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

Save this PDF as:

Size: px
Start display at page:

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

## Transcription

1 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, Combra, Portugal Tel.: ; Fax: Abstract The converson of geographcal enttes between the raster and vector data structures ntroduces errors n the enttes poston. The vector to raster converson results n a loss of nformaton, snce, when a Boolean classfcaton of each pxel s used, the enttes shape must follow the shape of the pxels. Thus, the nformaton about the poston of the enttes n the vector data structure s lost wth the converson. Ths loss can be mnmzed f, nstead of makng a Boolean classfcaton of the pxels, a fuzzy classfcaton s performed, buldng Fuzzy Geographcal Enttes. These enttes keep, for each pxel, the nformaton about the pxel area that was nsde the vector enttes and therefore only the nformaton about the poston of the enttes boundares nsde the pxels s lost. The Fuzzy Geographcal Enttes obtaned through the prevous converson may be converted back to the vector data structure. An algorthm was developed to reconstruct the boundares of the vector geographcal enttes usng the nformaton stored n the raster Fuzzy Geographcal Enttes. Snce the grades of membershp represent partal membershp of the pxels to the enttes, ths nformaton s valuable to reconstruct the enttes boundares n the vector data structure, generatng boundares of the obtaned vector enttes that are as close as possble to ther orgnal poston. Even though slght changes on the enttes shape and poston are expected after the re-converson to the vector data structure, when Fuzzy Geographcal Enttes are used, the enttes areas are always kept durng successve conversons between both structures. Moreover, f the converson and re-converson methods are successvely appled consderng the same pxel, sze, orgn and orentaton, the obtaned results are always dentcal, that s, the postonal errors ntroduce wll not propagate ndefntely. Keywords: vector to raster converson, raster to vector converson, fuzzy geographcal enttes, postonal error, error propagaton Introducton The geographcal nformaton may be stored usng the vector or the raster data structure. The use of ether structure depends on the methods used to collect the data and on the use that wll be gven the nformaton. Wthn ths paper only areal Geographcal Enttes (GEs) are consdered. In the vector data structure, the geographcal space s consdered contnuous and the prmtves used to represent the geographcal nformaton are ponts, lnes and areas. In the raster data structure, the geographcal space s dscrete and the prmtves used to represent the geographcal nformaton are cells (pxels), usually square, formng a tessellaton. Snce the pxels are homogeneous, that s, only one attrbute value can be assgned to each one, the shape 784

2 and poston of the GEs are condtoned to the shape, sze, orgn and orentaton of the cells formng the raster tessellaton. The converson between both data structures s sometmes necessary. The enttes shape and poston may be represented more accurately n the vector data structure than n the raster structure. The vector to raster converson corresponds to a dscretzaton of the geographcal space, and, snce a Boolean classfcaton s usually done, each pxel has to be classfed as belongng or not to the GE. When a grd correspondng to the pxels s overlad to the vector representaton, the boundares of the GEs wll cross some pxels, and therefore those pxels, known as mxed pxels, are partally outsde/nsde the entty. The oblgaton of assgnng to each mxed pxel full membershp or no membershp to the entty ntroduces errors n the GE poston and shape, whch depend on the crtera used to make the choce of weather the pxel s consdered to be nsde or outsde the GE. Examples of crtera used are: the pxel s consdered to belong to the entty or not dependng on the area occuped by the entty nsde t, or weather the entty occupes the pxel center (e.g. Burrough and MacDonnell, 998). The converson from the raster data structure to the vector data structure may be performed n two ways:. The segments contourng the geographcal entty correspond to the lmts of the cells n the raster structure. 2. The converson descrbed n the prevous pont may be smoothed to generate enttes wth a more real confguraton. The type of converson expressed n. s very smple and translates exactly the nformaton present n the raster structure, but the resultng enttes are represented wth a stepwse look, whch clearly does not correspond to ther real shape. The second approach results n enttes wth better look, but ther real poston, shape, and area s unknown, snce the smoothng methods don t use any nformaton about the real poston of the enttes, because that nformaton s not avalable n the raster structure. The problem assocated wth these conversons s that from the vector to the raster structure there s a loss of nformaton, and therefore, to convert the data back to the vector structure two choces are possble: or no more nformaton s added, and the result s a stepwse entty; or nformaton has to be created to smooth the entty boundary, but, snce no addtonal nformaton s avalable, t has to be created ndependently of the real characterstcs of the enttes. In ths paper, a method to mnmze the loss of nformaton when convertng GE from the vector to the raster structure s proposed, usng Fuzzy Geographcal Enttes (FGEs), as well as an algorthm to convert the obtaned fuzzy enttes back to the vector structure usng the supplementary nformaton stored. 2 Fuzzy Geographcal Enttes Defnton: A fuzzy geographcal entty (FGE) E s a geographcal entty whose poston n the geographcal space s defned by the fuzzy set E = { Regons belongng to geographcal entty E}, wth membershp functon µ E( p) [,] defned for every pont p n the space of nterest. The membershp value one represents full membershp. The membershp value zero represents no membershp, and the values n 785

