In Lecture 25, the noisy edge data was modeled as a series of connected straight lines plus iid N(0,

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Beverly Warren Perkins
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1 Lecture 26 A More Real-Life Edge oiy Meauremet Model (ad So Much More ) Oe ca ue all maer of tool to aalyze data But the way to bet aalyze data i to try to replicate it with a model I Lecture 25 we preeted the problem of edge predictio (actually, it wa etimatio ice typically predictio refer to future time) I cla, thi problem wa couched i the ettig where a mall autoomou droe flie alog hallway i a buildig, i order to obtai a map of the hallway that ca be ued to aid a evacuatio proce The droe ue GPS to idetify it poitio i pace, ad ue a IR beam to detect it ditace from the edge It alo ue feedbac cotrol to try to maitai a relatively fixed ditace from the edge I Lecture 25, the oiy edge data wa modeled a a erie of coected traight lie plu iid (, ) radom variable To motivate thi lecture, coider a imulatio of data=ormrd(,1,1,1) reulted i the plot below,,, 1 2 : The Matlab commad,,, ~ iid ormal(,1) Figure 1 Simulatio of 1 2 Viewed from afar, the plot at left doe appear to be a reaoable model of the oie But thi will deped directly o what the ource of the oie i I our example of a mall lightweight droe, probably the bigget ource of oie will be due to havig oly limited cotrol of it poitio a it progree alog a hallway Of coure, i a very moe-filled eviromet, the quality of the IR eor would alo be a igificat ource of oie The zoomed plot at right how completely radom variatio i poitio I other word were the oie level ow at ay time, uch iformatio would be uele i predict the oie iformatio at time +1 The quetio i: I thi a reaoable model? The awer to thi quetio will deped, at leat i part, o the peed at which the droe i movig Suppoe, for implicity, that the droe ide-to-ide perturbed motio were a imple iuoid: ( t) i( t) ow remember, we do ot have cotiuou-time data The droe will ue a aalog-to-digital (A2D) coverter that will ample the data at every Δ ecod The parameter 1/Δ i called the A2D amplig rate Hece, what the droe meaure i ( ) i( ) Deote the radial amplig frequecy a: 2 / The we have 2 / Hece, we ca write the ampled oie a: ( ) i[2 ( / )] (1) The figure below how (1) for two cae: / 1, ad / 1

2 Figure 2 Plot of ( ) i[2 ( / )] for / 1 (left) ad / 1 (right) Clearly, if we ue a high amplig rate, the ampled data poit are cloely paced Hece, if you were give a value at ay time, you would have a pretty good idea of that the value will be for ( 1) I other word, the radom variable ad 1 are highly correlated Thi will hold for pretty much ay cotiuou-time model for the oie; ot oly for a iuoid O the other had, it i reaoable to aume that the large the amplig iterval 2 / i (ie the lower the amplig frequecy i), the le correlated 2, oe could argue that it i eay to predict 1 from picture What i beig claimed i that there i o oie! ad 1 will be ow, i the cae of the right plot i Figure, ice all the ' are zero Thi i true But bac to the big The right figure wa icluded to demotrate ot a tatitical cocept, but a igal proceig cocept Thi cocept i ow a aliaig By amplig (1) at a amplig frequecy, the ampled iuoid appear to ugget to the viewer that it i a cotat; ot a iuoid I other word, it ha aliaed itelf Hece, the electio of a appropriate large amplig frequecy for the A2D i eetial i order to prevet aliaig While we will ot purue thi cocept i thi coure, the prevalece of ampled-data ytem mae it importat that every egieer have ome appreciatio for the cocept of aliaig, ad how to avoid it We will ummarize thi cocept i the followig theorem yquit Samplig Theorem- Suppoe that a cotiuou-time igal, t () ha zero eergy above a frequecy a igal i ampled with a amplig frequecy 2, the the ampled igal ( ) will have o aliaig If uch [By-ote: I jut had a eior i AERE iform me the other day that it wa hi geeral owledge of aliaig that laded him a iterhip ] Cocluio: How fat oe ample a cotiuou-time proce will determie the level of tatitical correlatio amog the radom variable cotitutig the ampled proce Thi lead u to the quetio: How ca a more real-life oie proce that exhibit correlatio be cotructed?

