The Basic Space Model

Transcription

1 The Basic Space Model Let x i be the ith idividual s (i=,, ) reported positio o the th issue ( =,, m) ad let X 0 be the by m matrix of observed data here the 0 subscript idicates that elemets are missig from the matrix -- ot all idividuals report their positios o all issues Let ψ ik be the ith idividual s positio o the kth (k =,, s) basic dimesio The model estimated is: X 0 = [ΨW' + J c'] 0 + E 0 (A) here Ψ is the by s matrix of coordiates of the idividuals o the basic dimesios, W is a m by s matrix of eights, c is a vector of costats of legth m, J is a legth vector of oes, ad E 0 is a by m matrix of error terms W ad c map the idividuals from the basic space oto the issue dimesios Equatio (A) ca be ritte as the product of partitioed matrices X W' Ψ c' (B) [ J ] = E here [ Ψ J ] is a by s+ matrix ad [ W c] 0 is a m by s+ matrix If > m ad there is o error or missig data, the the rak of X is s ad the rak of X J c is less tha or equal to s No Missig Data To solve () he there is o missig data, set c equal to the colum meas of X; that is c x = i= i = x

2 ad perform a sigular value decompositio of X J c : X J c = UΛV = ΨW here U is a by m matrix, Λ is a m by m matrix, ad V is a m by m matrix A simple solutio for Ψ ad W is Ψ = U Λ W = VΛ () here the diagoal elemets of Λ are the square roots of Λ Let I m be the m by m idetity matrix Equatio () implies that Ψ Ψ = W W That is: ad Ψ Ψ = Λ U U Λ = Λ I m Λ = Λ W W = Λ V V Λ = Λ I m Λ = Λ I additio, by costructio, J [X J c ] = 0, so that J U = J Ψ = 0, here 0 is a m legth vector of zeros Whe a s < m is preferred, the Eckart-Youg Theorem may be used i () to arrive at solutios for Ψ ad W That is, the s + to m sigular values are set equal to zero so that Ψ ad W from () are by s ad m by s matrices respectively Missig Data Because of the presece of missig data, SVD ad the Eckart-Youg Theorem caot be used directly Istead, I ork ith the loss fuctio ξ = m i i= = s {[ ψ ik k ] + c - xi} (3) k=

3 The otatio m i meas that the total of the summatio over may vary from s + to m depedig o ho may etries there are i the i th ro of X 0 That is, each idividual must report at least s + issue positios i order to be idetified Furthermore, the umber of missig etries i the colums of X 0 must also be restricted I most practical applicatios ill be much larger tha m Cosequetly, I ill adopt the covetio that there must be at least m etries i each colum of X 0 I lie ith the discussio above, the folloig to restrictios are applied to the loss fuctio: Ψ Ψ = W W ad J Ψ = 0 These restrictios produce the Lagragea multiplier problem µ = ξ + γ [ Ψ J ] + tr[φ(ψ Ψ - W W)] (4) here γ is a s legth vector of Lagragea multipliers ad Φ is a symmetric s by s matrix of Lagragea multipliers Give that the Lagragea multipliers are all zero, the partial derivatives of Ψ, W, ad c from equatios (3) ad (4) are idetical I particular: µ ψ ik µ k m s i ψ i + c xi = = = s = i= = (5A) k ψ ψ i + c x (5B) i ik µ c s = ψ i + c xi (5C) i= = here meas that the total of the summatio over i may vary from m to depedig upo ho may etries there are i the ith colum of X 0 3