3 between correspond to grades of membershp to E, decreasng from one to zero. (Fonte and Lodwck, 24) The degrees of membershp may have several semantc nterpretatons and may be computed consderng dfferent approaches (Fonte and Lodwck, 25). Ths allows the constructon of FGEs n many stuatons usng dfferent types of nformaton. For example, the degree of membershp of one element to a set may represent the uncertanty regardng the element s membershp to the set or partal membershp to t. 3 Converson between vector and raster data structures usng FGEs As explaned n secton, the problem assocated wth the consecutve converson between the vector and the raster data structures s that from the vector to the raster structure there s loss of nformaton and, wth the converson back to the vector structure, or a stepwse vector representaton s obtaned or a smoothng process s used, whch s ndependent from the real poston and shape of the entty, snce no further nformaton about ts poston s avalable. Moreover, f both processes are appled consecutvely transformatons are added each tme the converson s made, changng the entty s shape and poston. The consecutve converson between the vector and raster data structures generate the results 2 shown n Fgure. The area of the ntal GE s 38 m, and after the converson to the raster structure and ts re-converson to the vector structure, an area of 2 4 m was obtaned. a) b) Fgure a) The lghter vector GE was obtaned convertng the darker GE to the raster data structure and the resultng raster GE back to the vector structure. b) The lghter GE was obtaned convertng the darker GE between the vector and the raster data structures sx tmes. The phlosophy behnd the heren proposed work, t that, when convertng the GEs from the vector to the raster data structure, more nformaton about ther shape and poston should be kept, so that the errors ntroduced wth the converson are mnmzed. 786

4 3. Converson from vector GEs to raster FGEs The converson from the vector to the raster data structure wth a Boolean classfcaton assgns to each cell a value one or zero, meanng that the pxels belongs or not to the entty. If, nstead of makng a Boolean classfcaton of the pxels a fuzzy classfcaton s done, t s possble to assgn to each pxel a degree of membershp to the geographcal entty. In ths case, the grades of membershp of the pxels represent partal belongng (and not uncertanty), and translate the degree to whch the pxels belong to the entty. A semantc nterpretaton of the grades of membershp as degrees of smlarty s presented below, even though, n ths case, the computaton of the membershp functon s ntutve. The nterpretaton of degrees of membershp as degrees of smlarty s based on the concept of groupng elements nto sets characterzed by the propertes of one or several elements, consdered as deal, or by propertes representatve of the set. The degrees of membershp of the other elements to the set are computed evaluatng the smlarty of ther propertes to the propertes of the deal element or the propertes representatve of the set. The degree of smlarty between the elements or ther characterstcs may be evaluated consderng a dstance between each element of the set and the deal element, or a dstance between the characterstcs of the elements and the representatve characterstc (Blgç and Turksen, 999; Zmmermann and Zysno, 985). The dstances used to compute the degrees of membershp and the computaton of these degrees depend upon the applcaton. An example of applcaton of ths semantc nterpretaton may be found n Fonte and Lodwch, 25. The nterpretaton of the degrees of membershp as degrees of smlarty may be used to compute the degrees of membershp of the pxels to the raster FGE. The deal stuaton occurs when the pxels are completely occuped by the entty, that s, the pxels are nsde the GE. In ths case, the degree of membershp one s assgned to the pxels, whch form the core of the FGE. To compute the degree of membershp of the mxed pxels to the FGE t s necessary to evaluate the degree of smlarty between them and the deal pxels, whch s computed evaluatng a dstance between the characterstcs of the deal pxels and the mxed ones. In ths applcaton, the characterstc chosen to perform the comparson s the area occuped by the vector GE nsde each pxel. The quantfcaton of the smlarty between pxel p and the deal pxels s based on the dstance between the area occuped by the entty n the deal pxels, A Ideal, (whch corresponds to the total area of the pxels A Total, that s, AIdeal = ATotal ) and the A p (see Fgure 3). area occuped by the GE n pxel p, ( ) Occuped d A ( p ), A Occuped Ideal ( ) A Occuped Occuped p A Ideal Fgure 3 The dstance d AOccuped ( p ), A Ideal n pxel p, AOccuped ( p ), and the area of the deal pxels Ideal A s the dfference between the area occuped by the GE A. 787

