In recent years, the scientific basis of fingerprint analysis has been questioned,

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "In recent years, the scientific basis of fingerprint analysis has been questioned,"

Transcription

1 Can t Qute Put Our Fnger On It 231 Can t Qute Put Our Fnger On It Séamus Ó Ceallagh Alva Sheeley Adan Crangle Unversty College Cork Cork, Ireland Advsor: James J. Grannell Summary There are two man paths to dentfcaton through fngerprnts. Global analyss reles on the specfc arrangement and characterstcs of the rdges of a prnt. Local analyss assumes that the ndvdualty of a prnt s based on the poston and orentaton of the two basc types of mnutae. We subdvde a prnt nto a grd of square cells, consder the dstrbuton of rdge features, and calculate probabltes from combnatoral analyss. We make predctons by refnng parameters of the model, such as the number of mnutae requred for a postve match between two generc prnts, and the sze of a cell n the man grd. We compare our results to prevous studes and dscuss the relaton to DNA proflng. The smplcty of our model s ts key strength. We conclude that t s extremely unlkely that any two randomly selected people have, or have ever had, the same set of fngerprnts. Despte the apparently smplstc nature of fngerprntng, t s vastly more relable n dentfcaton than a DNA comparson test. Introducton In recent years, the scentfc bass of fngerprnt analyss has been questoned, n the U.S. Supreme Court rulng n the Daubert case that the relablty of expert scentfc testmony must be establshed along the followng fve crtera: The UMAP Journal 25 (3) (2004) c Copyrght 2004 by COMAP, Inc. All rghts reserved. Permsson to make dgtal or hard copes of part or all of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce. Abstractng wth credt s permtted, but copyrghts for components of ths work owned by others than COMAP must be honored. To copy otherwse, to republsh, to post on servers, or to redstrbute to lsts requres pror permsson from COMAP.

2 232 The UMAP Journal 25.3 (2004) 1. whether the partcular technque or methodology n queston has been subject to a statstcal hypothess testng; 2. whether ts error rate has been establshed; 3. whether the standards controllng the technque s operaton exst and have been mantaned; 4. whether t has been peer-revewed and publshed; and 5. whether t has a general wdespread acceptance. Our model tres to address the frst two ssues. We am to produce a probablstc method to measure the unqueness of a partcular prnt. Assumptons A thumbprnt s defned globally by rdge patterns and locally by a dstrbuton of mnutae, whch we refer to also as features. The area of nterest typcal thumbprnt s a 20 mm 20 mm square grd. There are two sgnfcant types of mnutae, the bfurcaton and the rdge endng: all other mnutae are compostons of these [Osterburg et al. 1977]. The probablty of a mnuta occurrng n a grd box s.234 [Osterburg 1977]. The orentaton of the mnutae was not taken nto account by Osterburg; we assgn a mnutae one of eght angles, from 0 to 157.5, n steps of When comparng two prnts, we know one prnt arbtrarly well. The number of people who have ever lved s [Haub 1995]. The Model Global Analyss Rdge Patterns And Orentaton Felds Global analyss concerns rdge patterns, whch dstngush prnts nto sx man pattern groups: Arch, Tented Arch, Left Loop, Rght Loop, Twn Loop and Whorl. Each pattern s determned by an orentaton feld, whch may have specfc statonary ponts, known as the the delta and the core. If a prnt contans 0 or 1 delta ponts and 0 or 1 core ponts, then t s classfed as Lasso, and Wrbel otherwse. The Lasso class conssts of arch, tented arch, rght loop, and left loop.

3 Can t Qute Put Our Fnger On It 233 If the fngerprnt has 0 delta ponts or 0 core ponts, then t s an arch. Otherwse, f the core pont and the delta pont are algned n the vertcal drecton, then the fngerprnt s an arch f the length between the core pont and the delta pont s less than 2.5 mm and a tented arch otherwse. Otherwse, f the core pont s to the rght of the delta pont, the fngerprnt s a rght loop. Otherwse, the fngerprnt s a left loop. The Wrbel class conssts of the whorl and the twn loop classes: If there are exactly two core ponts and exactly two delta ponts, then the fngerprnt s a whorl f the two core ponts are algned horzontally and a twn loop otherwse. Otherwse, the fngerprnt s a whorl. The man am of global analyss s a vector feld or orentaton feld to the rdge lnes of a fngerprnt. We must fnd sutable parameters for such functons that gve rse to the dfferent classes of rdge pattern. The most basc pattern wthout statonary ponts s the arch Fgure 1, modeled by the smple system dx dt = μy, dy dt = ν, wth parameters μ and ν. The orentaton felds for other rdge patterns are more complex, so the bulk of our model s drected at the prnt s local features. Local Analyss Estmates Fgure 1. Arch orentaton feld. For an ntal estmate of the probablty of any two people havng the same thumbprnt, we must consder:

4 234 The UMAP Journal 25.3 (2004) the total number of people who have ever lved, estmated to be [Haub 1995] (that s, about 6% of all people are alve rght now); and the total number of possble thumbprnts that can be classfed as dfferent. To decde whether or not one thumbprnt s the same as another, one must frst decde on what exactly s a thumbprnt. In our model, we take a typcal area of the prnt as 20 mm 20 mm, whch s large enough to encompass the area of nterest on any prnt. We dvde ths area nto boxes 1 mm on a sde, thus gvng 400 boxes, each of area 1 mm 2. In prncple, each box can be examned to determne whether or not t contans a mnuta. Mnutae are the features on a thumbprnt that are used by almost all dentfcaton technques to dstngush between prnts. There are from 10 [Galton 1892] to 16 [Optel Ltd. 2003] dfferent types of mnutae, but they are all composed of two fundamental types: rdge endngs and rdge bfurcatons. In our analyses, we consder only these two types. Also, f there are two mnutae n the same cell, t s mpossble to resolve them separately. Say that there are resolvable features on the prnt. The number of ways that we can nsert these features nto the 400 spaces n the grd s ( ) 400. Rememberng that there are two possbltes for each feature, the total number of combnatons s then ( ) What s the value of? The probablty that any box contans a feature we take as.234 [Osterburg et al. 1977]. Then has a bnomal dstrbuton: ( ) 400 P ( = x) = (.234) x (1.234) 400 x, x wth mean μ 94 and standard devaton σ 8, so that the average number of cells contanng features s 94. Thus, the total number of possble thumbprnts s 400 ( ) 400 N = 2. =0 The bnomal dstrbuton for, however, s concentrated manly n the regon μ σ<<μ+ σ,or94 8 << To be conservatve, we consder only ths range of numbers of mnutae; thus, there are approxmately N 102 =86 ( 400 )

