A HIERARCHICAL KALMAN FILTER

Transcription

1 A HIERARCHICAL KALMAN FILER Greg aylor aylor Fry Consulng Acuares Level 8, 3 Clarence Sree Sydney NSW Ausrala Professoral Assocae, Cenre for Acuaral Sudes Faculy of Economcs and Commerce Unversy of Melbourne Parkvlle VIC 35 Ausrala Phone: Fax: greg.aylor@aylorfry.com.au March 9

2 A Herarchcal Kalman Fler able of Conens Summary.... Inroducon.... Credbly esmaon Sandard (mul-dmensonal) credbly Kalman fler Herarchcal credbly A herarchcal Kalman fler Numercal example Concluson... 4 References...6

3 A Herarchcal Kalman Fler Summary Sund s herarchcal credbly model s generalsed o a dynamc form, e a form n whch parameers are assumed o evolve over me. hs s done by supermposng a Kalman fler on Sund s model. In he process s shown ha boh Sund s model and he Kalman fler may be derved as drec consequences of Hachemeser s credbly regresson model. A numercal example n Secon 7 llusraes he applcaon of he herarchcal Kalman fler o a herarchy of occupaonal groups. he example shows how parameer esmaes produced by he fler rack he rue values of parameers beer han he esmaes from a sac model when hose parameers evolve over me. Keywords: credbly, Hachemeser model, herarchcal credbly, Kalman fler.

4 A Herarchcal Kalman Fler. Inroducon Herarchcal credbly was nroduced by aylor (979). he parameers were scalars, as were he observaons on hem. hs framework was generalsed by Sund (979, 98) o nclude vecor parameers and observaons. In Sund s framework a lnear regresson model s placed a each level of he herarchy. he model s sac n he sense ha all parameers relae o jus a sngle epoch or me nerval. If, n fac, daa were colleced over mulple me perods, hey would be pooled on he assumpon ha he parameers remaned consan over me. Such poolng of daa renders he model sluggsh n s response o evolvng parameers. One srucure ha recognses parameer evoluon explcly s he Kalman fler (Kalman, 96; Jazwnsk, 97). he purpose of he presen paper s o adap Sund s herarchcal model o a dynamc form by supermposng a Kalman fler on. I s ancpaed ha he parameer esmaes yelded by he dynamc model wll rack he evolvng parameers beer han her sac counerpars. he Kalman fler s derved n Secon 4 and Sudn s resuls n Secon 5. s shown n he process ha boh can be vewed as consequences of Hachemeser s (975) mul-dmensonal credbly model. he herarchcal form of he Kalman fler s derved n Secon 6. Is use s llusraed n Secon 7, where s appled o he orgnal conex of herarchcal credbly (aylor, 979). In hs case he herarchy s a ree of occupaonal groups n a workers compensaon seng.. Credbly esmaon Le m be an unknown vecor random varable, and le x be an observable vecor random varable. A lnear credbly esmaor (or jus credbly esmaor) of m (based on x) s he lnear funcon L(x) of x whch mnmses E L( x) m. he lnear credbly esmaor L(x) s unbased for m. 3. Sandard (mul-dmensonal) credbly Le θ be an unknown random parameer and x an observable random vecor. Assume he followng: (3.) x= Yb( θ ) +ξ for some vecor-valued funcon b (.) Y, and cenred random vecor ξ ; (3.) θ and ξ are ndependen., deermnsc marx

5 A Herarchcal Kalman Fler 3 Le β= E b( θ) (3.) Λ= θ (3.) Var b( ) [ ] Var [ ] Φ= Var x θ = ξ (3.3) hs s he Hachemeser (975) framework, and he lnear credbly esmaor of b( θ ) s ( Z) Zbˆ b = β+ (3.4) where ( Z =ΛY Φ Y +ΛY Φ Y ) (3.5) and ˆb s he classcal regresson esmaor ( ) ˆ = Φ Φ x (3.6) b Y Y Y Here he upper denoes marx ransposon. Exenson. Relax he condon ha ξ be cenred. hen resuls (3.4) and (3.5) connue o hold bu wh (3.6) replaced as follows: ( ) [ ] ˆ b= Y Φ Y Y Φ x E ξ (3.6a) Proof. Wre u = x E[ ξ ] hen, by Assumpon (3.), u = Yb( θ ) +η where s he cenred random vecor η=ξ E[ ξ ] hen resuls (3.4) (3.6) hold wh x replaced by u.

