[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-127999-en":3,"doc-seo-127999-105":30,"detail-sidebar-cat-0-en-105":84},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":21,"language":22,"language_code":23,"site_id":24,"html_lang":23,"table_of_contents":25,"faqs":26,"seo_title":27,"seo_description":14,"update_tm":28,"read_time":29},127999,962084928904,"Asher","https://ap-avatar.wpscdn.com/davatar_155a257f0dc6eb9ab79c44ca47cae57d",8,"Research & Report","Modified Least Squares Method and a Review of Its Applications in Machine Learning and Fractional Differential/Integral Equations - paper","The modified least squares method is developed using fractional orthogonal polynomials from the M_n := span{1, x^alpha, x^{2alpha}, ..., x^{nalpha}} setting with alpha in (0, 2]. The approach is evaluated through numerical experiments on multiple problem classes, showing how to construct solutions and fit unknown parameters. Results indicate clear advantages over classical least squares. The work also reviews applications in fractional differential/integral equations and machine learning tasks such as regression analysis and classification.","arXiv :2405 .00382v1 [math .NA] 1 May 2024  \nModi􀀌ed least squares method and a review of its applications in  \nmachine learning and fractional di􀀋erential/integral equations Abhishek Kumar Singh 1,2 , Mani Mehra∗1, and Anatoly A. Alikhanov3  \n1 Deptartment of Mathematics, Indian Institute of Technology Delhi, India  \n2 Institute of Mathematics and Computer Science, Universit¨at Greifswald,  \nWalther-Rathenau-Stra􀀙e 47, 17489 Greifswald, Germany  \n3 North-Caucasus Center for Mathematical Research, North-Caucasus Federal University,  \nRussia  \n[assinghabhi@gmail. com](assinghabhi@gmail. com), [mmehra@maths. iitd. ac. in](mmehra@maths. iitd. ac. in), [aaalikhanov@gmail. com](aaalikhanov@gmail. com)  \nMay 2, 2024  \nAbstract  \nThe least squares method provides the best-􀀌t curve by minimizing the total squares error. In this work, we provide the modi􀀌ed least squares method based on the fractional orthogonal polynomials that belong to the space M􀀕n := span{1, x􀀕 , x2􀀕 , ... , xn􀀕 }, 􀀕 ∈ (0, 2] . Numerical experiments demonstrate how to solve di􀀋erent problems using the modi􀀌ed least squares method. Moreover, the results show the advantage of the modi􀀌ed least squares method compared to the classical least squares method. Furthermore, we discuss the various applications of the modi􀀌ed least squares method in the 􀀌elds like fractional di􀀋erential/integral equations and machine learning.  \nKeywords. Modi􀀌ed least squares method; M¨untz-Legendre polynomials; Machine learning; Fractional di􀀋erential/integral equations.  \n1 Introduction  \nThe least squares method is one of the oldest methods of modern statistics used to obtain the physical parameters from the experimental data. The 􀀌rst use of the least squares method is generally attributed to Gauss in 1795, although Legendre concurrently and independently used it [12] . Gauss invented the least squares method to estimate planets’ orbital motion from telescopic measurements. In modern statistics, Galton [2] was the 􀀌rst to use the least squares method in his work on the heritability of size, which laid down the foundations of correlation and regression analysis. Nowadays, the least squares method is widely used to 􀀌nd the best-􀀌t curve while 􀀌nding the parameter involved in the curve. There are many versions of the least squares method available in the literature. The simpler version is called the ordinary least squares method, and the more advanced one is the weighted least squares method, which performs better than the ordinary least squares method. The recent version of the least squares method is the moving least squares method [6], and the partial least squares method [27] .  \nOne of the areas where the least squares method is frequently used is machine learning, where we analyze data for regression analysis and classi􀀌cation [24] . Machine learning is a 􀀌eld of arti􀀌cial intelligence that allows computer systems to learn using available data. Recently machine learning algorithms (regression analysis and classi􀀌cation) have become very popular for analyzing data and making predictions. Another application of the least squares method is solving fractional di􀀋erential/integral equations. Fractional di􀀋erential/integral equations give an excellent way to deal with complex phenomena in nature, such as biological systems, control theory, 􀀌nance, signal and image processing, sub-di􀀋usion and  \n∗ Corresponding author  \nsuper-di􀀋usion process, viscoelastic 􀀍uid, electrochemical