5 Can t Qute Put Our Fnger On It 235 dfferent thumbprnts avalable for any actual thumb to hold. So very roughly, P (two people ever havng the same thumbprnt) = Comparson Ths fgure s the (approxmate) probablty that there have ever been two people who have had the same thumbprnt. How mght we take nto account the chance that, when compared, two prnts wll be judged to be the same? To do ths, we consder two hypothetcal prnts: A control prnt: an deal known prnt, n whch all features are seen. A sample prnt: a prnt wth more features than the n avalable for comparson. For two prnts that are compared n a realstc crcumstance, there wll be at least ( n) features that are not ncluded n the comparson. These features, n theory, could be n any combnaton of postons n the grd. The man queston s: How many prnts have n<features correspondng to a match? In other words, how many dfferent ways can the remanng ( n) features be nserted nto the grd and stll produce a match wth the control prnt? Knowng that, we can estmate how lkely t s that two thumbprnts not actually the same wll match. Incorrect Matchng We have ( n) features to dstrbute among (400 n) grd elements. The number of dfferent ways to do ths s, by prevous reasonng, 102 ( ) 400 n 2 n. n 86 In crmnal proceedngs, a matchng number of mnutae of anythng from 8 [Collns 1992] to 15 [Vacca 2002] are accepted as conclusve proof of dentfcaton. Our model predcts that for n =12, the total number of thumbprnts that could have the same set of matchng mnutae whle not beng the same prnt s N = But expressed as a fracton of the total number of possble prnts, the probablty of the prnt beng one of these, f t selected from them, s P (false match) = Ths s an extremely low probablty (1)

6 236 The UMAP Journal 25.3 (2004) Varyng Parameters The result (1) depends on the parameters, whch can be vared accordng to crcumstance and also as a way of refnng the model: p: the probablty of fndng a feature n a grd cell. We take p =.234 [Osterburg et al. 1977]. Others [Tha 2003; Kngston 1964; Stoney and Thornton 1987; Dankmejer et al ] gve values n the range.19 <p<.25. N: the number of cells n the grd. If there are more cells, on average, then more features wll be observed, snce p(feature) for a cell remans the same. n: the number of mnutae that one takes for comparson. We take n =12. : a varable, determned by p and N, that gves reasonable bounds for the summaton. F : the number of dfferent features that can appear n a grd cell. In our ntal estmate, we take F =2. L, A: the length of a sde, and the area, of a grd cell. We take L 1 mm, the average dstance between features [Tha 2003]. If we wsh to examne a thumbprnt more closely, we should consder smaller and smaller areas of the prnt. It s not meanngful, however, to take L less than 0.1 mm, snce ths s the typcal rdge wdth. The Dependence on L We rework the model, takng the wdth of the generc grd cell to be 0.5 mm. Takng the overall area of the prnt to be the same, there are now 1,600 grd cells to consder, each wth the same probablty of havng a feature. The bnomal expresson for, the number of features observed on the whole prnt, s now ( ) 1600 P ( = x) = (.234) x (1.234) 1600 x, x wth mean μ = 374 and standard devaton σ =17. Thus, the regon of relevance when summng s now = 357 <<391 = Ths means that when the thumbprnt s examned on a scale half that of the ntal, about four tmes as many mnutae wll be observed. Intutvely, the lkelhood of a false match wll decrease, snce there are more possbltes for the number of prnts: N 1600 = 391 =357 ( 1600 )

7 Can t Qute Put Our Fnger On It 237 The probablty that any of the 100 bllon people who have ever lved have had the same thumbprnt s P (2 people ever havng the same thumbprnt) = We now determne the number of ways n whch, when a certan number of mnutae are selected for comparson, the remanng mnutae can be arranged. Followng the same logc as before, ths fgure s 391 ( ) 1600 n 2 n, n 357 whch evaluates to for n =12. The probablty that any two compared thumbprnts, judged to be dentcal by the standards of comparson, are actually dfferent s therefore P (false match n =12,N = 1600) = Ths, nterestngly, s not much greater than the probablty for the prevous estmate. The result s not therefore acutely dependent on the value of L, nor, by assocaton, on the number N of grd cells. That sad, t easy to examne detals n any prnt to a scale of 0.5 mm. The Dependence on p The probablty of a feature n a cell s, as of yet, a purely emprcal fgure. The formaton of fngerprnts, and ther assocated characterstcs, s known: The foetus, at about 6.5 weeks, grows eleven volar pads pouches on varous locatons of the hand [Anonymous 2001]. These shrnk at about 11 weeks; and when they are gone, beneath where they lay are fngerprnts. However, the mechansm of formaton of the specfc features s unknown. Genetc nfluences are present, but the envronment s crucal also, evdenced by the fact that dentcal twns who have the same DNA genotype do not have the same fngerprnts. There s no way yet determned of predctng the frequency of occurrence of any type of mnuta on the prnt of a partcular person. Prevous studes, cted n Table 1, show varaton about 0.2 mnutae/mm 2. It s not unreasonable to propose that the densty depends on the prnt classfcaton (.e., whorl, loop, arch, etc.). The range s.204 <p<.246. For 1,600 boxes, we have P (false match) for p =.204, P (false match) for p =.246. The varaton of p changes the fnal predcton by no more than an order of magntude.

8 238 The UMAP Journal 25.3 (2004) Table 1. The multplcaton table of D 10. Source Number Mean densty of prnts (mnutae/mm 2 ) Osterburg et al. [1977].234 Dankmejer et al. [1980] 1, Stoney and Thornton [1987] Kngston [1964] Tha [2003] The Dependence on F In out ntal estmate, we take the number F of degrees of freedom of a feature n a prnt to be two: ether a rdge endng or a rdge bfurcaton. However, one can also consder the orentaton of a feature. Each mnuta les on a rdge, whch has a well-defned drecton. We dscretzed ths varable to one of eght possble drectons, angles from 0 to Thus each feature, nstead of havng 2 degrees of freedom, now has 16. The probablty of a false match, takng 1,600 grd cells and a probablty of occurrence p =.234, snow P (false match) = Takng n =12as before, we fnd that 391 =357 ( ) 1600 n 16 n n ( ) =357 P (false match) Ths s an astonshngly smaller probablty than the prevous estmate of The orentaton of a feature s no more dffcult to determne n practce than ts nature, so ncludng t n the comparson process s a great mprovement n effcacy wth a modest ncrease n effort. The Dependence on n It s crucal to determne how many matchng mnutae are necessary for a postve comparson. We have taken n =12n the precedng analyses; t s nstructve to consder the varaton of the probablty of a false match wth n. The graphs n Fgure 2 show that the probablty falls off sharply, even as n ncreases beyond 1. A value of n 5 s qute suffcent.

9 Can t Qute Put Our Fnger On It 239 Fgure 2. Probablty of false match (lnear and logarthmc scales) vs. n. Concluson The Model where Our model, n the most general form, s P (false match) = μ+σ =μ σ μ+σ =μ σ ( ) N n F n n (, N )F F s the number of degrees of freedom and N s the number of grd cells, μ and σ are the mean and standard devaton of the bnomal dstrbuton determned by N and p, p s the probablty of there beng a feature n a cell, and n s the number of mnutae beng used to make a comparson. The prelmnary result returned from our model, for n = 12 (a typcal threshold for postve dentfcaton n many countres), s P Further refnement of the parameters reduces ths to P We conclude that 12 s a very reasonable comparson crteron, and that n =5or 6 s qute damnng for any suspect so compared. Thus, we conclude that to a very hgh degree of certanty, not only that no two people, now lvng or havng ever lved n the past, have had the same thumbprnt, but also that there s a vanshngly small chance that two prnts are even close enough to be confused, gven a small fracton of mnutae from ther patterns to compare.