6 A Herarchcal Kalman Fler 4 4. Kalman fler 4. Sandard Kalman fler Le b, =, 3, ec be an unknown vecor parameer ha evolves randomly over me, and le x be an observable random vecor, where b and x sasfy he followng relaons. (4.) + + b = A b +ε +, =,, ec where A + s a deermnsc square marx and ε + a cenred random vecor, and he sysem s naed by a deermnsc vecor b. (4.) x = Y b + ξ, =,, ec where Y s a deermnsc marx and (4.) Each of he ses { ε } and { } ξ a cenred random vecor. ξ s muually sochascally ndependen, and furher he wo ses are sochascally ndependen of each oher. Le s b be he (unbased) credbly esmaor of b Specfcally, on accoun of Assumpon (4.), defne s based on {,,... } x x x. + = A + b (4.) b I s sraghforward o show ha on { x,... x }. s he credbly esmaor of b + based b + Le Φ = Var x b = Var ξ (4.) Ψ = Var ε (4.3) E b b (4.4) Λ = he credbly esmaor Λ. s he lnear funcon of x,, x - ha mnmses b By Assumpon (4.) and (4.), ( ) ( A) E Ab Ab Λ = +ε = AVar b + Ψ (4.5)

7 A Herarchcal Kalman Fler 5 where use has been made of Assumpon (4.) and he fac ha depend on ε. b does no he mahemacs of fndng he credbly esmaor Secon 3 bu wh he followng replacemens: b s he same as n x x Y Y b θ b ( ) ξ ξ β b b b he parallel of Assumpon (3.) here s ndependence of b and guaraneed by Assumpon (4.). ξ, whch s I hen follows ha Λ Λ [from (3.) and (4.4)] Φ Φ [from (3.3) and (4.)] Subsuon of hese replacemens no (3.4) and (3.5) yelds ( ) b = K b + K b ˆ (4.6) where ( ) ( ) = + = + (4.7) K L L L L L ( ) ( ) =Λ Y Φ Y (4.8) ( ) ( ) ( ) ( ) bˆ = Y Φ Y Y Φ x (4.9) Snce ˆ b depends on jus x and does no, (4.6) gves b ( ) ( ) ˆ ( ) Var b K Var b = K + K Var b K (4.) = ( K ) Λ ( K ) + K ( Y ) ( Φ ) Y ( K ) o smplfy hs, emporarly suppress he upper. Subsue (4.7) and (4.8) no (4.), nong ha K = ( + L) -, o oban

8 A Herarchcal Kalman Fler 6 ( ) { }( ) Var b = + L Λ+ L Y Φ Y L + L ( L) = + Λ Wh he upper resored ( K ) (4.) Var b = Λ Equaons (4.), (4.5) (4.9) and (4.) collecvely show how o proceed from b o b and, n dong so, consue a sngle eraon of he Kalman fler (Kalman, 96; Jazwnsk, 97). 4. Exenson + Relax he requremen n Assumpon (4.) ha ε be cenred. hen he resuls of Secon 4. connue o hold wh he sngle change ha (4.) s replaced by he followng: = + ε (4.a) b A b E Proof. Defne he sequence a, =,, ec by he recurson a = A a + E ε a = (4.) hen defne c b a = (4.3) By (4.3) and Assumpon (4.) c A c = +η (4.4) where + η s he cenred random vecor η =ε E ε (4.5) By (4.3) and Assumpon (4.), x = Y c + ζ (4.6) wh ζ = Ya+ξ (4.7)

9 A Herarchcal Kalman Fler 7 Noe ha (4.4) and (4.6) are of he same form as Assumpons (4.) and (4.) wh he followng replacemens: b c ε η ξ ζ and where ζ s no cenred. he dervaon of he Kalman fler for hs sysem s by exacly he same argumen as gven above n relaon o Assumpons (4.) and (4.) excep ha appeal s now made o he exenson n Secon 3 n order o accommodae he non-cenredness of ζ. Noe ha a s deermnsc, and so he varance quanes Φ, Ψ, Λ are all lef unchanged by he above replacemens. he only change requred o he Kalman fler s he replacemen of (4.) by (4.a). 5. Herarchcal credbly Consder he herarchcal credbly model of Sund (979, 98). Here he unknown parameer and observable random vecor x of Secon 3 are replaced by unknown random parameers θ and observable random vecors x, =,,,n, wh denong he level whn a mul-level herarchy. he model s defned by he followng assumpons. (5.) For each =,, n, x θ,..., θ = Yb( θ,..., θ ) +ξ( θ,..., θ ) b., deermnsc marx, and for some vecor-valued funcon ( ) random vecor ξ (,..., θ θ ) sasfyng E ξ ( θ,..., θ ) θ,..., θ =. (5.) For each =,, n, b θ..., θ = W b θ..., θ +η θ..., θ for some deermnsc ( ) ( ) ( ) W and random vecor (..., ) η ( θ,..., θ ) θ,..., θ = marx E. For he case =, b θ =β+η θ ) ( ) ( η θ θ sasfyng wh β some known parameer and ( ) (5.) ξ ( θ θ ) and ( ),..., E η θ =. η,..., θ θ do no depend on θ j, j >. Y