4 Settig (5A) to zero ad collectig the s partial derivatives of the ith ro of Ψ ito a vector ad dividig by produces [W* W*]ψ i - W* [x oi - c o ] = 0 here W* is a m i by s matrix ith the appropriate ros correspodig to missig etries i X o removed, ψ i is the ith ro of Ψ, x oi is the ith ro of X 0 ad is of legth m i, c o is the m i legth vector of costats correspodig to the elemets of x oi, ad 0 is a s legth vector of zeroes If W* W* is osigular, the ψ i = (W* W*) - W* [x oi - c o ] (6) ad the ros of Ψ ca be estimated through ordiary least squares The s partial derivatives of the th ro of W from equatio (5B) ad the partial derivative for c from (5C) ca be collected ito the vector * * * [ Ψ ' Ψ ] - Ψ 'x = 0 c here Ψ * = [Ψ o J o ] is a by s + matrix (the matrix Ψ ith the appropriate ros correspodig to missig data removed ad the bordered by oes), is the s legth o vector of the th ro elemets of W, c is the th elemet of c, x o is the th colum of X o ad is of legth, ad 0 is a s+ legth vector of zeroes If Ψ * Ψ * is osigular, the c * * - * = ( Ψ ' Ψ ) Ψ ' x (7) o ad the ros of W ad the elemets of c ca be estimated through ordiary least squares 4

5 The easiest ay to estimate W ad Ψ is to select some suitable startig estimate of either matrix ad the iterate betee (6) ad (7) util covergece is achieved The costraits o W ad Ψ ca be met at ay stage of the iteratio by simply settig the colum meas of Ψ equal to zero, formig the matrix product Ψ W, ad performig the sigular value decompositio: Ψ W = UΛV here Λ is a s by s diagoal matrix cotaiig the s sigular values i descedig order, ad U ad V are by s ad s by s matrices respectively such that U U = V V = I s Settig Ψ = U Λ ad W = VΛ as i () satisfies the costraits A simple ay to proceed ith the estimatio is to exploit the orthogoality of Ψ ad estimate oe colum of Ψ ad W at a time This is motivated by the fact that if the are close to, Ψ * Ψ * i (7) ill be very close to a diagoal matrix 5

6 Table Summary of the Estimatio Procedure ) Obtai startig estimates of c, deoted by c ( ), usig the colum meas of X 0 Obtai startig estimates of, deoted by ( ), by fidig the vector of plus ad mius oes that maximizes the umber of positive elemets i the covariace matrix [X o J c ] [X o J c ] (see Appedix B) ) Use c ( ) ad ( i equatio (8) to obtai a startig estimate of ψ, deoted by ψ ( ), ad set the mea of ( equal to zero 3) Use ψ ( ) i equatio (7) to obtai a secod estimate of c ad -- c ( ) ad ( respectively 4) Use c ( ) ad ( i equatio (6) to obtai a secod estimate of ψ, ( Set the mea of ( equal to zero ad set the sum of squares of ( equal to the sum of squares of ψ ( ) ; that is 5) Repeat steps (3) ad (4) util covergece 6) Compute E ˆ 0 = X0 - ψ ŵ' - Jĉ' i= () () ψˆ = ψˆ 7) Obtai startig estimates of, ( ), by fidig the vector of plus ad mius oes that maximizes the umber of positive elemets i the covariace ' matrix E E 0 0 i i= i 8) Use ( i equatio (0) to obtai startig estimates of ψ, ( 6

7 9) Use ψ ( ) i equatio () to obtai ( 0) Use ( i equatio () to obtai ψ ( ) Set the mea of ( equal to zero ad set the sum of squares of ( as i step (4) above equal to the sum of squares of ˆψ () ) Repeat steps (9) ad (0) util covergece ) Compute E0 = X0 - ψ ˆ ŵ' ˆ - J ˆ ĉ'- ψ ŵ' = E0 - ψ ŵ' 3) Repeat steps (7) - () to estimate remaiig dimesios; that is: 3 ad ψ 3, 4 ad ψ 4,, ad s ad ψ s 4) Use the full by s matrix Ψ i equatio (7) to obtai the full m by s matrix W ad the m legth vector of costats c 5) Use W ad c i equatio (6) to obtai a e estimate of Ψ 6) Repeat steps (4) ad (5) util covergece 7