3 I thee ote we will addre oe uch cotructio Specifically, coider the followig dicrete-time oie model: W where { W} 1 ~ iid (, ), where 1 1 W, ad where ~ (, ) (2) It ca be how [ee homewor 5] that: (i) E ( ) ad (ii) Cov(, ) (3a) m 2 m if we et (1 ) (3b) W Sice either the mea or the covariace deped o the time idex, the collectio of radom variable { } i called a wide ee tatioary (w) radom proce Actually uch a proce mut allow, 1, 2, It i oly becaue we wated to begi at zero that we obtaied (3) Had we begu at -, we would the have: m 2 Cov(, m) R ( m) (4) The fuctio (4) i called the autocorrelatio fuctio for the radom proce { } 9 ad for 2 The autocorrelatio fuctio (4) for i how at right The iformatio i thee plot [ad give by (4)] iclude the followig: (i)the variace of ay radom variable : Cov R 2 2 (, ) () (ii)the covariace betwee ay two radom variable mfor a choe m: ad Cov R m m 2 (, m) ( ) I Figure 3 we ee that both procee have the ame variace Figure 3 R ( m) for 9 ad for 2 However, for mall value of the lag m (ie the ditace betwee ay 9 i much troger tha that aociated with 2 ad m ), the correlatio for the proce with At thi poit, the role of the A2D amplig iterval Δ hould be clear If Δ i mall, R ( m) will be larger tha it would be were Δ large Hece, we have the followig cocluio: Cocluio 1: If the A2D amplig iterval Δ i ufficietly mall, the we ca o loger aume that there i o correlatio amog the elemet of 1, 2,, O the other had, if Δ i ufficietly large, uch that, the we ca aume there i o correlatio amog the elemet of 1, 2,,

Example 1 The Matlab Dryde Wid Turbulece Model (Dicrete) [ http://wwwmathworcom/help/aerobl/drydewidturbulecemodeldicretehtml ] The logitudial dicrete-time model for turbulece i related to Figure 4

W where { } ~ iid (, ) (5) V Thi i exactly the ame model a (2) otice that from Figure 4 ad (5) we have 1 T, where V i the peed Lu of the plae, T i the A2D amplig iterval, ad L u i a patial correlatio

4 Example 1 The Matlab Dryde Wid Turbulece Model (Dicrete) [ ] The logitudial dicrete-time model for turbulece i related to Figure 4 below Figure 4 The elemet of the dicrete-time model for wid turbulece { } The iformatio i Figure (4) i to be iterpreted a havig the form: Let W U 1 The (5) become: U U where { } ~ iid (,1) (5) 1 U U W where { } ~ iid (, ) (5) V Thi i exactly the ame model a (2) otice that from Figure 4 ad (5) we have 1 T, where V i the peed Lu of the plae, T i the A2D amplig iterval, ad L u i a patial correlatio legth parameter Hece, we ee that at low peed ad/or large A2D T value, 1 I word, the turbulece will be highly correlated We will ow retur to the droe edge-etimatio problem The oie model ued to obtai Figure 1 wa,,, ~ iid (,1) The autocorrelatio fuctio i, for m therefore, R ( m) Let ow uppoe that the droe A2D i amplig ufficietly fat that for m ( ) 9 m R m Our oie model i the: 9 1 W, where from (3b): W (1 ) 19 The followig Matlab code wa ued to imulate the oie record how at right %EDGE OISE MODEL: var=1; %oie variace a=9; %oie correlatio parameter varw=(1-a^2)*var; %Drivig white oie variace =zero(1,); %Iitialize oie array (1)=ormrd(,var^5); %Model iitial coditio for =2: ()=a*(-1)+ormrd(,varw^5); ed figure(2) plot(vec,) title('simulatio of Edge oie') Figure 5 Simulatio of edge oie

