Ch. 1 Introduction to Estimation 1/15

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1 Ch. Itrducti t stimati /5

2 ample stimati Prblem: DSB R S f M f s f f f ; f, φ m tcsπf t + φ t f lectrics dds ise wt usually white BPF & mp t s t + w t st. lg. f & φ X udi mp cs π f + φ t Oscillatr w/ f & φ M f f Gal: Give t s t; f, φ w t + Fid stimates that are ptimal i sme sese Describe with Prbability Mdel: PDF & Crrelati /5

3 Discrete-Time stimati Prblem These days, almst always wrk with samples f the bserved sigal sigal plus ise: s ; f, φ + w Our Thught Mdel: ach time yu bserve it ctais same s but differet realizati f ise w, s the estimate is differet each time. f & φ are RVs Our Jb: Give fiite data set,, - Fid estimatr fuctis that map data it estimates: f φ g g,,,,,, g g These are RVs eed t describe w/ prbability mdel 3/5

4 PDF f stimate Because estimates are RVs we describe them with a PDF p f Will deped :. structure f s. prbability mdel f w 3. frm f est. fucti g Mea measures cetrid Std. Dev. & Variace measure spread f f Desire: { f } f { f f } small f 4/5

5 . Mathematical stimati Prblem Geeral Mathematical Statemet f stimati Prblem: Fr Measured Data - Ukw Parameter p is t Radm is a -dimesial radm data vectr Q: What captures all the statistical ifrmati eeded fr a estimati prblem? : eed the -dimesial PDF f the data, parameterized by I practice, t give PDF!!! Chse a suitable mdel p ; Captures ssece f Reality Leads t Tractable swer We ll use p; t fid g 5/5

6 . stimatig a DC Level i Zer Mea WG Csider a sigle data pit is bserved S the eeded parameterized PDF is: + w ~, Gaussia zer mea variace p; which is Gaussia with mea f S i this case the parameterizati chages the data PDF mea: p; p; p; 3 3 6/5

7 . Mdelig Data with Liear Tred See Fig..6 i Tet Lkig at the figure we see what lks like a liear tred perturbed by sme ise S the egieer prpses sigal ad ise mdels: + B + w %"$"# s ;, B Sigal Mdel: Liear Tred ise Mdel: WG w/ zer mea WG dditive White Gaussia ise { } White ad m are ucrrelated fr m w w w w I T 7/5

8 Typical ssumptis fr ise Mdel W ad G is always easiest t aalyze Usually assumed uless yu have reas t believe therwise Whiteess is usually first assumpti remved Gaussia is less fte remved due t the validity f Cetral Limit Thm Zer Mea is a early uiversal assumpti Mst practical cases have zer mea But if t w wzm + µ -Zer Mea f µ Zer Mea w grup it sigal mdel Variace f ise des t always have t be kw t make a estimate BUT, must kw t assess epected gdess f the estimate Usually perfrm gdess aalysis as a fucti f ise variace r SR Sigal-t-ise Rati ise variace sets the SR level f the prblem 8/5

9 Classical vs. Bayesia stimati ppraches If we view parameter t estimate as -Radm Classical stimati Prvides way t iclude a priri ifrmati abut If we view parameter t estimate as Radm Bayesia stimati llws use f sme a priri PDF The first part f the curse: Classical Methds Miimum Variace, Maimum Likelihd, Least Squares Last part f the curse: Bayesia Methds MMS, MP, Wieer filter, Kalma Filter 9/5

10 .3 ssessig stimatr Perfrmace Ca ly d this whe the value f is kw: Theretical alysis, Simulatis, Field Tests, etc. Recall that the estimate g is a radm variable Thus it has a PDF f its w ad that PDF cmpletely displays the quality f the estimate. Illustrate with -D parameter case Ofte just capture quality thrugh mea ad variace f g p Desire: m { } { } small If this is true: say estimate is ubiased /5

11 quivalet View f ssessig Perfrmace Defie estimati errr: e + RV RV t RV Cmpletely describe estimatr quality with errr PDF: pe pe e e Desire: m e e { e} { } e { e} small If this is true: say estimate is ubiased /5

12 /5 ample: DC Level i WG Mdel:,,,, + w Gaussia, zer mea, variace White ucrrelated sample-t-sample PDF f a idividual data sample: ep π i i p Ucrrelated Gaussia RVs are Idepedet s jit PDF is the prduct f the idividual PDFs: / ep ep π π p prperty: prd f ep s gives sum iside ep

13 3/5 ach data sample has the same mea, which is the thig we are tryig t estimate s, we ca imagie tryig t estimate by fidig the sample mea f the data: Statistics Prb. Thery Let s aalyze the quality f this estimatr Is it ubiased? i { } } { { } %$# Yes! Ubiased! var var var var Ca make var small by icreasig!!! Due t Idep. white & Gauss. Idep. Ca we get a small variace?

14 Theretical alysis vs. Simulatis Ideally we d like t be always be able t theretically aalyze the prblem t fid the bias ad variace f the estimatr Theretical results shw hw perfrmace depeds the prblem specificatis But smetimes we make use f simulatis t verify that ur theretical aalysis is crrect smetimes ca t fid theretical results 4/5

15 5/5 Curse Gal Fid Optimal stimatrs There are several differet defiitis r criteria fr ptimality! Mst Lgical: Miimum MS Mea-Square-rrr See Sect..4 T see this result: var{ } b mse + { } var{ } { } { } { } { } b b b mse " %"$"# } { b Bias lthugh MS makes sese, estimates usually rely