High-fat diet and acute myocardial infarction,
\\n\\n
IntechOpen Book Series will also publish a program of research-driven Thematic Edited Volumes that focus on specific areas and allow for a more in-depth overview of a particular subject.
\\n\\nIntechOpen Book Series will be launching regularly to offer our authors and editors exciting opportunities to publish their research Open Access. We will begin by relaunching some of our existing Book Series in this innovative book format, and will expand in 2022 into rapidly growing research fields that are driving and advancing society.
\\n\\nLaunching 2021
\\n\\nArtificial Intelligence, ISSN 2633-1403
\\n\\nVeterinary Medicine and Science, ISSN 2632-0517
\\n\\nBiochemistry, ISSN 2632-0983
\\n\\nBiomedical Engineering, ISSN 2631-5343
\\n\\nInfectious Diseases, ISSN 2631-6188
\\n\\nPhysiology (Coming Soon)
\\n\\nDentistry (Coming Soon)
\\n\\nWe invite you to explore our IntechOpen Book Series, find the right publishing program for you and reach your desired audience in record time.
\\n\\nNote: Edited in October 2021
\\n"}]',published:!0,mainMedia:{caption:"",originalUrl:"/media/original/132"}},components:[{type:"htmlEditorComponent",content:'With the desire to make book publishing more relevant for the digital age and offer innovative Open Access publishing options, we are thrilled to announce the launch of our new publishing format: IntechOpen Book Series.
\n\nDesigned to cover fast-moving research fields in rapidly expanding areas, our Book Series feature a Topic structure allowing us to present the most relevant sub-disciplines. Book Series are headed by Series Editors, and a team of Topic Editors supported by international Editorial Board members. Topics are always open for submissions, with an Annual Volume published each calendar year.
\n\nAfter a robust peer-review process, accepted works are published quickly, thanks to Online First, ensuring research is made available to the scientific community without delay.
\n\nOur innovative Book Series format brings you:
\n\nIntechOpen Book Series will also publish a program of research-driven Thematic Edited Volumes that focus on specific areas and allow for a more in-depth overview of a particular subject.
\n\nIntechOpen Book Series will be launching regularly to offer our authors and editors exciting opportunities to publish their research Open Access. We will begin by relaunching some of our existing Book Series in this innovative book format, and will expand in 2022 into rapidly growing research fields that are driving and advancing society.
\n\nLaunching 2021
\n\nArtificial Intelligence, ISSN 2633-1403
\n\nVeterinary Medicine and Science, ISSN 2632-0517
\n\nBiochemistry, ISSN 2632-0983
\n\nBiomedical Engineering, ISSN 2631-5343
\n\nInfectious Diseases, ISSN 2631-6188
\n\nPhysiology (Coming Soon)
\n\nDentistry (Coming Soon)
\n\nWe invite you to explore our IntechOpen Book Series, find the right publishing program for you and reach your desired audience in record time.
\n\nNote: Edited in October 2021
\n'}],latestNews:[{slug:"webinar-introduction-to-open-science-wednesday-18-may-1-pm-cest-20220518",title:"Webinar: Introduction to Open Science | Wednesday 18 May, 1 PM CEST"},{slug:"step-in-the-right-direction-intechopen-launches-a-portfolio-of-open-science-journals-20220414",title:"Step in the Right Direction: IntechOpen Launches a Portfolio of Open Science Journals"},{slug:"let-s-meet-at-london-book-fair-5-7-april-2022-olympia-london-20220321",title:"Let’s meet at London Book Fair, 5-7 April 2022, Olympia London"},{slug:"50-books-published-as-part-of-intechopen-and-knowledge-unlatched-ku-collaboration-20220316",title:"50 Books published as part of IntechOpen and Knowledge Unlatched (KU) Collaboration"},{slug:"intechopen-joins-the-united-nations-sustainable-development-goals-publishers-compact-20221702",title:"IntechOpen joins the United Nations Sustainable Development Goals Publishers Compact"},{slug:"intechopen-signs-exclusive-representation-agreement-with-lsr-libros-servicios-y-representaciones-s-a-de-c-v-20211123",title:"IntechOpen Signs Exclusive Representation Agreement with LSR Libros Servicios y Representaciones S.A. de C.V"},{slug:"intechopen-expands-partnership-with-research4life-20211110",title:"IntechOpen Expands Partnership with Research4Life"},{slug:"introducing-intechopen-book-series-a-new-publishing-format-for-oa-books-20210915",title:"Introducing IntechOpen Book Series - A New Publishing Format for OA Books"}]},book:{item:{type:"book",id:"7458",leadTitle:null,fullTitle:"Accuracy of GNSS Methods",title:"Accuracy of GNSS Methods",subtitle:null,reviewType:"peer-reviewed",abstract:"Following the GPS, new GNSS techniques are emerging today. Various surveying and processing methods are available for the analysis of GNSS data. Equipment and software are also varied. The orbit quality, controlled by the system designer, and the IGS are continuously improved. The user is mainly interested in the quality of position and of the deformation rates produced by the GNSS. Hence, research needs to guide the user in terms of selecting the best combination of the available methods and instrumentation to produce the desired accuracy. This book reviews the current available accuracy obtainable using the GNSS methods. In fact, the main aim of this book is to make an impact on young researchers so that they keep updating the accuracy of GNSS for future generations.",isbn:"978-1-78984-926-4",printIsbn:"978-1-78984-925-7",pdfIsbn:"978-1-83881-824-1",doi:"10.5772/intechopen.75424",price:119,priceEur:129,priceUsd:155,slug:"accuracy-of-gnss-methods",numberOfPages:150,isOpenForSubmission:!1,isInWos:1,isInBkci:!1,hash:"a14e33dccec2e13f34de6b9374364a5c",bookSignature:"Dogan Ugur Sanli",publishedDate:"January 16th 2019",coverURL:"https://cdn.intechopen.com/books/images_new/7458.jpg",numberOfDownloads:7806,numberOfWosCitations:5,numberOfCrossrefCitations:3,numberOfCrossrefCitationsByBook:0,numberOfDimensionsCitations:8,numberOfDimensionsCitationsByBook:1,hasAltmetrics:1,numberOfTotalCitations:16,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"February 21st 2018",dateEndSecondStepPublish:"March 14th 2018",dateEndThirdStepPublish:"May 13th 2018",dateEndFourthStepPublish:"August 1st 2018",dateEndFifthStepPublish:"September 30th 2018",currentStepOfPublishingProcess:5,indexedIn:"1,2,3,4,5,6,7",editedByType:"Edited by",kuFlag:!1,featuredMarkup:null,editors:[{id:"245723",title:"Dr.",name:"Dogan Ugur",middleName:null,surname:"Sanli",slug:"dogan-ugur-sanli",fullName:"Dogan Ugur Sanli",profilePictureURL:"https://mts.intechopen.com/storage/users/245723/images/6878_n.jpg",biography:"Obtained his PhD from Newcastle University, UK. Studied sea level rise using GPS in his PhD thesis. In particular, specialized in vertical crustal deformation monitoring using GPS. Then moved into the area of GPS positioning accuracy. Managed to model both vertical and horizontal accuracy for relative-PPP and PPP. He also studied GPS positioning accuracy for rapid-static positioning for BERNESE and OPUS-RS software. On the other hand, he continued co-authoring sea level studies. He was involved in NEAMTWS Tsunami Warning efforts under the auspices of UNESCO. He was a member of Sea Level Working Group 3. Currently he is interested in assessing the accuracy of GPS positioning velocities with the focus on PPP derived positions from repeated GPS surveys.",institutionString:null,position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"1",institution:{name:"Yıldız Technical University",institutionURL:null,country:{name:"Turkey"}}}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"653",title:"Remote Sensing",slug:"geology-and-geophysics-remote-sensing"}],chapters:[{id:"63737",title:"Introductory Chapter: The Philosophy Behind the Accuracy Assessment of GNSS Methods",doi:"10.5772/intechopen.81288",slug:"introductory-chapter-the-philosophy-behind-the-accuracy-assessment-of-gnss-methods",totalDownloads:877,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,abstract:null,signatures:"Dogan Ugur Sanli",downloadPdfUrl:"/chapter/pdf-download/63737",previewPdfUrl:"/chapter/pdf-preview/63737",authors:[{id:"245723",title:"Dr.",name:"Dogan Ugur",surname:"Sanli",slug:"dogan-ugur-sanli",fullName:"Dogan Ugur Sanli"}],corrections:null},{id:"63574",title:"Evaluation Methods of Satellite Navigation System Performance",doi:"10.5772/intechopen.81034",slug:"evaluation-methods-of-satellite-navigation-system-performance",totalDownloads:1047,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,abstract:"With the development of global satellite navigation system, for example, global positioning system (GPS) and so on, some regional navigation systems and augmentation systems are developing rapidly. The continuous development of satellite navigation system has attracted the users’ attention to satellite navigation performance, which makes the navigation system performance become the key of satellite navigation system competition in the field of GNSS applications. The signal in space (SIS) continuity evaluation model based on the reliability is established, and the mean time between failures (MTBF) is used to characterize the probability that there is no continuity loss in unit time. Aiming at the incompleteness of the current availability model, a per-satellite availability evaluation models based on Markov process is established. Moreover, the constellation availability evaluation model is proposed by combining the satellite failure rate, repair rate and backup situation. By analyzing the measured data, the probability of the continuity and availability of GPS and BeiDou Navigation Satellite System (BDS) are calculated respectively. The results are instructive for the study of the availability performance monitoring and the evaluation of global BDS.",signatures:"Ershen Wang, He He and Chaoying Jia",downloadPdfUrl:"/chapter/pdf-download/63574",previewPdfUrl:"/chapter/pdf-preview/63574",authors:[{id:"251168",title:"Dr.",name:"Dr.",surname:"Wang",slug:"dr.