5 That s, ( ) = ( ), = ( ) d p d AOccuped p AIdeal AIdeal AOccuped p () The grades of membershp of the mxed pxels p to the FGE are then computed as a functon of ths dstance, that s: ( p ) f d( p ) µ E = where, f ( ) =, f ( A ) = and d( p ) Ideal AIdeal. Consderng a lnear varaton between the grades of membershp and the dstance, functon f corresponds to a straght lne and the grades of membershp are computed as shown n Equaton 2 (see Fgure 4). µ µ E ( p ) f d( p ) E ( p ) Ideal ( ) AIdeal d p = = (2) A d A Ideal Fgure 4 Membershp functon of pxels p to the FGE E. Accordng to Equaton A d( p ) A ( p ) =, and therefore, from Equaton 2 Ideal Occuped Total ( p ) AOccuped µ E ( p) = (3) A Equaton 3 proves that the grades of membershp may be computed dvdng the area of the pxel occuped by the entty nsde each pxel by the area of the pxels, whch s the normalzed area of the pxel occuped by the GE. The GE shown n Fgure 5a) was converted to a raster FGE consderng the grd represented n Fgure 5b). The obtaned FGE s shown n Fgure

6 Y Y a) X b) X Fgure 5 a) GE represented n the vector data structure. b) The vector GE overlad wth a grd of cells Fgure 6 Raster FGE correspondng to the vector GE shown s Fgure 5. If the Rosenfeld operator to compute the area of FGEs (Fonte and Lodwck, 24) s used to compute de area of the FGE generated by the converson, the obtaned value s equal to the area of the orgnal entty represented n the vector data structure. In ths way, even though the nformaton regardng the locaton of the orgnal border of the entty s lost, the nformaton about ts area s kept, as well as the area occuped by the entty nsde each pxel. 3.2 Converson of raster FGEs to the vector data structure The converson of a raster FGE back to the vector data structure requres the dentfcaton of the segments formng the entty boudary. In the prevous secton, Equaton 3 was used to compute the grade of membershp of each pxel to the FGE. Now, the grades of membershp and the total area of the pxels are known and therefore AOccuped ( p ) = µ E ( p ) ATotal may be computed. The segments formng the boundary are found, such that the area occuped by the 789

7 entty nsde each pxel s correct. An algorthm was developed consderng the followng steps:. Identfy the ponts and segments belongng to the boundary of the entty. Whenever there are neghbor pxels wth grades of membershp zero and one, the common pont or segment belongs to the entty boundary (see Fgure 7) Fgure 7 Example of ponts and segments belongng to the border of the GE. If the grades of membershp are consdered wth only one or two decmal places these ponts are easy to fnd, snce the roundng of the grades of membershp wll turn some of them nto zero and one. If no pont s found, the pxel wth the smallest grade of membershp wth two contnuous neghbors wth membershp equal to zero s chosen and two ponts are dentfed such that a rectangular trangle s formed wth equal cathetus, such that the area occuped by the GE nsde the pxel s correct (see Fgure 8) Fgure 8 Example of ponts consdered to belong to the border of the GE. 2. Choose one of the ponts found n the prevous step. 3. Identfy n whch drecton (to each pxel) the boundary wll head, consderng that the fronter s followed s the clockwse drecton. Once the drecton to follow the boundary s chosen, whenever the locaton of a pont of the boundary over a pxel border s known, t s possble to know whch pxel the boundary wll cross. For example, n Fgure 9a), b) and c), the border wll have to cross the central pxels and head respectvely to the pxels wth µ E =.7, µ E =.3 and µ E =.5. a) b) c).4 P P. P Fgure 9 The arrows represent the drecton taken by the entty boundary when leavng pont P, f the boundary s followed n the clockwse drecton. 79

8 4. Identty the next pont of the fronter. The poston of the next pont s determned such that the area of the pxel beng crossed equals A p = µ p A. ( ) ( ) Occuped E Total If one of the ponts found n step one, besdes pont P, s on the fronter of the pxel beng crossed, pont Q, two cases may occur: the grade of membershp of the crossed pxel s.5, and n that case Q s the next pont of the boundary; f the grade of membershp s dfferent from.5, an ntermedate pont I has to be found, located on the dagonal of the pxel (see Fgure ). a) b).7 Q.7 Q I.5.3 P P.. Fgure In a) the next pont of the boundary s Q. In b), to occupy only the area of the pxel A p t s necessary to consder an ntermedate pont I between P and Q. correspondng to ( ) Occuped If the prevous stuaton does not verfy, t s necessary to dentfy to whch pxel the boundary should head to determne over whch pxel edge the next pont Q wll be P = { p, p2, p3, p4} located. If s the set of the four neghbors of the pxel to be < µ E ( p) < crossed, the chosen pxel wll be the one satsfyng and µ ( p ) = mn{ µ ( p )} E c p E P. The coordnates of pont Q are computed respectng the area occuped by the entty nsde the pxel. Intermedate ponts may be consdered whenever necessary. a) b).4 P P Q Q. c) P.6.8 Q Fgure The next pont Q of the boundary s dentfed, respectng the area occuped by the entty nsde the crossed pxel. 5. The pont or ponts determned n the prevous step are now used to determne another pont R usng smlar rules, and the process s repeated untl the frst pont s reached and the entty closed. 79

