D 1 through D -1 are therefore solated by matched bases B 0 and B. Furthermore, all solated bases are varants D = B 0 c c = Br 0 = Br = B that are -1D B -for 1 all between 1 and -1, whch follows wth the help of Equaton 2: D = c r = B he equaton above completes ths proof. If a substrng of colors s consstent wth a -base change, assume the leadng base s not a varant and apply heorem 1. o annotate color calls, note that a color call s consstent wth a -base change f t s part of a color strng that s consstent wth a -base change. heorem 2: color call that s consstent wth a -base change s not also consstent wth an m-base change for m. Proof: ssume the opposte, so that there s a color call c that s part of a color strng c that s consstent wth a -base change, and also part of a color strng d that s consstent wth an m-base change. here are two substrngs of bases, of dfferng lengths +2 and m+2, where the outer two bases are matches and the nner and m form the solated varants. ssume wthout loss of generalty that the -base sequence overlaps the m-base sequence, at color c, on the left. he extreme case s setched below, n whch a matched base s a 1 and a msmatched base s an x. Recall that color calls appear only between bases. By heorem 1, poston s compatble wth m solated varants at lne brea. By heorem 2, there s no need to contnue searchng, because wll not be compatble wth any other number. Note the effcency of ths algorthm: f s compatble wth m varants, the algorthm maes the least number of tests possble before breang from the loop. On the other hand, all the tests that t does mae are requred by heorem 1 because f t sps any test, an adversary could devse an nput on whch the algorthm would gve the wrong answer. See ppendx 2 for an mplementaton n Java. Fgure 6 shows a small porton of a vsualzaton of a FF fle annotated wth ths method. lgorthm: getompatablty Input: Read colors, Reference colors R, Query poston, Maxmum acceptable varants M Output: Number of compatble solated base varants m. Method: //summ s the group theoretc sum of m read colors c startng at =. //sumrm s the group theoretc sum of m read colors r startng at =. Intalze: m = summ = sumdm = 0; n = sze(c) whle m M, do { summ = addtonable[summ][c]; sumrm = addtonable[sumrm][d]; f (summ == sumrm) brea; // Substrng s = c c +m s compatble wth an solated group of m adacent varants. m = m + 1 } // for m // If the loop termnates wth m = M+1, then s not compatble wth M or less solated varants. f (m > M) return Incompatble else return m 1xxx1 1xxxx1 -base msmatches on the left m-base msmatches he frst base of the m-base sequence, a match, concdes wth a msmatch n the -base sequence, whch contradcts the fact that a base cannot smultaneously match and msmatch the reference. hs contradcton proves the theorem. he lgorthm he pseudo-code functon getompatblty, shown here, returns m f the color call at poston s compatble wth an solated group of m msmatched bases (varants), where m s between 0 and a preset maxmum acceptable number M. For readablty, ths pseudo-code gnores read edge effects.

10 oncluson he exstng 2 base color code for SOLD sequencng follows from a small set of requrements. hs color code s essentally the only one that satsfes these requrements. If colors are treated as transformers of bases, these requrements lead to a formulaton of color combnatons as the well-nown Klen four-group. he group theoretc propertes lead to effcent and systematc methods for annotatng color reads from the SOLD System. In partcular, they allow for a smple Java mplementaton that annotates color-calls wth any gven number of DN varants. Frst Base Fgure 7. permuted color addton table. ppendx 1: lternatve 2 Base olor odng Schemes hs appendx explores alternatve color codng schemes that exst, and ther effects on subsequent group propertes. Mantanng the Requrements here s essentally only one codng scheme that satsfes all the requrements. Once the frst row has been decded, the requrements determne the rest of the table, as shown n Fgure 3. Each of the 4! = 24 ways to lay down the frst row (gven requrements 1 and 2) corresponds to a dfferent table. But these are all somorphc to one another. Even though there can only be one table, the colors can be permuted to obtan 23 other tables. For example, the permutaton: Fgure 6. Vsualzaton of an nnotated FF fle Usng the getompatblty lgorthm. References 1. osta, na, et al. pplcatons of Next eneraton Sequencng Usng Stepwse ycled-lgaton, Product/Sold_Knowledge/PDF/SH_2007_2791.pdf generates the table n Fgure Pecham, Heather E. et al. SOLD Sequencng and 2 Base Encodng, Product/Sold_Knowledge/PDF/SH_2007_2624.pdf 3. Rhodes M. SOLD System Data 2 Base Encodng, Webnar, 2be/ /