-wang",fullName:"Dr. Wang"}],corrections:null},{id:"63070",title:"Robust GNSS Positioning in Urban Environment",doi:"10.5772/intechopen.80412",slug:"robust-gnss-positioning-in-urban-environment",totalDownloads:1213,totalCrossrefCites:1,totalDimensionsCites:1,hasAltmetrics:0,abstract:"In the past years, global navigation satellite systems (GNSS) have gained the core position concerning the geolocalization applications and services in urban environments. The major issue of the GNSS-based urban application expansion is related to the positioning service quality assurance, expressed in terms of accuracy, integrity, availability, and continuity of the localization service. The dense urban environments, such as city centers, are challenging to the GNSS signal reception causing the frequent blockage of the line-of-sight (LOS) signals and the multipath phenomenon, referred to as the reception of the diffracted/reflected echoes of the transmitted signal. These effects severely affect the pseudo-range and Doppler measurements, used by a GNSS receiver for the user’s position computation, which will further induce the computation of an erroneous positioning solution by the navigation processor down to a positioning loss in the presence of limited satellite visibility and few provided measurements. Therefore, advanced signal processing techniques do represent viable solutions aiming at the mitigation of these undesired effects in order to foster the accuracy and availability of the localization solution. This chapter will address in details the GNSS vector tracking (VT) receiver’s configuration able to cope with the urban environment-induced effects.",signatures:"Enik Shytermeja",downloadPdfUrl:"/chapter/pdf-download/63070",previewPdfUrl:"/chapter/pdf-preview/63070",authors:[{id:"247893",title:"Dr.",name:"Enik",surname:"Shytermeja",slug:"enik-shytermeja",fullName:"Enik Shytermeja"}],corrections:null},{id:"62312",title:"GPS Scintillations and Total Electron Content Climatology in the Southern American Sector",doi:"10.5772/intechopen.79218",slug:"gps-scintillations-and-total-electron-content-climatology-in-the-southern-american-sector",totalDownloads:1208,totalCrossrefCites:1,totalDimensionsCites:5,hasAltmetrics:0,abstract:"The radio communication and navigation systems can be strongly affected by the ionospheric conditions, which are controlled by solar phenomena associated with radiation variations and solar wind disturbances. These phenomena can generate ionospheric large-scale plasma redistribution and irregularities with scale sizes varying from centimeters to hundred kilometers. These ionospheric irregularities can produce rapid fluctuations in the amplitude and phase of global navigation satellite system (GNSS) signals, degrading the accuracy of GNSS measurements. Here we give a short review of the ionospheric variations associated with solar phenomena, and the actual state of art in the investigations of long-term (seasonal and solar cycle scales) TEC variations and climatology of scintillations, with focus on the southern American sector. It also presented a new TEC calibration procedure when applied to single-frequency PPP.",signatures:"Emília Correia, Marcio Tadeu de Assis Honorato Muella, Lucilla\nAlfonsi, Fabricio dos Santos Prol and Paulo de Oliveira Camargo",downloadPdfUrl:"/chapter/pdf-download/62312",previewPdfUrl:"/chapter/pdf-preview/62312",authors:[{id:"249305",title:"Dr.",name:"Emilia",surname:"Correia",slug:"emilia-correia",fullName:"Emilia Correia"},{id:"259537",title:"Dr.",name:"Marcio",surname:"Muella",slug:"marcio-muella",fullName:"Marcio Muella"},{id:"259539",title:"Dr.",name:"Lucilla",surname:"Alfonsi",slug:"lucilla-alfonsi",fullName:"Lucilla Alfonsi"},{id:"259540",title:"MSc.",name:"Fabricio",surname:"Prol",slug:"fabricio-prol",fullName:"Fabricio Prol"},{id:"259541",title:"Dr.",name:"Paulo",surname:"Camargo",slug:"paulo-camargo",fullName:"Paulo Camargo"}],corrections:null},{id:"62255",title:"Evaluation of GNSS Data with Internet Based Services: The Case of HRUH Station",doi:"10.5772/intechopen.79064",slug:"evaluation-of-gnss-data-with-internet-based-services-the-case-of-hruh-station",totalDownloads:960,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,abstract:"Nowadays, as the Internet services are developed, it becomes possible to offer services for many applications in the engineering field online via the Internet. One of these services is the evaluation of online GPS data. The most important feature of Internet-based applications is that these services are free and easy to use. In this study, the data of Global Navigation Satellite Systems (GNSS) of different periods belonging to newly established HRUH permanent GNSS station in Harran University were evaluated through Internet-based services. The evaluation strategy of GNSS data was conducted in 1-, 2-, 6-, 12-, and 24-h campaigns, and the results were compared between different Internet site solution results. When the results obtained are examined, it can be said that the accuracy of the data obtained from these services can be used in many applications requiring precision in centimeter levels and is capable of satisfying the expectancies.",signatures:"Mustafa Ulukavak",downloadPdfUrl:"/chapter/pdf-download/62255",previewPdfUrl:"/chapter/pdf-preview/62255",authors:[{id:"247738",title:"Dr.",name:"Mustafa",surname:"Ulukavak",slug:"mustafa-ulukavak",fullName:"Mustafa Ulukavak"}],corrections:null},{id:"63924",title:"Comparative Study of Some Online GNSS Post-Processing Services at Selected Permanent GNSS Sites in Nigeria",doi:"10.5772/intechopen.79924",slug:"comparative-study-of-some-online-gnss-post-processing-services-at-selected-permanent-gnss-sites-in-n",totalDownloads:1480,totalCrossrefCites:1,totalDimensionsCites:2,hasAltmetrics:1,abstract:"Many applications in surveying and mapping have been made simpler and more precise due to the advent of GNSS, and thus, the demand for using cutting-edge GNSS techniques in surveying and mapping applications has become indispensable. Online GNSS post-processing services are now available to provide support for users in need of precise point positioning or conventional differential positioning services and without requiring a prior knowledge of GNSS processing software. This study evaluates the performance of some online GNSS facilities with emphasis on observation duration (i.e. 1hr, 2hr, 6hr 12hr and 24hr observations). Three of these online facilities (AUSPOS, GAPS and magic-GNSS) were chosen based on their mode of operation and were evaluated at the location of five permanent GNSS stations in Nigeria. The study cut across two epochs in the year 2014 (i.e. seven days each in the months of January and July). Results in this study indicate that users can expect reliable results from these online services and their accuracy is within allowable limits for mapping applications in Nigeria. The similarity of the results between all of the services used is amazing, thus further demonstrates the robustness of the algorithms and processes employed by the different online facilities.",signatures:"Olalekan Adekunle Isioye, Mefe Moses and Lukman Abdulmumin",downloadPdfUrl:"/chapter/pdf-download/63924",previewPdfUrl:"/chapter/pdf-preview/63924",authors:[{id:"246437",title:"Dr.",name:"Olalekan",surname:"Isioye",slug:"olalekan-isioye",fullName:"Olalekan Isioye"},{id:"260387",title:"Mr.",name:"Mefe",surname:"Moses",slug:"mefe-moses",fullName:"Mefe Moses"},{id:"260388",title:"BSc.",name:"Abdulmumin",surname:"Lukman",slug:"abdulmumin-lukman",fullName:"Abdulmumin Lukman"}],corrections:null},{id:"62619",title:"Development of Recurrent Method with Rotation for Combined Adjustment of Terrestrial Geodetic and GNSS Networks in National Spatial Reference System",doi:"10.5772/intechopen.78770",slug:"development-of-recurrent-method-with-rotation-for-combined-adjustment-of-terrestrial-geodetic-and-gn",totalDownloads:1021,totalCrossrefCites:0,totalDimensionsCites:0,hasAltmetrics:0,abstract:"A construction of national spatial reference systems (NSRS) is promoted in many countries due to modern achievements of Global Navigation Satellite System (GNSS) methods and results of building of high accurate geoid/quasi-geoid models at centimeter level of accuracy. One of the most popular methods used for the construction of the NSRS is related to Helmert block adjustment method, by which we ought to solve techno-scientific task of a separate adjustment of GNSS network in International Terrestrial Reference Frame (ITRF) and next combination of a results of adjustment of the terrestrial geodetic and GNSS networks in the NSRS. In this chapter, we carry out a research on the usage of a recurrent adjustment method with Givens rotation for solving the abovementioned task on an account of its advantages of being effective for application of a technique of sparse matrix, outlier detection and very simple for solving the subsystem of observation equations, created based on the transformation of the results of the separate adjustment of the GNSS network from the ITRF into the NSRS. The experiment results of solving the abovementioned task for the GPS network in the North Vietnam had shown that the horizontal and vertical position accuracy of the GPS points in VN2000–3D had reached the few centimeter level.",signatures:"Ha Minh Hoa",downloadPdfUrl:"/chapter/pdf-download/62619",previewPdfUrl:"/chapter/pdf-preview/62619",authors:[{id:"247148",title:"Dr.",name:"Ha",surname:"Hoa",slug:"ha-hoa",fullName:"Ha Hoa"}],corrections:null}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},subseries:null,tags:null},relatedBooks:[{type:"book",id:"3343",title:"Advances in Geoscience and Remote Sensing",subtitle:null,isOpenForSubmission:!1,hash:"d40150daaa27a3f6aa378ff979c402b7",slug:"advances-in-geoscience-and-remote-sensing",bookSignature:"Gary