10 A Herarchcal Kalman Fler 8 (5.v) (a) ξ (,..., θ θ ) and ξ (,..., j θ θ j ) are condonally ndependen for j >, gven θ,..., θ η,..., θ θ and η (,..., j θ θ j ) are condonally ndependen for (b) ( ) (c) j >, gven θ,..., θ ξ (,..., θ θ ) and η (,..., j θ θ j ) are ndependen for all,j. Remark 5. I follows from hese assumpons ha (5.v) x and θ j are ndependen for j > (5.v) For each =,..., n, ( x,..., x ) and (,..., x x n ) ndependen gven θ,..., θ. + are condonally he assumpons (5.) (5.v) are n fac sronger han hose of Sund, who assumed (5.v) and (5.v) n place of (5.) and (5.v). Le ( ) Λ = EVar b θ,..., θ θ,..., θ, =,,..., n (5.) [ ] Φ = EVar x θ,..., θ, =,,..., n (5.) Now defne x = x,..., x, =,,..., n ( n ) (5.3) and defne he vecors ξ, η smlarly. When hese vecors are wren as condoned by θ, each componen wll be undersood o be condoned by all relevan nformaon, e n x, x wll be condoned by θ,..., θ ; x + by θ,..., θ+, ec. Equvalenly, each componen s condoned by θ,..., θn. By Assumpons (5.) and (5.), parcularly he lneary of he relaons here, x θ= Y b θ,..., θ + Nη θ+ξ θ (5.4) ( ) for some marces Y and N. o calculae hese marces, noe ha x θ= Y b θ,..., θ + N η θ+ξ θ ( + ) ( + ) + ( + ) + ( + ) ( + ) = Y( ) W b( θ,..., θ ) + Y( ), N ( ) ( +) η θ+ξ θ (5.5) by Assumpon (5.).

11 A Herarchcal Kalman Fler 9 Combnng Assumpon (5.) wh (5.5) and equang he resul o (5.4) yelds Y Y = Y( ) W +, =,,..., n (5.6) N = Y( ) N + +, =,,..., n (5.7) where he zero sub-marces have he same numbers of rows as x. he recurson (5.6) and (5.7) s naed wh Y = Y N = ( ) n, n n (5.8) obaned from (5.4) wh = n. Denoe Φ = EVar,..., x θ θ Π = EVar x,..., θ θ (5.9) (5.) Now [ θ θ ] EVar x,..., Φ = E x( ) θ,..., θ + (5.) Φ = Π ( + ), =,,..., n where he ndependence assumpon (5.v) has been used n he frs equaly. he recurson (5.) s naed wh Φ ( n) =Φ n (5.) Π he marces may be evaluaed by reference o he frs equaly n (5.5) wh + replaced by, whence {,...,,..., } E E x Y b ( ) Π = θ θ θ θ θ,..., θ θ where s convenen for he me beng o show explcly he varaes wh respec o whch expecaons are aken.

