MARSS Reference Sheet

Transcription

1 MARSS Reference Shee The defaul MARSS model (form="marxss") is wrien as follows: x = B x 1 + u + C c + w where w MVN( Q ) y = Z x + a + D d + v where v MVN( R ) x 1 MVN(π Λ) or x MVN(π Λ) c and d are inpus (numeric) and mus have no missing values. The MARSS package is designed o handle linear consrains wihin he parameer marices (he B u C Q Z a D R π and Λ)). Linear consrain means you can wrie he elemens of he marix as a linear equaion of all he oher elemens alhough ypically each marix elemen is jus a fixed or esimaed value. The following shows an example of a mean-revering random walk model wih hree observaion ime series wrien as a MARSS model: = b b 1 + w1 w 2 w1 w 2 MVN y v 1 y 2 = 1 + v y v 3 ( v 1 v 2 v 3 q11 q 12 q 12 q 22 ) a 1 MV N MVN r 11 r r ( ) 1 1 To fi his wih MARSS we ranslae his model ino equivalen marices (or arrays if ime-varying) in R: B1=marix(lis("b""b")22) U1=marix(21) Q1=marix(c("q11""q12""q12""q22")22) Z1=marix(c(1111)32) A1=marix(lis("a1")31) R1=marix(lis("r11""r""r")33) pi1=marix(21); V1=diag(12) model.lis=lis(b=b1u=u1q=q1z=z1a=a1r=r1x=pi1v=v1inix=) 1 (1)

2 Defauls and shorcus for model specificaion B defaul ideniy ideniy marix shorcu unconsrained all elemens esimaed shorcu diagonal and equal diagonal marix wih one value on he diagonal shorcu diagonal and unequal diagonal marix wih unique values on he diagonal numeric marix diag(.82) specify as marix wih numbers char marix marix(c( a b )22) esimaed marix wih only 2 esimaed parameers lis marix marix(lis( a 21 b )22) combine numeric (fixed) and esimaed values U and x defaul unequal all u s differen shorcu unconsrained same as unequal shorcu zero all zero numeric marix marix(121) specify as marix wih numbers char marix marix(c( a a )21) all esimaed parameers lis marix marix(lis( a )21) combine numeric (fixed) and esimaed values 2

3 Q and R All variance-covariance marices are symmeric defaul diagonal and unequal diagonal marix wih unique values on he diagonal; independen wih differen variances shorcu diagonal and equal diagonal wih one value on diagonal; i.i.d. shorcu unconsrained unconsrained variance-covariance marix shorcu equalvarcov one variance and one covariance shorcu ideniy i.i.d wih variance of 1 numeric marix diag(.82) specify as marix wih numbers char marix marix(c( a c c b )22) specify consrained wihin a var-cov marix; careful marix mus be a valid var-cov srucure lis marix marix(lis(1 b )22) combine numeric (fixed) and esimaed values; careful marix mus be a valid var-cov srucure Z defaul numeric marix ideniy marix(c(111)32) Z num of columns will specify he num of x; columns of Z mus mach rows in u B Q (oherwise Z specifies a differen num of x han u B Q) ideniy marix; means one x will be esimaed for each y specify as marix wih numbers; example is a design marix wih s and 1s (assigns each x o a y) shorcu unconsrained all elemens esimaed; rarely used shorcu diagonal and equal diagonal marix wih one value on he diagonal shorcu diagonal and unequal diagonal marix wih unique values on he diagonal char marix marix(c( a b )22) esimaed marix wih only 2 esimaed parameers lis marix marix(lis( a 21 b )22) combine numeric (fixed) and esimaed values 3

4 A defaul scalar esimaes an inercep; ses he firs scalar of y assoc wih an x o shorcu zero all zero numeric marix marix(121) specify as marix wih numbers lis marix marix(lis( a )21) inix is x a = or =1 defaul a = 1 a =1 Inpus - se o zero by defaul c defaul zero if no inpus if passed in marix(1at) combine numeric (fixed) and esimaed values; make sure marix does no lead o a indeerminan model inpus; mus be numeric; no missing values allowed; dimension is q x T (number of columns in y) if c passed in mus be a numeric marix wih same num of columns as y C if passed in; dimension is m (rows in x) q(rowsinc) defaul zero if no c passed in hen defaul is zero defaul unconsrained if c passed in corresponding C has all elemens esimaed. numeric marix marix(1mq) fixed values; specify as marix wih numbers char marix marix(lis( b a )mq) all esimaed values bu some shared (equal) lis marix marix(lis( a )mq) combine numeric (fixed) and esimaed values 4

5 d defaul zero if no inpus if passed in marix(1pt) inpus; mus be numeric; no missing values allowed; dimension is p x T (number of columns in y) if c or d passed in mus be a numeric marix wih same num of columns as y D if passed in; dimension is n (rows in y) p(rowsind) defaul zero if no d passed in hen defaul is zero defaul unconsrained if d passed in corresponding D has all elemens esimaed. numeric marix marix(1np) fixed values; specify as marix wih numbers char marix marix(lis( b a )np) all esimaed values bu some shared (equal) lis marix marix(lis( a )np) combine numeric (fixed) and esimaed values 5