11 Relaxng the onstrants Dfferent codng schemes can be generated by relaxng the constrants. Unfortunately, these alternatve schemes are not guaranteed to satsfy the group theoretc propertes that are needed for annotaton, error detecton, and correcton. Here are four of the requrements: For all bases b, d, e n B: 1. he avalable colors are 0, 1, 2, and 3: color (bd ) {0,1,2,3}. 2. wo dfferent d-bases that nevertheless have the same frst base get dfferent colors: color (bd ) color (be) f d e. For example, color () color (). 3. d-base and ts reverse get the same color: color (bd ) = color (db) For example, color() = color(). 4. Monodbases get the same color: color (bb) = color (dd) hat s, color () = color () = color () = color (). prvate nt getompatablty( Strng readolors, Strng refolors, nt ) { //summ s the group theoretc sum of ref color c from = to +m. nt m = 0; // Number of color msmatches. nt summ = 0; nt sumdm = 0; nt n = Math.mn(readolors.length(), refolors.length()); nt numremanng = n - - 1; // he number of postons past. // maxm s the largest bloc sze m to chec. nt maxm = (MX_DJEN_VRINS < numremanng)? MX_DJEN_VRINS : numremanng; for (m=0; m<=maxm; m++) { nt c = haracter.getnumercvalue(refolors.chart(+m)); nt d = haracter.getnumercvalue(readolors.chart(+m)); // dd n the new colors. // ssume mssng data f ether s out of range. f (0 <= c && c <=3 && 0 <= d && d <= 3) // n range. { summ = addolors[summ][c]; sumdm = addolors[sumdm][d]; f (summ == sumdm) brea; // ompatble wth solated group of m adacent varants. } else // out of range. ermnate ths test as f we ran over maxn. { m = maxm + 1; } } // for m Frst Base Frst Base // Return m f t s n range. Otherwse, 0 means ncompatble and solated, // and -1 means ncompatble but not solated. f (m < 1 m > maxm) { m = sisolated(, readolors, refolors)? 0 : -1; } return m; } // getompatablty() Panel Panel B Fgure 8. Droppng olor ode Requrements Wll Not Satsfy ll roup Propertes. () color code that satsfes only requrements 1, 2, and 3, now does not allow 01 to behave le a sngle color. (B) color code that satsfes only requrements 1, 2, and 4, now does not allow 23 to behave le a sngle color. an requrement 4 be dropped? One such table s shown n Fgure 8, Panel, but t no longer satsfes the group propertes. For example 01 does not behave le a sngle color anymore: 01 = and 1 =, but 01 = and 2 =. So color 0 1 behaves le color 1 for, but t behaves le color 2 for. an we drop ust requrement 3? Such a table s shown n Fgure 8, Panel B, but t no longer satsfes the group propertes. For example 23 does not behave le a sngle color anymore: 23 = and 1 =, but 23 = and 0 =. So color 2 3 behaves le color 1 for, but t behaves le color 0 for. ppendx 2: Java Implementaton of getompatblty he Java functon below, getompatblty, mplements the pseudo-code n the Secton he lgorthm.

12 For Research Use Only. Not for use n dagnostc procedures. Notce to Purchaser: Lcense Dsclamer ppled Bosystems. ll rghts reserved. ll other trademars are the property of ther respectve owners. pplera, ppled Bosystems, and B (Desgn) are regstered trademars and SOLD s a trademar of pplera orporaton or ts subsdares n the U.S. and/or certan other countres. Prnted n the US. 07/2010 Publcaton 139WP01-02 O13982 Headquarters 850 Lncoln entre Drve Foster ty, US Phone oll Free Internatonal Sales For our offce locatons please call the dvson headquarters or refer to our Web ste at

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..

More information

Extending Probabilistic Dynamic Epistemic Logic

Extending Probabilistic Dynamic Epistemic Logic Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set

More information

BERNSTEIN POLYNOMIALS

BERNSTEIN POLYNOMIALS On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful

More information

Recurrence. 1 Definitions and main statements

Recurrence. 1 Definitions and main statements Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.

More information

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by

8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by 6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng

More information

An Alternative Way to Measure Private Equity Performance

An Alternative Way to Measure Private Equity Performance An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate

More information

Section 5.4 Annuities, Present Value, and Amortization

Section 5.4 Annuities, Present Value, and Amortization Secton 5.4 Annutes, Present Value, and Amortzaton Present Value In Secton 5.2, we saw that the present value of A dollars at nterest rate per perod for n perods s the amount that must be deposted today

More information

1 Example 1: Axis-aligned rectangles

1 Example 1: Axis-aligned rectangles COS 511: Theoretcal Machne Learnng Lecturer: Rob Schapre Lecture # 6 Scrbe: Aaron Schld February 21, 2013 Last class, we dscussed an analogue for Occam s Razor for nfnte hypothess spaces that, n conjuncton