12 A Herarchcal Kalman Fler he las relaon may be expanded as follows: {{,...,,..., } Π = E ( ),..., E θ θ x E Y b θ θ θ θ θ { }} + Y E b θ,..., θ θ,..., θ E b θ,..., θ θ,..., θ θ,..., θ ( ) ( ) =Φ + Y Λ Y by (5.) and (5.9). Le (5.3) b denoe he credbly esmaor of b (,..., θ θ ) based on x. hs means ha b s he lnear funcon of x ha mnmses { ( θ,..., θ ) θ,..., θ} E E θ θ b b... Sund (98) shows ha, by Assumpon (5.), W b s he credbly esmaor of E b ( θ,..., θ ) θ,..., θ, and ha, by Assumpon (5.v), b depends on jus he x par of x. he mahemacs of fndng hs esmaor s once agan he same as n Secon 3 bu wh he followng replacemens: x x Y Y b θ b,..., θ θ ξ ξ β W b b b ( ) ( ) he expecaon n he mnmsaon creron s also replaced. he objecve funcon E L( x) m s now replaced by E E b b { (,..., ) θ θ θ,..., θ } θ... θ θ All expecaons on whch he credbly esmaor depends are replaced correspondngly. As a resul

13 A Herarchcal Kalman Fler ( ) Λ Eθ,..., θ Var b θ θ,..., θ θ,..., θ =Λ Φ Φ { } bˆ = Y Φ Y Y Φ x (5.4) Wh hese replacemens, (3.4) and (3.5) become b = Z W b + Z bˆ (5.5) ( ) and Z Y Y =Λ Φ +Λ Y Φ Y (5.6) 6. A herarchcal Kalman fler Secon 5 defned he herarchy of parameers b( ) observaons { :,..., } vecor over me. {,..., :,..., } θ θ = n and x = n. Secon 4 defned he evoluon of a parameer he presen secon combnes he wo conceps no an evoluonary herarchy b : =,..., n; =,,... and observaon vecors of parameer vecors { } { x :,..., n;,,... } = =. he herarchy s subjec o he followng assumpons. (6.) For each =,, and each =,, n, x b,..., b = Y b + ξ for some parameer vecor, deermnsc marx, and cenred random vecor. ξ b (6.) For each =,, and each =,, n, defne c = b W b for some deermnsc marx W. For he case =, defne c = b W β where β = E b and W s some deermnsc marx. he vecors c and β evolve accordng o c = A c +ε, =,..., n; =,, β = A, β +ε =,,... Y for deermnsc marces A + + and cenred random vecors ε. s (6.) he random vecors ξ, ε are muually sochascally ndependen. j

14 A Herarchcal Kalman Fler Remark 6.. he random vecors b are no longer expressed as funcons of absrac laen parameers θ as n Secon 5. Insead, he explc relaons beween hese vecors are spulaed n Assumpon (6.). Remark 6.. By Assumpon (6.), he evoluon of he b s accordng o b = W b + A b A W b +ε, =,..., n; =,,... (6.) For he case =, hs relaon connues o hold f β, β +. b, b + are replaced by he followng wll use a noaon parallel o ha of Secon 5 bu augmened by he superscrp represenng me. For example, x wll denoe he vecor x, defned by (5.4), as occurs n me perod. he followng wll also use he s superscrp noaon from Secon 4 wh he s same meanng as here. For example, b wll denoe he lnearsed esmaor of based on daa x,..., x. s b he objecve s o fnd esmaors b for =,,, n; =, 3,. hs s done by applcaon of he Kalman fler o he vecors x. Begn by nong ha he evoluon of he b may be expressed n he form + = + j j j j j= j= b M b N ε (6.) for marces M + j, N + j defned below, and where b n nerpreed as β. hs relaon may be proved by nducon. I s rue n he case =, by Assumpon (6.). Here M = A, N = (6.3) For he case >, use (6.) o subsue for b + n (6.). hs yelds b W M b N = j j + j j j= j= + A b A W b +ε ε (6.4) Equaly of (6.4) wh (6.) yelds he recurson

15 A Herarchcal Kalman Fler M = W M, j =,,..., j j M + = A M = W M A W N = N j = W N j, j =,,..., (6.5) he relaon (6.) for =,,, n may be summarsed n he form b = M b + N ε (6.6) + ( ) ( ) ( ) ( ) ( ) where he marx M + ( ) akes he block form M M M ( ) M = n + n + n + M M Mn (6.7) and N + s smlarly consruced from he blocks N +. ( ) j Y [] Y Now defne (dsnc from ) as he block dagonal marx wh blocks Y,..., Y. As n Secon 5, where he vecors n x are wren whou as condoned by b, each componen wll be undersood o be condoned by all relevan nformaon, e n x, x wll be condoned by,..., b b ; x + by,..., b b +, ec. Equvalenly, each componen s condoned by b,..., b n. By Assumpon (6.), x b = Y b + (6.8) ξ [] Noe ha (6.6) and (6.8) wh = are of he same form as Assumpons (4.) and (4.) and so he Kalman fler, conssng of (4.), (4.6) (4.9) and (4.), may be appled wh he followng replacemens: b b ε ( ) A M x Y x ξ ξ Y ( ) N [ ] ( ) ( ) ( ) ( ) ε

