GAMsetup {mgcv} | R Documentation |
Sets up design matrix X, penalty matrices S_i and linear equality constraint matrix C for a GAM defined in terms of
penalized regression splines. Various other information characterising the bases used is also returned.
The output is such that the model can be fitted and
smoothing parameters estimated by the method of Wood (2000) as implemented in routine
mgcv()
. This is usually called by gam
.
GAMsetup(G)
G |
is the single argument to this function: it is a list containing at least the elements listed below: |
m |
the number of smooth terms in the model |
df |
an array of G$m integers specifying the maximum d.f. for each spline
term. |
n |
the number of data to be modelled |
nsdf |
the number of user supplied columns of the design matrix for any parametric model parts |
dim |
An array of dimensions for the smooths. dim[i] is the
number of covariates that smooth i is a function of. |
fix |
An array of logicals indicating whether each smooth term has fixed degrees of freedom or not. |
s.type |
An array giving the type of basis used for each term. 0 for cubic regression spline, 1 for t.p.r.s |
p.order |
An array giving the order of the penalty for each term. 0 for auto selection. |
x |
an array of G$n element arrays of data and (optionally) design matrix
columns. The first G$nsdf elements of G$x should contain the elements of
the columns of the design matrix corresponding to the parametric part of the model. The
remaining G$m elements of G$x are the values of the covariates that are
arguments of the spline terms. Note that the smooths will be centred and no intercept term
will be added unless an array of 1's is supplied as part of in
G$x |
vnames |
Array of variable names, including the constant, if present. |
w |
prior weights on response data. |
by |
a 2-d array of by variables (i.e. covariates that multiply a
smooth term) by[i,j] is the jth value for the ith by
variable. There are only as many rows of this array as there are
by variables in the model (often 0). The rownames of by
give the by variable names. |
by.exists |
an array of logicals: by.exists[i] is
TRUE if the ith smooth has a by variable associated with
it, FALSE otherwise. |
knots |
a compact array of user supplied knot locations for each smooth, in the order corresponding
to the row order in G$x . There are G$dim[i] arrays of length G$n.knots[i] for the ith
smooth - all these arrays are packed end to end in 1-d array G$knots -
zero length 1 for no knots. |
n.knots |
array giving number of user supplied knots of basis for each smooth term 0's for none supplied. |
A list H
, containing the elements of G
(the input list) plus the
following:
X |
the full design matrix. |
S |
A one dimensional array containing the non-zero elements of the
penalty matrices. Let start[k+1]<-start[k]+H$df[1:(k-1)]^2 and
start[1]<-0 . Then penalty matrix k has
H$S[start[k]+i+H$df[i]*(j-1) on its ith row and jth column.
To get the kth full penalty matrix the matrix so obtained would be
inserted into a full matrix of zeroes with it's 1,1 element at H$off[k],H$off[k] .
|
off |
is an array of offsets, used to facilitate efficient storage of the penalty
matrices and to indicate where in the overall parameter vector the parameters of the ith
spline reside (e.g. first parameter of ith spline is at p[off[i]+1] ). |
C |
a matrix defining the linear equality constraints on the parameters used to define the the model (i.e. C in Cp=0). |
UZ |
Array containing matrices, which transform from a t.p.r.s. basis to the
equivalent t.p.s. basis (for t.p.r.s. terms only). The packing method
is as follows: set start[1]<-0 and
start[k+1]<-start[k]+(M[k]+n)*tp.bs[k] where n is number
of data, M[k] is penalty null space dimension and
tp.bs[k] is zero for a cubic regression spline and the basis
dimension for a t.p.r.s. Then element i,j of the UZ matrix for
model term k is:UZ[start[k]+i+(j=1)*(M[k]+n)] . |
Xu |
Set of unique covariate combinations for each term. The packing method
is as follows: set start[1]<-0 and
start[k+1]<-start[k]+(xu.length[k])*tp.dim[k] where xu.length[k] is number
of unique covariate combinations and tp.dim[k] is zero for a
cubic regression spline
and the dimension of the smooth (i.e. number of covariates it is a
function of) for a t.p.r.s. Then element i,j of the Xu matrix for
model term k is:Xu[start[k]+i+(j=1)*(xu.length[k])] . |
xu.length |
Number of unique covariate combinations for each t.p.r.s. term. |
covariate.shift |
All covariates are centred around zero before bases are constructed - this is an array of the applied shifts. |
xp |
matrix whose rows contain the covariate values corresponding to the parameters of each cubic regression spline - the cubic regression splines are parameterized using their y- values at a series of x values - these vectors contain those x values! Note that these will be covariate shifted. |
Simon N. Wood simon@stats.gla.ac.uk
Wood, S.N. (2000) Modelling and smoothing parameter estimation with multiple quadratic penalties JRSSB 62(2):413-428
Wood, S.N. (2003) Thin plate regression splines. JRSSB 65(1):95-114.
# This example modified from routine SANtest() set.seed(0) n<-100 # number of observations to simulate x <- runif(5 * n, 0, 1) # simulate covariates x <- array(x, dim = c(5, n)) # put into array for passing to GAMsetup pi <- asin(1) * 2 # begin simulating some data y <- 2 * sin(pi * x[2, ]) y <- y + exp(2 * x[3, ]) - 3.75887 y <- y + 0.2 * x[4, ]^11 * (10 * (1 - x[4, ]))^6 + 10 * (10 * x[4, ])^3 * (1 - x[4, ])^10 - 1.396 sig2<- -1 # set magnitude of variance e <- rnorm(n, 0, sqrt(abs(sig2))) y <- y + e # simulated data w <- matrix(1, n, 1) # weight matrix par(mfrow = c(2, 2)) # scatter plots of simulated data plot(x[2, ], y);plot(x[3, ], y);plot(x[4, ], y);plot(x[5, ], y) x[1,]<-1 # create list for passing to GAMsetup.... G <- list(m = 4, n = n, nsdf = 0, df = c(15, 15, 15, 15),dim=c(1,1,1,1), s.type=c(0,0,0,0),by=0,by.exists=c(FALSE,FALSE,FALSE,FALSE), p.order=c(0,0,0,0),x = x,n.knots=rep(0,4)) H <- GAMsetup(G) H$y <- y # add data to H H$sig2 <- sig2 # add variance (signalling GCV use in this case) to H H$w <- w # add weights to H H$sp<-array(-1,H$m) H$fix<-array(FALSE,H$m) H$conv.tol<-1e-6;H$max.half<-15 H$min.edf<-5;H$fixed.sp<-0 H <- mgcv(H) # select smoothing parameters and fit model