More information

21 Vectors: The Cross Product & Torque

21 Vectors: The Cross Product & Torque 21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl

More information

8 Algorithm for Binary Searching in Trees

8 Algorithm for Binary Searching in Trees 8 Algorthm for Bnary Searchng n Trees In ths secton we present our algorthm for bnary searchng n trees. A crucal observaton employed by the algorthm s that ths problem can be effcently solved when the

More information

What is Candidate Sampling

What is Candidate Sampling What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble

More information

Implementation of Deutsch's Algorithm Using Mathcad

Implementation of Deutsch's Algorithm Using Mathcad Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"

More information

A Probabilistic Theory of Coherence

A Probabilistic Theory of Coherence A Probablstc Theory of Coherence BRANDEN FITELSON. The Coherence Measure C Let E be a set of n propostons E,..., E n. We seek a probablstc measure C(E) of the degree of coherence of E. Intutvely, we want

More information

Support Vector Machines

Support Vector Machines Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.

More information

v a 1 b 1 i, a 2 b 2 i,..., a n b n i.

v a 1 b 1 i, a 2 b 2 i,..., a n b n i. SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are

More information

+ + + - - This circuit than can be reduced to a planar circuit

+ + + - - This circuit than can be reduced to a planar circuit MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to

More information

Conversion between the vector and raster data structures using Fuzzy Geographical Entities

Conversion between the vector and raster data structures using Fuzzy Geographical Entities Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,

More information

greatest common divisor

greatest common divisor 4. GCD 1 The greatest common dvsor of two ntegers a and b (not both zero) s the largest nteger whch s a common factor of both a and b. We denote ths number by gcd(a, b), or smply (a, b) when there s no

More information

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic

Institute of Informatics, Faculty of Business and Management, Brno University of Technology,Czech Republic Lagrange Multplers as Quanttatve Indcators n Economcs Ivan Mezník Insttute of Informatcs, Faculty of Busness and Management, Brno Unversty of TechnologCzech Republc Abstract The quanttatve role of Lagrange

More information

Luby s Alg. for Maximal Independent Sets using Pairwise Independence

Luby s Alg. for Maximal Independent Sets using Pairwise Independence Lecture Notes for Randomzed Algorthms Luby s Alg. for Maxmal Independent Sets usng Parwse Independence Last Updated by Erc Vgoda on February, 006 8. Maxmal Independent Sets For a graph G = (V, E), an ndependent

More information

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits

Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.

More information

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS

IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS IDENTIFICATION AND CORRECTION OF A COMMON ERROR IN GENERAL ANNUITY CALCULATIONS Chrs Deeley* Last revsed: September 22, 200 * Chrs Deeley s a Senor Lecturer n the School of Accountng, Charles Sturt Unversty,

More information

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date

MAPP. MERIS level 3 cloud and water vapour products. Issue: 1. Revision: 0. Date: 9.12.1998. Function Name Organisation Signature Date Ttel: Project: Doc. No.: MERIS level 3 cloud and water vapour products MAPP MAPP-ATBD-ClWVL3 Issue: 1 Revson: 0 Date: 9.12.1998 Functon Name Organsaton Sgnature Date Author: Bennartz FUB Preusker FUB Schüller

More information

Calculating the high frequency transmission line parameters of power cables

Calculating the high frequency transmission line parameters of power cables < ' Calculatng the hgh frequency transmsson lne parameters of power cables Authors: Dr. John Dcknson, Laboratory Servces Manager, N 0 RW E B Communcatons Mr. Peter J. Ncholson, Project Assgnment Manager,

More information

DEFINING %COMPLETE IN MICROSOFT PROJECT

DEFINING %COMPLETE IN MICROSOFT PROJECT CelersSystems DEFINING %COMPLETE IN MICROSOFT PROJECT PREPARED BY James E Aksel, PMP, PMI-SP, MVP For Addtonal Informaton about Earned Value Management Systems and reportng, please contact: CelersSystems,

More information

The OC Curve of Attribute Acceptance Plans

The OC Curve of Attribute Acceptance Plans The OC Curve of Attrbute Acceptance Plans The Operatng Characterstc (OC) curve descrbes the probablty of acceptng a lot as a functon of the lot s qualty. Fgure 1 shows a typcal OC Curve. 10 8 6 4 1 3 4

More information

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis

The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.

More information

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy

Answer: A). There is a flatter IS curve in the high MPC economy. Original LM LM after increase in M. IS curve for low MPC economy 4.02 Quz Solutons Fall 2004 Multple-Choce Questons (30/00 ponts) Please, crcle the correct answer for each of the followng 0 multple-choce questons. For each queston, only one of the answers s correct.