9 a) b).5 R.7.9 P Q..4 P Q R c) P.6.8 Q.5 R.8.2 Fgure 2 Pont R of the boundary s dentfed respectng the area occuped by the entty nsde the crossed pxel. The GE obtaned after the converson of the FGE represented n Fgure 6 to the vector data structure n shown n Fgure 3, overlad wth the orgnal vector GE. The area of the obtaned GE s equal to the area of the ntal vector entty, and only small varatons n the entty poston are found. In ths case, only one ntermedate pont was used nsde each pxel, but more ponts can be consdered f eventual spkes are to be avoded. Fgure 3 The vector GE obtaned from the re-converson of the FGE back to the vector data structure s represented n grey, overlad wth the orgnal vector GE (n black). 4 Conclusons The conversons between the vector and raster data structures are useful for many applcatons, therefore t s useful to mnmze the errors and uncertanty ntroduced n the geographcal poston of the enttes durng the converson between both data structures. The converson from the vector to the raster data structure results n a loss of nformaton, snce, when a Boolean classfcaton of each pxel s consdered, the enttes boundares must follow the shape of the pxels. Thus, nformaton about the poston of the enttes n the vector data structure, whch was orgnally avalable, s lost wth the converson. A method to 792

10 mnmze ths loss s presented, whch conssts n, nstead of makng a Boolean classfcaton of the pxels, performng a fuzzy classfcaton, buldng Fuzzy Geographcal Enttes. These enttes keep, for each pxel, the nformaton about the pxel area that was nsde the vector enttes and therefore the total area of the GE s also kept. Wth ths approach only the nformaton about the poston of the enttes boundares nsde the pxels s lost. The obtaned Fuzzy Geographcal Enttes may be used for analyss and may be reconverted back to the vector data structure. The converson of GEs from the raster to the vector data structure results n GEs wth stepwse aspect. To overcome ths dsadvantage, the boundares of the obtaned entty can be smoothed, but, snce no further nformaton s avalable about the poston of the entty, ths smoothng s smply a geometrc operaton, transformng once agan the shape, poston and geometrc characterstcs of the entty, such as ts area. On the other hand, the converson of raster FGEs to the vector data structure allows a more accurate result, snce there s more nformaton avalable to determne the poston of the segments formng the entty boundary. An algorthm was developed to reconstruct the boundares of the vector geographcal enttes from the nformaton stored n the raster FGEs. Snce the grades of membershp represent partal membershp of the pxels to the enttes, ths nformaton s valuable to reconstruct the enttes boundares n the vector data structure, so that the boundares of the obtaned vector enttes are as close as possble to ther orgnal poston. Even though slght changes on the enttes boundares are expected after the re-converson to the vector data structure, when Fuzzy Geographcal Enttes are used, the enttes areas are kept. If successve conversons between both structures are done consderng the same pxel sze, orgn and orentaton, the same results are obtaned, snce the areas occuped by the entty nsde the pxels are always the same, that s, the postonal errors ntroduced wll not propagate ndefntely. References Blgç T, Türkşen I, 2, Measurement of Membershp functons: Theoretcal and emprcal work. In: Fundamentals of Fuzzy Sets, Dubos D, Prade H (Eds) The Handbook of Fuzzy Sets Seres, Kluwer Acad. Publ., Fonte, C. and Lodwck, W, 24, Areas of fuzzy geographcal enttes. Internatonal Journal of Geographcal Informaton Scence, 8 (2), pp Fonte, C. and Lodwck, W, 25, Modellng the Fuzzy Spatal Extent of Geographcal Enttes. In Fuzzy Modellng wth Spatal Informaton for Geographc Problems, F. E. Petry, V. B. Robnson and M. A. Cobb (Eds.), pp. 2-42, Berln: Sprnger-Verlag. Zmmermann, H. and Zysno, P., 985, Quantfyng vagueness n decson models. European Journal of Operatonal Research, Vol. 22, pp

