Matlab_AR模型阶数确定

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1、自回归(AR)模型理论模型自回归(AutoRegressive,AR)模型又称为时间序列模型，数学表达式为y()ay()ay(na)，e()na其中，e(t)为均值为0方差为某值的白噪声信号。MatlabToolbox(5Par-amertricModelingp-Asrcov-finwmodei.pornrre-tersusingiavortoncemethodr-Aarmcctv-Estin向tARvmMd.pa阳啓etrsusingmodifie(fcovcnancemethodh-Aaryule-imoteAR)cfi-poiemodetusingYute-Woiitrmethoti!

2、Ainyfreqs-/c/entifycantinwa/s-OjtkfilterparamefersfromfreqxrKypespansedatap-Ainvfreqz-Ideatifydismte-timefitterparmetersfromfrequencyrepnsedato,/jfprony-?methodfordesignLAstmcb-Compufe(ineGrrnotfusingitciotion研究表明，采用Yule-Walker方法可得到优化的AR模型1，故采用aryule程序估计模型参数。m,refl=ar(y,n,approach,window)模型阶数的确定有几种方

3、法来确定。如Shin提出基于SVD的方法，而AIC和FPE方法是目前应用最广泛的方法。若计算出的AIC较小，例如小于-20,则该误差可能对应于损失函数的10-10级别,则这时阶次可以看成是系统合适的阶次。(3ModelAnalysisfaadvice-山口口如百占andrecammFAaic-AikeInfomTationCriterionforeitQiTijLErfmadel-i-faarxdata-ARXparametersfrommuttiple-outputmodelswitnvorioneeinformation-haired-ReducemodElo)vier(retfuires

4、C-ontrulSystemToolboxproductfxbode-Computeandplatfrs-quencyresponsemagnitu-deandphaseforlogaiithmicfretfuencies-ficompare-Comparemotfeioutputandmeosudoutputp-faffplot-Computeandplotfrequencyreiponmognitudeontfp/ioieforfineorfrequenciesp-freqre&p-rsquencyresponseJotofrom(inearmodels-fifpe-Finai.PsrEd

5、ictionEmr/crestimoferfmodel.am=aic(modell,model2,.)fp=fpe(Modell,Model2,Model3,.)AR预测yp=predict(m,y,k)m表示预测模型；y为实际输出；k预测区间；yp为预测输出。y(l),y(2),.,y(t-k-1),y(t-k),.,y(t-2),y(t-1),y(t)当klnf时，yp(t)为模型m与y(1,2,t-k)的预测值；当k=lnf时，yp(t)为模型m的纯仿真值；默认情况下，k=1。在计算AR模型预测时，k应取1,原因参照AR模型理论公式。compare(y,m,k)yh,fit,x0=com

6、pare(y,m,k)Compare的预测原理与predict相同，但其对预测进行了比较。fit1001IIy一yhII，、11y一11丿AR误差女p勺-口馆虫卅苗门EfrondssocttiiEt/mihm-odelun-dJofase-lp-Aplat-PlottoormoJeiobjectsh-Apolydata-fromsingle-inputohjIsingie-outputpolynomial,morfei；卜predict-jredtrioufputkstepsoheoaApr亡dictCidnlara)-Pjtc/ictdutput片5tepsaheadfornontinear

7、AfiXmodet.ApredirtCidnlgrey-PrsdictoulUl/rstepsohend/ornnIineorODEmodeI?-Apredictfidnlhw)-Prsdkt口utputk$t即srhforHommerstein-WienermodelApresent-DiEpinynodetin/hrmotian,includingestimoteiiuncertaiFity卜女pzmap-Pi.oizerosonJpoleswithconfidenceuit-erv-ni.Aresid-Computeandtestmodeiresiduals(predictionemor

8、-ii_一一_-e=pe(m,data)pe误差计算。采用yh=predict(m,data,1)进行预测，然后计算误差e=data-yh;e,r=resid(m,data,mode,lags);resid(r)resid计算并检验误差。采用pe计算误差；在无输出的情况下，绘出误差图，误差曲线应足够小，黄色区域为99%的置信区间，误差曲线在该区域内表明通过检验。Matlab确定模型阶数采用ASCEbenchmark模型120DOF,选取y方向的响应，共8个。首先，对响应数据进行标准化处理；其次，将标准化处理后的数据建立AR模型；最后，确定合适的模型阶次，通过选取一系列阶数，分别计算对应的AIC

9、值，从图中可以看出，阶次80以后的AIC值变化不大，因此，合适的阶次选择为80。ARorderAR模型预测sensor2.(1-steppred)4-486202-6-)2s/m(2rosnes52121511110511AR误差计算0-2esensor221-11.051.11.151.21.251.31.351.4Timelag附录MATLAB代码1)%ARmodelorderclc;clear;addpath(genpath(pwd),1);data=load(Case2Damage0_1_0.01_0.001_40_10_150_123_1.mat);dofy=2,4,6,8,10,1

10、2,14,16;x=zscore(data.acc(:,dofy);order=10:10:130;fori=1:length(order)forj=1:size(x,2)m=ar(x(:,j),order(i),yw);am(i,j)=aic(m);endendplot(order,am,LineWidth,2,Marker,*)xlabel(ARorder),ylabel(AIC)legend(sensor2,sensor4,sensor6,sensor8,.sensor10,sensor12,sensor14,sensor16)clc;clear;addpath(genpath(pwd)

11、,1);data=load(Case2Damage0_1_0.01_0.001_40_10_150_123_1.mat);dofy=2;4;6;8;10;12;14;16;order=80;ts=zscore(data.acc(:,dofy);k=1;name=sensor2,sensor4,sensor6,sensor8,.sensor10,sensor12,sensor14,sensor16;Ounit=repmat(m/s2,length(dofy),1);ts=iddata(zscore(data.acc(:,dofy),data.dt,.OutputName,name,OutputU

12、nit,Ounit,.Name,ASCE-benchmark120DOF);m=ar(ts(:,k),order,yw);compare(ts(:,k),m,r-.,1,1000:1300);resid(m,ts(:,k),corr,25)e,r=resid(m,ts(:,k),corr,40);resid(r)plot(e(1000:1300)参考文献Matlabhelp1DASILVAS,DIASJNIORM,LOPESJUNIORV.DamagedetectioninabenchmarkstructureusingAR-ARXmodelsandstatisticalpatternrecognitionJ.JBrazSocMechSciEng,2007,29(2):174-84.