In1801,ItalianastronomerGiuseppePiazzidiscoveredthefirstasteroidCeres.After40daysoftracking,PiazzilostthepositionofCeresbecauseCeresmovedtothebackofthesun.ThenscientistsallovertheworldusedPiazzi'sobservationaldatatostartsearchingforCeres,butsearchingforCeresbasedontheresultsofmostpeople'scalculationshasnoresults.TheorbitofCerescalculatedbyGauss,whowasonly24yearsold,wasconfirmedbytheobservationsoftheAustrianastronomerHeinrichAlbers,allowingtheastronomicalcommunitytopredicttheprecisepositionofCeres.ThesamemethodalsoproducedmanyastronomicalresultssuchasHalley'sComet.ThemethodusedbyGaussistheleastsquaresmethod,whichwaspublishedinhisbook"OntheMovementofCelestialBodies"in1809.Infact,theFrenchscientistLegendreindependentlyinventedthe"leastsquaresmethod"in1806,butitwasunknownbecauseitwasunknowntotheworld.
In1829,Gaussprovidedproofthattheoptimizationeffectoftheleastsquaresmethodisstrongerthanothermethods.
Определение
Theleastsquaremethod(alsoknownastheleastsquaremethod)isamathematicaloptimizationtechnique.Itfindsthebestfunctionmatchofthedatabyminimizingthesumofsquaresoftheerror.Theleastsquaresmethodcanbeusedtoeasilyobtainunknowndataandminimizethesumofsquarederrorsbetweentheobtaineddataandtheactualdata.
Theleastsquaremethodcanalsobeusedforcurvefitting,andsomeotheroptimizationproblemscanalsobeexpressedbytheleastsquaremethodbyminimizingenergyormaximizingentropy.
Basicidea
Theleastsquaremethodisthemostcommonlyusedmethodtosolvecurvefittingproblems.Thebasicideais:Let
Amongthem,isasetoflinearlyindependentfunctionsselectedinadvance,istheundeterminedcoefficient,andthefittingcriterionistominimizethesumofsquaresofthedistancebetweenand,Calledtheleastsquarescriterion.
Основен принцип
Да предположим, че (x,y) е двойка наблюдения и отговаря на следните теоретични функции:
където е параметър за определяне.
Inordertofindtheoptimalestimatedvalueoftheparameterofthefunction,foragivengroup(usuallysection>)Observationdata,solvetheobjectivefunction
Taketheminimumparameter.Thistypeofproblemtobesolvediscalledaleastsquaresproblem,andthegeometriclanguageofthemethodtosolvethisproblemiscalledleastsquaresfitting.
Forunconstrainedoptimizationproblems,thegeneralformoftheleastsquaresmethodis:
whereCalledtheresidualfunction.Whenisalinearfunctionof,itiscalledalinearleastsquaresproblem,otherwiseitiscalledanonlinearleastsquaresproblem.
Проблем с оптимизацията на най-малките квадрати
Inunconstrainedoptimizationproblems,therearesomeimportantspecialcases,suchastheobjectivefunctionconsistingofthesumofthesquaresofseveralfunctions.ThistypeoffunctioncangenerallybeWrittenas:
Сред тях обикновено се изисква ≥n. Минимизираме проблема на този тип функция:
Itiscalledtheleastsquaresoptimizationproblem.Leastsquaresoptimizationisaspecialkindofoptimizationproblem.
Оценител на характеристиките на най-малките квадрати
Accordingtothesampledata,theleastsquaresestimatorcanbeusedtoobtaintheestimatorofthesimplelinearregressionmodelparameters.Buthowclosetheestimatorparameteristotheoveralltrueparameter?Whetherthereareotherbetterestimationformulas?Thisinvolvestheleastsquaresestimationformulaortheminimumvariance(orbest)(Best)oftheestimator,linearity(Linear)Andunbiased(Unbiased),referredtoasBLUcharacteristics.Thisisthemainreasonforthewidespreaduseofordinaryleastsquarestoestimateeconometricmodels.Thefollowingprovesthattheordinaryleastsquaresestimatorhastheabovethreecharacteristics.
1. Линейна характеристика
Theso-calledlinearcharacteristicmeansthattheestimatoristhelinearfunctionofthesampleobservationvalue,thatis,thelinearcombinationoftheestimatorandtheobservationvalue.
2. Безпристрастност
Unbiasednessmeansthattheexpectedvaluesofparameterestimatorsareequaltotheoveralltrueparameters.
3. Свойство на минимална вариация
Theso-calledminimumvariancepropertyreferstotheminimumvarianceoftheestimatorcomparedwiththeestimatorobtainedbyothermethods,thatis,thebest.Theminimumvarianceisalsocalledeffectiveness.ThispropertyisthefamousGauss-Markov(Gauss-Markov)theorem.Thistheoremstatesthattheordinaryleastsquaresestimatoristhebestcomparedwithanylinearunbiasedestimatorobtainedbyothermethods.