En k\u00fc\u00e7\u00fck kareler y\u00f6ntemi, birbirine ba\u011fl\u0131 olarak de\u011fi\u015fen iki fiziksel b\u00fcy\u00fckl\u00fck aras\u0131ndaki matematiksel ba\u011flant\u0131y\u0131, m\u00fcmk\u00fcn oldu\u011funca ger\u00e7e\u011fe uygun bir denklem olarak yazmak i\u00e7in kullan\u0131lan, standart bir regresyon y\u00f6ntemidir. Bir ba\u015fka deyi\u015fle bu y\u00f6ntem, \u00f6l\u00e7\u00fcm sonucu elde edilmi\u015f veri noktalar\u0131na “m\u00fcmk\u00fcn oldu\u011fu kadar yak\u0131n” ge\u00e7ecek bir fonksiyon e\u011frisi bulmaya yarar. Gauss-Markov Teoremi’ne g\u00f6re en k\u00fc\u00e7\u00fck kareler y\u00f6ntemi, regresyon i\u00e7in optimal y\u00f6ntemdir.<\/p>\n
Tarihi
\n Bilindi\u011fi kadar\u0131yla, en k\u00fc\u00e7\u00fck kareler y\u00f6ntemi ilk olarak 1795’te Carl Friedrich Gauss taraf\u0131ndan geli\u015ftirilmi\u015ftir. Gauss 1801 y\u0131l\u0131nda bu y\u00f6ntemi kullanarak, ke\u015ffinden k\u0131sa s\u00fcre sonra kaybedilen Ceres asteroidinin tekrar g\u00f6zlemlenebilece\u011fi pozisyonu hesaplayabilmi\u015f, bu ba\u015far\u0131s\u0131yla b\u00fcy\u00fck \u00fcne kavu\u015fmu\u015ftur. Gauss bu y\u00f6ntemi ilk olarak 1809’da yay\u0131mlam\u0131\u015ft\u0131r. 1806’da Frans\u0131z matematik\u00e7i Adrien-Marie Legendre ve 1808’de Amerikal\u0131 matematik\u00e7i Robert Adrain, Gauss’tan (ve muhtemelen birbirlerinden) ba\u011f\u0131ms\u0131z olarak bu y\u00f6ntemi geli\u015ftirip kullanm\u0131\u015flard\u0131r.
\n En k\u00fc\u00e7\u00fck kareler y\u00f6ntemi, bug\u00fcn neredeyse t\u00fcm bilim dallar\u0131nda ve m\u00fchendislikte yayg\u0131n olarak kullan\u0131lmaktad\u0131r.<\/p>\n
\u00c7izgisel (Do\u011frusal) \u00d6rnek<\/p>\n
K\u0131rm\u0131z\u0131 noktalar \u00f6l\u00e7\u00fcmle elde edilmi\u015f veri noktalar\u0131n\u0131, mavi \u00e7izgi ise en k\u00fc\u00e7\u00fck kareler y\u00f6ntemi ile bulunmu\u015f teorik ba\u011flant\u0131y\u0131 ifade eder.
\nBasit bir \u00f6rnek vermek gerekirse, aralar\u0131nda \u00e7izgisel (lineer) bir ba\u011flant\u0131 olan, X ve Y ad\u0131nda iki fiziksel b\u00fcy\u00fckl\u00fck d\u00fc\u015f\u00fcnelim. (Mesela, X belli bir a\u011fa\u00e7 t\u00fcr\u00fcn\u00fcn ya\u015f\u0131, Y ayn\u0131 t\u00fcr a\u011fac\u0131n g\u00f6vde \u00e7ap\u0131 olabilir.) Y ‘yi X ‘in fonksiyonu olarak yazmak istiyoruz. Bu iki b\u00fcy\u00fckl\u00fck aras\u0131ndaki ba\u011flant\u0131 \u00e7izgisel oldu\u011funa g\u00f6re, \u015f\u00f6yle bir denklem halinde ifade edilebilir:<\/p>\n
Bizim arad\u0131\u011f\u0131m\u0131z \u015fey, bu denklemdeki a ve b say\u0131lar\u0131 i\u00e7in m\u00fcmk\u00fcn olan en do\u011fru de\u011ferlerdir. Bu de\u011ferleri belirlemek i\u00e7in bir dizi \u00f6l\u00e7\u00fcm yapt\u0131\u011f\u0131m\u0131z\u0131 d\u00fc\u015f\u00fcnelim. (A\u011fa\u00e7 \u00f6rne\u011fine d\u00f6nersek, ilgilendi\u011fimiz t\u00fcrden pek \u00e7ok a\u011fac\u0131n ya\u015f\u0131n\u0131 ve g\u00f6vde \u00e7ap\u0131n\u0131 \u00f6l\u00e7elim.) Bu \u00f6l\u00e7\u00fcmler bize bir dizi (Xi, Yi) \u00e7ifti verecektir. Bir kartezyen d\u00fczlem \u00fczerinde bu \u00e7iftlere kar\u015f\u0131l\u0131k gelen noktalar\u0131 tek tek i\u015faretlersek, kabaca d\u00fcz bir \u00e7izgi \u00fczerinde yay\u0131lm\u0131\u015f bir “noktalar bulutu” elde ederiz. Noktalar, \u00e7e\u015fitli sebeplerden dolay\u0131 (\u00f6l\u00e7\u00fcm hatalar\u0131, istisnai durumlar, modele kat\u0131lmayan d\u0131\u015f etkiler, vs) kusursuz bir \u00e7izgi \u00fczerinde \u00e7\u0131kmayacakt\u0131r.