Jedlovec",coverURL:"https://cdn.intechopen.com/books/images_new/3343.jpg",editedByType:"Edited by",editors:[{id:"4191",title:"Dr.",name:"Gary",surname:"Jedlovec",slug:"gary-jedlovec",fullName:"Gary Jedlovec"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"2280",title:"Remote Sensing",subtitle:"Applications",isOpenForSubmission:!1,hash:"ceea9c29b9b1f05fc128ae2cf564f110",slug:"remote-sensing-applications",bookSignature:"Boris Escalante-Ramirez",coverURL:"https://cdn.intechopen.com/books/images_new/2280.jpg",editedByType:"Edited by",editors:[{id:"111500",title:"Dr.",name:"Boris",surname:"Escalante",slug:"boris-escalante",fullName:"Boris Escalante"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"1340",title:"Remote Sensing",subtitle:"Advanced Techniques and Platforms",isOpenForSubmission:!1,hash:"336dcc589420ec91a5fdc29720442313",slug:"remote-sensing-advanced-techniques-and-platforms",bookSignature:"Boris Escalante-Ramirez",coverURL:"https://cdn.intechopen.com/books/images_new/1340.jpg",editedByType:"Edited by",editors:[{id:"111500",title:"Dr.",name:"Boris",surname:"Escalante",slug:"boris-escalante",fullName:"Boris Escalante"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"3838",title:"Advanced Geoscience Remote Sensing",subtitle:null,isOpenForSubmission:!1,hash:"74f648b4e5f6fb290baeb0642c037c1d",slug:"advanced-geoscience-remote-sensing",bookSignature:"Maged Marghany",coverURL:"https://cdn.intechopen.com/books/images_new/3838.jpg",editedByType:"Edited by",editors:[{id:"96666",title:"Prof.",name:"Prof.Dr. Maged",surname:"Marghany",slug:"prof.dr.-maged-marghany",fullName:"Prof.Dr. Maged Marghany"}],equalEditorOne:null,equalEditorTwo:null,equalEditorThree:null,productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"6540",title:"Multifunctional Operation and Application of GPS",subtitle:null,isOpenForSubmission:!1,hash:"e8e4255f626679a5ba02da035d8c0aea",slug:"multifunctional-operation-and-application-of-gps",bookSignature:"Rustam B. 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They are an intersection set between life science (for example medicine, microbiology, and comfort or strain) and technical applications (textile chemistry, engineering, and testing and certification). Developments in one of these areas affect the other one; for example, the invention of superabsorbent and gel-forming materials affected the production of a new type of baby diapers. Next-generation textiles can also be considered an important part of technical textiles, being used for different purposes such as chemical and biohazard protection. They present an important aspect from an economic point of view and the necessity for their production has been increasing; for example, a huge necessity for smart medical textiles comes from the increase of the elderly population in developed countries. In the last few decades, the rapid development of command cotton fabrics also occurred. This affects all textile sectors, for example, biodegradable fibers for implantations, three-dimension spacer fabrics, and reduction of bacterial growth by using silver ion-based textiles finishing. In this and other ways, the fields concerning the next-generation textiles have been growing rapidly and are becoming a more complex area to understand.
",isbn:"978-1-80355-883-7",printIsbn:"978-1-80355-882-0",pdfIsbn:"978-1-80355-884-4",doi:null,price:0,priceEur:0,priceUsd:0,slug:null,numberOfPages:0,isOpenForSubmission:!1,isSalesforceBook:!1,hash:"093f9e26bb829b8d414d13626aea1086",bookSignature:"Dr. Hassan Ibrahim",publishedDate:null,coverURL:"https://cdn.intechopen.com/books/images_new/11124.jpg",keywords:"Next-Generation Textile, Intelligent Textile, Smart Textile, Technical Textile, Next-Generation Material, Medical Textile, Sustainable Textile, Nanofiber, Fabric, Smart Material, Biodegradable Fiber, Technological Innovation",numberOfDownloads:27,numberOfWosCitations:0,numberOfCrossrefCitations:0,numberOfDimensionsCitations:0,numberOfTotalCitations:0,isAvailableForWebshopOrdering:!0,dateEndFirstStepPublish:"November 5th 2021",dateEndSecondStepPublish:"February 24th 2022",dateEndThirdStepPublish:"April 25th 2022",dateEndFourthStepPublish:"July 14th 2022",dateEndFifthStepPublish:"September 12th 2022",remainingDaysToSecondStep:"3 months",secondStepPassed:!0,currentStepOfPublishingProcess:4,editedByType:null,kuFlag:!1,biosketch:"Dr. Hassan Ibrahim was the Egyptian National Representative of the Chemistry and Human Health Division Committee (VII) at the International Union of Pure and Applied Chemistry (IUPAC) in 2018-2019 and is currently a member of several national committees of pure and applied chemistry. He has been awarded the best Ph.D. thesis in 2010 from the national research center, Cairo, Egypt.",coeditorOneBiosketch:null,coeditorTwoBiosketch:null,coeditorThreeBiosketch:null,coeditorFourBiosketch:null,coeditorFiveBiosketch:null,editors:[{id:"90645",title:"Dr.",name:"Hassan",middleName:null,surname:"Ibrahim",slug:"hassan-ibrahim",fullName:"Hassan Ibrahim",profilePictureURL:"https://mts.intechopen.com/storage/users/90645/images/system/90645.jpeg",biography:"Dr. Hassan Ibrahim is an associate professor of Textile Chemistry and Technology at National Research Center, Cairo, Egypt. He received his Ph.D in Organic Chemistry in 2011 from Ain Shams University. He published 24 technical papers, one review article, and one book chapter with international publishers. He supervised 8 PhD and M.Sc. thesis, and participated in 14 national and international projects dealing with organic and environmental chemistry, hazardous wastes, medical textiles, nanotechnology, and electrospun nanofibers formation. He has expertise in applied chemistry and technology of organic chemistry, especially in carbohydrates, polymers, pollution prevention, preparation, and applications of nanoparticles (polymer chemistry, chemistry of chitosan, chitosan modification, nanoparticles preparation, and electrospinning technique). He built this model after years of research and teaching at university and research centers. He was the Egyptian National Representative of the Chemistry and Human Health Division Committee (VII) at the International Union of Pure and Applied Chemistry (IUPAC) in 2018-2019, and is currently a member of several national committees of pure and applied chemistry. He has been awarded for the best PhD thesis in 2010 from national research center, Cairo, Egypt.",institutionString:"National Research Centre",position:null,outsideEditionCount:0,totalCites:0,totalAuthoredChapters:"1",totalChapterViews:"0",totalEditedBooks:"0",institution:{name:"National Research Centre",institutionURL:null,country:{name:"Egypt"}}}],coeditorOne:null,coeditorTwo:null,coeditorThree:null,coeditorFour:null,coeditorFive:null,topics:[{id:"14",title:"Materials Science",slug:"materials-science"}],chapters:[{id:"81032",title:"Healthcare and Hygiene Products Application in Medical Textile",slug:"healthcare-and-hygiene-products-application-in-medical-textile",totalDownloads:16,totalCrossrefCites:0,authors:[{id:"336997",title:"Dr.",name:"Ramratan",surname:"Guru",slug:"ramratan-guru",fullName:"Ramratan Guru"},{id:"337607",title:"Prof.",name:"Anupam",surname:"Kumar",slug:"anupam-kumar",fullName:"Anupam Kumar"},{id:"337608",title:"Mr.",name:"Rohit",surname:"Kumar",slug:"rohit-kumar",fullName:"Rohit Kumar"}]},{id:"81855",title:"Progress of Recycled Polyester in Rheological Performance in Molding, and Economic Analysis of Recycled Fibers in Fashion and Textile Industry",slug:"progress-of-recycled-polyester-in-rheological-performance-in-molding-and-economic-analysis-of-recycl",totalDownloads:0,totalCrossrefCites:null,authors:[null]},{id:"81618",title:"To Study the Implantable and Non-Implantable Application in Medical Textile",slug:"to-study-the-implantable-and-non-implantable-application-in-medical-textile",totalDownloads:11,totalCrossrefCites:0,authors:[null]}],productType:{id:"1",title:"Edited Volume",chapterContentType:"chapter",authoredCaption:"Edited by"},personalPublishingAssistant:{id:"444312",firstName:"Sara",lastName:"Tikel",middleName:null,title:"Ms.",imageUrl:"https://mts.intechopen.com/storage/users/444312/images/20015_n.jpg",email:"sara.t@intechopen.com",biography:"As an Author Service Manager, my responsibilities include monitoring and facilitating all publishing activities for authors and editors. From chapter submission and review to approval and revision, copyediting and design, until final publication, I work closely with authors and editors to ensure a simple and easy publishing process. I maintain constant and effective communication with authors, editors and reviewers, which allows for a level of personal support that enables contributors to fully commit and concentrate on the chapters they are writing, editing, or reviewing. I assist authors in the preparation of their full chapter submissions and track important deadlines and ensure they are met. I help to coordinate internal processes such as linguistic review and monitor the technical aspects of the process. As an ASM I am also involved in the acquisition of editors. Whether that be identifying an exceptional author and proposing an editorship collaboration, or contacting researchers who would like the opportunity to work with IntechOpen, I establish and help manage author and editor acquisition and contact."}},relatedBooks:[{type:"book",id:"6320",title:"Advances in Glass Science and Technology",subtitle:null,isOpenForSubmission:!1,hash:"6d0a32a0cf9806bccd04101a8b6e1b95",slug:"advances-in-glass-science-and-technology",bookSignature:"Vincenzo M. 