More information

Canon NTSC Help Desk Documentation

Canon NTSC Help Desk Documentation Canon NTSC Help Desk Documentaton READ THIS BEFORE PROCEEDING Before revewng ths documentaton, Canon Busness Solutons, Inc. ( CBS ) hereby refers you, the customer or customer s representatve or agent

More information

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).

benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ). REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or

More information

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2

6. EIGENVALUES AND EIGENVECTORS 3 = 3 2 EIGENVALUES AND EIGENVECTORS The Characterstc Polynomal If A s a square matrx and v s a non-zero vector such that Av v we say that v s an egenvector of A and s the correspondng egenvalue Av v Example :

More information

Lecture 3: Force of Interest, Real Interest Rate, Annuity

Lecture 3: Force of Interest, Real Interest Rate, Annuity Lecture 3: Force of Interest, Real Interest Rate, Annuty Goals: Study contnuous compoundng and force of nterest Dscuss real nterest rate Learn annuty-mmedate, and ts present value Study annuty-due, and

More information

where the coordinates are related to those in the old frame as follows.

where the coordinates are related to those in the old frame as follows. Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product

More information

QUANTUM MECHANICS, BRAS AND KETS

QUANTUM MECHANICS, BRAS AND KETS PH575 SPRING QUANTUM MECHANICS, BRAS AND KETS The followng summares the man relatons and defntons from quantum mechancs that we wll be usng. State of a phscal sstem: The state of a phscal sstem s represented

More information

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting

Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of

More information

Lecture 2: Single Layer Perceptrons Kevin Swingler

Lecture 2: Single Layer Perceptrons Kevin Swingler Lecture 2: Sngle Layer Perceptrons Kevn Sngler kms@cs.str.ac.uk Recap: McCulloch-Ptts Neuron Ths vastly smplfed model of real neurons s also knon as a Threshold Logc Unt: W 2 A Y 3 n W n. A set of synapses

More information

Loop Parallelization

Loop Parallelization - - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze

More information

From Selective to Full Security: Semi-Generic Transformations in the Standard Model

From Selective to Full Security: Semi-Generic Transformations in the Standard Model An extended abstract of ths work appears n the proceedngs of PKC 2012 From Selectve to Full Securty: Sem-Generc Transformatons n the Standard Model Mchel Abdalla 1 Daro Fore 2 Vadm Lyubashevsky 1 1 Département

More information

Project Networks With Mixed-Time Constraints

Project Networks With Mixed-Time Constraints Project Networs Wth Mxed-Tme Constrants L Caccetta and B Wattananon Western Australan Centre of Excellence n Industral Optmsaton (WACEIO) Curtn Unversty of Technology GPO Box U1987 Perth Western Australa

More information

2.4 Bivariate distributions

2.4 Bivariate distributions page 28 2.4 Bvarate dstrbutons 2.4.1 Defntons Let X and Y be dscrete r.v.s defned on the same probablty space (S, F, P). Instead of treatng them separately, t s often necessary to thnk of them actng together

More information

A Secure Password-Authenticated Key Agreement Using Smart Cards

A Secure Password-Authenticated Key Agreement Using Smart Cards A Secure Password-Authentcated Key Agreement Usng Smart Cards Ka Chan 1, Wen-Chung Kuo 2 and Jn-Chou Cheng 3 1 Department of Computer and Informaton Scence, R.O.C. Mltary Academy, Kaohsung 83059, Tawan,

More information

Calculation of Sampling Weights

Calculation of Sampling Weights Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample

More information

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm

A hybrid global optimization algorithm based on parallel chaos optimization and outlook algorithm Avalable onlne www.ocpr.com Journal of Chemcal and Pharmaceutcal Research, 2014, 6(7):1884-1889 Research Artcle ISSN : 0975-7384 CODEN(USA) : JCPRC5 A hybrd global optmzaton algorthm based on parallel

More information

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2)

n + d + q = 24 and.05n +.1d +.25q = 2 { n + d + q = 24 (3) n + 2d + 5q = 40 (2) MATH 16T Exam 1 : Part I (In-Class) Solutons 1. (0 pts) A pggy bank contans 4 cons, all of whch are nckels (5 ), dmes (10 ) or quarters (5 ). The pggy bank also contans a con of each denomnaton. The total

More information

Simple Interest Loans (Section 5.1) :

Simple Interest Loans (Section 5.1) : Chapter 5 Fnance The frst part of ths revew wll explan the dfferent nterest and nvestment equatons you learned n secton 5.1 through 5.4 of your textbook and go through several examples. The second part

More information

IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM

IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM Abstract IDENTIFICATION AND CONTROL OF A FLEXIBLE TRANSMISSION SYSTEM Alca Esparza Pedro Dept. Sstemas y Automátca, Unversdad Poltécnca de Valenca, Span alespe@sa.upv.es The dentfcaton and control of a