10 240 The UMAP Journal 25.3 (2004) DNA Analyss DNA dentty testng s based on aspects of the DNA patterns called loc. For a 100 % match, the FBI [Thompson et al. n.d.] recommends that 13 loc be used. Usng STR (Short Tandem Repeat) markers ensures that the nhertance profle at one locaton does not nfluence the nhertance at other locatons. Each loc has two alleles, so 26 alleles must match. The FBI says that the possblty of a false match s whle other sources quote between 10 9 and For two people chosen at random, the probablty of a match based on the four most frequently analyzed alleles s between and Ths s sgnfcantly hgher than our estmated probablty or a match for thumbprnts. Hence, thumbprntng remans the most accurate form of bometrc securty known. Strengths and Weaknesses Strengths of the Model Smplcty. Our model s based on easly understood prncples and smply expressed assumptons. Realstc assumptons. Parameters. The parameters n the model, such as the sze of the grd-box, the total area, and the number of mnutae needed to match two thumbprnts, can be easly vared. Degrees of freedom. In specfyng two dfferent knds of possble mnutae and 8 orentaton ranges for each one, the number of degrees of freedom s 16, greatly ncreasng the number of possble confguratons of thumbprnts and so mnmsng the probablty of msdentfcaton. Other studes [Osterburg et al. 1977; Galton 1892] do not take nto account the orentaton of the mnutae. By dscretzng the drectons of the features, we agan keep the model smple. Corroboraton. The probabltes returned by our model te n wth those gven by prevous studes by experts n the feld (Table 2). Table 2. Comparson probabltes of studes. Galton [1892] Osterburg et al. [1977] Stoney and Thornton [1987] our model to 10 22

11 Can t Qute Put Our Fnger On It 241 Weaknesses of the Model Multple entres. We assumed that n any gven grd-box only one mnuta can be present, whch s suffcently accurate for most types of mnutae. For example, both the brdge (consstng of two rdge bfurcatons) and the spur (consstng of a rdge bfurcaton and a rdge endng) have been defned [Osterburg et al. 1977] as beng less than 2 mm n length. Thus f the brdge or spur s more than mm n length, ther consttuent endngs and bfurcatons appear n dfferent boxes and are counted as two separate mnutae. However, for mnutae consstng of rdge endngs and rdge bfurcatons n very close proxmty, there s a chance that each wll not be caught n a dfferent box. An example s a dot. The dstance between the two rdge endngs that make up a dot s so small that t s unlkely that our model would catch these two occurrences of rdge endngs n dfferent boxes. A dot has been defned [Osterburg et al. 1977] as beng large enough to encompass one pore, whose sze ranges from mm to 0.22 mm [Roddy and Stosz 1997]. Therefore, the two rdge endngs wll not appear n dfferent boxes but wll nstead be msdentfed as a sngle mnuta. Independence of mnuta occurrence. We assume that the placement of a mnuta s completely unrelated to the placement of any others. Ths s not qute the case; there s a slght tendency for mnutae not to occur n drect proxmty to each other. Global analyss. The overall rdge pattern of a thumbprnt s entrely dstnctve n ts own rght. We have not quantfed ths factor n our model. Appendx: Classfcaton Mnutae The rdges n a fngerprnt or thumbprnt form varous patterns, those patterns beng called mnutae. Ten dfferent types are shown n Fgure A1. Rdge Endng. A rdge endng occurs when a rdge ends abruptly. We defne the orentaton of a rdge endng as the drecton the rdge came from. Bfurcaton. A bfurcaton s formed when two dfferent rdges merge. We defne the orentaton as beng the drecton n whch the merged rdge came from. Island. An sland s a short rdge, comprsed of two rdge endngs whose orentatons are n opposng drectons. Two rdge endngs occurrng n neghbourng boxes wth opposte confguraton ndcate the presence of an sland.

12 242 The UMAP Journal 25.3 (2004) Fgure A1. Ten types of mnuta. Dot. A dot s an sland but on a smaller scale. Brdge. A brdge s when a rdge branches out and merges wth another rdge wthn a short regon. It s composed of two bfurcatons. Spur. A spur s when a rdge branches out and does not merge wth another rdge. It s composed of one bfurcaton and one rdge endng. Eye. An eye s formed by a rdge branchng out nto two rdges, and then recombnng agan a short dstance later. It conssts of two bfurcatons. Double Bfurcaton. As the name suggests, ths type of mnuta contans two bfurcatons n successon. Delta. Ths type of mnuta s composed of a dot and bfurcaton, where the dot s between the mergng rdges. Trfurcaton. A trfurcaton s a rdge that splts nto 3 separate branches. It can be thought of as two bfurcatons occurrng n the same place. Rdge Patterns Fgure A2 shows examples of the dfferent classfcatons of a fngerprnt, ncludng the presence of cores and delta ponts.

13 Can t Qute Put Our Fnger On It 243 Fgure A2. Features n a fngerprnt. References Anonymous Frcton skn growth. homestead.com/frcton_skn_growth.html. Ballan, Meltem, F. Ayhan Sakarya, and Bran L. Evans A fngerprnt classfcaton technque usng drectonal mages. In Proceedngs of the IEEE Aslomar Conference on Sgnals, Systems, and Computers (Nov. 3 5, 1997, Pacfc Grove, CA), vol. 1, papers/1997/fngerprnts/. Beeton, Mary Scentfc methodology and the frcton rdge dentfcaton process. Identfcaton Canada 25 (3) (September 2002). Reprnted n Georga Forensc News 32 (3) (November 2002): 1, rdgesandfurrows.homestead.com/fles/scentfc_methodology.pdf. Collns, Martn W. Marty Realzng the full value of latent prnts. Calforna Identfcaton Dgest (March 1992): com/realzng_the_full_value_of_late.htm. Dankmejer, J., J.M. Waltman, and A.G. de Wlde Bologcal foundatons for forensc dentfcaton based on fngerprnts. Acta Morphologca Nederlando-scandnavca 18 (1) (March 1980): nh.gov/entrez/query.fcg?cmd=retreve&db=pubmed&lst_uds= &dopt= Ctaton. Galton, F Fnger Prnts. London: Macmllan.