### Identifying Workloads in Mixed Applications

, pp.395-400 http://dx.do.org/0.4257/astl.203.29.8 Identfyng Workloads n Mxed Applcatons Jeong Seok Oh, Hyo Jung Bang, Yong Do Cho, Insttute of Gas Safety R&D, Korea Gas Safety Corporaton, Shghung-Sh,

### 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.

### Graph Theory and Cayley s Formula

Graph Theory and Cayley s Formula Chad Casarotto August 10, 2006 Contents 1 Introducton 1 2 Bascs and Defntons 1 Cayley s Formula 4 4 Prüfer Encodng A Forest of Trees 7 1 Introducton In ths paper, I wll

### The Magnetic Field. Concepts and Principles. Moving Charges. Permanent Magnets

. The Magnetc Feld Concepts and Prncples Movng Charges All charged partcles create electrc felds, and these felds can be detected by other charged partcles resultng n electrc force. However, a completely

### 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

### 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

### Communication Networks II Contents

8 / 1 -- Communcaton Networs II (Görg) -- www.comnets.un-bremen.de Communcaton Networs II Contents 1 Fundamentals of probablty theory 2 Traffc n communcaton networs 3 Stochastc & Marovan Processes (SP

### 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

### 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..

### Multivariate EWMA Control Chart

Multvarate EWMA Control Chart Summary The Multvarate EWMA Control Chart procedure creates control charts for two or more numerc varables. Examnng the varables n a multvarate sense s extremely mportant

### 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.

### Ring structure of splines on triangulations

www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon

### 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

### 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

### 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

### 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

### 9.1 The Cumulative Sum Control Chart

Learnng Objectves 9.1 The Cumulatve Sum Control Chart 9.1.1 Basc Prncples: Cusum Control Chart for Montorng the Process Mean If s the target for the process mean, then the cumulatve sum control chart s

### Feature selection for intrusion detection. Slobodan Petrović NISlab, Gjøvik University College

Feature selecton for ntruson detecton Slobodan Petrovć NISlab, Gjøvk Unversty College Contents The feature selecton problem Intruson detecton Traffc features relevant for IDS The CFS measure The mrmr measure

### Chapter 7. Random-Variate Generation 7.1. Prof. Dr. Mesut Güneş Ch. 7 Random-Variate Generation

Chapter 7 Random-Varate Generaton 7. Contents Inverse-transform Technque Acceptance-Rejecton Technque Specal Propertes 7. Purpose & Overvew Develop understandng of generatng samples from a specfed dstrbuton

### + + + - - 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

### 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

### IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS 1

Nov Sad J. Math. Vol. 36, No. 2, 2006, 0-09 IMPROVEMENT OF CONVERGENCE CONDITION OF THE SQUARE-ROOT INTERVAL METHOD FOR MULTIPLE ZEROS Modrag S. Petkovć 2, Dušan M. Mloševć 3 Abstract. A new theorem concerned

### Nonlinear data mapping by neural networks

Nonlnear data mappng by neural networks R.P.W. Dun Delft Unversty of Technology, Netherlands Abstract A revew s gven of the use of neural networks for nonlnear mappng of hgh dmensonal data on lower dmensonal

### State function: eigenfunctions of hermitian operators-> normalization, orthogonality completeness

Schroednger equaton Basc postulates of quantum mechancs. Operators: Hermtan operators, commutators State functon: egenfunctons of hermtan operators-> normalzaton, orthogonalty completeness egenvalues and

### PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 12

14 The Ch-squared dstrbuton PSYCHOLOGICAL RESEARCH (PYC 304-C) Lecture 1 If a normal varable X, havng mean µ and varance σ, s standardsed, the new varable Z has a mean 0 and varance 1. When ths standardsed

### Aryabhata s Root Extraction Methods. Abhishek Parakh Louisiana State University Aug 31 st 2006

Aryabhata s Root Extracton Methods Abhshek Parakh Lousana State Unversty Aug 1 st 1 Introducton Ths artcle presents an analyss of the root extracton algorthms of Aryabhata gven n hs book Āryabhatīya [1,

### Generator Warm-Up Characteristics

NO. REV. NO. : ; ~ Generator Warm-Up Characterstcs PAGE OF Ths document descrbes the warm-up process of the SNAP-27 Generator Assembly after the sotope capsule s nserted. Several nqures have recently been

### A DATA MINING APPLICATION IN A STUDENT DATABASE

JOURNAL OF AERONAUTICS AND SPACE TECHNOLOGIES JULY 005 VOLUME NUMBER (53-57) A DATA MINING APPLICATION IN A STUDENT DATABASE Şenol Zafer ERDOĞAN Maltepe Ünversty Faculty of Engneerng Büyükbakkalköy-Istanbul