More information

1 Approximation Algorithms

1 Approximation Algorithms CME 305: Dscrete Mathematcs and Algorthms 1 Approxmaton Algorthms In lght of the apparent ntractablty of the problems we beleve not to le n P, t makes sense to pursue deas other than complete solutons

More information

Brigid Mullany, Ph.D University of North Carolina, Charlotte

Brigid Mullany, Ph.D University of North Carolina, Charlotte Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte

More information

CHAPTER 14 MORE ABOUT REGRESSION

CHAPTER 14 MORE ABOUT REGRESSION CHAPTER 14 MORE ABOUT REGRESSION We learned n Chapter 5 that often a straght lne descrbes the pattern of a relatonshp between two quanttatve varables. For nstance, n Example 5.1 we explored the relatonshp

More information

7.5. Present Value of an Annuity. Investigate

7.5. Present Value of an Annuity. Investigate 7.5 Present Value of an Annuty Owen and Anna are approachng retrement and are puttng ther fnances n order. They have worked hard and nvested ther earnngs so that they now have a large amount of money on

More information

Activity Scheduling for Cost-Time Investment Optimization in Project Management

Activity Scheduling for Cost-Time Investment Optimization in Project Management PROJECT MANAGEMENT 4 th Internatonal Conference on Industral Engneerng and Industral Management XIV Congreso de Ingenería de Organzacón Donosta- San Sebastán, September 8 th -10 th 010 Actvty Schedulng

More information

Trivial lump sum R5.0

Trivial lump sum R5.0 Optons form Once you have flled n ths form, please return t wth your orgnal brth certfcate to: Premer PO Box 2067 Croydon CR90 9ND. Fll n ths form usng BLOCK CAPITALS and black nk. Mark all answers wth

More information

The Analysis of Outliers in Statistical Data

The Analysis of Outliers in Statistical Data THALES Project No. xxxx The Analyss of Outlers n Statstcal Data Research Team Chrysses Caron, Assocate Professor (P.I.) Vaslk Karot, Doctoral canddate Polychrons Economou, Chrstna Perrakou, Postgraduate

More information

Joe Pimbley, unpublished, 2005. Yield Curve Calculations

Joe Pimbley, unpublished, 2005. Yield Curve Calculations Joe Pmbley, unpublshed, 005. Yeld Curve Calculatons Background: Everythng s dscount factors Yeld curve calculatons nclude valuaton of forward rate agreements (FRAs), swaps, nterest rate optons, and forward

More information

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001.

Inequality and The Accounting Period. Quentin Wodon and Shlomo Yitzhaki. World Bank and Hebrew University. September 2001. Inequalty and The Accountng Perod Quentn Wodon and Shlomo Ytzha World Ban and Hebrew Unversty September Abstract Income nequalty typcally declnes wth the length of tme taen nto account for measurement.

More information

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence

How Sets of Coherent Probabilities May Serve as Models for Degrees of Incoherence 1 st Internatonal Symposum on Imprecse Probabltes and Ther Applcatons, Ghent, Belgum, 29 June 2 July 1999 How Sets of Coherent Probabltes May Serve as Models for Degrees of Incoherence Mar J. Schervsh

More information

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression

A Novel Methodology of Working Capital Management for Large. Public Constructions by Using Fuzzy S-curve Regression Novel Methodology of Workng Captal Management for Large Publc Constructons by Usng Fuzzy S-curve Regresson Cheng-Wu Chen, Morrs H. L. Wang and Tng-Ya Hseh Department of Cvl Engneerng, Natonal Central Unversty,

More information

PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB.

PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. PRACTICE 1: MUTUAL FUNDS EVALUATION USING MATLAB. INDEX 1. Load data usng the Edtor wndow and m-fle 2. Learnng to save results from the Edtor wndow. 3. Computng the Sharpe Rato 4. Obtanng the Treynor Rato

More information

A machine vision approach for detecting and inspecting circular parts

A machine vision approach for detecting and inspecting circular parts A machne vson approach for detectng and nspectng crcular parts Du-Mng Tsa Machne Vson Lab. Department of Industral Engneerng and Management Yuan-Ze Unversty, Chung-L, Tawan, R.O.C. E-mal: edmtsa@saturn.yzu.edu.tw

More information

PERRON FROBENIUS THEOREM

PERRON FROBENIUS THEOREM PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()

More information

Enterprise Master Patient Index

Enterprise Master Patient Index Enterprse Master Patent Index Healthcare data are captured n many dfferent settngs such as hosptals, clncs, labs, and physcan offces. Accordng to a report by the CDC, patents n the Unted States made an

