Cooper and McGillem Chapter 4: Moments Linear Regression
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1 Cooper d McGllem Chpter 4: Momets Ler Regresso Chpter 4: lemets of Sttstcs 4-6 Curve Fttg d Ler Regresso 4-7 Correlto Betwee Two Sets of Dt Cocepts How close re the smple vlues to the uderlg pdf vlues? Prctcl curve fttg, usg NTC resstor to mesure temperture. Sttstcs Defto: The scece of ssemlg, clssfg, tultg, d lzg dt or fcts: Descrptve sttstcs the collectg, groupg d presetg dt w tht c e esl uderstood or ssmlted. Iductve sttstcs or sttstcl ferece use dt to drw coclusos out or estmte prmeters of, the evromet from whch the dt cme from. Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 of 7 C 3800
2 4-6 Curve Fttg d Ler Regresso Fttg les/curves to sctter plots. Dt provded s (,) prs. Is there fucto tht goes through ll the pots? es If ou wt to use poloml of degree - for prs! But we usull wt smple curves to represet the dt, lke les or prols, etc. where or c To ft the curve we wt to mmze the followg fucto (the squred error): For ler regresso ( le), we hve err To mmze for the vlues d, tke the dervtves d set them equl to zero. The solve for d : Solvg results d err d d err d 0 0 d Wht hppes whe we tke epected vlues? Notce sttstcl terms? Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 of 7 C 3800
3 Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 3 of 7 C 3800 Proof: 0 d err d Workg o 0 d err d Susttutg for the computto for Isoltg Now tht s determed sed o the vlues, retur to Susttutg for the computto for to
4 Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 4 of 7 C 3800 Therefore ecomes Alterte formulto usg the computed smple mes of d Notce covrce d vrce terms s well.
5 Ler regresso emple p. 80. Fgure 4-5. %% % Fgure 4_5 % cler; close ll; =(0:0.5:0)'; % Ler Curve vlues =*c+ =; =4; ref = +*; % Rdom ose dded to the le dt = ref + 5*rd(sze()); fgure plot(,dt,'',,ref) leged('dt','ref Le') me=me(); me=me(dt); mesq = me(.^); mesq = me(dt.^); mecorr = me(.*dt); Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 5 of 7 C 3800
6 est_equ = (me*mesq-me*mecorr)/(mesq-me^); est_equ = (mecorr-me*me)/(mesq-me^); est_equ = est_equ + est_equ*; p=polft(,dt,); est = p(); est = p(); est = polvl(p,); fgure plot(,dt,'o',,ref,'k',,est,'r',,est_equ,'m'); leged('dt','ref Le','Polft Le','qu Le') fprtf('computto error\') m(s(est_equ-est)) r = mecorr/sqrt(mesq*mesq) Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 6 of 7 C 3800
7 No-ler estmto usg poloml ft mple: Tkg the dt from Tle 4-3 o p T, VB, Fgure V= * * Brekdow Voltge Temperture ( C) p=polft(,,); = p(3); = p(); c = p(); z = + * + c*.^; fgure plot(,,'o',,z,'r'); lel('temperture ( C)') lel('brekdow Voltge') ttle('fgure 4-6') grd tt=sprtf('v=%g+%g*+%g*^',,,c); tet(50,375,tt); Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 7 of 7 C 3800
8 Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 8 of 7 C Correlto of dscrete rdom vrles For sgle rdom vrle, we hve defed mesures of the reltoshp of oe smple or evet d the et. These re the mes d momets d the vrce. Me or st Momet d Momet d f, d f, d d R d Cetrl Momet d f, d C The vrce s mesure of the smlrt of successve smples or evets wth ech other. How close or correlted wth the others would evet e epected to e? R C
9 Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 9 of 7 C 3800 Correlto etwee dscrete rdom vrles d For two sequeces or pred groupgs (,). If we ssume tht ever (,) pr s equll lkel, the pmf of the fuctos hs the sme vlue for ever pr. Repeted prs smpl sum the prolt t the pot. So, for correlto, d d f, for, prs, to we c defe pmf for ech smple pot s /. Therefore, R Defg the cross correlto d d f, C
10 Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 0 of 7 C 3800 The Dscrete Correlto coeffcet For two sequeces or pred groupgs (,). If we ssume tht ever (,) pr s equll lkel, the pmf of the fuctos hs the sme vlue for ever pr, /. Repeted prs smpl sum the prolt t the pot. So, d d f, r C r or mkg t full dt drve r The tet defes ths s the Perso s r sttstcl mesure, the ler correlto coeffcet etwee two sets of dt! from Wkped Perso product-momet correlto coeffcet:
11 Bsed o the dscrete terms, ler estmto ecomes The, R R R R C R d R R C C Pvlov codtog for smpled dt lws compute the followg wth dt! : Me, d momet, vrce (, R, d ) : Me, d momet, vrce (, R, d ) d : For more formto: R, C, d R C R R C C Alerto Leo-Grc, Prolt, Sttstcs, d Rdom Processes For lectrcl geerg, 3rd ed., Perso Pretce Hll, Upper Sddle Rver, NJ, 008. Chp. 8. Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 of 7 C 3800
12 Prctcl mple: NTC Resstor Temperture Mesuremets Suseeker Bsed o Vsh BCCompoets, Resstor Products Applcto Note, Documet Numer: 9053, 4 M, 0. Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 of 7 C 3800
13 Note: The B costt wll e clled K costt the followg mterl. Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 3 of 7 C 3800
14 The dt s epoetl curve wth respect to temperture. NTC Resstors tpcll refereced to 5 C or 98.5 K. R T R T ep K For the st order ppromto, ssume T T Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 4 of 7 C 3800
15 Plottg the resst versus temperture sed o the dt d some ppromtos, we hve Suseeker NTC Resstor Temperture Curves Resstce (ohms) Dt Sheet K5/85 K5/ Temperture (deg C) See cel Spred Sheet for vlues Tpcll, dt sheets provde K vlues sed o 5 C d 85 C. K R T l R T T 490 T Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 5 of 7 C 3800
16 For etter ccurc wth crtcl rego, the K c e computed to oud desred temperture opertg pots. For Suseeker, ke tempertures for tter operto re 45 C d 60 C. Therefore K sed o 5 C d 60 C s suffcet for operto. Ths resulted porto of the spred sheet lss. Desgg wth NTC Thermstor. PCOS NTC Thermstor Applcto Notes, Fe A referece curret or voltge s requred. I ths cse kow voltge s provded to resstor dvder d the output voltge s dctve of the temperture. The resultg curve s hghl o-ler due to the epoetl ture of the devce. To lerze the curve d reduce the steepest prt of the curve, plce the NTC prllel wth lrge resstor. Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 6 of 7 C 3800
17 For Suseeker: A.5 Vref drves the resstor dvder. The Upper vlue used s 00kΩ d the resstor prllel wth the NTC thermstor s 330kΩ. A vertg op-mp s ot used, we re drectl coected to 4-t ADC. The resultg voltge to temperture curve s NTC See the spred sheet for the epected ADC outputs d hedecml dgtl vlues. Notes d fgures re sed o or tke from mterls the course tetook: Prolstc Methods of Sgl d Sstem Alss (3rd ed.) George R. Cooper d Clre D. McGllem; Oford Press, 999. ISBN: B.J. Bzu, Fll 06 7 of 7 C 3800
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