processes, and so on [3, 13 , 28 , 26 , 18] . The fractional di􀀋erential equations are equivalent to the Hammerstein form of Volterra’s second kind integral equations for the speci􀀌c choice of kernel (for more details see [9]) . Due to the importance of fractional di􀀋erential/integral equations, people are interested in solving them numerically because of the non-availability of exact solutions. Many numerical methods are available in the literature to solve fractional di􀀋erential/integral equations, such as 􀀌nite d","cbCaicW7ooYPSx5S","https://ap.wps.com/l/cbCaicW7ooYPSx5S","pdf",264555,1,19,"English","en",105,"# Abstract\n# Introduction\n# Preliminaries\n## Muntz space and Muntz-Szsz theorem\n# Modified least squares method\n# Numerical results\n# Applications\n# Conclusion","[{\"question\":\"Where are the applications of the modified least squares method discussed?\",\"answer\":\"The applications are reviewed mainly in fractional differential/integral equations and in machine learning, including regression analysis and classification.\"}]","Modified Least Squares Method and a Review of Its Applications in Machine Learning and Fractional Differential/Integral Equations - paper | PDF",1785943743,48,{"code":4,"msg":31,"data":32},"ok",{"site_id":24,"language":23,"slug":33,"title":13,"keywords":34,"description":14,"schema_data":35,"social_meta":79,"head_meta":81,"extra_data":83,"updated_unix":28},"modified-least-squares-method-and-a-review-of-its-applications-in-machine-learning-and-fractional-differentialintegral-equations-paper","",{"@graph":36,"@context":78},[37,54,69],{"@type":38,"itemListElement":39},"BreadcrumbList",[40,44,48,51],{"item":41,"name":42,"@type":43,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":45,"name":46,"@type":43,"position":47},"https://docshare.wps.com/document/","Document",2,{"item":49,"name":12,"@type":43,"position":50},"https://docshare.wps.com/document/research-report/",3,{"item":52,"name":13,"@type":43,"position":53},"https://docshare.wps.com/document/modified-least-squares-method-and-a-review-of-its-applications-in-machine-learning-and-fractional-differentialintegral-equations-paper/127999/",4,{"url":52,"name":13,"@type":55,"author":56,"headline":13,"publisher":58,"fileFormat":61,"inLanguage":23,"description":14,"dateModified":62,"datePublished":63,"encodingFormat":61,"isAccessibleForFree":64,"interactionStatistic":65},"DigitalDocument",{"name":9,"@type":57},"Person",{"url":41,"name":59,"@type":60},"DocShare","Organization","application/pdf","2026-08-22","2026-08-05",true,{"@type":66,"interactionType":67,"userInteractionCount":20},"InteractionCounter",{"@type":68},"ViewAction",{"@type":70,"mainEntity":71},"FAQPage",[72],{"name":73,"@type":74,"acceptedAnswer":75},"Where are the applications of the modified least squares method discussed?","Question",{"text":76,"@type":77},"The applications are reviewed mainly in fractional differential/integral equations and in machine learning, including regression analysis and classification.","Answer","https://schema.org",{"og:url":52,"og:type":80,"og:title":13,"og:site_name":59,"og:description":14},"article",{"robots":82,"canonical":52},"index,follow",{"doc_id":7,"site_id":24},{"code":4,"msg":5,"data":85},[86,90,94,98,103,108,113,116,121,124,128],{"id":20,"doc_module":4,"doc_module_name":46,"category_name":87,"show_sort_weight":88,"slug":89},"Story & Novel",90,"story-novel",{"id":47,"doc_module":4,"doc_module_name":46,"category_name":91,"show_sort_weight":92,"slug":93},"Literature",80,"literature",{"id":53,"doc_module":4,"doc_module_name":46,"category_name":95,"show_sort_weight":96,"slug":97},"Exam",70,"exam",{"id":99,"doc_module":4,"doc_module_name":46,"category_name":100,"show_sort_weight":101,"slug":102},5,"Comic",60,"comic",{"id":104,"doc_module":4,"doc_module_name":46,"category_name":105,"show_sort_weight":106,"slug":107},6,"Technology",50,"technology",{"id":109,"doc_module":4,"doc_module_name":46,"category_name":110,"show_sort_weight":111,"slug":112},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":46,"category_name":12,"show_sort_weight":114,"slug":115},30,"research-report",{"id":117,"doc_module":4,"doc_module_name":46,"category_name":118,"show_sort_weight":119,"slug":120},9,"Religion & Spirituality",20,"religion-spirituality",{"id":119,"doc_module":4,"doc_module_name":46,"category_name":122,"show_sort_weight":119,"slug":123},"World Cup","world-cup",{"id":125,"doc_module":4,"doc_module_name":46,"category_name":126,"show_sort_weight":125,"slug":127},10,"Lifestyle","lifestyle",{"id":21,"doc_module":4,"doc_module_name":46,"category_name":129,"show_sort_weight":99,"slug":130},"General","general"]