More information

Multiplication Algorithms for Radix-2 RN-Codings and Two s Complement Numbers

Multiplication Algorithms for Radix-2 RN-Codings and Two s Complement Numbers Multplcaton Algorthms for Radx- RN-Codngs and Two s Complement Numbers Jean-Luc Beuchat Projet Arénare, LIP, ENS Lyon 46, Allée d Itale F 69364 Lyon Cedex 07 jean-luc.beuchat@ens-lyon.fr Jean-Mchel Muller

More information

Multiple discount and forward curves

Multiple discount and forward curves Multple dscount and forward curves TopQuants presentaton 21 ovember 2012 Ton Broekhuzen, Head Market Rsk and Basel coordnator, IBC Ths presentaton reflects personal vews and not necessarly the vews of

More information

The k-binomial Transforms and the Hankel Transform

The k-binomial Transforms and the Hankel Transform 1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 9 (2006, Artcle 06.1.1 The k-bnomal Transforms and the Hankel Transform Mchael Z. Spvey Department of Mathematcs and Computer Scence Unversty of Puget

More information

POLYSA: A Polynomial Algorithm for Non-binary Constraint Satisfaction Problems with and

POLYSA: A Polynomial Algorithm for Non-binary Constraint Satisfaction Problems with and POLYSA: A Polynomal Algorthm for Non-bnary Constrant Satsfacton Problems wth and Mguel A. Saldo, Federco Barber Dpto. Sstemas Informátcos y Computacón Unversdad Poltécnca de Valenca, Camno de Vera s/n

More information

Texas Instruments 30X IIS Calculator

Texas Instruments 30X IIS Calculator Texas Instruments 30X IIS Calculator Keystrokes for the TI-30X IIS are shown for a few topcs n whch keystrokes are unque. Start by readng the Quk Start secton. Then, before begnnng a specfc unt of the

More information

Time Value of Money Module

Time Value of Money Module Tme Value of Money Module O BJECTIVES After readng ths Module, you wll be able to: Understand smple nterest and compound nterest. 2 Compute and use the future value of a sngle sum. 3 Compute and use the

More information

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT

Chapter 4 ECONOMIC DISPATCH AND UNIT COMMITMENT Chapter 4 ECOOMIC DISATCH AD UIT COMMITMET ITRODUCTIO A power system has several power plants. Each power plant has several generatng unts. At any pont of tme, the total load n the system s met by the

More information

Vembu StoreGrid Windows Client Installation Guide

Vembu StoreGrid Windows Client Installation Guide Ser v cepr ov dered t on Cl enti nst al l at ongu de W ndows Vembu StoreGrd Wndows Clent Installaton Gude Download the Wndows nstaller, VembuStoreGrd_4_2_0_SP_Clent_Only.exe To nstall StoreGrd clent on

More information

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek

THE DISTRIBUTION OF LOAN PORTFOLIO VALUE * Oldrich Alfons Vasicek HE DISRIBUION OF LOAN PORFOLIO VALUE * Oldrch Alfons Vascek he amount of captal necessary to support a portfolo of debt securtes depends on the probablty dstrbuton of the portfolo loss. Consder a portfolo

More information

Vision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION

Vision Mouse. Saurabh Sarkar a* University of Cincinnati, Cincinnati, USA ABSTRACT 1. INTRODUCTION Vson Mouse Saurabh Sarkar a* a Unversty of Cncnnat, Cncnnat, USA ABSTRACT The report dscusses a vson based approach towards trackng of eyes and fngers. The report descrbes the process of locatng the possble

More information

Section 5.3 Annuities, Future Value, and Sinking Funds

Section 5.3 Annuities, Future Value, and Sinking Funds Secton 5.3 Annutes, Future Value, and Snkng Funds Ordnary Annutes A sequence of equal payments made at equal perods of tme s called an annuty. The tme between payments s the payment perod, and the tme

More information

1.1 The University may award Higher Doctorate degrees as specified from time-to-time in UPR AS11 1.

1.1 The University may award Higher Doctorate degrees as specified from time-to-time in UPR AS11 1. HIGHER DOCTORATE DEGREES SUMMARY OF PRINCIPAL CHANGES General changes None Secton 3.2 Refer to text (Amendments to verson 03.0, UPR AS02 are shown n talcs.) 1 INTRODUCTION 1.1 The Unversty may award Hgher

More information

HÜCKEL MOLECULAR ORBITAL THEORY

HÜCKEL MOLECULAR ORBITAL THEORY 1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ

More information

Lecture 2 Sequence Alignment. Burr Settles IBS Summer Research Program 2008 bsettles@cs.wisc.edu www.cs.wisc.edu/~bsettles/ibs08/