16 A Herarchcal Kalman Fler 4 { } { } Assumpon (6.) ensures ha he ses N( ) ε ( ) and ξ ( ) are muually sochascally ndependen, as requred by Assumpon (4.). I wll be necessary for hs fler o map an esmaor of o an esmaor of, raher han. hs s because here are no drec observaons Y on b ( ) ˆ and so no esmaor b for use n (6.7). Defne N : ( ) b b b o be he marx obaned by deleng he frs row of M ( : ) M ( ) smlarly. hen (6.6) gves ( ). Defne b = M b + N ε ( ) ( :) ( ) ( :) ( ) (6.9) In correspondence wh (4.) (4.4), defne Φ = Var x b = Var ξ (6.) Ψ = ε (6.) Var Λ = E b b (6.) Noe ha he muual ndependence of he ξ mples ha dagonal srucure. Also defne Φ has block Γ = Var b and noe ha, n correspondence wh (4.5), (6.3) { } Λ = E M b + N ε M b ( ) ( :) ( ) ( :) ( ) ( :) ( ) (6.4) = M Γ M N + Ψ N ( : ) ( ) ( : ) ( : ) ( ) ( : ) (6.5) where, n (6.4), use has been made of (6.9) and (6.6) has been ancpaed. Use has also been made of Assumpon (6.). hen he Kalman fler, wh equaons now wren n operaonal order, s as follows: b M b (6.6) = : ( ) ( )

17 A Herarchcal Kalman Fler 5 ( ) ( ) [] ( []) ( ) ˆ b = Y [] Φ Y Y Φ x (6.7) Λ accordng o (6.5) L [] ( Y ) ( ) [] =Λ Φ Y (6.8) ( ) K L L = + (6.9) ( ) ˆ b = K b + K b (6.) ( K ) Γ = Λ (6.) I remans o calculae b and hence b ( ), and hen Γ ( ). Begn by nong, from (6.), ha ( ) b = W b + A b W b + ε (6.) Now he marx W may no be square. I wll be assumed here o have a leas as many rows as columns and o be of full rank, hs beng he more common case n pracce (eg see example n Secon 7). As shown n he appendx, s always possble o ransform W as follows: R W = U (6.3) where R s defned n he appendx, and U akes he block dagonal form U u = u (6.4) wh each u j a column vecor wh all componens uny. By (6.) and (6.3), ( ) R b = U b + R A b W b + R ε Pre-mulply boh sdes of hs relaon by he block dagonal marx

18 A Herarchcal Kalman Fler 6 Ω = ω ω (6.5) where each ω j s a row vecor and ω u = (6.6) j j hs gves ( ) bo =ΩR b A b W b ε I follows from he unbasedness of ε ha an unbased esmae of b s ( =Ω ) b R b A b W b (6.7) Ω he marx has been consraned by (6.5) and (6.6) bu s oherwse sll free. I may be chosen so ha b s mnmum varance as well as unbased. hs may be done componen of b by componen. By (6.5), each of hese componens akes he form b =ω α (6.8) where he subscrp on ω has been suppressed and α denoes R [ ] n (6.7) and, by (6.6), ω s subjec o he consran ω u = (6.9) By (6.8), Vω (6.3) Var b = ω where V denoes Var [ α ]. I s sraghforward o show ha he choce of ω mnmsng (6.3) subjec o (6.9) s ω= ( uv u) V u (6.3) hs may be evaluaed f V s known. Now V s a submarx of he covarance marx