\nX ve Y aras\u0131ndaki ba\u011flant\u0131y\u0131 tek bir \u00e7izgisel denklem olarak ifade etmek istiyorsak, bu noktalara m\u00fcmk\u00fcn oldu\u011funca yak\u0131n ge\u00e7ecek bir \u00e7izgi bulmal\u0131y\u0131z. Bir ba\u015fka deyi\u015fle, yukar\u0131daki denklemde a ve b’yi \u00f6yle se\u00e7meliyiz ki, ortaya \u00e7\u0131kan \u00e7izgi veri noktalar\u0131na m\u00fcmk\u00fcn oldu\u011funca yak\u0131n olsun.
\n En k\u00fc\u00e7\u00fck kareler y\u00f6ntemi, denklemin verdi\u011fi (teorik) Y de\u011ferleri ile \u00f6l\u00e7\u00fcmlerin verdi\u011fi (ger\u00e7ek) Y de\u011ferleri aras\u0131ndaki farklar\u0131n karelerinin toplam\u0131n\u0131 k\u00fc\u00e7\u00fcltme fikrine dayan\u0131r. Bu y\u00f6ntem, denklemdeki a ve b say\u0131lar\u0131n\u0131, bahsedilen kareler toplam\u0131n\u0131 en k\u00fc\u00e7\u00fck yapacak \u015fekilde se\u00e7er (ve ad\u0131n\u0131 da buradan al\u0131r).<\/p>\n
E\u011frisel (Do\u011frusal Olmayan) \u00d6rnek
\n Aralar\u0131nda do\u011frusal olmayan (non-lineer) bir ba\u011flant\u0131 olan, fiziksel b\u00fcy\u00fckl\u00fckler i\u00e7in de benzer \u015fekilde en k\u00fc\u00e7\u00fck kareler (EKK) y\u00f6ntemi kullan\u0131labilir.
\n EKK y\u00f6ntemi denklem formunun bilinmesini gerektir. Bu formda ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin \u00fcsleri ile birlikte birden \u00e7ok ba\u011f\u0131ms\u0131z de\u011fi\u015fkenin \u00e7e\u015fitli bi\u00e7imleri bulunabilir.<\/p>\n
EKK’n\u0131n i\u015fe yaramas\u0131 i\u00e7in de\u011fi\u015fkenler aras\u0131ndaki ili\u015fkiyi g\u00f6steren formun katsay\u0131lardan ba\u011f\u0131ms\u0131z olarak biliniyor olmas\u0131 gerekir. Bunun i\u00e7in Ekonometri biliminde \u00e7ok \u00e7e\u015fitli y\u00f6ntemler mevcuttur.
\n Formun nas\u0131l olaca\u011f\u0131na karar verdikten sonra katsay\u0131lar bulunur. T\u00fcm \u00f6rnek sonu\u00e7lar\u0131na bak\u0131larak hata terimlerinin karelerini en d\u00fc\u015f\u00fck yapan katsay\u0131lar t\u00fcrev yard\u0131m\u0131yla bulunur. Burada t\u00fcrevin s\u0131f\u0131r oldu\u011fu noktan\u0131n en k\u00fc\u00e7\u00fck de\u011fer olmas\u0131 kural\u0131ndan faydalan\u0131l\u0131r.<\/p>\n","protected":false},"excerpt":{"rendered":"
En k\u00fc\u00e7\u00fck kareler y\u00f6ntemi, birbirine ba\u011fl\u0131 olarak de\u011fi\u015fen iki fiziksel b\u00fcy\u00fckl\u00fck aras\u0131ndaki matematiksel ba\u011flant\u0131y\u0131, m\u00fcmk\u00fcn oldu\u011funca ger\u00e7e\u011fe uygun bir denklem olarak yazmak i\u00e7in kullan\u0131lan, standart bir regresyon y\u00f6ntemidir. Bir ba\u015fka deyi\u015fle bu y\u00f6ntem, \u00f6l\u00e7\u00fcm sonucu elde edilmi\u015f veri noktalar\u0131na “m\u00fcmk\u00fcn oldu\u011fu kadar yak\u0131n” ge\u00e7ecek bir fonksiyon e\u011frisi bulmaya yarar. Gauss-Markov Teoremi’ne g\u00f6re en k\u00fc\u00e7\u00fck kareler …<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[1404,1403],"tags":[7480],"class_list":["post-3236","post","type-post","status-publish","format-standard","hentry","category-matematik-odevleri","category-odevler","tag-en-kucuk-kareler-yontemi"],"_links":{"self":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3236","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/comments?post=3236"}],"version-history":[{"count":0,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/posts\/3236\/revisions"}],"wp:attachment":[{"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/media?parent=3236"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/categories?post=3236"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.islamidavet.com\/kutuphane\/wp-json\/wp\/v2\/tags?post=3236"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}