Lecture 2 Sequence Alignment. Burr Settles IBS Summer Research Program 2008 bsettles@cs.wisc.edu www.cs.wisc.edu/~bsettles/ibs08/ Lecture 2 Sequence lgnment Burr Settles IBS Summer Research Program 2008 bsettles@cs.wsc.edu www.cs.wsc.edu/~bsettles/bs08/ Sequence lgnment: Task Defnton gven: a par of sequences DN or proten) a method

More information

1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits

1E6 Electrical Engineering AC Circuit Analysis and Power Lecture 12: Parallel Resonant Circuits E6 Electrcal Engneerng A rcut Analyss and Power ecture : Parallel esonant rcuts. Introducton There are equvalent crcuts to the seres combnatons examned whch exst n parallel confguratons. The ssues surroundng

More information

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA )

Hollinger Canadian Publishing Holdings Co. ( HCPH ) proceeding under the Companies Creditors Arrangement Act ( CCAA ) February 17, 2011 Andrew J. Hatnay ahatnay@kmlaw.ca Dear Sr/Madam: Re: Re: Hollnger Canadan Publshng Holdngs Co. ( HCPH ) proceedng under the Companes Credtors Arrangement Act ( CCAA ) Update on CCAA Proceedngs

More information

APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT

APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT APPLICATION OF PROBE DATA COLLECTED VIA INFRARED BEACONS TO TRAFFIC MANEGEMENT Toshhko Oda (1), Kochro Iwaoka (2) (1), (2) Infrastructure Systems Busness Unt, Panasonc System Networks Co., Ltd. Saedo-cho

More information

Using Series to Analyze Financial Situations: Present Value

Using Series to Analyze Financial Situations: Present Value 2.8 Usng Seres to Analyze Fnancal Stuatons: Present Value In the prevous secton, you learned how to calculate the amount, or future value, of an ordnary smple annuty. The amount s the sum of the accumulated

More information

Multiple-Period Attribution: Residuals and Compounding

Multiple-Period Attribution: Residuals and Compounding Multple-Perod Attrbuton: Resduals and Compoundng Our revewer gave these authors full marks for dealng wth an ssue that performance measurers and vendors often regard as propretary nformaton. In 1994, Dens

More information

J. Parallel Distrib. Comput.

J. Parallel Distrib. Comput. J. Parallel Dstrb. Comput. 71 (2011) 62 76 Contents lsts avalable at ScenceDrect J. Parallel Dstrb. Comput. journal homepage: www.elsever.com/locate/jpdc Optmzng server placement n dstrbuted systems n

More information

The Greedy Method. Introduction. 0/1 Knapsack Problem

The Greedy Method. Introduction. 0/1 Knapsack Problem The Greedy Method Introducton We have completed data structures. We now are gong to look at algorthm desgn methods. Often we are lookng at optmzaton problems whose performance s exponental. For an optmzaton

More information

Fast degree elevation and knot insertion for B-spline curves

Fast degree elevation and knot insertion for B-spline curves Computer Aded Geometrc Desgn 22 (2005) 183 197 www.elsever.com/locate/cagd Fast degree elevaton and knot nserton for B-splne curves Q-Xng Huang a,sh-mnhu a,, Ralph R. Martn b a Department of Computer Scence

More information

Nordea G10 Alpha Carry Index

Nordea G10 Alpha Carry Index Nordea G10 Alpha Carry Index Index Rules v1.1 Verson as of 10/10/2013 1 (6) Page 1 Index Descrpton The G10 Alpha Carry Index, the Index, follows the development of a rule based strategy whch nvests and

More information

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)

1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP) 6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes

More information

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression.

Lecture 3: Annuity. Study annuities whose payments form a geometric progression or a arithmetic progression. Lecture 3: Annuty Goals: Learn contnuous annuty and perpetuty. Study annutes whose payments form a geometrc progresson or a arthmetc progresson. Dscuss yeld rates. Introduce Amortzaton Suggested Textbook

More information

Frequency Selective IQ Phase and IQ Amplitude Imbalance Adjustments for OFDM Direct Conversion Transmitters

Frequency Selective IQ Phase and IQ Amplitude Imbalance Adjustments for OFDM Direct Conversion Transmitters Frequency Selectve IQ Phase and IQ Ampltude Imbalance Adjustments for OFDM Drect Converson ransmtters Edmund Coersmeer, Ernst Zelnsk Noka, Meesmannstrasse 103, 44807 Bochum, Germany edmund.coersmeer@noka.com,