19 A Herarchcal Kalman Fler 7 ( ) + ( ) ( ) ( ) ( ) R Var b R R AVar b A R + R A W Var b W A R + ( ) ( ) ( ) ( ) ( ) R A Cov b b W A R, RCovb, b A R ( ) ( ) ( ) RCovb, b W A R (6.3) { } where Cov [b,c] denoes E b E[ b] c E[ c] and <.> denoes he symmersaon of he argumen, e, ACov b c B = ACov b, c B + BCov c, b A. [ ] [ ] [ ] Of he sx Var and Cov erms on he rgh, he frs s obanable from Γ n (6.), and he nex hree from Γ ( ). he las wo may be obaned by calculang Cov b, b ( ). By (6.6) and (6.), ( ) ˆ ( ) ( ) ( ) b( ) b = K M b + K : (6.33) ˆ Noe ha depends on daa relang o only me whereas depends on b b ( ) daa relang o only pror perods. he wo are herefore sochascally ndependen, and so (6.33) yelds ( ) ( ) ( ) Cov b, b ( ) = K M Γ (6.34) : Remark 6.3. he ransformaon (6.3) of W no he form (6.4), whle generally applcable, has been seleced here because s convenen for he example n Secon 7. In fac, oher ransformaons could have been chosen, eg (6.3) wh m x n marx U akng he block form U = An alernave choce of U would lead of course o a dfferen R n (6.3) and a dfferen Ω sasfyng (6.6). he remander of he reasonng (6.) (6.34) would be unchanged.

20 A Herarchcal Kalman Fler 8 7. Numercal example 7. Daa he followng example consders workers compensaon clam coss whn he herarchcal occupaonal srucure llusraed n Fgure 7.. Fgure 7. Occupaonal srucure Level oal pool of rsks Level Occupaonal Group A Occupaonal Group B Level Occupaonal sub-group A Occupaonal sub-group A Occupaonal sub-group A3 Occupaonal sub-group B Occupaonal sub-group B Occupaonal sub-group B3 Clams experence has been smulaed over 6 years for each of he occupaonal sub-groups. Each observaon s a realsaon of a gamma, dsrbuon wh parameers as se ou n able 7.. CoV denoes coeffcen of varaon n he able. able 7. Clam cos parameers Segmen Exposure Year Mean CoV Mean CoV Mean CoV Mean CoV Mean CoV Mean CoV $M % % % % % % % % % % % % Pool A B A A A B B Fgure 7. plos he mean clam cos per un exposure for he dfferen occupaonal groups. Noe ha he underlyng clam cos of B undergoes a sep ncrease n Year 4, whle ha of A ramps up over Years 3 and 4.

21 A Herarchcal Kalman Fler 9 Fgure 7. Mean clam cos per un exposure 4.5 Mean clam cos per un exposure (%) Year A A A3 B B he smulaed clams coss are se ou n able 7.. able 7. Clams daa Occupaonal Clam cos per un exposure observed n year group % % % % % % A B A A A B B Sac herarchcal credbly rang For comparson, he occupaonal groups are frs raed by means of Sund s (98) herarchcal credbly, as descrbed n Secon 5. hs s a sac credbly sysem n he sense ha he underlyng clam cos parameers are assumed consan over me. When hey are n fac varable, as n able 7., hs credbly sysem canno be expeced o rack he changes well.

22 A Herarchcal Kalman Fler Snce he sysem s sac, s naural applcaon a he end of years s o he accumulaed clams experence over hose years. Observaons on hs bass are se ou n able 7.3. able 7.3 Accumulang clams coss per un exposure Occupaonal Average clam cos per un exposure observed over years group o o 3 o 4 o 5 o 6 % % % % % % Pool A B A A A B B he paramerc srucure assumed for he occupaonal herarchy s as follows. Le b denoe he vecor of mean clam coss per un exposure a Level of he herarchy. hese vecors wll be of dmensons, and 5 for =, and respecvely. Le x denoe he 5-vecor of observaons a Level. he herarchcal srucure, n he noaon of Secon 5, s as follows: W = W = Y = 5x 5 un marx Y s null as here are no observaons a Level. A leas no drec observaons, hose shown a Levels and n able 7.3 beng merely weghed averages of he observaons a Level. he assumed parameers are as se ou n able 7.4.

23 A Herarchcal Kalman Fler able 7.4 Sac herarchcal credbly parameers Parameer Value β.% (.%) x un marx (x) Λ Λ Φ (.5%) x un marx (5x5) (.5%) x wegh marx (5x5) he wegh marx referred o n able 7.4 s he dagonal marx whose (k,k) elemen s equal o e / ek where ek s he exposure of he k-h occupaonal group a Level. ha s, he varances n Φ are nversely proporonal o he occupaonal group exposures n able 7.. he credbly sysem (5.5) and (5.6) s appled o he observaons, yeldng he resuls se ou n able 7.5. able 7.5 Sac herarchcal credbly rang Occupaonal Sac herarchcal credbly rang group % % % % % % A B A A A B B As seen n Fgure 7., occupaonal groups A and B are subjec o dramac change n rsk over he 6-year perod. Fgure 7.3 plos he credbly esmaes of mean clam cos per un exposure agans he rue parameers. As expeced, he sac naure of he model produces poor rackng of changng rsk parameers.