### 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

### 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

### 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,

### 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

### 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.

### I. SCOPE, APPLICABILITY AND PARAMETERS Scope

D Executve Board Annex 9 Page A/R ethodologcal Tool alculaton of the number of sample plots for measurements wthn A/R D project actvtes (Verson 0) I. SOPE, PIABIITY AD PARAETERS Scope. Ths tool s applcable

### 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

### An interactive system for structure-based ASCII art creation

An nteractve system for structure-based ASCII art creaton Katsunor Myake Henry Johan Tomoyuk Nshta The Unversty of Tokyo Nanyang Technologcal Unversty Abstract Non-Photorealstc Renderng (NPR), whose am

### Traffic State Estimation in the Traffic Management Center of Berlin

Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D-763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,

### 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.

### 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

### CS 2750 Machine Learning. Lecture 17a. Clustering. CS 2750 Machine Learning. Clustering

Lecture 7a Clusterng Mlos Hauskrecht mlos@cs.ptt.edu 539 Sennott Square Clusterng Groups together smlar nstances n the data sample Basc clusterng problem: dstrbute data nto k dfferent groups such that

### 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

### SPEE Recommended Evaluation Practice #6 Definition of Decline Curve Parameters Background:

SPEE Recommended Evaluaton Practce #6 efnton of eclne Curve Parameters Background: The producton hstores of ol and gas wells can be analyzed to estmate reserves and future ol and gas producton rates and

### Comparing Class Level Chain Drift for Different Elementary Aggregate Formulae Using Locally Collected CPI Data

Comparng Class Level Chan Drft for Dfferent Elementary Aggregate Formulae Usng Gareth Clews 1, Anselma Dobson-McKttrck 2 and Joseph Wnton Summary The Consumer Prces Index (CPI) s a measure of consumer

### 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

### SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA

SIX WAYS TO SOLVE A SIMPLE PROBLEM: FITTING A STRAIGHT LINE TO MEASUREMENT DATA E. LAGENDIJK Department of Appled Physcs, Delft Unversty of Technology Lorentzweg 1, 68 CJ, The Netherlands E-mal: e.lagendjk@tnw.tudelft.nl

### Introduction: Analysis of Electronic Circuits

/30/008 ntroducton / ntroducton: Analyss of Electronc Crcuts Readng Assgnment: KVL and KCL text from EECS Just lke EECS, the majorty of problems (hw and exam) n EECS 3 wll be crcut analyss problems. Thus,

### Introduction to Regression

Introducton to Regresson Regresson a means of predctng a dependent varable based one or more ndependent varables. -Ths s done by fttng a lne or surface to the data ponts that mnmzes the total error. -

### An Enhanced Super-Resolution System with Improved Image Registration, Automatic Image Selection, and Image Enhancement

An Enhanced Super-Resoluton System wth Improved Image Regstraton, Automatc Image Selecton, and Image Enhancement Yu-Chuan Kuo ( ), Chen-Yu Chen ( ), and Chou-Shann Fuh ( ) Department of Computer Scence

### EE201 Circuit Theory I 2015 Spring. Dr. Yılmaz KALKAN

EE201 Crcut Theory I 2015 Sprng Dr. Yılmaz KALKAN 1. Basc Concepts (Chapter 1 of Nlsson - 3 Hrs.) Introducton, Current and Voltage, Power and Energy 2. Basc Laws (Chapter 2&3 of Nlsson - 6 Hrs.) Voltage

### "Research Note" APPLICATION OF CHARGE SIMULATION METHOD TO ELECTRIC FIELD CALCULATION IN THE POWER CABLES *

Iranan Journal of Scence & Technology, Transacton B, Engneerng, ol. 30, No. B6, 789-794 rnted n The Islamc Republc of Iran, 006 Shraz Unversty "Research Note" ALICATION OF CHARGE SIMULATION METHOD TO ELECTRIC

### Semiconductor sensors of temperature

Semconductor sensors of temperature he measurement objectve 1. Identfy the unknown bead type thermstor. Desgn the crcutry for lnearzaton of ts transfer curve.. Fnd the dependence of forward voltage drop

### Logistic Regression. Lecture 4: More classifiers and classes. Logistic regression. Adaboost. Optimization. Multiple class classification

Lecture 4: More classfers and classes C4B Machne Learnng Hlary 20 A. Zsserman Logstc regresson Loss functons revsted Adaboost Loss functons revsted Optmzaton Multple class classfcaton Logstc Regresson