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Kawsar Alam"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"6851",title:"New Uses of Micro and Nanomaterials",subtitle:null,isOpenForSubmission:!1,hash:"49e0ab8961c52c159da40dd3ec039be0",slug:"new-uses-of-micro-and-nanomaterials",bookSignature:"Marcelo Rubén Pagnola, Jairo Useche Vivero and Andres Guillermo Marrugo",coverURL:"https://cdn.intechopen.com/books/images_new/6851.jpg",editedByType:"Edited by",editors:[{id:"112233",title:"Dr.Ing.",name:"Marcelo Rubén",surname:"Pagnola",slug:"marcelo-ruben-pagnola",fullName:"Marcelo Rubén Pagnola"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9393",title:"Engineering Steels and High Entropy-Alloys",subtitle:null,isOpenForSubmission:!1,hash:"d33466a3272f97353a6bf6d76d7512a5",slug:"engineering-steels-and-high-entropy-alloys",bookSignature:"Ashutosh Sharma, Zoia Duriagina, Sanjeev Kumar",coverURL:"https://cdn.intechopen.com/books/images_new/9393.jpg",editedByType:"Edited by",editors:[{id:"145236",title:"Dr.",name:"Ashutosh",surname:"Sharma",slug:"ashutosh-sharma",fullName:"Ashutosh Sharma"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"7360",title:"Fillers",subtitle:"Synthesis, Characterization and Industrial Application",isOpenForSubmission:!1,hash:"4cb5f0dcdfc23d6ec4c1d5f72f726ab4",slug:"fillers-synthesis-characterization-and-industrial-application",bookSignature:"Amar Patnaik",coverURL:"https://cdn.intechopen.com/books/images_new/7360.jpg",editedByType:"Edited by",editors:[{id:"43660",title:"Associate Prof.",name:"Amar",surname:"Patnaik",slug:"amar-patnaik",fullName:"Amar Patnaik"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}},{type:"book",id:"9360",title:"Perovskite Materials, Devices and Integration",subtitle:null,isOpenForSubmission:!1,hash:"4068d570500b274823e17413e3547ff8",slug:"perovskite-materials-devices-and-integration",bookSignature:"He Tian",coverURL:"https://cdn.intechopen.com/books/images_new/9360.jpg",editedByType:"Edited by",editors:[{id:"259466",title:"Prof.",name:"He",surname:"Tian",slug:"he-tian",fullName:"He Tian"}],productType:{id:"1",chapterContentType:"chapter",authoredCaption:"Edited by"}}]},chapter:{item:{type:"chapter",id:"59393",title:"Limitations and Biases in Cohort Studies",doi:"10.5772/intechopen.74324",slug:"limitations-and-biases-in-cohort-studies",body:'\nThe external validity of the results of an analytical study (including cohort studies) is determined by the possibility that the results can be extrapolated to larger populations, making the representativeness and randomness of the sample(s) important. However, there is controversy about the real need of representativeness when other situations are more relevant in the study, for example, some practical reasons, restrictions in the selection criteria or focus in certain population groups [1].
\nInternal validity, however, is determined by a series of factors that can lead to systematic errors or biases [2]. Bias can originate both in the design stage of the study, such as sample selection, data collection or analysis, but can be minimized with good planning of the study protocol or using statistical analysis techniques in this phase of the study [3]. The sample size will determine the validity in terms of the statistical power necessary to reject or approve the working hypothesis. An adequate sample size will make it easier to avoid random errors in the results of the study.
\nAlthough cohort studies have a lower risk of presenting biases than other types of epidemiological studies (ecological, cross-sectional or prevalence studies, cases and controls), they are not free of them. This chapter highlights the types of biases, their origin, their effects on the validity of the study and ways to avoid or minimize them. The chapter also gives examples that allow better understanding of the concepts as well as practical advice when carrying out a cohort study.
\nStudy protocols should always adhere to the evaluation of duly accredited Scientific Ethics Committees. Ethical principles indicate that all participants must adhere to informed consent before beginning to participate in the study, being able to understand all the implications of participating and to decline his/her participation at any moment. Authorizations of the managers in charge of the administration of any institution (healthcare centers, schools, municipalities, hospitals or others) are usually required to access the registered data or to collect the health information of the users. In studies of occupational health, authorization of the workplaces is required to perform the evaluations of jobs and workers exposed to occupational hazards. In studies about infants or children, framed in the educational sector, the assent of the minors is required, in addition to informed consent of the parents/guardians/proxies, and authorization of the executives of participating educational facilities. Collaboration agreements, purchase of laboratory services, transport, locations, surveyors, data analysis, computer support and other technical and logistical requirements that involve carrying out a follow-up study of people, usually for several years, must also be managed. When a large research team is involved, protocols must be in place for recruitment, evaluations, transporting and storage of samples and materials, laboratory procedures, recording data, backing up information and so on.
\nCertainly, among analytical epidemiological research, cohort studies are less prone to have bias than the case-control ones, specifically regarding memory bias. But as any other epidemiological study, several biases could be present in cohort studies. In this sense, researchers must be aware of those biases in advance and take them into account at the moment of selecting participants, designing the study (collection tools/instruments), when registering the data during field work (data base design) and, later on, at the moment of analyzing and interpreting the data (statistical analysis).
\nWe understand bias as systematic errors that can lead to mistaken results or interpretation regarding the association under study, when the purpose of a study is assessing the association of certain factors toward supporting the causality of a health event or outcome [4].
\nThere are several ways of classification of biases. For academic proposes, we will use the following classification [2, 4]:
\nSelection bias: originated from the way the participants of the study are selected or followed and can affect the apparent association between the exposure and outcome.
\nInformation biases: could originate in the observed individuals, in the observers or in the instruments used to assess the outcomes.
\nConfusion bias: their origin is in the relationship that other variables that are not the exposition are related to the outcome, and can modulate the effect(s) of the exposition, contributing to a spurious association.
\nWe will now review each kind of bias in detail and with some examples.
\nIn cohort studies, the researcher must select exposed and nonexposed individuals. In the first place, it should be understood that both groups are representative of the general population from where they are taken, in order to facilitate the external validity of the study (basic condition to generalize the results in order to support causality). This condition, however, would not necessarily affect the internal validity. In other words, the internal validity is due to systematic errors sourced in stubborn participation of individuals.
\nThe appropriated assessment of the exposure is the first crucial step. Auto-selection is one of the circumstances that could lead to inaccurate selection. As an example, a study conducted among pregnant women in Norway intended to evaluate auto-selection bias by comparing two cohorts; one group was taken from the Medical Birth Registry (2000–2006) as a population-based cohort, and the second group was from women who agreed to participate in the Norwegian Mother and Child Cohort Study. The results suggested that the prevalence estimates of exposures and the outcomes were biased due to self-selection in the Norwegian Mother and Child Cohort Study. Nevertheless, the estimates of exposure-outcome associations were not biased [5]. But in other cases, the associations could also be flawed.
\nAnother example for selection bias might occur when the compared cohorts are part of a population who receive public health interventions, so the exposure can be misled by this influence. That is the example given by researchers who studied the association of bad water quality (measured by
Selection criteria must be clearly defined from the beginning of the study in a way that ensures that biases are avoided. For example, a research was conducted with the objective of assessing the association between exposure to pesticides and neurocognitive impairment, including fine motor coordination [7]. The researchers used Purdue Pegboard and MOART reaction time tests to measure the outcome. If any right- or left-handed people were selected, bias may be introduced when evaluating the outcomes due to the way that the tests are performed. Both tests have separated evaluation of left and right hands, giving certain scoring to the performance. Then, one important inclusion criterion to consider was right-handed people only; so the responses were standardized under the same criteria, and bias was eluded.
\nAnother example of selection bias can happen in large multicenter cohort studies evaluating the association between diet and cancer. In this case, systematic errors may originate in the measurement of the exposure by dietary questionnaires that are not easy to standardize for all locations. Researchers suggest to use the calibration approach for such cases [8].
\nAnother type of selection bias is known as the nonresponders or no-participation bias, which are less frequent in prospective cohort studies due to the need for strict following-up of the participants, strengthening the evaluations during the follow-up visits, encouraging participants and evaluators (observers) to always respond and/or register the records properly. Nevertheless, missing data could be present in retrospective cohort studies, where previously registered data are used. This will be explained in detail later, related to the information biases (Section 2.2).