14 244 The UMAP Journal 25.3 (2004) Haub, C How many people have ever lved on Earth? Populaton Today 23 (2) (February): 4 5. Reprnted. 30 (8) (November/December 2002): ct-Dec02/How_Many_People_Have_Ever_Lved_on_Earth_.htm. Kngston, C.R Probablstc analyss of partal fngerprnt patterns. D.Crm. dssertaton, Unversty of Calforna, Berkeley. Lndsay, Don Don Lndsay archve. org/creaton/defntons.htm. Nlsson, Kenneth, and Josef Bgun Localzaton of correspondng ponts n fngerprnts by complex flterng. Pattern Recognton Letters 24 (13) (September 2003): nlsson03prl.pdf. Optel Ltd New numercal methods of fngerprnts recognton based on mathematcal descrpton of arrangement of dermatoglyphcs and creaton of mnutea [sc]. Osterburg, James W., T. Parthasarathy, T.E.S. Raghavan and Stanley L. Sclove Development of a mathematcal formula for the calculaton of fngerprnt probabltes based on ndvdual characterstcs. Journal of the Amercan Statstcal Assocaton 72: Pankant, Sharath, Sall Prabhakar, and Anl K. Jan On the ndvdualty of fngerprnts. Roddy, A.R., and J.D. Stosz Fngerprnt features Statstcal analyss and system performances estmates. Proceedngs of the IEEE 85 (9): Sclove, Stanley L The occurrence of fngerprnt characterstcs as a twodmensonal process. Communcatons n Statstcs (A) 9: http: //tgger.uc.edu/~slsclove/papers.htm. Stoney, Davd A., and John I. Thornton A systematc study of epdermal rdge mnutae. Journal of Forensc Scences. Tha, Raymond Fngerprnt mage enhancement and mnutae extracton. Honours thess, School of Computer Scence and Software Engneerng, Unversty of Western Australa. studentprojects/raymondtha/. Vacca, J Bometrc securty solutons. Excerpt from Identty Theft Indanapols, IN: Prentce Hall PTR.http://www.nformt.com/artcles/ artcle.asp?p=29801&redr=1. Thompson, Wllam C., Smon Ford, Travs Doom, Mchael Raymer, and Dan Krane. n.d. Evaluatng forensc DNA evdence: Essental elements of a competent defense revew. SIAC03-Krane2.PDF.

Department of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6

Department of Quantitative Methods & Information Systems. Time Series and Their Components QMIS 320. Chapter 6 Department of Quanttatve Methods & Informaton Systems Tme Seres and Ther Components QMIS 30 Chapter 6 Fall 00 Dr. Mohammad Zanal These sldes were modfed from ther orgnal source for educatonal purpose only.

More information

Psychology 282 Lecture #24 Outline Regression Diagnostics: Outliers

Psychology 282 Lecture #24 Outline Regression Diagnostics: Outliers Psychology 282 Lecture #24 Outlne Regresson Dagnostcs: Outlers In an earler lecture we studed the statstcal assumptons underlyng the regresson model, ncludng the followng ponts: Formal statement of assumptons.

More information

Chapter 13: Multiple Regression

Chapter 13: Multiple Regression Chapter 13: Multple Regresson 13.1 Developng the multple-regresson Model The general model can be descrbed as: It smplfes for two ndependent varables: The sample ft parameter b 0, b 1, and b are used to

More information

STAT 3008 Applied Regression Analysis

STAT 3008 Applied Regression Analysis STAT 3008 Appled Regresson Analyss Tutoral : Smple Lnear Regresson LAI Chun He Department of Statstcs, The Chnese Unversty of Hong Kong 1 Model Assumpton To quantfy the relatonshp between two factors,

More information

Negative Binomial Regression

Negative Binomial Regression STATGRAPHICS Rev. 9/16/2013 Negatve Bnomal Regresson Summary... 1 Data Input... 3 Statstcal Model... 3 Analyss Summary... 4 Analyss Optons... 7 Plot of Ftted Model... 8 Observed Versus Predcted... 10 Predctons...

More information

NUMERICAL DIFFERENTIATION

NUMERICAL DIFFERENTIATION NUMERICAL DIFFERENTIATION 1 Introducton Dfferentaton s a method to compute the rate at whch a dependent output y changes wth respect to the change n the ndependent nput x. Ths rate of change s called the

More information

Code_Aster. Identification of the model of Weibull

Code_Aster. Identification of the model of Weibull Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : /8 Responsable : PARROT Aurore Clé : R70209 Révson : Identfcaton of the model of Webull Summary One tackles here the problem of the

More information

Basic Business Statistics, 10/e

Basic Business Statistics, 10/e Chapter 13 13-1 Basc Busness Statstcs 11 th Edton Chapter 13 Smple Lnear Regresson Basc Busness Statstcs, 11e 009 Prentce-Hall, Inc. Chap 13-1 Learnng Objectves In ths chapter, you learn: How to use regresson

More information

Statistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation

Statistics for Managers Using Microsoft Excel/SPSS Chapter 13 The Simple Linear Regression Model and Correlation Statstcs for Managers Usng Mcrosoft Excel/SPSS Chapter 13 The Smple Lnear Regresson Model and Correlaton 1999 Prentce-Hall, Inc. Chap. 13-1 Chapter Topcs Types of Regresson Models Determnng the Smple Lnear

More information

Comparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method

Comparison of the Population Variance Estimators. of 2-Parameter Exponential Distribution Based on. Multiple Criteria Decision Making Method Appled Mathematcal Scences, Vol. 7, 0, no. 47, 07-0 HIARI Ltd, www.m-hkar.com Comparson of the Populaton Varance Estmators of -Parameter Exponental Dstrbuton Based on Multple Crtera Decson Makng Method

More information

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity

Week3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle

More information

Joint Statistical Meetings - Biopharmaceutical Section

Joint Statistical Meetings - Biopharmaceutical Section Iteratve Ch-Square Test for Equvalence of Multple Treatment Groups Te-Hua Ng*, U.S. Food and Drug Admnstraton 1401 Rockvlle Pke, #200S, HFM-217, Rockvlle, MD 20852-1448 Key Words: Equvalence Testng; Actve

More information

Department of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution

Department of Statistics University of Toronto STA305H1S / 1004 HS Design and Analysis of Experiments Term Test - Winter Solution Department of Statstcs Unversty of Toronto STA35HS / HS Desgn and Analyss of Experments Term Test - Wnter - Soluton February, Last Name: Frst Name: Student Number: Instructons: Tme: hours. Ads: a non-programmable

More information

A PROBABILITY-DRIVEN SEARCH ALGORITHM FOR SOLVING MULTI-OBJECTIVE OPTIMIZATION PROBLEMS

A PROBABILITY-DRIVEN SEARCH ALGORITHM FOR SOLVING MULTI-OBJECTIVE OPTIMIZATION PROBLEMS HCMC Unversty of Pedagogy Thong Nguyen Huu et al. A PROBABILITY-DRIVEN SEARCH ALGORITHM FOR SOLVING MULTI-OBJECTIVE OPTIMIZATION PROBLEMS Thong Nguyen Huu and Hao Tran Van Department of mathematcs-nformaton,

More information

Section 8.3 Polar Form of Complex Numbers

Section 8.3 Polar Form of Complex Numbers 80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the

More information

x yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting.

x yi In chapter 14, we want to perform inference (i.e. calculate confidence intervals and perform tests of significance) in this setting. The Practce of Statstcs, nd ed. Chapter 14 Inference for Regresson Introducton In chapter 3 we used a least-squares regresson lne (LSRL) to represent a lnear relatonshp etween two quanttatve explanator

More information

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur

Module 3 LOSSY IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur Module 3 LOSSY IMAGE COMPRESSION SYSTEMS Verson ECE IIT, Kharagpur Lesson 6 Theory of Quantzaton Verson ECE IIT, Kharagpur Instructonal Objectves At the end of ths lesson, the students should be able to:

More information

Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for P Charts. Dr. Wayne A. Taylor

Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for P Charts. Dr. Wayne A. Taylor Taylor Enterprses, Inc. Control Lmts for P Charts Copyrght 2017 by Taylor Enterprses, Inc., All Rghts Reserved. Control Lmts for P Charts Dr. Wayne A. Taylor Abstract: P charts are used for count data

More information

Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law:

Introduction to Vapor/Liquid Equilibrium, part 2. Raoult s Law: CE304, Sprng 2004 Lecture 4 Introducton to Vapor/Lqud Equlbrum, part 2 Raoult s Law: The smplest model that allows us do VLE calculatons s obtaned when we assume that the vapor phase s an deal gas, and

More information

Case A. P k = Ni ( 2L i k 1 ) + (# big cells) 10d 2 P k.

