最小二乘法的應(yīng)用研究最小二乘法的應(yīng)用研究摘要最小二乘法是從誤差擬合角度對(duì)回歸模型進(jìn)行參數(shù)估計(jì)或系統(tǒng)辨識(shí),并在參數(shù)估計(jì)、系統(tǒng)辨識(shí)以及預(yù)測(cè)、預(yù)報(bào)等眾多領(lǐng)域中得到極為廣泛的應(yīng)用.然而,最小二乘法因其抽象、難懂常常不能被準(zhǔn)確理解.本文探討了最小二乘法的基本原理及其各種變形的擬合方法,其中包括:一元線性最小二乘法擬合、多元線性擬合、多項(xiàng)式擬合、非線性擬合,并且討論了用鏡像映射和切比雪夫多項(xiàng)式解“病態(tài)”矛盾方程組的基本原理和方法,在此基礎(chǔ)上給出了幾種最小二乘法程序的設(shè)計(jì)原理.關(guān)鍵詞：最小二乘法,線性擬合,曲線擬合,切比雪夫多項(xiàng)式StudyontheApplicationaboutMethodofLeastSquareAbstractLeastsquarewasusedtoestimateparametersandidentifysystemofregressionmodel,bythepointoferrorfitting.Andithaswidelyapplicationintheparametersestimate,systemidentification,prediction,forecastingandotherfields.However,theleastsquaremethodbecauseofitsabstractanddifficult，oftencannotbeaccuratelyunderstanding.Theleastsquaremethodsprincipleandthevariouskindsoffittingmethodssuchasthelinearleastsquarefitting,multiplelinearfitting,polynomialfittinganonlinearfittingaredealtwith.AnddiscussedusingmirrorandChebyshevpolynomialsolutionpathologicalcontradictoryequationsbasicprinciplesandmethods.Finallysomekindsoftheprincipleoftheprogramsontheleastsquaremethodaregiven.KeyWords：leastsquaremethod,linearfitting,curvefitting,Chebyshevpolynomial目錄一、最小二乘法的統(tǒng)計(jì)學(xué)原理1二、曲線擬合21.一元線性擬合22.多元線性擬合43.多項(xiàng)式擬合54.非線性最小二乘法擬合65.多項(xiàng)式回歸的高精度快速算法7三、應(yīng)用最小二乘法的幾個(gè)問(wèn)題9四、程序設(shè)計(jì)原理101.線性擬合程序的設(shè)計(jì)原理102.多元線性擬合程序的設(shè)計(jì)原理103.Shehata方程12ksksuksks的擬合程序設(shè)計(jì)原理11結(jié)束語(yǔ)11參考文獻(xiàn)121一、最小二乘法的統(tǒng)計(jì)學(xué)原理1基本最小二乘法,其統(tǒng)計(jì)學(xué)原理是:設(shè)物理量y與l個(gè)變量12,lxxx間的依賴關(guān)系式為1201(,)lnyfxxxaaa,其中01,naaa是方程中需要確定的1n個(gè)參數(shù).最小二乘法就是通過(guò)1mmn個(gè)實(shí)驗(yàn)點(diǎn)12(,)(1,2,)iiilixxxyim,確定出一組參數(shù)值01(,)naaa,使由這組參數(shù)得出的函數(shù)值1201=(,)iiilnyfxxxaaa與實(shí)驗(yàn)值iy間的偏差平方和2011(,)()mniisaaayy取得極小值.在設(shè)計(jì)實(shí)驗(yàn)時(shí),為了減小隨機(jī)誤差,一般進(jìn)行多點(diǎn)測(cè)量,使方程式個(gè)數(shù)大于待求參數(shù)的個(gè)數(shù),即1mn.這時(shí)構(gòu)成的方程組叫做矛盾方程組.通過(guò)用最小二乘法進(jìn)行統(tǒng)計(jì)處理,將矛盾方程組轉(zhuǎn)換成未知數(shù)個(gè)數(shù)和方程個(gè)數(shù)相等的正規(guī)方程組,再進(jìn)行求解得出01,naaa.由微分學(xué)的求極值方法可知01,naaa應(yīng)滿足下列方程組:0iya(1,2,)in,這樣就實(shí)現(xiàn)矛盾方程組向正規(guī)方程組的轉(zhuǎn)換.2二、曲線擬合1.一元線性擬合2設(shè)變量y與x成線性關(guān)系,即01yaax.現(xiàn)在已知m個(gè)實(shí)驗(yàn)點(diǎn),iixy(1,2,)im,求兩個(gè)未知參數(shù)01,aa.方法一由最小二乘法原理,參數(shù)01,aa應(yīng)使201011(,)()miiisaayaax取得極小值.根據(jù)極小值的求法,0a和1a應(yīng)滿足011001112()02()0miiimiiiisyaaxasyaaxxa,10112011111mmiiiimmmiiiiiiiaaxymmaxaxxy,這就是含有兩個(gè)未知數(shù)和兩個(gè)方程的正規(guī)方程組.從中解得01,aa,即2211101()/()mmiiiiiaxymxyxmxayax(1)其中1111,mmiiiixxyymm,線性相關(guān)系數(shù)/xyxxyyRlll,式中32222111,mmmxyiixxiyyiiiilxymxylxmxlymy,相關(guān)系數(shù)是用來(lái)衡量實(shí)驗(yàn)點(diǎn)的線性特性.方法二將,(1,2,)iixyim代入01yaax得矛盾方程組1011201201mmyaaxyaaxyaax(2)令12111mxxAx,12myyBy,則（2）式可寫成01aBAa,則有01TTaABAAa,所以011()TTaAAABa.其中A稱為結(jié)構(gòu)矩陣,B稱為數(shù)據(jù)矩陣