24 A Herarchcal Kalman Fler Fgure 7.3 Sac herarchcal credbly esmaes rackng parameers Mean clam cos per un exposure (%) Year A parameer A credbly esmae B parameer B credbly esmae 7.3 Dynamc herarchcal credbly rang In hs sub-secon he herarchcal Kalman fler developed n Secon 6 s appled o he observaons se ou n able 7.. he herarchcal srucure s as n Secon 7., wh here akng he same form for each as W n ha earler sub-secon. Each A s a un marx, meanng ha he parameers evolve accordng o ndependen random walks. Furher, Y akes he same form as Y n Secon 7. and R from (6.3) s he 5 5 un marx. W he assumed dsperson parameers are se ou n able 7.6. able 7.6 Dsperson parameers for herarchcal Kalman flers Parameer Value Φ As for Φ n Secon 7. Ψ (.%) Ψ Ψ (.5%) x un marx (x) (.5%) x un marx (5x5) he herarchcal fler (6.6) (6.) s appled o he observaons. I needs o be adaped slghly o he presen herarchy n order o apply o suffx raher han. he resul (6.7) requres smlar adapaon o produce b and b raher han jus he laer.

25 A Herarchcal Kalman Fler 3 he resuls produced by he fler are se ou n able 7.7, whch also reproduces, for comparson, rue parameers, observaons, and sac herarchcal credbly esmaes. able 7.7 Herarchcal Kalman fler credbly rang Occupaonal Observaons and credbly rangs a end of year group % % % % % % Observaons A B A A A B B Sac herarchcal credbly rang A B A A A B B Herarchcal Kalman fler rang A B A A A B B rue parameer A B A A A B B

26 A Herarchcal Kalman Fler 4 Fgure 7.4 provdes a graphcal comparson of rue parameers wh sac and dynamc credbly esmaes for he wo occupaonal groups, A and B, whose parameers have changed over me. he dynamc esmaes (doed lnes) are seen o rack he parameers (sold lnes) far beer han he sac esmaes (dashed lnes). Fgure 7.4 Sac and dynamc herarchcal credbly esmaes rackng parameers Mean clam cos per un exposure (%) Year A parameer A credbly esmae A Kalman esmae B parameer B credbly esmae B Kalman esmae 8. Concluson When he parameers of a herarchy evolve hrough me, he convenonal (sac) herarchcal credbly esmaes of Sund (979, 98) (Secon 5) are lkely o rack poorly. he dynamc equvalen of Sund s model explcly recognses parameer evoluon. hs s done by supermposng a Kalman fler on he herarchcal framework (Secon 6). hs s lkely o lead o much beer rackng of he evolvng parameers by he credbly esmaes. hs s llusraed numercally n Secon 7. Appendx. Proof of (6.3). Suppose W s of dmenson m x n wh m n. Snce W s of full rank, s possble o denfy n lnearly ndependen rows. A row permuaon, represened by m x n marx P, wll poson hese n he frs n rows, e

27 A Herarchcal Kalman Fler 5 PW X = Y where X s he n x n marx conssng of he lnearly ndependen rows, and s herefore non-sngular. hen APW = Y where X A = A sequence of elemenary row operaons, represened by he marx M, may be appled so ha Y s replaced by a marx, each of whose rows consss of a sngle un elemen and he res zero. Fnally, a permuaon of rows, represened by m x m marx Q gves QMAPW of he form (6.4). hen R = QMAP.

28 A Herarchcal Kalman Fler 6 References Hachemeser, C.A. (975). Credbly for regresson models wh applcaon o rend. Appears n Kahn (975, 9-63). Jazwnsk, A.H. (97). Sochasc processes and flerng heory. Academc Press, New York, NY. Kalman, R.E. (96). A new approach o lnear flerng and predcon sysems. Journal of Basc Engneerng, 8, Sund, B. (979). A herarchcal credbly regresson model. Scandnavan Acuaral Journal, 7-4. Sund, B. (98). A mul-level herarchcal credbly regresson model. Scandnavan Acuaral Journal, 5-3. aylor, G.C. (979). Credbly analyss of a general herarchcal model. Scandnavan Acuaral Journal, -.