Case A. P k = Ni ( 2L i k 1 ) + (# big cells) 10d 2 P k. THE CELLULAR METHOD In ths lecture, we ntroduce the cellular method as an approach to ncdence geometry theorems lke the Szemeréd-Trotter theorem. The method was ntroduced n the paper Combnatoral complexty

More information

Lecture 12: Classification

Lecture 12: Classification Lecture : Classfcaton g Dscrmnant functons g The optmal Bayes classfer g Quadratc classfers g Eucldean and Mahalanobs metrcs g K Nearest Neghbor Classfers Intellgent Sensor Systems Rcardo Guterrez-Osuna

More information

Indeterminate pin-jointed frames (trusses)

Indeterminate pin-jointed frames (trusses) Indetermnate pn-jonted frames (trusses) Calculaton of member forces usng force method I. Statcal determnacy. The degree of freedom of any truss can be derved as: w= k d a =, where k s the number of all

More information

Chapter 8. Potential Energy and Conservation of Energy

Chapter 8. Potential Energy and Conservation of Energy Chapter 8 Potental Energy and Conservaton of Energy In ths chapter we wll ntroduce the followng concepts: Potental Energy Conservatve and non-conservatve forces Mechancal Energy Conservaton of Mechancal

More information

Code_Aster. Identification of the Summarized

Code_Aster. Identification of the Summarized Verson Ttre : Identfcaton du modèle de Webull Date : 2/09/2009 Page : /8 Responsable : Aurore PARROT Clé : R70209 Révson : 609 Identfcaton of the Summarzed Webull model One tackles here the problem of

More information

Bayesian predictive Configural Frequency Analysis

Bayesian predictive Configural Frequency Analysis Psychologcal Test and Assessment Modelng, Volume 54, 2012 (3), 285-292 Bayesan predctve Confgural Frequency Analyss Eduardo Gutérrez-Peña 1 Abstract Confgural Frequency Analyss s a method for cell-wse

More information

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4)

Econ107 Applied Econometrics Topic 3: Classical Model (Studenmund, Chapter 4) I. Classcal Assumptons Econ7 Appled Econometrcs Topc 3: Classcal Model (Studenmund, Chapter 4) We have defned OLS and studed some algebrac propertes of OLS. In ths topc we wll study statstcal propertes

More information

= z 20 z n. (k 20) + 4 z k = 4

= z 20 z n. (k 20) + 4 z k = 4 Problem Set #7 solutons 7.2.. (a Fnd the coeffcent of z k n (z + z 5 + z 6 + z 7 + 5, k 20. We use the known seres expanson ( n+l ( z l l z n below: (z + z 5 + z 6 + z 7 + 5 (z 5 ( + z + z 2 + z + 5 5

More information

Lecture 12: Discrete Laplacian

Lecture 12: Discrete Laplacian Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly

More information

Statistics for Business and Economics

Statistics for Business and Economics Statstcs for Busness and Economcs Chapter 11 Smple Regresson Copyrght 010 Pearson Educaton, Inc. Publshng as Prentce Hall Ch. 11-1 11.1 Overvew of Lnear Models n An equaton can be ft to show the best lnear

More information

STAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression

STAT 405 BIOSTATISTICS (Fall 2016) Handout 15 Introduction to Logistic Regression STAT 45 BIOSTATISTICS (Fall 26) Handout 5 Introducton to Logstc Regresson Ths handout covers materal found n Secton 3.7 of your text. You may also want to revew regresson technques n Chapter. In ths handout,

More information

Predictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore

Predictive Analytics : QM901.1x Prof U Dinesh Kumar, IIMB. All Rights Reserved, Indian Institute of Management Bangalore Sesson Outlne Introducton to classfcaton problems and dscrete choce models. Introducton to Logstcs Regresson. Logstc functon and Logt functon. Maxmum Lkelhood Estmator (MLE) for estmaton of LR parameters.

More information

ANOVA. The Observations y ij

ANOVA. The Observations y ij ANOVA Stands for ANalyss Of VArance But t s a test of dfferences n means The dea: The Observatons y j Treatment group = 1 = 2 = k y 11 y 21 y k,1 y 12 y 22 y k,2 y 1, n1 y 2, n2 y k, nk means: m 1 m 2

More information

Uncertainty in measurements of power and energy on power networks

Uncertainty in measurements of power and energy on power networks Uncertanty n measurements of power and energy on power networks E. Manov, N. Kolev Department of Measurement and Instrumentaton, Techncal Unversty Sofa, bul. Klment Ohrdsk No8, bl., 000 Sofa, Bulgara Tel./fax:

More information

Laboratory 3: Method of Least Squares

Laboratory 3: Method of Least Squares Laboratory 3: Method of Least Squares Introducton Consder the graph of expermental data n Fgure 1. In ths experment x s the ndependent varable and y the dependent varable. Clearly they are correlated wth

More information

Remarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence

Remarks on the Properties of a Quasi-Fibonacci-like Polynomial Sequence Remarks on the Propertes of a Quas-Fbonacc-lke Polynomal Sequence Brce Merwne LIU Brooklyn Ilan Wenschelbaum Wesleyan Unversty Abstract Consder the Quas-Fbonacc-lke Polynomal Sequence gven by F 0 = 1,

More information

Lecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding

Lecture 9: Linear regression: centering, hypothesis testing, multiple covariates, and confounding Recall: man dea of lnear regresson Lecture 9: Lnear regresson: centerng, hypothess testng, multple covarates, and confoundng Sandy Eckel seckel@jhsph.edu 6 May 8 Lnear regresson can be used to study an

More information

Cokriging Partial Grades - Application to Block Modeling of Copper Deposits

Cokriging Partial Grades - Application to Block Modeling of Copper Deposits Cokrgng Partal Grades - Applcaton to Block Modelng of Copper Deposts Serge Séguret 1, Julo Benscell 2 and Pablo Carrasco 2 Abstract Ths work concerns mneral deposts made of geologcal bodes such as breccas

More information

Bayesian Learning. Smart Home Health Analytics Spring Nirmalya Roy Department of Information Systems University of Maryland Baltimore County

Bayesian Learning. Smart Home Health Analytics Spring Nirmalya Roy Department of Information Systems University of Maryland Baltimore County Smart Home Health Analytcs Sprng 2018 Bayesan Learnng Nrmalya Roy Department of Informaton Systems Unversty of Maryland Baltmore ounty www.umbc.edu Bayesan Learnng ombnes pror knowledge wth evdence to

More information

The Feynman path integral

The Feynman path integral The Feynman path ntegral Aprl 3, 205 Hesenberg and Schrödnger pctures The Schrödnger wave functon places the tme dependence of a physcal system n the state, ψ, t, where the state s a vector n Hlbert space

More information

Pulse Coded Modulation

Pulse Coded Modulation Pulse Coded Modulaton PCM (Pulse Coded Modulaton) s a voce codng technque defned by the ITU-T G.711 standard and t s used n dgtal telephony to encode the voce sgnal. The frst step n the analog to dgtal

More information

The Geometry of Logit and Probit

The Geometry of Logit and Probit The Geometry of Logt and Probt Ths short note s meant as a supplement to Chapters and 3 of Spatal Models of Parlamentary Votng and the notaton and reference to fgures n the text below s to those two chapters.