\nIn prospective cohort studies, loss of follow-up may occur, giving rise to selection bias. Loss of follow-up bias is caused by the loss of individuals from one or more exposure groups. Because cohort studies take normally several months or years of following the participants, it is expected that life situations will vary from time to time, causing some of the participants to get lost during the development of the study. Individuals can be lost homogeneously in the groups to be compared, causing bias of poor global miss-classification, which generally leads the estimate toward the null value [9]. Or individuals from a single group can be lost, causing bias of poor differential miss-classification. In the first case, the estimated risk would not be severely affected, because the incidence rates would keep similar in both groups, but the power of the results may be lost. In the latter case, the results to be obtained may be underestimating or overestimating the association. For example, if the people who are exposed and develop the outcome (disease) are lost, the incidence rate may be lower among the exposed individuals and the relative risk (RR) would be underestimated. On the other hand, if people who are not exposed and do not get the disease after time of follow-up get lost, then the incidence rate among the nonexposed will be higher and the RR would be overestimated.
\nHere is a hypothetic example showing the four possibilities of losing individuals:
\nSize of the exposed cohort = 1000.
\nSize of the nonexposed cohort = 1000.
\nNumber of individuals with the outcome among the exposed = 100.
\nNumber of individuals with the outcome among the nonexposed = 10.
\nIncidence rate in exposed = 100/1000 = 0.1.
\nIncidence rate in nonexposed = 10/1000 = 0.01.
\nRelative risk = 10.
\nIncidence rate in exposed = 50/1000 = 0.05.
\nIncidence rate in nonexposed = 10/1000 = 0.01.
\nRelative risk = 5.
\nL
Incidence rate in exposed = 100/1000 = 0.1.
\nIncidence rate in nonexposed = 5/1000 = 0.005.
\nRelative risk = 20.
\nIncidence rate in exposed = 100/800 = 0.125.
\nIncidence rate in nonexposed = 10/1000 = 0.01.
\nRelative risk = 12.5.
\nIncidence rate in exposed = 100/1000 = 0.1.
\nIncidence rate in nonexposed = 10/800 = 0.0125.
\nRelative risk = 8.
\nAs you can see, the estimated association variation is given by the number of people who completed the follow-up schedule. The general recommendation is that 60–80% of the individuals complete the timeframe defined originally, but a study that simulated a cohort of 500 observations with 1000 replications in computer found utterly biased estimates of the risks with low ranks of loss to follow-up [10]. On the other hand, as was already said, the results of the diminution in the number of subjects can also affect the statistical power of the results. Then, in the design of the study, at least 10% sample loss must be considered, so this proportion must be added to the minimum calculated sample size for the study. During the field work phase, measures need to be taken in advance in order to avoid losing individuals. To ensure the permanence of the individuals during the follow-up time, it is suggested to include incentives for the participants. These incentives do not necessarily have to be monetary, and a food and transportation voucher can be offered for those who attend scheduled evaluations.
\nIn addition, it may happen that nonexposed individuals enter into the exposed group or vice versa. An example of this could occur when studying the association of tobacco consumption and a certain outcome. Then, during the study, people who smoke can leave the consumption and/or people who do not smoke can start smoking. In those cases, it is suggested to use the
The incidence density is calculated as follows:
\nIt is important to mention that the person-time unit is not in all occasions equivalent to the person-time of all individuals. For example, one person-year could represent one person being followed for 1 year or two people being followed for 6 months. But in any case, this is a way of measuring incidence that is very useful in cohort studies because it avoids the issue of subjects shifting form one exposure group to the other.
\nFinally, we have the selective survival bias. This bias is known in occupational health as the
Loss during follow-up may cause information bias that was already explained in detail in Section 2.1 [10].
\nUsually in prospective cohorts, information bias is easy to elude, because measures may be taken during the design by including all variables in the registration forms (instruments), in order to not miss variables of interest. On the other hand, in retrospective cohorts, already existing records may be used. In that case, there could be missing data due to poor registration quality or due to variables that were not considered to be registered in advance. In both cases, the origin of missing information can lead to information bias. To minimize this effect on large population-based cohorts, it is possible to exclude individuals who have missing data from the analyses. But, this is a decision that researchers can take when the size of the remaining cohort still allows for sufficient statistical power to validate the results. That was the case presented in a large study conducted among the Danish population assessing the association between lifestyle and colorectal cancer [16]. From a total of 160,725 potential participants, several hundreds were not included due to nonresponse, cancer diagnosis and missing data (N = 997). Finally, a cohort of over 55,000 people was included in the investigation.
\nOne important source of bias in cohort studies can occur when diagnosing the health event or outcome. It is necessary to apply the same protocol for measuring or evaluating the health outcomes in exposed and nonexposed individuals in order to avoid the biases of misclassification [9]. Similarly to what happen in the previously explained bias caused by loss of follow-up, the final effect of misclassification will depend on whether the inaccuracy in the evaluated outcome influences both exposure groups (global misclassification bias) or only affects one of them (differential misclassification bias).
\nLet us have a look at a hypothetical example in a study that evaluates the risk of having myocardial infarction due to exposure to a high-fat diet. Table 1 shows the correct classification.
\nHigh-fat diet | \nMyocardial infarction | \n|
---|---|---|
Disease | \nNo disease | \n|
Exposed | \n250 | \n450 | \n
Nonexposed | \n100 | \n900 | \n
High-fat diet and acute myocardial infarction,
RR = (250/700)/(100/1000) = 0.357/0.1 = 3.57.
If the evaluation of the exposure is misled in both groups due to the mistakes in the daily food register, this results in a non-differential misclassification. Imagining that 20% of the exposed people go to the nonexposed group and 20% of nonexposed goes to the exposed group, we could have the following situation (Table 2).
\nHigh-fat diet | \nMyocardial infarction | \n|
---|---|---|
Disease | \nNo disease | \n|
Exposed | \n290 | \n410 | \n
Nonexposed | \n260 | \n740 | \n
High-fat diet and acute myocardial infarction, non-differential misclassification.
RR = (290/700)/(260/1000) = 0.414/0.26 = 1.59.
In that case, the relative risk is diminished due to a higher incidence among the nonexposed group.
\nNow, suppose that the evaluators applied two diagnostic tests to the exposed that resulted in an increased diagnosis of myocardial infarction among the exposed group. This will result in a differential misclassification due to the mistaken diagnosis in the outcomes (Table 3).
\nHigh-fat diet | \nMyocardial infarction | \n|
---|---|---|
Disease | \nNo disease | \n|
Exposed | \n295 | \n405 | \n
Nonexposed | \n100 | \n900 | \n
High-fat diet and acute myocardial infarction, differential misclassification.
RR = (295/700)/(100/1000) = 0.421/0.1 = 4.21.
In this last case, when 10% of the exposed people without myocardial infarction moved to the disease group, the result is a higher relative risk due to a higher incidence among the exposed group.
\nA good example of this kind of misclassification bias could be given regarding the use of mortality records, which are frequently used in epidemiological studies. The registered codes of the diagnoses may be mistaken and lead to misclassification of the outcomes. That was studied recently by Deckert, who reported the results of a simulation study based on real data of cardiovascular disease mortality [17]. He reported that non-differential bias can to lead to a null hypothesis, whereas differential misclassification leads the observed Standardized Mortality Ratios to be incorrect, in either direction or magnitude. Differences were from 10 to 30%, depending on the sensitivity and specificity characteristics of the diagnosis of cardiovascular disease [17]. Statistic techniques like quantitative bias analysis (QBA) or bootstrapping disease status imputation could be used to correct misclassification bias due to correct diagnostic codes [18]. Although statistical adjustments are possible to do in cases where standard information is available, these techniques are not always enough to overcome the bias. An example is reported by Candice Johnson et al., related to the misclassification of self-reported obesity and diabetes, adjusted by the National Health and Nutrition Examination Survey [19].
\nRegarding the accuracy in gathering information during the follow-up visits, one could have the temptation to assess more strictly the exposed individuals than the nonexposed or to evaluate the exposed persons more frequently than the nonexposed. The advice is to apply the same protocol and instruments to both groups of people, and in that way, bias introduced by the observer or the instruments is avoided. We understand as instruments the questionnaires, weighting scale, sphygmomanometer, altimeter, laboratory tests/techniques and others. Additionally, if the person(s) who observe and diagnose the outcome are aware of the exposure status, a preconception may lead to overdiagnose the exposed people and/or underdiagnose the nonexposed people. Alike in randomized trials, the best way to avoid this bias is blinding the observers.
\nThere is also a possibility that bias may originate from the observed individuals. It can happen that people, who know they are under observation, change their behavior. This has been called the Hawthorne effect and is due to the effects that the research can produce in the participants (observers and/or studied individuals). This was first described in a factory near Chicago between 1924 and 1936, in which a group of workers who knew they were under strict supervision significantly improved their productivity, compared to workers who were not aware of being observed [20]. There is still some controversy about the real predisposition effect of the participant’s observation and the amount of bias that could cause. Some studies have found such an effect, but others have not [20, 21]. For example, a study conducted in Tanzania regarding malaria treatment did find a modest suggestion that the health professionals maintained better practice during the study [22].
