Department of Mathematics UNIVERSITY OF OSLO. FORMULAS FOR STK4040 (version 1, September 12th, 2011) A - Vectors and matrices

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1 Deartet of Matheatcs UNIVERSITY OF OSLO FORMULAS FOR STK4040 (verso Seteber th 0) A - Vectors ad atrces A) For a x atrx A ad a x atrx B we have ( AB) BA A) For osgular square atrces A ad B we have ( ) ( ) A A ad ( ) AB B A A3) A x atrx Q s orthogoal f QQ Q Q I e f Q Q A4) For x atrces A ad B we have AB A B A5) For a dagoal atrx A dag{ a a a } we have A a A6) Let A be a syetrc x atrx ad x a desoal vector The xa x a x x s deoted a quadratc for A7) A syetrc x atrx A s oegatve defte f xax 0 for all desoal vectors x It s ostve defte f xax 0 for all x 0 A8) A x atrx A has egevalue wth corresodg egevector e 0 f Ae e A9) A egevalue s a soluto to the characterstc equato AI 0 A0) For a x atrx A the trace s gve by tr( A ) a A) tr( A) where the s are the egevalues of A A) For a x atrx B ad a x atrx C we have tr( BC) tr( CB )

2 A3) For a ostve defte x atrx A we have the sectral decoosto A e e e e e e Here 0are the egevalues of A ad e e e are the corresodg orthogoal ad oralzed egevectors (e e e ad e e 0 for ) A4) For a ostve defte x atrx A the square root atrx of A ad ts verse are defed as / / A ee ad A e e A5) Let B be a ostve defte atrx ad d be a gve vector The for a arbtrary ozero ( xd ) vector x we have ax db d wth the axu attaed whe x cb d x0 xbx for ay costat c 0 A6) Let B be a ostve defte x atrx wth egevalues 0 ad corresodg orthogoal ad oralzed egevectors e e e The xbx ax (attaed whe xe) x0 xx Moreover for xbx ax (attaed whe xe ) xe e e xx A7) Gve a ostve defte x atrx B ad a scalar b > 0 we have b b ex tr ( b) e b b Σ B Σ B for ay ostve defte x atrx Σ wth equalty f ad oly f Σ (/ b) B B - Rado vectors ad atrces B) For rado atrces X ad Y of the sae deso ad A ad B atrces of costats we have that E( XY) E( X) E( Y ) E AXB A E X ) B ( ) ( B) For a rado vector X wth ea vector the covarace atrx s gve by Cov( X) E{( Xμ)( Xμ ) } B3) For a rado vector X wth ea vector ad covarace atrx ad a atrx C of costats we have that E( CX) CE( X) Cμ ad Cov( CX) CΣC

3 B4) Let deote the correlato atrx The ρ ρ V V Σ ad V ΣV / / / / / where =dag V s the stadard devato atrx C - The ultvarate oral dstrbuto ad related dstrbutos C) A -varate rado vector X wth ea vector ad covarace atrx s ultvarate orally dstrbuted X ~ ( μσ ) f ts desty taes the for f N ( x) ex ( ) ( ) / / x μ Σ x μ ( ) Σ C) X ~ N ( μσ ) f ad oly f ax ~ N( aμ a Σa ) for all -desoal vectors a C3) Let X ~ ( μσ ) the the robablty s that X taes values the ellsod N x x μ Σ x μ : ( ) ( ) ( ) C4) Let X ~ ( μσ ) ad let A be a qx atrx ad d a q-desoal vector N The AX d ~ N ( Aμ d A ΣA ) q C5) Assue that X ~ N μ Σ Σ where X s a q desoal vector X μ Σ Σ The X X x ~ ( μ Σ Σ ( x μ ) Σ Σ Σ Σ ) N q C6) Assue that X X X are deedet X ~ N ( μ Σ ) ad let V cx cx cx V b X b X b X where the c 's ad b 's are costats The ad c V μ ~ N ccσ bc Σ V b μ bcσ b b Σ C7) Assue that Z Z are d N ( 0 Σ ) The dstrbuto of ZZ s called the Wshart dstrbuto wth degrees of freedo (ad x covarace atrx Σ ) deoted W ( Σ ) C8) Proertes of the Wshart dstrbuto: If W ~ W ( Σ ) ad W ~ W ( Σ ) are deedet the W W ~ W ( ) Σ If W ~ W ( Σ ) ad C s a qx atrx the CWC~ W q ( CΣC ) 3

4 C9) Let Z~ N ( 0 Σ ) ad W ~ W ( Σ ) be deedet The W Z Z s dstrbuted as F D Estato for ultvarate dstrbutos D) Assue that X X X are d wth ea vector μ ad covarace atrx Σ Ubased estators for μ ad Σ are X X ad S X X X X D) Assue that X X X are d N ( μσ ) The we have the followg results X ad S are deedet X ~ N μ Σ ( ) S ~ W ( Σ ) E Prcal cooets E) Let X be a -varate rado vector wth ea vector μ ad covarace atrx Σ The frst oulato rcal cooet s the lear cobato ax that axzes Var( ax ) subect to aa The secod oulato rcal cooet s the lear cobato ax that axzes Var( ax ) subect to aa ad Cov( a X ax ) 0 Etc E) Let X a -varate rado vector wth ea vector μ ad covarace atrx Σ ad assue that Σ has egevalue-egevector ars ( e) ( e) ( e ) wth 0 The we have the followg results: The -th oulato rcal cooet s Y ex Var( Y ) Var( X ) Var( Y) e corr( Y X) 4

5 F Factor aalyss F) Let X a -varate rado vector wth ea vector μ The orthogoal factor odel assues that X μ LF ε where { } L s a x atrx of factor loadgs l F F F F s a -dsoal vector of coo factors ad ε s a -vector of errors (secfc factors) F) The uobservable rado vectors F ad ε the factor odel satsfy: E( F) 0 Cov( F) I E() ε 0 Cov( ε) Ψ dag{ } Cov( ε F) 0 G Dscrato ad classfcato G) We have two oulatos deoted ad wth ror robabltes ad If a rado vector X s selected fro t has desty f ( x ) whle t has desty f ( x ) f t s selected fro The cost of sclassfyg a observato fro as cog fro s c ( ) whle the cost s c ( ) for sclassfyg a observato fro as cog fro The the exected cost of sclassfcato s zed f a observato x s allocated to rovded that f ( x) c( ) f ( x) c( ) ad allocated to otherwse G) If the destes f( x ) are ultvarate oral wth ea vectors μ ad coo covarace atrx Σ the the exected cost of sclassfcato s zed f a observato x s allocated to rovded that c( ) ( μ μ ) Σ x ( μ μ ) Σ ( μ μ ) l 0 c( ) ad allocated to otherwse 5