More information

Singular Value Decomposition: Theory and Applications

Singular Value Decomposition: Theory and Applications Sngular Value Decomposton: Theory and Applcatons Danel Khashab Sprng 2015 Last Update: March 2, 2015 1 Introducton A = UDV where columns of U and V are orthonormal and matrx D s dagonal wth postve real

More information

Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for U Charts. Dr. Wayne A. Taylor

Copyright 2017 by Taylor Enterprises, Inc., All Rights Reserved. Adjusted Control Limits for U Charts. Dr. Wayne A. Taylor Taylor Enterprses, Inc. Adjusted Control Lmts for U Charts Copyrght 207 by Taylor Enterprses, Inc., All Rghts Reserved. Adjusted Control Lmts for U Charts Dr. Wayne A. Taylor Abstract: U charts are used

More information

The Study of Teaching-learning-based Optimization Algorithm

The Study of Teaching-learning-based Optimization Algorithm Advanced Scence and Technology Letters Vol. (AST 06), pp.05- http://dx.do.org/0.57/astl.06. The Study of Teachng-learnng-based Optmzaton Algorthm u Sun, Yan fu, Lele Kong, Haolang Q,, Helongang Insttute

More information

Professor Terje Haukaas University of British Columbia, Vancouver The Q4 Element

Professor Terje Haukaas University of British Columbia, Vancouver  The Q4 Element Professor Terje Haukaas Unversty of Brtsh Columba, ancouver www.nrsk.ubc.ca The Q Element Ths document consders fnte elements that carry load only n ther plane. These elements are sometmes referred to

More information

1 Generating functions, continued

1 Generating functions, continued Generatng functons, contnued. Generatng functons and parttons We can make use of generatng functons to answer some questons a bt more restrctve than we ve done so far: Queston : Fnd a generatng functon

More information

Andreas C. Drichoutis Agriculural University of Athens. Abstract

Andreas C. Drichoutis Agriculural University of Athens. Abstract Heteroskedastcty, the sngle crossng property and ordered response models Andreas C. Drchouts Agrculural Unversty of Athens Panagots Lazards Agrculural Unversty of Athens Rodolfo M. Nayga, Jr. Texas AMUnversty

More information

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity

LINEAR REGRESSION ANALYSIS. MODULE IX Lecture Multicollinearity LINEAR REGRESSION ANALYSIS MODULE IX Lecture - 30 Multcollnearty Dr. Shalabh Department of Mathematcs and Statstcs Indan Insttute of Technology Kanpur 2 Remedes for multcollnearty Varous technques have

More information

Tracking with Kalman Filter

Tracking with Kalman Filter Trackng wth Kalman Flter Scott T. Acton Vrgna Image and Vdeo Analyss (VIVA), Charles L. Brown Department of Electrcal and Computer Engneerng Department of Bomedcal Engneerng Unversty of Vrgna, Charlottesvlle,

More information

III. Econometric Methodology Regression Analysis

III. Econometric Methodology Regression Analysis Page Econ07 Appled Econometrcs Topc : An Overvew of Regresson Analyss (Studenmund, Chapter ) I. The Nature and Scope of Econometrcs. Lot s of defntons of econometrcs. Nobel Prze Commttee Paul Samuelson,

More information

Chapter 14 Simple Linear Regression

Chapter 14 Simple Linear Regression Chapter 4 Smple Lnear Regresson Chapter 4 - Smple Lnear Regresson Manageral decsons often are based on the relatonshp between two or more varables. Regresson analss can be used to develop an equaton showng

More information

18. SIMPLE LINEAR REGRESSION III

18. SIMPLE LINEAR REGRESSION III 8. SIMPLE LINEAR REGRESSION III US Domestc Beers: Calores vs. % Alcohol Ftted Values and Resduals To each observed x, there corresponds a y-value on the ftted lne, y ˆ ˆ = α + x. The are called ftted values.

More information

Weighted Voting Systems

Weighted Voting Systems Weghted Votng Systems Elssa Brown Chrssy Donovan Charles Noneman June 5, 2009 Votng systems can be deceptve. For nstance, a votng system mght consst of four people, n whch three of the people have 2 votes,

More information

RELIABILITY ASSESSMENT

RELIABILITY ASSESSMENT CHAPTER Rsk Analyss n Engneerng and Economcs RELIABILITY ASSESSMENT A. J. Clark School of Engneerng Department of Cvl and Envronmental Engneerng 4a CHAPMAN HALL/CRC Rsk Analyss for Engneerng Department

More information

Lecture 3 Stat102, Spring 2007

Lecture 3 Stat102, Spring 2007 Lecture 3 Stat0, Sprng 007 Chapter 3. 3.: Introducton to regresson analyss Lnear regresson as a descrptve technque The least-squares equatons Chapter 3.3 Samplng dstrbuton of b 0, b. Contnued n net lecture

More information

Note 10. Modeling and Simulation of Dynamic Systems

Note 10. Modeling and Simulation of Dynamic Systems Lecture Notes of ME 475: Introducton to Mechatroncs Note 0 Modelng and Smulaton of Dynamc Systems Department of Mechancal Engneerng, Unversty Of Saskatchewan, 57 Campus Drve, Saskatoon, SK S7N 5A9, Canada

More information

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM

ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look

More information

Lossy Compression. Compromise accuracy of reconstruction for increased compression.

Lossy Compression. Compromise accuracy of reconstruction for increased compression. Lossy Compresson Compromse accuracy of reconstructon for ncreased compresson. The reconstructon s usually vsbly ndstngushable from the orgnal mage. Typcally, one can get up to 0:1 compresson wth almost

More information

System in Weibull Distribution

System in Weibull Distribution Internatonal Matheatcal Foru 4 9 no. 9 94-95 Relablty Equvalence Factors of a Seres-Parallel Syste n Webull Dstrbuton M. A. El-Dacese Matheatcs Departent Faculty of Scence Tanta Unversty Tanta Egypt eldacese@yahoo.co

More information

Topic 23 - Randomized Complete Block Designs (RCBD)

Topic 23 - Randomized Complete Block Designs (RCBD) Topc 3 ANOVA (III) 3-1 Topc 3 - Randomzed Complete Block Desgns (RCBD) Defn: A Randomzed Complete Block Desgn s a varant of the completely randomzed desgn (CRD) that we recently learned. In ths desgn,

More information

A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS. Dr. Derald E. Wentzien, Wesley College, (302) ,