\nFinally, all types of epidemiological studies may be affected by partiality in the phase of the analysis. The way to avoid this analytical bias is by
We understand a confounder as a variable that is associated with the exposure as well as to the health event or outcome, but not being necessarily a cause of the event. For example, an inaccurate causal inference can be made between drinking coffee and pancreatic cancer, when drinking coffee has been associated with a smoking habit [4]. This is known as
The most common confusion variables to be considered during the design of any epidemiological study are gender and age. As cohort studies are observational, people are not randomly assigned to the exposure and nonexposure group; it is not always possible to match both groups by certain variables such as sex, age, or other confounders. Depending on the exposure or events being studied, other variables could work as confounders; therefore, before designing any study, it is important for researchers to read previous studies and develop the design with all evidence that highlight confounders.
\nSeveral examples can be given in this matter: (1) In the aforementioned study about acetylsalicylic acid exposure and major bleeding, confounders considered were age, sex, previous hospitalization for alcoholism, non-bleeding ulcer disease, other non-bleeding conditions, and comorbidities [15]; then the researchers could adjust the risk ratios according to those variables. (2) In the study relating endometriosis and infertility, the considered confounders were menstrual cycle pattern, hirsutism, participant’s birthweight, race, household income, husband’s education, BMI at age 18 years, alcohol consumption, oral contraception use, any analgesic use, health screening behavior, personal history of cardiovascular disease, and personal history of diabetes [23]. In that case, researchers could evaluate if any of those variables truly acted as confounders or not. (3) In a study relating air pollution and mortality risks, past exposure to pollution and habit of tobacco consumption were not considered; because they may act as confounders for the results of the association and causal inference could be misled [24]. For a better understanding, please refer to Figure 1.
\nScheme of confounding (smoking) in relation to the exposure (air pollution) and the outcome (acute myocardial infarction).
There are some cases in which the dose of exposure may introduce confusion. Such is the case of age, smoking, or drinking alcohol. Age involves the quantity of years (or months) itself, and for the analysis, it will be possible to use as a continuous variable or create ranges of age for stratification analysis. On the other hand, for consumption, it is recommended to consider registering the quantities being consumed by the individuals, so strata can be made during the analysis phase. Tobacco may be registered by number of cigarettes per day. The case of alcohol is rather difficult. The suggestion is to ask for quantity (number of glasses) and types of drinks consumed and then transform it into grams of pure alcohol consumed daily or weekly.
\nConfounding variables can be controlled in several ways: restriction, matching, stratification and more sophisticated multivariate techniques [2].
\nRestriction is a simple way of avoiding the introduction of already known confounders, by excluding people who present that factor from the beginning. The problem is that this could limit recruitment and the representativeness of certain population groups. So, while increasing the internal validity, this may reduce external validity [2].
\nAs it was said before, matching is rather difficult to do in prospective studies because the first enrolment criterion is the exposure. Matching normally is used in case control studies, but researchers could emphasize that the proportion of women and men would be 50% each or that a ratio of young/old people was similar in the exposure groups.
\nStratification is a simple statistic technique that could be used during analysis, but that requires forethought concerning the possible confusion variables and registering them.
\nThe technique consists of separating the analysis of association, according to strata of the confusing factor, for example, perform separate analyses of men and women when it is suspected that gender may be a confounder. Then, when the difference between the calculated raw risk and the risk calculated by strata is over 15%, we could say that confusion is present.
\nLet’s see an example. In a rural area, a study proposed to evaluate the relationship between indoor exposure to smoke—from the combustion of wood stoves—and the occurrence of tuberculosis (TB). The results obtained are presented in Table 4.
\nIndoor exposure to smoke | \nTuberculosis | \n\n | |
---|---|---|---|
Disease | \nNo disease | \nTotal | \n|
Exposed | \n50 | \n21 | \n71 | \n
Nonexposed | \n238 | \n524 | \n762 | \n
Total | \n288 | \n545 | \n833 | \n
Tuberculosis and indoor exposure to smoke from wood burning.
The RR calculation is presented as: RR = (50/71)/(238/762) = 0.704/0.312 = 2.25.
Therefore, the factor
Given that there is a suspicion that cigarette smoking could modify the effect of indoor contamination on the risk of acquiring tuberculosis, smoking habits were considered. Then, it was possible to assess if this condition acted as a confounder in the association between tuberculosis and indoor smoke exposure using stratification.
\nThe stratification is shown below, where the smoking habit was coded as “never” or “past or present” (Tables 5 and 6).
\nIndoor exposure to smoke | \nTuberculosis | \n\n | |
---|---|---|---|
Disease | \nNo disease | \nTotal | \n|
Exposed | \n33 | \n17 | \n50 | \n
Nonexposed | \n186 | \n411 | \n597 | \n
TB and indoor exposure to smoke from wood burning of
Risk ratio calculation among
Indoor exposure to smoke | \nTuberculosis | \n\n | |
---|---|---|---|
Disease | \nNo disease | \nTotal | \n|
Exposed | \n17 | \n4 | \n21 | \n
Nonexposed | \n52 | \n113 | \n165 | \n
TB and indoor exposure to smoke from wood burning of
Risk ratio calculation among
As a conclusion of the stratification results, the factor
In addition to confusion, we have the concept of
Let us try to find interaction in the same example.
\nIn order to assess interaction, it will be necessary to calculate the association between smoking and tuberculosis alone (without the indoor exposure to wood burning smoke) Table 7.
\nSmoking past or present | \nTuberculosis | \n\n | |
---|---|---|---|
Disease | \nNo disease | \nTotal | \n|
Yes | \n52 | \n113 | \n165 | \n
No smoking | \n186 | \n411 | \n597 | \n
TB and smoking habits (without indoor exposure to smoke from wood burning).
Risk calculation of smoking among nonexposed to smoke from wood burning: RR = (52/165)/(186/597) = 0.315/0.311 = 1.01.
The result shows that the relative risk of developing TB due exclusively to the habit of smoking is almost nil. But, to know if there is interaction, we should estimate if the presence of both exposures together differs or not from the expected effects if the two exposures were simply the sum of both.
\nFrom the previous tables and calculations, we have that the incidences are the following:
Incidence rate of TB without any smoke exposure = 31.1%
Incidence rate of TB with smoking only = 31.5%
Incidence rate of TB with indoor pollution only = 66%
Incidence rate of TB with smoking and indoor pollution = 81%
In order to know whether interaction is present, we should clear the incidences from the underground risk of developing TB (baseline incidence). Then, we should start by calculating the attributable risks (ARs), as follows:
AR to smoking = (TB incidence due to smoking−baseline TB incidence) = 31.5–31.1 = 0.4.
AR to indoor pollution = (TB incidence if indoor pollution−baseline TB incidence) = 66–31.1 = 34.9.
The expected attributable risk to both factors would be the addition of the TB incidence of (smoking + indoor contamination) = 34.9 + 0.4 = 35.3%. Then, the expected incidence will be (31.1 + 35.3) = 66.4%.
\nBut the real TB incidence with both exposure factors was 81%. The difference between 81 and 66.4 would be attributable to the interaction, which is 14.6%.
\nIn other way, the incidence when both factors are together is higher than the addition of incidences when the factors are alone, taking into consideration that we have to clear the underground risk (incidence of TB in population free of exposures).
\nEffectively, we have shown that interaction is present, because the incidences of both exposures together differ from the expected effects if the two exposures were simply the sum of both. As a conclusion, the indoor pollution is a risk factor to develop TB in that setting, but this risk increases substantially more if people smoke indoors. For a better understanding, please refer to Figure 2.
\nIncidence rates and attributable risk to factors related to TB incidence and their interaction.
Coming back to the control of confusion bias, adjustment techniques using statistical models require computer training and have the advantage of working with two or more possible confounding variables; opposite to stratification that permits working on one factor only. When using modeling multivariate techniques, logistic regression or proportional hazard regression might be used, but researchers must be aware of how to interpret the results properly [2].
\nAn important comment about the confusion is that finding a confounder is not always an issue to be worried about. It could also be useful. For example, in the mentioned study about pesticide exposure in agricultural workers and cognitive impairment, gender turned out to be a confounder [7]. This resulted from the type of work performed differing between men and women. Men used to perform tasks like mixing, blending and applying pesticides; while women proned to collecting fruits, so men were directly exposed to the toxins. Then, knowing that men were more exposed and, consequently, more susceptible to the health damage, the preventive measures may be oriented by strengthening them toward men, but still keeping care on women.
\nFinally, confounders are not a mistake in the research, but a phenomenon that is present must be understood by the investigators in order to finally consider them when interpreting the results of the study [4].
\nAs shown, biases can be present in any study, originating from multiple steps of the investigation. Their presence should not be grounds for rejection of the results due to the poor quality of the study, but careful attention is required when interpreting the results. To the extent that the researcher is able to recognize the biases, he/she can be proactive in mitigating them, either by way of improving the design or applying statistical techniques (stratification or multivariate adjustment) when analyzing the results. Therefore, when clinicians or researchers look for good quality of articles to read and use as references, they must recognize them when interpreting the results and acknowledge the limitations that the studies may have.
\nIt should be noted that biases are more frequent among retrospective cohort, given by missing information when using existing records (information bias) or by selection bias, because individuals are selected after the outcome has occurred, so both conditions (exposure and outcome) are present at the moment of enrollment. In that case, it is easier that exposed or unexposed subjects would be related to the result of interest, causing selection bias. On the other hand, a prospective cohort design could be affected by the loss of follow-up. Both types of cohort studies may be influenced by information bias, confusion or interaction.