### 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

### Uncertain Data Mining: A New Research Direction

Uncertan Data Mnng: A New Research Drecton Mchael Chau 1, Reynold Cheng, and Ben Kao 3 1: School of Busness, The Unversty of Hong Kong, Pokfulam, Hong Kong : Department of Computng, Hong Kong Polytechnc

### 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

### 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

### Forecasting the Direction and Strength of Stock Market Movement

Forecastng the Drecton and Strength of Stock Market Movement Jngwe Chen Mng Chen Nan Ye cjngwe@stanford.edu mchen5@stanford.edu nanye@stanford.edu Abstract - Stock market s one of the most complcated systems

### 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"

### MONITORING METHODOLOGY TO ASSESS THE PERFORMANCE OF GSM NETWORKS

Electronc Communcatons Commttee (ECC) wthn the European Conference of Postal and Telecommuncatons Admnstratons (CEPT) MONITORING METHODOLOGY TO ASSESS THE PERFORMANCE OF GSM NETWORKS Athens, February 2008

### 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

### A Performance Analysis of View Maintenance Techniques for Data Warehouses

A Performance Analyss of Vew Mantenance Technques for Data Warehouses Xng Wang Dell Computer Corporaton Round Roc, Texas Le Gruenwald The nversty of Olahoma School of Computer Scence orman, OK 739 Guangtao

### Time Series Analysis in Studies of AGN Variability. Bradley M. Peterson The Ohio State University

Tme Seres Analyss n Studes of AGN Varablty Bradley M. Peterson The Oho State Unversty 1 Lnear Correlaton Degree to whch two parameters are lnearly correlated can be expressed n terms of the lnear correlaton

### HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION

HYPOTHESIS TESTING OF PARAMETERS FOR ORDINARY LINEAR CIRCULAR REGRESSION Abdul Ghapor Hussn Centre for Foundaton Studes n Scence Unversty of Malaya 563 KUALA LUMPUR E-mal: ghapor@umedumy Abstract Ths paper

### LETTER IMAGE RECOGNITION

LETTER IMAGE RECOGNITION 1. Introducton. 1. Introducton. Objectve: desgn classfers for letter mage recognton. consder accuracy and tme n takng the decson. 20,000 samples: Startng set: mages based on 20

### An Integrated Semantically Correct 2.5D Object Oriented TIN. Andreas Koch

An Integrated Semantcally Correct 2.5D Object Orented TIN Andreas Koch Unverstät Hannover Insttut für Photogrammetre und GeoInformaton Contents Introducton Integraton of a DTM and 2D GIS data Semantcs

### 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,

### 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

### Questions that we may have about the variables

Antono Olmos, 01 Multple Regresson Problem: we want to determne the effect of Desre for control, Famly support, Number of frends, and Score on the BDI test on Perceved Support of Latno women. Dependent

### 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

### A Note on the Decomposition of a Random Sample Size

A Note on the Decomposton of a Random Sample Sze Klaus Th. Hess Insttut für Mathematsche Stochastk Technsche Unverstät Dresden Abstract Ths note addresses some results of Hess 2000) on the decomposton

### A Simple Approach to Clustering in Excel

A Smple Approach to Clusterng n Excel Aravnd H Center for Computatonal Engneerng and Networng Amrta Vshwa Vdyapeetham, Combatore, Inda C Rajgopal Center for Computatonal Engneerng and Networng Amrta Vshwa

### Level Annuities with Payments Less Frequent than Each Interest Period

Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Level Annutes wth Payments Less Frequent than Each Interest Perod 1 Annuty-mmedate 2 Annuty-due Symoblc approach

### An Evaluation of the Extended Logistic, Simple Logistic, and Gompertz Models for Forecasting Short Lifecycle Products and Services

An Evaluaton of the Extended Logstc, Smple Logstc, and Gompertz Models for Forecastng Short Lfecycle Products and Servces Charles V. Trappey a,1, Hsn-yng Wu b a Professor (Management Scence), Natonal Chao

### 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,

### Control Charts for Means (Simulation)

Chapter 290 Control Charts for Means (Smulaton) Introducton Ths procedure allows you to study the run length dstrbuton of Shewhart (Xbar), Cusum, FIR Cusum, and EWMA process control charts for means usng

### 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

### Lecture 18: Clustering & classification

O CPS260/BGT204. Algorthms n Computatonal Bology October 30, 2003 Lecturer: Pana K. Agarwal Lecture 8: Clusterng & classfcaton Scrbe: Daun Hou Open Problem In HomeWor 2, problem 5 has an open problem whch