A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS. Dr. Derald E. Wentzien, Wesley College, (302) , A LINEAR PROGRAM TO COMPARE MULTIPLE GROSS CREDIT LOSS FORECASTS Dr. Derald E. Wentzen, Wesley College, (302) 736-2574, wentzde@wesley.edu ABSTRACT A lnear programmng model s developed and used to compare

More information

Electrical double layer: revisit based on boundary conditions

Electrical double layer: revisit based on boundary conditions Electrcal double layer: revst based on boundary condtons Jong U. Km Department of Electrcal and Computer Engneerng, Texas A&M Unversty College Staton, TX 77843-318, USA Abstract The electrcal double layer

More information

Einstein-Podolsky-Rosen Paradox

Einstein-Podolsky-Rosen Paradox H 45 Quantum Measurement and Spn Wnter 003 Ensten-odolsky-Rosen aradox The Ensten-odolsky-Rosen aradox s a gedanken experment desgned to show that quantum mechancs s an ncomplete descrpton of realty. The

More information

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. . For P such independent random variables (aka degrees of freedom): 1 =

Fall 2012 Analysis of Experimental Measurements B. Eisenstein/rev. S. Errede. . For P such independent random variables (aka degrees of freedom): 1 = Fall Analss of Epermental Measurements B. Esensten/rev. S. Errede More on : The dstrbuton s the.d.f. for a (normalzed sum of squares of ndependent random varables, each one of whch s dstrbuted as N (,.

More information

since [1-( 0+ 1x1i+ 2x2 i)] [ 0+ 1x1i+ assumed to be a reasonable approximation

since [1-( 0+ 1x1i+ 2x2 i)] [ 0+ 1x1i+ assumed to be a reasonable approximation Econ 388 R. Butler 204 revsons Lecture 4 Dummy Dependent Varables I. Lnear Probablty Model: the Regresson model wth a dummy varables as the dependent varable assumpton, mplcaton regular multple regresson

More information

Goodness of fit and Wilks theorem

Goodness of fit and Wilks theorem DRAFT 0.0 Glen Cowan 3 June, 2013 Goodness of ft and Wlks theorem Suppose we model data y wth a lkelhood L(µ) that depends on a set of N parameters µ = (µ 1,...,µ N ). Defne the statstc t µ ln L(µ) L(ˆµ),

More information

CHAPTER 4. Vector Spaces

CHAPTER 4. Vector Spaces man 2007/2/16 page 234 CHAPTER 4 Vector Spaces To crtcze mathematcs for ts abstracton s to mss the pont entrel. Abstracton s what makes mathematcs work. Ian Stewart The man am of ths tet s to stud lnear

More information

INF 5860 Machine learning for image classification. Lecture 3 : Image classification and regression part II Anne Solberg January 31, 2018

INF 5860 Machine learning for image classification. Lecture 3 : Image classification and regression part II Anne Solberg January 31, 2018 INF 5860 Machne learnng for mage classfcaton Lecture 3 : Image classfcaton and regresson part II Anne Solberg January 3, 08 Today s topcs Multclass logstc regresson and softma Regularzaton Image classfcaton

More information

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens

THE CHINESE REMAINDER THEOREM. We should thank the Chinese for their wonderful remainder theorem. Glenn Stevens THE CHINESE REMAINDER THEOREM KEITH CONRAD We should thank the Chnese for ther wonderful remander theorem. Glenn Stevens 1. Introducton The Chnese remander theorem says we can unquely solve any par of

More information

A new construction of 3-separable matrices via an improved decoding of Macula s construction

A new construction of 3-separable matrices via an improved decoding of Macula s construction Dscrete Optmzaton 5 008 700 704 Contents lsts avalable at ScenceDrect Dscrete Optmzaton journal homepage: wwwelsevercom/locate/dsopt A new constructon of 3-separable matrces va an mproved decodng of Macula

More information

Irregular vibrations in multi-mass discrete-continuous systems torsionally deformed

Irregular vibrations in multi-mass discrete-continuous systems torsionally deformed (2) 4 48 Irregular vbratons n mult-mass dscrete-contnuous systems torsonally deformed Abstract In the paper rregular vbratons of dscrete-contnuous systems consstng of an arbtrary number rgd bodes connected

More information

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry

Workshop: Approximating energies and wave functions Quantum aspects of physical chemistry Workshop: Approxmatng energes and wave functons Quantum aspects of physcal chemstry http://quantum.bu.edu/pltl/6/6.pdf Last updated Thursday, November 7, 25 7:9:5-5: Copyrght 25 Dan Dll (dan@bu.edu) Department

More information

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS

8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 493 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces you have studed thus far n the text are real vector spaces because the scalars

More information

Statistical Evaluation of WATFLOOD

Statistical Evaluation of WATFLOOD tatstcal Evaluaton of WATFLD By: Angela MacLean, Dept. of Cvl & Envronmental Engneerng, Unversty of Waterloo, n. ctober, 005 The statstcs program assocated wth WATFLD uses spl.csv fle that s produced wth

More information

CS 468 Lecture 16: Isometry Invariance and Spectral Techniques

CS 468 Lecture 16: Isometry Invariance and Spectral Techniques CS 468 Lecture 16: Isometry Invarance and Spectral Technques Justn Solomon Scrbe: Evan Gawlk Introducton. In geometry processng, t s often desrable to characterze the shape of an object n a manner that

More information

Modeling of Dynamic Systems

Modeling of Dynamic Systems Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how

More information

Comparative Studies of Law of Conservation of Energy. and Law Clusters of Conservation of Generalized Energy

Comparative Studies of Law of Conservation of Energy. and Law Clusters of Conservation of Generalized Energy Comparatve Studes of Law of Conservaton of Energy and Law Clusters of Conservaton of Generalzed Energy No.3 of Comparatve Physcs Seres Papers Fu Yuhua (CNOOC Research Insttute, E-mal:fuyh1945@sna.com)

More information

CSC 411 / CSC D11 / CSC C11

CSC 411 / CSC D11 / CSC C11 18 Boostng s a general strategy for learnng classfers by combnng smpler ones. The dea of boostng s to take a weak classfer that s, any classfer that wll do at least slghtly better than chance and use t

More information

Exercises of Chapter 2

Exercises of Chapter 2 Exercses of Chapter Chuang-Cheh Ln Department of Computer Scence and Informaton Engneerng, Natonal Chung Cheng Unversty, Mng-Hsung, Chay 61, Tawan. Exercse.6. Suppose that we ndependently roll two standard

More information

Games of Threats. Elon Kohlberg Abraham Neyman. Working Paper

Games of Threats. Elon Kohlberg Abraham Neyman. Working Paper Games of Threats Elon Kohlberg Abraham Neyman Workng Paper 18-023 Games of Threats Elon Kohlberg Harvard Busness School Abraham Neyman The Hebrew Unversty of Jerusalem Workng Paper 18-023 Copyrght 2017

More information

Problem Set 9 Solutions

Problem Set 9 Solutions Desgn and Analyss of Algorthms May 4, 2015 Massachusetts Insttute of Technology 6.046J/18.410J Profs. Erk Demane, Srn Devadas, and Nancy Lynch Problem Set 9 Solutons Problem Set 9 Solutons Ths problem

More information

As is less than , there is insufficient evidence to reject H 0 at the 5% level. The data may be modelled by Po(2).

