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1、[lectureNOTES]DongYuandLiDengSolvingNonlinearEstimationProblemsUsingSplinesedescribetheusealgorithmsbyfindingadirectrelation-withtheconstraintthatthefunctionisofsplinesforsolv-shipbetweentheinterpolatedvaluesandcontinuousinthefirst-andsecond-orderingnonlinearthoseatthesplinek
2、nots(i.e.,thegivenderivativesatallknotsmodelestimationortobeestimatedinput/outputpoints)Wproblems,inwhichforanarbitrarynonlinearfunction.Thisy1xi25yinonlinearfunctionswithunknowndirectrelationshipcanbeformulatedasfori51,2,c,N(2)shapesandvaluesareinvolved,bycon-aninnerproducto
3、flocation-dependentyi115y1xi1125fi1xi112vertingthenonlinearestimationweightsandthevaluesattheknots.The5fi111xi112problemsintolinearonesatahigher-approachpresentedheremayfindprac-fori51,2,c,N21(3)rr1x25fr1x2dimensionalspace.Thiscontrastswithticalapplicationswithinpatternyi115y
4、i11ii11r1x2thetypicaluseofthesplines[1]–[3]forrecognition,classification,systemcom-5fi11i11functioninterpolationwherethefunc-bination,speechrecognition,andsignalfori51,2,c,N21(4)s1x25ys1x25fs1x2tionalvaluesatsomeinputpointsareprocessing.Someexistingapplicationsyi11i11i11ii11s
5、1x2givenandthevaluescorrespondingtoarebrieflydiscussed.5fi11i11otherinputpointsaresoughtforviafori51,2,c,N21(5)interpolation.ThetechniquedescribedPREREQUISITESinthiscolumnappliestoarbitrarynon-Theprerequisitesconsistofbasiccalcu-Itcanbeeasilyverifiedthattheabovelinearestimati
6、onproblemswhereoneorlusandbasiclinearalgebra.Optimizationformulationcanberewrittenas[1],[2]moreone-dimensionalnonlinearfunc-techniquescouldalsobeusefulbutnotsstionsareinvolvedandcanbeextendednecessary.y1x25ayj1byj111cyj1dyj11,(6)tocaseswherehigher-dimensionalnon-linearfunctio
7、nsareused.BACKGROUNDwhere3xj,xj114isthesegmenttowhichThebenefitofusingtheapproachSplinesarepiecewiseormultiple-seg-theinputvaluexbelongsdescribedhereisobvious.Manyreal-mentfunctionswithpiecesorsegmentsxj112xworldsystemscanonlybeappropriatelyconnectedsmoothlywitheachother,a5,(
8、7)xj112xjmodeledwithnonlinearfunctions,whilewheretheconnectingpoints