5 The Matlab code to arrive at the edge meauremet how at right i give i the Appedix Clearly, thi i a more realitic model tha were oe to ue a white oie (ie iid oie) model The oie tadard deviatio ha bee reduced by ~5% Furthermore, the etimate ued abolutely o prior owledge of the oie correlatio tructure I other word, ot much thought wa give to icorporatig a oie model Figure 6 5-poit lidig widow etimate of the edge We will ow illutrate what ca be accomplihed if oe ha prior owledge of the oie correlatio tructure The figure at right how what i called the Kalma Filter (KF) etimate of the edge Thi etimate i clearly better tha the lidig widow etimate, a it hould be ice it ue iformatio about the oie The developmet of the KF model i beyod the cope of thi coure It i covered i AERE/ME/EE 573 The major prerequiite for the coure are liear ytem ad tatitic Ay tudet who fiihe thi STAT 35 coure with a grade of B or better would defiitely have a olid tatitic bacgroud [The KF code i icluded i the Appedix for thoe who have the iteret] Figure 7 Kalma filter etimate of the edge

6 Appedix Matlab Code %PROGRAM AME: edgev2m %HALLWAY COSTRUCTIO: ec=3; %umber of ectio =[1,1,1]; %Sectio legth =um();%total umber of poit th=[45,-3]; %ectio lope i Degree z=zero(1,); %Iitialize edge array z(1:(1))=1*oe(1,(1)); 12=1:(2); z((1)+1:(1)+(2))=z((1))+ 1*12*id(th(1)); 23=1:(3); z((1)+(2)+1:)=z((1)+(2)) + 1*23*id(th(2)); vec=1:; figure(1) plot(vec,z,'') %=========================================== %EDGE OISE MODEL: var=1; %oie variace a=9; %oie correlatio parameter varw=(1-a^2)*var; %Drivig white oie variace =zero(1,); %Iitialize oie array (1)=ormrd(,var^5); %Model iitial coditio for =2: ()=a*(-1)+ormrd(,varw^5); ed figure(2) plot(vec,) title('simulatio of Edge oie') %======================================== %PLOT OF EDGE MEASUREMET DATA: x=z+; figure(1) hold o plot(vec,x,'b') title('edge Meauremet Data') leged('edge','edge Meauremet') ave 'edgedata' 'x' %========================================= %========================================= %EDGE ESTIMATIO SLIDIG WIDOW ALGORITHM: load edgedatatxt % Thi file i the array 'z' load lietxt %Lie egmet without oie tmax = legth(x); w = 5; % Width of lidig widow xhat = zero(1,tmax); for t = w:tmax xhat(t) = mea(x(t-w+1:t)); ed xx = x(w:tmax); %Reduce data legth by w poit xxhat = xhat(w:tmax); zz=z(w:tmax); tt = 1:tmax - w+1; tt = tt'; figure(3) plot(tt,zz,'','liewidth',2) hold o plot(tt,xx,'b') plot(tt,xxhat,'r','liewidth',2) xlabel('time') title('oiy Edge(BLUE),Slidig Widow Etimatio(RED) WI SIZE = 5') leged('edge','meauremet','etimate') %============================================= %============================================= %Kalma Filter Etimatio of Edge: I=eye(3); F=[1 ;1 1 ; a]; H=[ 1 1]; R=;

7 varu=1^-7; %Slope white oie tuig variace Q=diag([varU varw]); P_old=diag([varU varu varw]); xhat_old=[;1;]; %Iitial tate that ue owledge of x() K=[]; xhat=[]; for =1: K= P_old*H'*(H*P_old*H' + R)^(-1); K=[K,K]; xhat=xhat_old + K*(x()-H*xhat_old); xhat=[xhat,xhat]; P=(I-K*H)*P_old; xhat_old=f*xhat; P_old=F*P*F' + Q; ed xhat2=xhat(2,:); % figure(4) plot(vec,z,'','liewidth',2) hold o plot(vec,x,'b') plot(vec,xhat2,'r','liewidth',2) xlabel('time') title('oiy Edge(BLUE),Kalma Filter Etimatio(RED)') leged('edge','meauremet','etimate')

Statistical Inference Procedures

Statistical Inference Procedures Statitical Iferece Procedure Cofidece Iterval Hypothei Tet Statitical iferece produce awer to pecific quetio about the populatio of iteret baed o the iformatio i a ample. Iferece procedure mut iclude a