\nInteresting tools for weighting quality and predisposition to unfairness in observational studies have been gathered and reported by Sanderson et al. [25]. Those included items for selection methods, measurement of study variables, design-specific sources of bias, control of confounding variables and use of statistics.
\nFinally, it is considered that cohort studies are used normally as a source of information of systematic reviews and meta-analysis. In those cases, publication bias and outcome reporting bias must be taken into consideration. This is because the journals are prone to publish positive results rather than negative ones, a situation that has been shown [26].
\nThe author of this chapter declares no conflict of interest.
Optimization is a very old subject of a great interest; we can search deep into a human history to find important examples of applying optimization in the usual life of a human being, for example, the need of finding the best way to produce food yielded finding the best piece of land for producing, as well as (later on, how the time was going) the best ways of treatment of the chosen land and the chosen seedlings to get the best results.
\nFrom the very beginning of manufacturing, the manufacturers were trying to find the ways to get maximum income with minimum expenses.
\nThere are plenty of examples of optimization processes in pharmacology (for determination of the geometry of a molecule), in meteorology, in optimization of a trajectory of a deep-water vehicle, in optimization of power management (optimization of the production of electrical power plants), etc.
\nOptimization presents an important tool in decision theory and analysis of physical systems.
\nOptimization theory is a very developed area with its wide application in science, engineering, business management, military, and space technology.
\nOptimization can be defined as the process of finding the best solution to a problem in a certain sense and under certain conditions.
\nAlong with the passage of time, optimization was evolving. Optimization became an independent area of mathematics in 1940, when Dantzig presented the so-called simplex algorithm for linear programming.
\nThe development of nonlinear programming became great after presentation of conjugate gradient methods and quasi-Newton methods in the 1950s.
\nToday, there exist many modern optimization methods which are made to solve a variety of optimization problems. Now, they present the necessary tool for solving problems in diverse fields.
\nAt the beginning, it is necessary to define an objective function, which, for example, could be a technical expense, profit or purity of materials, time, potential energy, etc.
\nThe object function depends on certain characteristics of the system, which are known as variables. The goal is to find the values of those variables, for which the object function reaches its best value, which we call an extremum or an optimum.
\nIt can happen that those variables are chosen in such a way that they satisfy certain conditions, i.e., restrictions.
\nThe process of identifying the object function, variables, and restrictions for the given problem is called
The first and the most important step in an optimization process is the construction of the appropriate model, and this step can be the problem by itself. Namely, in the case that the model is too much simplified, it cannot be a faithful reflection of the practical problem. By the other side, if the constructed model is too complicated, then solving the problem is also too complicated.
\nAfter the construction of the appropriate model, it is necessary to apply the appropriate algorithm to solve the problem. It is no need to emphasize that there does not exist a universal algorithm for solving the set problem.
\nSometimes, in the applications, the set of input parameters is bounded, i.e., the input parameters have values within the allowed space of input parameters \n
Except (1), the next conditions can also be imposed:
\nOptimization task is to find the minimum (maximum) of the objective function \n
If the object function is linear, and the functions \n
Unconstrained optimization problem can be presented as
\nwhere \n
Problem (4) is, in fact, the unconstrained minimization problem. But, it is well known that the unconstrained minimization problem is equivalent to an unconstrained maximization problem, i.e.
\nas well as
\nThe ideal situation is finding a global minimizer of \n
So, we could be satisfied by finding the local minimizer of the function \n
Formal definitions of local weak and strict minimizer of the function \n
Considering backward definitions 1.1.2 and 1.1.3, the procedure of finding local minimizer (weak or strict) does not seem such easy; it seems that we should examine all points from the neighborhood of \n
Fortunately, if the object function \n
For example, we can assume that the object function \n
All algorithms for unconstrained minimization require the user to start from a certain point, so-called the starting point, which we usually denote by \n
There exist two important classes of iterative methods—
In this chapter, at first, we discuss different kinds of line search. Then, we consider some line search optimization methods in details, i.e., we study steepest descent method, Barzilai-Borwein gradient method, Newton method, and quasi-Newton method.
\nAlso, we try to give some of the most recent results in these areas.
\nNow, let us consider the problem
\nwhere \n
There exists a great number of methods made in the aim to solve the problem (7).
\nThe optimization methods based on line search utilize the next iterative scheme:
\nwhere \n
At first, we consider the monotone line search.
\nNow, we give the iterative scheme of this kind of search.
\nStep 1. If \n
Step 2. Find the descent direction \n
Step 3. Find the step size \n
Step 4. Set \n
Step 5. Take \n
Denote
\nTrying to solve the minimization problem, we are going to search for the step size \n
That procedure is called the monotone line search.
\nWe can search for the step size \n
i.e.
\nor we can use the next formula:
\nIn this case we are talking about
By the other side, instead of using the relation (9), or the relation (11), we can be satisfied by searching for such \n
Then, we are talking about
There are several reasons to use the inexact instead of the exact line search. One of them is that the exact line search is expensive. Further, in the cases when the iteration is far from the solution, the exact line search is not efficient. Next, in the practice, the convergence rate of many optimization methods (such as Newton or quasi-Newton) does not depend on the exact line search.
\nFirst, we are going to mention so-called basic and, by the way, very well-known inexact line searches.
\nAssumptions: \n
Step 1. \n
Step 2. While \n
Step 3. Set \n
Now, we describe the Armijo rule.
\nwhere \n
Next, we describe the Goldstein rule [2].
\nThe step size \n
where \n
Now, Wolfe line search rules follow [3], [4].
\nStandard Wolfe line search conditions are
\nwhere \n
This efficient strategy means that we should accept a positive step length \n
Strong Wolfe line search conditions consist of (12) and the next, stronger version of (13):
\nIn the generalized Wolfe line search conditions, the absolute value in (14) is replaced by the inequalities:
\nBy the other side, in the approximate Wolfe line search conditions, the inequalities (15) are changed into the next ones:
\nThe next lemma is very important.
\nBy the other side, the introduction of the non-monotone line search is motivated by the existence of the problems where the search direction does not have to be a descent direction. This can happen, for example, in stochastic optimization [6].
\nNext, some efficient quasi-Newton methods, for example, \n
Further, some efficient methods like spectral are not monotone at all.
\nSome numerical results given in [7, 8, 9, 10, 11] show that non-monotone techniques are better than the monotone ones if the problem is to find the global optimal values of the object function.
\nAlgorithms of the non-monotone line search do not insist on a descent of the object function in every step. But, even these algorithms require the reduction of the object function after a predetermined number of iterations.
\nThe first non-monotone line search technique is presented in [12]. Namely, in [12], the problem is to find the step size which satisfies
\nwhere \n
This strategy is in fact the generalization of Armijo line search. In the same work, the authors suppose that the search directions satisfy the next conditions for some positive constants \n
The next non-monotone line search is described in [11].
\nLet \n
Let \n
The step size has to satisfy the next conditions:
\nThe value \n
Non-monotone rules which contain the sequence of nonnegative parameters \n
and the corresponding rule is
\nNow, we give the non-monotone line search algorithm, shortly \n
Assumptions: \n
Set \n
Step 1. If \n
Step 2. Set \n
Step 3. Choose \n
Step 4. Set \n
We can notice [11] that \n
If \n
If \n
This study would be very incomplete unless we mention that there are many modifications of the abovementioned line searches. All these modifications are made to improve the previous results.
\nFor example, in [15], the new inexact line search is described by the next way.
\nLet \n
Further, in [16], a new inexact line search rule is presented. This rule is a modified version of the classical Armijo line search rule. We describe it now.
\nLet \n
Find a step size \n
holds, where \n
Next, in [17], a new, modified Wolfe line search is given in the next way.
\nFind \n
where \n
More recent results on this topic can be found, for example, in [18, 19, 20, 21, 22, 23].
\nThe classical steepest descent method which is designed by Cauchy [24] can be considered as one among the most important procedures for minimization of real-valued function defined on \n
Steepest descent is one of the simplest minimization methods for unconstrained optimization. Since it uses the negative gradient as its search direction, it is known also as the gradient method.
\nIt has low computational cost and low matrix storage requirement, because it does not need the computations of the second derivatives to be solved to calculate the search direction [25].
\nSuppose that \n
Using Taylor expansion of the function \n
The iterative scheme of the \n
The classical steepest descent method uses the exact line search.
\nNow, we give the algorithm of the steepest descent method which refers to the exact as well as to the inexact line search.
\nAssumptions: \n
Step 1. If \n
Step 2. Find the step size \n
else find the step size \n
Step 3. Set \n
Step 4. Set \n
The classical and the oldest steepest descent step size \n
where \n
More information about the convergence of the
Although known as the first unconstrained optimization method, this method is still a theme considered by scientists.
\nDifferent modifications of this method are made, for example, see [25, 28, 29, 30, 31, 32].
\nIn [28], the authors presented a new search direction from Cauchy’s method in the form of two parameters known as
So, in [28], a new modification of \n
In [25], a new scaled search direction of \n
The method proposed in [25] is known as
where \n
Further, in [25], a comparison among \n
It is interesting that the exact line search is used in [25].
\nIn [34], the properties of steepest descent method from the literature are reviewed together with advantages and disadvantages of each step size procedure.