### Single and multiple stage classifiers implementing logistic discrimination

Sngle and multple stage classfers mplementng logstc dscrmnaton Hélo Radke Bttencourt 1 Dens Alter de Olvera Moraes 2 Vctor Haertel 2 1 Pontfíca Unversdade Católca do Ro Grande do Sul - PUCRS Av. Ipranga,

### PLANAR GRAPHS. Plane graph (or embedded graph) A graph that is drawn on the plane without edge crossing, is called a Plane graph

PLANAR GRAPHS Basc defntons Isomorphc graphs Two graphs G(V,E) and G2(V2,E2) are somorphc f there s a one-to-one correspondence F of ther vertces such that the followng holds: - u,v V, uv E, => F(u)F(v)

### 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

### 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

### 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

### Sensitivity Analysis in a Generic Multi-Attribute Decision Support System

Senstvty Analyss n a Generc Mult-Attrbute Decson Support System Sxto Ríos-Insua, Antono Jménez and Alfonso Mateos Department of Artfcal Intellgence, Madrd Techncal Unversty Campus de Montegancedo s/n,

### Research Article Enhanced Two-Step Method via Relaxed Order of α-satisfactory Degrees for Fuzzy Multiobjective Optimization

Hndaw Publshng Corporaton Mathematcal Problems n Engneerng Artcle ID 867836 pages http://dxdoorg/055/204/867836 Research Artcle Enhanced Two-Step Method va Relaxed Order of α-satsfactory Degrees for Fuzzy

### Online Learning from Experts: Minimax Regret

E0 370 tatstcal Learnng Theory Lecture 2 Nov 24, 20) Onlne Learnng from Experts: Mn Regret Lecturer: hvan garwal crbe: Nkhl Vdhan Introducton In the last three lectures we have been dscussng the onlne

### 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

### A Suspect Vehicle Tracking System Based on Video

3rd Internatonal Conference on Multmeda Technology ICMT 2013) A Suspect Vehcle Trackng System Based on Vdeo Yad Chen 1, Tuo Wang Abstract. Vdeo survellance systems are wdely used n securty feld. The large

### SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW.

SUPPLIER FINANCING AND STOCK MANAGEMENT. A JOINT VIEW. Lucía Isabel García Cebrán Departamento de Economía y Dreccón de Empresas Unversdad de Zaragoza Gran Vía, 2 50.005 Zaragoza (Span) Phone: 976-76-10-00

### FREQUENCY OF OCCURRENCE OF CERTAIN CHEMICAL CLASSES OF GSR FROM VARIOUS AMMUNITION TYPES

FREQUENCY OF OCCURRENCE OF CERTAIN CHEMICAL CLASSES OF GSR FROM VARIOUS AMMUNITION TYPES Zuzanna BRO EK-MUCHA, Grzegorz ZADORA, 2 Insttute of Forensc Research, Cracow, Poland 2 Faculty of Chemstry, Jagellonan

### 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

### 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

### Analysis of Small-signal Transistor Amplifiers

Analyss of Small-sgnal Transstor Amplfers On completon of ths chapter you should be able to predct the behaour of gen transstor amplfer crcuts by usng equatons and/or equalent crcuts that represent the

### A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION. Michael E. Kuhl Radhamés A. Tolentino-Peña

Proceedngs of the 2008 Wnter Smulaton Conference S. J. Mason, R. R. Hll, L. Mönch, O. Rose, T. Jefferson, J. W. Fowler eds. A DYNAMIC CRASHING METHOD FOR PROJECT MANAGEMENT USING SIMULATION-BASED OPTIMIZATION

### On-Line Fault Detection in Wind Turbine Transmission System using Adaptive Filter and Robust Statistical Features

On-Lne Fault Detecton n Wnd Turbne Transmsson System usng Adaptve Flter and Robust Statstcal Features Ruoyu L Remote Dagnostcs Center SKF USA Inc. 3443 N. Sam Houston Pkwy., Houston TX 77086 Emal: ruoyu.l@skf.com

### 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

### Transients Analysis of a Nuclear Power Plant Component for Fault Diagnosis

A publcaton of CHEMICAL ENGINEERING TRANSACTIONS VOL. 33, 213 Guest Edtors: Enrco Zo, Pero Barald Copyrght 213, AIDIC Servz S.r.l., ISBN 978-88-9568-24-2; ISSN 1974-9791 The Italan Assocaton of Chemcal

### ErrorPropagation.nb 1. Error Propagation

ErrorPropagaton.nb Error Propagaton Suppose that we make observatons of a quantty x that s subject to random fluctuatons or measurement errors. Our best estmate of the true value for ths quantty s then