More information

HALL EFFECT SENSORS AND COMMUTATION

HALL EFFECT SENSORS AND COMMUTATION OEM770 5 Hall Effect ensors H P T E R 5 Hall Effect ensors The OEM770 works wth three-phase brushless motors equpped wth Hall effect sensors or equvalent feedback sgnals. In ths chapter we wll explan how

More information

General Auction Mechanism for Search Advertising

General Auction Mechanism for Search Advertising General Aucton Mechansm for Search Advertsng Gagan Aggarwal S. Muthukrshnan Dávd Pál Martn Pál Keywords game theory, onlne auctons, stable matchngs ABSTRACT Internet search advertsng s often sold by an

More information

SEVERAL trends are opening up the era of Cloud

SEVERAL trends are opening up the era of Cloud 1 Towards Secure and Dependable Storage Servces n Cloud Computng Cong Wang, Student Member, IEEE, Qan Wang, Student Member, IEEE, Ku Ren, Member, IEEE, Nng Cao, Student Member, IEEE, and Wenjng Lou, Senor

More information

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6)

Passive Filters. References: Barbow (pp 265-275), Hayes & Horowitz (pp 32-60), Rizzoni (Chap. 6) Passve Flters eferences: Barbow (pp 6575), Hayes & Horowtz (pp 360), zzon (Chap. 6) Frequencyselectve or flter crcuts pass to the output only those nput sgnals that are n a desred range of frequences (called

More information

PRO- CRIMPER* III Hand

PRO- CRIMPER* III Hand PRO- CRIMPER* III Hand Instructon Sheet Crmpng Tool Assembly 58571-1 408-4135 wth De Assembly 58571-2 06 NOV 09 PROPER USE GUIDELINES Cumulatve Trauma Dsorders can result from the prolonged use of manually

More information

On the Optimal Control of a Cascade of Hydro-Electric Power Stations

On the Optimal Control of a Cascade of Hydro-Electric Power Stations On the Optmal Control of a Cascade of Hydro-Electrc Power Statons M.C.M. Guedes a, A.F. Rbero a, G.V. Smrnov b and S. Vlela c a Department of Mathematcs, School of Scences, Unversty of Porto, Portugal;

More information

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol

CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK. Sample Stability Protocol CHOLESTEROL REFERENCE METHOD LABORATORY NETWORK Sample Stablty Protocol Background The Cholesterol Reference Method Laboratory Network (CRMLN) developed certfcaton protocols for total cholesterol, HDL

More information

We assume your students are learning about self-regulation (how to change how alert they feel) through the Alert Program with its three stages:

We assume your students are learning about self-regulation (how to change how alert they feel) through the Alert Program with its three stages: Welcome to ALERT BINGO, a fun-flled and educatonal way to learn the fve ways to change engnes levels (Put somethng n your Mouth, Move, Touch, Look, and Lsten) as descrbed n the How Does Your Engne Run?

More information

Lecture 7 March 20, 2002

Lecture 7 March 20, 2002 MIT 8.996: Topc n TCS: Internet Research Problems Sprng 2002 Lecture 7 March 20, 2002 Lecturer: Bran Dean Global Load Balancng Scrbe: John Kogel, Ben Leong In today s lecture, we dscuss global load balancng

More information

Efficient Project Portfolio as a tool for Enterprise Risk Management

Efficient Project Portfolio as a tool for Enterprise Risk Management Effcent Proect Portfolo as a tool for Enterprse Rsk Management Valentn O. Nkonov Ural State Techncal Unversty Growth Traectory Consultng Company January 5, 27 Effcent Proect Portfolo as a tool for Enterprse

More information

Joint Scheduling of Processing and Shuffle Phases in MapReduce Systems

Joint Scheduling of Processing and Shuffle Phases in MapReduce Systems Jont Schedulng of Processng and Shuffle Phases n MapReduce Systems Fangfe Chen, Mural Kodalam, T. V. Lakshman Department of Computer Scence and Engneerng, The Penn State Unversty Bell Laboratores, Alcatel-Lucent

More information

Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching)

Face Verification Problem. Face Recognition Problem. Application: Access Control. Biometric Authentication. Face Verification (1:1 matching) Face Recognton Problem Face Verfcaton Problem Face Verfcaton (1:1 matchng) Querymage face query Face Recognton (1:N matchng) database Applcaton: Access Control www.vsage.com www.vsoncs.com Bometrc Authentcaton

More information

SEVERAL trends are opening up the era of Cloud

SEVERAL trends are opening up the era of Cloud IEEE Transactons on Cloud Computng Date of Publcaton: Aprl-June 2012 Volume: 5, Issue: 2 1 Towards Secure and Dependable Storage Servces n Cloud Computng Cong Wang, Student Member, IEEE, Qan Wang, Student

More information