\nNamely, the step size procedures, which are compared in this paper, are:
\n1. \n
2. Given \n
3. Given \n
4. \n
5. \n
The comparison is based on time execution, number of total iteration, total percentage of function, gradient and Hessian evaluation, and the most decreased value of objective function obtained.
\nFrom the numerical results, the authors conclude that the \n
Further, in [34], the general conclusions about the steepest descent method are given:
This method is sensitive to the initial point.
This method has a descent property, and it is a logical starting procedure for all gradient based methods.
\n\n
In [35], in the aim to achieve fast convergence and the monotone property, a new step size for the steepest descent method is suggested.
\nIn [36], for quadratic positive definite problems, an over-relaxation has been considered. Namely, Raydan and Svaiter [36] proved that the poor behavior of the steepest descent method is due to the optimal Cauchy choice of step size and not to the choice of the search direction. These results are extended in [29] to convex, well-conditioned functions. Further, in [29], it is shown that a simple modification of the step length by means of a random variable uniformly distributed in \n
where \n
In the recent years, the steepest descent method has been applied in many branches of science; one can be inspired, for example, by [37, 38, 39, 40, 41, 42, 43].
\nRemind to the fact that \n
Also, remind to the fact that this poor behavior of \n
Barzilai and Borwein presented [44] a two-point step size gradient method, which is well known as \n
The step size is derived from a two-point approximation to the secant equation.
\nConsider the gradient iteration form:
\nIt can be rewritten as \n
To make the matrix \n
This yields that
\nBut, using symmetry, we may minimize \n
Now, we give the algorithm of \n
Assumptions: \n
Step 1. If \n
Step 2. If \n
Step 3. Set \n
Step 4. Set \n
Considering Algorithm 1.2.5, we can conclude that this method does not require any matrix computation or any line search.
\nThe Barzilai-Borwein method is in fact the gradient method, which requires less computational work than \n
In the general non-quadratic case, a globalization strategy based on non-monotone line search is applied in this method.
\nIn this general case, \n
Using the iteration
\nwith
\nwe get
\nNow, we give the algorithm of the Barzilai-Borwein method with non-monotone line search.
\nAssumptions: \n
Step 1. If \n
Step 2. If \n
Step 3. Set \n
Step 4. (non-monotone line search) If
\nthen set
\nand go to Step 6.
\nStep 5. Choose \n
Step 6. Set \n
Obviously, the above algorithm is globally convergent.
\nSeveral authors paid attention to the Barzilai-Borwein method, and they proposed some variants of this method.
\nIn [8], the globally convergent Barzilai-Borwein method is proposed by using non-monotone line search by Grippo et al. [12]. In the same paper, Raydan proves the global convergence of the non-monotone Barzilai-Borwein method.
\nFurther, Grippo and Sciandrone [45] propose another type of the non-monotone Barzilai-Borwein method.
\nDai [7] gives the basic analysis of the non-monotone line search strategy.
\nMoreover, in [46] numerical results are presented, using
\nand
\nwhere for \n
Many researchers study the gradient method for minimizing a strictly convex quadratic function, namely,
\nwhere \n
and \n
In [50], the authors extend the Barzilai-Borwein method, and they give
and
\nwhere \n
Also, an application of algorithm \n
Following Raydan [8], the authors [50] further combine the non-monotone line search and algorithm \n
The Barzilai-Borwein method and its related methods are reviewed by Dai and Yuan [51] and Fletcher [52].
\nIn [53], a new concept of the approximate optimal step size for gradient method is introduced and used to interpret the \n
The approximate optimal step size is different from the steepest descent step size, which will lead to the expensive computational cost. The approximate optimal step size is generally calculated easily, and it can be applied to unconstrained optimization.
\nDue to the effectiveness of \n
This is the purpose of work [53]. Further, if the objective function \n
If \n
If \n
In [54], derivative-free iterative scheme that uses the residual vector as search direction for solving large-scale systems of nonlinear monotone equations is presented.
\nThe Barzilai-Borwein method is widely used; some interesting results can be found in [55, 56, 57].
\nThe basic idea of Newton method for unconstrained optimization is the iterative usage of the quadratic approximation \n
Let \n
We model \n
Minimization of \n
which is known as Newton formula.
\nDenote \n
Then, we have a simpler form:
\nA
We have supposed that \n
Now, we give the algorithm of the Newton method.
\nAssumptions: \n
Step 1. If \n
Step 2. Solve \n
Step 3. Set \n
Step 4. \n
The next theorem shows the local convergence and the quadratic convergence rate of Newton method.
\nwhere \n
But, in spite of this quadratic rate, the Newton method is a local method: when the starting point is far away from the solution, there is a possibility that \n
So, to guarantee the global convergence, we can use Newton method with line search. We can remind to the fact that only when the step size sequence \n
Newton iteration with line search is as follows:
\nNow, we give the algorithm.
\nAssumptions: \n
Step 1. If \n
Step 2. Solve \n
Step 3. Line search step: find \n
or find \n
Step 4. Set \n
The next theorems claim that Algorithm 1.2.8 with the exact line search, as well as Algorithm 1.2.8 with the inexact line search, are globally convergent.
\nwhere \n
When \n
When \n
Note that the next relation holds from the standard Wolfe line search:
\nwhere the constant \n
and \n
The main problem in Newton method could be the fact that the Hessian \n
So, many modified schemes are made. Now, we describe the next two methods shortly.
\nIn [58], Goldstein and Price use the steepest descent method when \n
where \n
In [59], the authors present another modified Newton method. When \n
To consider the other modified Newton methods, such as finite difference Newton method, negative curvature direction method, Gill-Murray stable Newton method, etc., one can see [27], for example.
\nBy the other side, because of the high cost of the exact Newton method, especially when the dimension \n
Consider solving the nonlinear equations:
\nwhere \n
Remind that the basic Newton step is obtained by solving
\nand setting
\nThe inexact Newton method means that we solve
\nwhere
\nSet
\nHere, \n
Now, we give two theorems; the first of them claims the linear convergence, and the second claims the superlinear convergence of the inexact Newton method.
\nThe relation
\npresents the secant method.
\nIn [60], a modification of the classical secant method for solving nonlinear, univariate, and unconstrained optimization problems based on the development of the cubic approximation is presented. The iteration formula including an approximation of the third derivative of \n
This modified secant method is constructed in [60], having in view, as it is emphasized, that it is possible to construct a cubic function which agrees with \n
In [61], the authors propose an inexact Newton-like conditional gradient method for solving constrained systems of nonlinear equations. The local convergence of the new method as well as results on its rate is established by using a general majorant condition.
\nConsider the Newton method.
\nFor various practical problems, the computation of Hessian may be very expensive, or difficult, or Hessian can be unavailable analytically. So, the class of so-called quasi-Newton methods is formed, such that it uses only the objective function values and the gradients of the objective function and it is close to Newton method. Quasi-Newton method is such a class of methods which does not compute Hessian, but it generates a sequence of Hessian approximations and maintains a fast rate of convergence.
\nSo, we would like to construct Hessian approximation \n
Now, let \n
Finding the derivatives, we get
\nSetting \n
Relation (41) transforms into the next one if \n
Let \n
Let \n
is also the quasi-Newton equation.
\nIf
\nthen the matrix \n
Assumptions: \n
Step 1. If \n
Step 2. Compute \n
Step 3. Find \n
Step 4. Update \n
Step 5. Set \n
In Algorithm 1.2.9, usually we take \n
Sometimes, instead of \n
Then,
By the other side,
Let \n
where \n
where \n
wherefrom
\nFurther, from (46) and (47), we have
\nHaving in view that the inverse Hessian approximation \n
More information about SR1 update can be found.
\nThere exists another type of update, which is a rank-two update. In fact, we get \n
where \n
Using quasi-Newton equation (43), we can get
\nThe values of \n
Now, from (50), we get:
\nHence, we get the formula
\nwhich is \n
\n\n
\n\n
\n\n
\n\n
The \n
In [62], an adaptive scaled \n
where \n
Namely, in [70], the next function was introduced:
\nwhich is defined on positive definite matrices.
\nThis function is a measure of matrices involving all the eigenvalues of \n
Observe that function \n
Further, in [62], a double-parameter scaling \n
where \n
In [62], the next proposition is proved.
\nFrom (55), it can be seen that \n
Further, the next values of the scaling parameters \n
Consider the relation
\nwhere \n
where the matrix \n
The next theorems are also given in [62].
\nNext, in [72], using chain rule, a modified secant equation is given, to get a more accurate approximation of the second curvature of the objective function. Then, based on this modified secant equation, a new \n
Some interesting applications of Newton, modified Newton, inexact Newton, and quasi-Newton methods can be found, for example, in [73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83], etc.
\nA very interesting paper is [84].
\nAn interesting application of \n
Today, the modifications of the line search techniques are very actual and all in the aim to create new, better optimization methods.
\nFurther, following recent trends in unconstrained optimization, we can notice that almost all optimization methods, which are considered in this chapter, are still actual.
\nThey are applied in the other areas of Mathematics, as well as in practice. Also, different modifications of these methods are made, in the aim to improve them.
\nLet us emphasize that \n
Our business values are based on those any scientist applies to their research. The values of our business are based on the same ones that all good scientists apply to their research. We have created a culture of respect and collaboration within a relaxed, friendly, and progressive atmosphere, while maintaining academic rigour.
\n\nPlease check out our job board for open positions.
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