As is less than , there is insufficient evidence to reject H 0 at the 5% level. The data may be modelled by Po(2). Ch-squared tests 6D 1 a H 0 : The data can be modelled by a Po() dstrbuton. H 1 : The data cannot be modelled by Po() dstrbuton. The observed and expected results are shown n the table. The last two columns

More information

A PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY.

A PROCEDURE FOR SIMULATING THE NONLINEAR CONDUCTION HEAT TRANSFER IN A BODY WITH TEMPERATURE DEPENDENT THERMAL CONDUCTIVITY. Proceedngs of the th Brazlan Congress of Thermal Scences and Engneerng -- ENCIT 006 Braz. Soc. of Mechancal Scences and Engneerng -- ABCM, Curtba, Brazl,- Dec. 5-8, 006 A PROCEDURE FOR SIMULATING THE NONLINEAR

More information

Explicit constructions of all separable two-qubits density matrices and related problems for three-qubits systems

Explicit constructions of all separable two-qubits density matrices and related problems for three-qubits systems Explct constructons of all separable two-qubts densty matrces and related problems for three-qubts systems Y. en-ryeh and. Mann Physcs Department, Technon-Israel Insttute of Technology, Hafa 2000, Israel

More information

Sampling Theory MODULE VII LECTURE - 23 VARYING PROBABILITY SAMPLING

Sampling Theory MODULE VII LECTURE - 23 VARYING PROBABILITY SAMPLING Samplng heory MODULE VII LECURE - 3 VARYIG PROBABILIY SAMPLIG DR. SHALABH DEPARME OF MAHEMAICS AD SAISICS IDIA ISIUE OF ECHOLOGY KAPUR he smple random samplng scheme provdes a random sample where every

More information

Fundamental loop-current method using virtual voltage sources technique for special cases

Fundamental loop-current method using virtual voltage sources technique for special cases Fundamental loop-current method usng vrtual voltage sources technque for specal cases George E. Chatzaraks, 1 Marna D. Tortorel 1 and Anastasos D. Tzolas 1 Electrcal and Electroncs Engneerng Departments,

More information

Case Study of Markov Chains Ray-Knight Compactification

Case Study of Markov Chains Ray-Knight Compactification Internatonal Journal of Contemporary Mathematcal Scences Vol. 9, 24, no. 6, 753-76 HIKAI Ltd, www.m-har.com http://dx.do.org/.2988/cms.24.46 Case Study of Marov Chans ay-knght Compactfcaton HaXa Du and

More information

Non-Mixture Cure Model for Interval Censored Data: Simulation Study ABSTRACT

Non-Mixture Cure Model for Interval Censored Data: Simulation Study ABSTRACT Malaysan Journal of Mathematcal Scences 8(S): 37-44 (2014) Specal Issue: Internatonal Conference on Mathematcal Scences and Statstcs 2013 (ICMSS2013) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal

More information

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur

Module 2. Random Processes. Version 2 ECE IIT, Kharagpur Module Random Processes Lesson 6 Functons of Random Varables After readng ths lesson, ou wll learn about cdf of functon of a random varable. Formula for determnng the pdf of a random varable. Let, X be

More information

Econ Statistical Properties of the OLS estimator. Sanjaya DeSilva

Econ Statistical Properties of the OLS estimator. Sanjaya DeSilva Econ 39 - Statstcal Propertes of the OLS estmator Sanjaya DeSlva September, 008 1 Overvew Recall that the true regresson model s Y = β 0 + β 1 X + u (1) Applyng the OLS method to a sample of data, we estmate

More information

Online Classification: Perceptron and Winnow

Online Classification: Perceptron and Winnow E0 370 Statstcal Learnng Theory Lecture 18 Nov 8, 011 Onlne Classfcaton: Perceptron and Wnnow Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton In ths lecture we wll start to study the onlne learnng

More information

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force.

A particle in a state of uniform motion remain in that state of motion unless acted upon by external force. The fundamental prncples of classcal mechancs were lad down by Galleo and Newton n the 16th and 17th centures. In 1686, Newton wrote the Prncpa where he gave us three laws of moton, one law of gravty,

More information

Quantum Mechanics for Scientists and Engineers. David Miller

Quantum Mechanics for Scientists and Engineers. David Miller Quantum Mechancs for Scentsts and Engneers Davd Mller Types of lnear operators Types of lnear operators Blnear expanson of operators Blnear expanson of lnear operators We know that we can expand functons

More information

Excess Error, Approximation Error, and Estimation Error

Excess Error, Approximation Error, and Estimation Error E0 370 Statstcal Learnng Theory Lecture 10 Sep 15, 011 Excess Error, Approxaton Error, and Estaton Error Lecturer: Shvan Agarwal Scrbe: Shvan Agarwal 1 Introducton So far, we have consdered the fnte saple

More information

Markov Chain Monte Carlo Lecture 6

Markov Chain Monte Carlo Lecture 6 where (x 1,..., x N ) X N, N s called the populaton sze, f(x) f (x) for at least one {1, 2,..., N}, and those dfferent from f(x) are called the tral dstrbutons n terms of mportance samplng. Dfferent ways

More information

Adiabatic Sorption of Ammonia-Water System and Depicting in p-t-x Diagram

Adiabatic Sorption of Ammonia-Water System and Depicting in p-t-x Diagram Adabatc Sorpton of Ammona-Water System and Depctng n p-t-x Dagram J. POSPISIL, Z. SKALA Faculty of Mechancal Engneerng Brno Unversty of Technology Techncka 2, Brno 61669 CZECH REPUBLIC Abstract: - Absorpton

More information

Maximizing Overlap of Large Primary Sampling Units in Repeated Sampling: A comparison of Ernst s Method with Ohlsson s Method

Maximizing Overlap of Large Primary Sampling Units in Repeated Sampling: A comparison of Ernst s Method with Ohlsson s Method Maxmzng Overlap of Large Prmary Samplng Unts n Repeated Samplng: A comparson of Ernst s Method wth Ohlsson s Method Red Rottach and Padrac Murphy 1 U.S. Census Bureau 4600 Slver Hll Road, Washngton DC

More information

Physics 5153 Classical Mechanics. Principle of Virtual Work-1

Physics 5153 Classical Mechanics. Principle of Virtual Work-1 P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal

More information

Systematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal

Systematic Error Illustration of Bias. Sources of Systematic Errors. Effects of Systematic Errors 9/23/2009. Instrument Errors Method Errors Personal 9/3/009 Sstematc Error Illustraton of Bas Sources of Sstematc Errors Instrument Errors Method Errors Personal Prejudce Preconceved noton of true value umber bas Prefer 0/5 Small over large Even over odd

More information

UNR Joint Economics Working Paper Series Working Paper No Further Analysis of the Zipf Law: Does the Rank-Size Rule Really Exist?

UNR Joint Economics Working Paper Series Working Paper No Further Analysis of the Zipf Law: Does the Rank-Size Rule Really Exist? UNR Jont Economcs Workng Paper Seres Workng Paper No. 08-005 Further Analyss of the Zpf Law: Does the Rank-Sze Rule Really Exst? Fungsa Nota and Shunfeng Song Department of Economcs /030 Unversty of Nevada,

More information